“We have long known, thanks to Ernest McClain, that the ancients were obsessed with harmonic numbers and that the Bible encodes these from beginning to end. Now new evidence appears, as these numbers correlate with the planetary periods, and their discovery is pushed far back into the prehistoric era. Richard Heath’s work, based not on speculation but on objective data, challenges all accepted notions of cultural evolution and religious origins.”
–JOSCELYN GODWIN, author of Harmonies of Heaven and Earth and Atlantis and the Cycles of Time
As modern humans first walked the Earth roughly 70,000 years ago, the moon’s orbit came into harmonic resonance with the outer planets of Jupiter, Saturn, and Uranus. The common denominators underlying these harmonic relationships are the earliest prime numbers of the Fibonacci series–two, three, and five–the same numbers that interact to give us the harmonic relationships of music.
Exploring the simple mathematical relationships that underlie the cycles of the solar system and the music of Earth, Richard Heath reveals how Neolithic astronomers discovered these ratios using megalithic monuments like Stonehenge and the Carnac stones, discoveries that informed later myths and stories, including the Epic of Gilgamesh, the Resurrection of Osiris, the Rg Veda, the Hebrew Bible, Homer’s epic tales, and the Return of Quetzalcoatl. He explains how this harmonic planetary knowledge formed the basis of the earliest religious systems, in which planets were seen as gods, and shows how they spread through Sumer, Egypt, and India into Babylon, Judea, Mexico, and archaic Greece. He exposes how the secret knowledge encoded within the Bible’s god YHWH was lost as Greek logic and reason steadily weakened mythological beliefs.
Revealing the mysteries of the octave and of our musical scales, Heath shows how the orbits of the outer and inner planets gave a structure to time, which our moon’s orbit could then turn into a harmonic matrix. He explains how planetary time came to function as a finely tuned musical instrument, leading to the rise of intelligent life on our planet. He demonstrates how this harmonic science of numbers can be read in the secret symbolism and sacred geometry of ancient cities such as Teotihuacan and in temples such as the Parthenon, connecting the higher worlds of planetary time and harmonics with the spiritual and physical life on Earth.
Recasting our understanding of the solar system, Heath seeks to reawaken humanity’s understanding of how sacred numbers structure reality, offering an opportunity to recover this lost harmonic doctrine and reclaim our intended role in the outer life of our planet.
Harmonic Origins of the World Contents (272 pages, 100 b&w illustrations) Preface Introduction: The Significance of Planetary Harmony (5) PART 1: RECOVERING LOST KNOWLEDGE OF THE WORLD SOUL 1 Climbing the Harmonic Mountain (20) 2 Heroic Gods of the Tritone (19) 3 YHWH Rejects the Gods (15) 4 Plato’s Dilemma (22) PART 2: A COSMICALLY CREATIVE HARMONY 5 The Quest for Apollo’s Lyre (25) 6 Life on the Mountain (23) PART 3 THE WAR IN HEAVEN 7 Gilgamesh Kills the Stone Men (16) 8 Quetzalcoatl’s Brave New World (31) 9 YHWH’s Matrix of Creation (19) 10 The Abrahamic Incarnation (15) Postscript: Intelligent Star Systems APPENDIX 1: Astronomical Periods and Their Matrix Equivalents APPENDIX 2: Ancient Use of Tone Circles (11) Notes Bibliography Index
RICHARD HEATH is a development engineer with degrees in systems science and computer-aided design. His interest in megalithic astronomy and ancient metrology has resulted in 6 books, including, in January 2021, Sacred Geometry: Language of the Angels. He lives in the Preseli hills of West Wales.
Over the last seven thousand years, hunter-gathering humans have been transformed into the “modern” norms of citizens (city dwellers) through a series of metamorphoses during which the intellect developed ever-larger descriptions of the world. Past civilizations and even some tribal groups have left wonders in their wake, a result of uncanny skills – mental and physical – which, being hard to repeat today, cannot be considered primitive. Buildings such as Stonehenge and the Great Pyramid of Giza are felt anomalous, because of the mathematics implied by their construction. Our notational mathematics only arose much later and so, a different maths must have preceded ours.
We have also inherited texts from ancient times. Spoken language evolved before there was any writing with which to create texts. Writing developed in three main ways: (1) Pictographic writing evolved into hieroglyphs, like those of Egyptian texts, carved on stone or inked onto papyrus, (2) the Sumerians used cross-hatched lines on clay tablets, to make symbols representing the syllables within speech. Cuneiform allowed the many languages of the ancient Near East to be recorded, since all spoken language is made of syllables, (3) the Phoenicians developed the alphabet, which was perfected in Iron Age Greece through identifying more phonemes, including the vowels. The Greek language enabled individual writers to think new thoughts through writing down their ideas; a new habit that competed with information passed down through the oral tradition. Ironically though, writing down oral stories allowed their survival, as the oral tradition became more-or-less extinct. And surviving oral texts give otherwise missing insights into the intellectual life behind prehistoric monuments.
The texts and iconography of the ancient world once encrypted the special numbers used to create ancient and pre-historic monuments, using a numeracy which modelled the earth and sky using the invariant numbers found in celestial time, and in the world of number itself. Oral stories spoke from a unified construct, connecting the people to their gods. Buildings were echoes of an original Building, whose dimensions came to form a canon within metrologyThe application of units of length to problems of measurement, design, comparison or calculation.; the ancient science of measure. But the language of the gods within this Building was seen to be that of musical tuning theory, the number science which concerns us here. The gods in question are primarily the planetary and calendric periods seen from earth, and it was only through the astronomy associated with the earliest, megalithic buildings that the ancient maths could have naturally evolved.
To see musical harmony in the sky, time was counted as lengths of time between visible astronomical events such as sun rise, moon set, or full moon. Geometry evolved to set alignmentsA name special to Carnac’s three successive groups of parallel rows of stones, starting above Carnac called Le Menec, Kermario, and Kerlescan and another found near Erdevan. to horizon events, such as the solsticeThe extreme points of sunrise and sunset in the year. In midwinter the sun is to the south of the celestial equator (the reverse in the southern hemisphere) and in midsummer the sun is north of that equator, which is above the geographical Equator). sun or, to place long lengths of day-counting within geometries such as the trigonometric triangle. Megalithic astronomy (chapter one) consisted of a set of quantified lengths of time and the geometrical relationship between them. It would have discovered that some of the ratios between time periods were especially simple: most significantly the two outer planets Jupiter and Saturn, related as 9/8 (a musical whole tone interval) and 16/15 (a semitone interval) to the lunar year. In each ratio the lunar year is the denominator and the planetary synods are the numerators. If we make the denominators the same (by multiplying the ratios by each other’s denominator) we obtain (times 15) 135/120 and (times 8) 128/120. Because the lunar year is 12 lunar months long, the lunar month must comprise ten sub-units of time; the Jupiter synod must be 13.5 months long; and Saturn’s synod must be 12.8 months long.
The idea that astronomy could have caused the ancient world to have any great interest in musical tuning theory runs against the standard musicological model of history in which, it was the making of music which drove the Babylonian tuning texts to appear on Cuneiform tablets from Nippur and other places. However, lists of regular numbers and tables of reciprocals counting down from sixty to the power of four (12,960,000) hardly seem relevant to practical music. 12,960,000 is a significant number belonging in my work to Venus, the bright planet of the inner solar system, in its synod relative to the lunar year. The number is a large one because she is higher in “heaven”, becoming Quetzelcoatl in the Olmec’s cult of astronomical time inherited by the Maya and Aztec cultures (chapter seven). Tuning theory must have found its way to Mexico before the devastating Bronze Age collapse circa 1200 BC; a date when Mexico’s likely contact with the eastern Mediterranean would have ceased. The future of European tuning theory in the ensuing Iron Age then lay in the hands of the Archaic Greeks (Homer and Hesiod) and surprisingly, the Jewish school responsible for the early Bible (chapter five).
Whenever civilizations fall they pass on information. When megalithic astronomy died, it bequeathed the idea that the planets were gods related to the Moon through musical harmony, also leaving the ancient world their metrology. When temples were built or stories to the planetary gods passed on, these could express musical numbers and ratios within architecture, iconography and myth. In classical Greece, the power of writing had won over the oral world whereupon Athens enshrined musical harmony in the Parthenon and in Plato’s writings about the ancient tradition of musical tuning theory (chapter six).
I first noticed the musical resonances (of Jupiter and Saturn to the lunar year) in 2000, for which I could find no traditional setting except mythology [Heath 2004]. The extensive works of the Pythagorean tradition for instance, concerned with planetary harmony, are complex and appear more influenced by Greek mathematics than by the ancient world. After some decades though, understanding came through the work of Ernest G. McClain, and through my collaboration with him in the last years of his life. These outer planetary resonances slotted perfectly into McClain’s frameworks for ancient tuning theory. The primary sources for McClain’s work were the surviving texts of the ancient world [McClain, 1976] but his key to these texts were Plato’s dialogues, for which he had provided a definitive interpretation [McClain, 1978], as being a cryptic textbook for ancient tuning theory.
McClain found harmonic numbers (*which only have factors of two, three and five) referred to (as if arbitrarily) in various guises within ancient stories, allowing the initiated to reconstitute a much larger array of harmonic numbers belonging to the god or to a spiritual locale, which the story was intended to animate. (**This resembles the aboriginal habit of recitations before painted caves or cliffs, where an initiate recounts the story illustrated by the paintings, thus “joining the dots”. Our word esoteric perhaps hails from this practice of leaving cryptic clues within texts, in this case linking to musical tuning theory.) In his popular work, The Myth of Invariance, Ernest McClain American Cryptologist and Pythagorean Musicologist who decoded Plato’s cryptic numerical ciphers in The Pythagorean Plato. The Myth of Invariance showed limiting numbers had been an ancient way of defining the onset of key musical tuning realities, then coded into many religious texts. Wikipedia. recreated many otherwise hidden harmonic worlds from number references within texts; from India, Mesopotamia, Egypt, Greece and the New World.
It became obvious to me that the common denominators (see earlier ratios of the Jupiter and Saturn synodic The recurring time cycle of a given celestial phenomenon seen from the Earth. periods) would “place” them in the corner of McClain’s “holy mountains” (**his arrays of regular numbers which could be inferred from a single number limiting the array as for the high do for an octave). More and more “characters” from astronomical time started appearing “on the mountain”, in parallel with McClain’s own interpretations from the Bible, Homer, Babylonian texts, the RgVeda, etc.
The astronomical significance of harmonic numbers left in ancient texts explains the mystery of why they should be there in the first place and it confirms the important role texts have played in carrying a whole system of knowledge, within an oral tradition. In their heyday, texts were only in the heads of reciters and listeners of all sorts – some hearing a good story and others learning new facets of harmonic knowledge. Such a tradition evidently thought the world had come into existence due to musical harmony (chapter eight) and that relationships to the gods were organised according to harmonic laws. Indeed, the astrology that so obsessed the Babylonians was probably rooted in a harmonic model of fate involving planets and calendars. One can see how stories such as Gilgamesh reveal the Sumerians knew of it (c. 3000 BC), placing planetary gods like Iaana/Ishtar/Venus in heroic stories that make better sense if referring to “holy mountains” (chapter two) located in a harmonic heaven.
Click here to view the publisher page , where extra information on this and my other books, including reviews, can be found. The contents of Harmonic Origins of the World are:
Preface
Introduction: The Significance of Planetary Harmony
PART 1 RECOVERING LOST KNOWLEDGE OF THE WORLD SOUL
Plato’s description of how the Creator designed the world using only the intervals of musical fifth (3/2), whole tone (9/8) and fourth (4/3), within a purely numerical framework (6 8 9 12).
Authors Colette Caillat, Ravi Kumar Publisher Ravi Kumar, 2004 ISBN 1588860582, 9781588860583 Length 196 pages
Cosmology Old & New, Being a Modern Commentary on the Fifth Chapter of Tattvārthādhigama Sūtra
Volume 5 of Jñānapīṭha Mūrtidevī granthamālā.Jñānapīṭha Mūrtīdevī granthamālā. English series Volume 5 of Jñānapīṭha Mūrtidevī granthamālā: English series Volume 5 of Murtidevī granthamālā
Author G. R. Jain Publisher Bharatiya Jnanpith Publication, 1975 Original from the University of Michigan Digitized Oct 6, 2006 Length 219 pages
Victorious Ones: Jain Images of Perfection
Author Phyllis Emily Granoff
Editor Phyllis Emily Granoff Contributor Rubin Museum of Art (New York, N.Y.) Edition illustrated Publisher Rubin Museum of Art, 2009 ISBN 0944142834, 9780944142837 Length 308 pages
Elements of Jaina Geography: The Jambūdvīpasaṃgrahaṇī of Haribhadra Sūri : Critically Edited and Translated with the Commentary of Prabhānanda Sūri
Authors Haribhadrasūri, Frank van den Bossche
Editor Frank van den Bossche Compiled by Frank van den Bossche Publisher Motilal Banarsidass, 2007 Original from the University of Michigan Digitized Jun 22, 2009 ISBN 8120829344, 9788120829343 Length 327 pages
The circle of fifths developed in the late 1600s and early 1700s to theorize the modulation of the Baroque era (see § Baroque era).
The first circle of fifths diagram appears in the Grammatika (1677) of the composer and theorist Nikolay Diletsky, who intended to present music theory as a tool for composition.[7] It was “the first of its kind, aimed at teaching a Russian audience how to write Western-style polyphonic compositions.”
A circle of fifths diagram was independently created by German composer and theorist Johann David Heinichen in his Neu erfundene und gründliche Anweisung (1711),[8] which he called the “Musical Circle” (German: Musicalischer Circul).[9][10] This was also published in his Der General-Bass in der Composition (1728).
Heinichen placed the relative minor key next to the major key, which did not reflect the actual proximity of keys. Johann Mattheson (1735) and others attempted to improve this—David Kellner (1737) proposed having the major keys on one circle, and the relative minor keys on a second, inner circle. This was later developed into chordal space, incorporating the parallel minor as well.[11]
Some sources imply that the circle of fifths was known in antiquity, by Pythagoras.[12][13][14] This is a misunderstanding and an anachronism.[15]Tuning by fifths (so-called Pythagorean tuning) dates to Ancient Mesopotamia;[16] see Music of Mesopotamia § Music theory, though they did not extend this to a twelve note scale, stopping at seven. The Pythagorean comma was calculated by Euclid and by Chinese mathematicians (in the Huainanzi); see Pythagorean comma § History. Thus, it was known in antiquity that a cycle of twelve fifths was almost exactly seven octaves (more practically, alternating ascending fifths and descending fourths was almost exactly an octave). However, this was theoretical knowledge, and was not used to construct a repeating twelve-tone scale, nor to modulate. This was done later in meantone temperament and twelve-tone equal temperament, which allowed modulation while still being in tune, but did not develop in Europe until about 1500. Although popularized as the circle of fifths, its Anglo-Saxon etymological origins trace back to the name “wheel of fifths.”
The system dates to Ancient Mesopotamia,[4] and consisted of alternating ascending fifths and descending fourths; see Music of Mesopotamia § Music theory. Within Ancient Greek music, the system had been mainly attributed to Pythagoras (who lived around 500 BCE) by modern authors of music theory; Ancient Greeks borrowed much of their music theory from Mesopotamia, including the diatonic scale, Pythagorean tuning, and modes. The Chinese Shí-èr-lǜ scale uses the same intervals as the Pythagorean scale and was invented between 600 BCE and 240 CE.[2][9]
Because of the wolf interval when using a 12-tone Pythagorean temperament, this tuning is rarely used today, although it is thought to have been widespread. In music which does not change key very often, or which is not very harmonically adventurous, the wolf interval is unlikely to be a problem, as not all the possible fifths will be heard in such pieces. In extended Pythagorean tuning there is no wolf interval, all perfect fifths are exactly 3:2.
Because most fifths in 12-tone Pythagorean temperament are in the simple ratio of 3:2, they sound very “smooth” and consonant. The thirds, by contrast, most of which are in the relatively complex ratios of 81:64 (for major thirds) and 32:27 (for minor thirds), sound less smooth depending on the instrument.[10]
From about 1510 onward, as thirds came to be treated as consonances, meantone temperament, and particularly quarter-comma meantone, which tunes thirds to the relatively simple ratio of 5:4, became the most popular system for tuning keyboards. At the same time, syntonic-diatonic just intonation was posited first by Ramos and then by Zarlino as the normal tuning for singers.
However, meantone presented its own harmonic challenges. Its wolf intervals proved to be even worse than those of the Pythagorean tuning (so much so that it often required 19 keys to the octave as opposed to the 12 in Pythagorean tuning). As a consequence, meantone was not suitable for all music. From around the 18th century, as the desire grew for instruments to change key, and therefore to avoid a wolf interval, this led to the widespread use of well temperaments and eventually equal temperament.
Pythagorean temperament can still be heard in some parts of modern classical music from singers and from instruments with no fixed tuning such as the violin family. Where a performer has an unaccompanied passage based on scales, they will tend towards using Pythagorean intonation as that will make the scale sound best in tune, then reverting to other temperaments for other passages (just intonation for chordal or arpeggiated figures, and equal temperament when accompanied with piano or orchestra). Such changes are never explicitly notated and are scarcely noticeable to the audience, just sounding ‘in tune’.
According to Ervin Laszlo, the coherence of the atom and the galaxies is the same coherence that keeps living cells together, cooperating to form life. When a complex system made up of many interacting parts is operating, sometimes an unexpected jump to a new level of complex organization happens. Our human body is made up of many such levels, each formed by another jump in complexity. Our lowest level of the cell jumps up a level to body tissue, to body organ, to body system and to the whole body. We are therefore formed with many onion skin like levels that all cooperate in complex ways to make one whole human being. It is really amazing how it all fits together.
At the same time, in the human realm of consciousness, we are – as far as we know – the only creatures able to contemplate who we are, why we are here and how we fit into the universe. We can even contemplate on the fact that we can contemplate about who we are and how we fit into the universe. This coherence also allows evolution to happen and that has enabled us to evolve from a microscopic bacterium right through to the complex beings that we are with all our mental, physical, emotional and spiritual capabilities.
Ervin Laszlo presents a theory that helps to tie both together. He proposes that the quantum vacuum –which we know contains all the information of our history from the Big Bang to now – is also consciousness. Everything in the universe therefore has consciousness; from a pebble to a tree, to a cloud, to a person. While this goes against the view of mainstream science, there are some highly respected scientists such as Freeman Dyson, David Bohm and Fritjof Capra, who support the idea that the universe is in fact conscious. Ervin Laszlo says that life happens because it comes from the quantum vacuum.
What is consciousness? Consciousness is about being aware of our own existence and the environment in which we live. So if one sub-atomic particle reacts in line with another particle somewhere else in the universe, we could say it is aware of what the other one is doing. In a way it is aware of itself in the universe. So, the question is: Is it enough to say all particles in the universe are conscious?
We are conscious of our existence and have evolved a brain able to access and use the consciousness held in quantum vacuum. Consciousness is yet another manifestation of coherence allowing a mass of nerve cells to co-operate and form a unified sense of self.
Ervin Laszlo equates this quantum vacuum with the Akashic Field of ancient Hindu spiritual tradition. The Hindu say the Akashic record is a field from which all the universe is formed and which holds all that ever was, is or will be. The Hindu also say that the Big Bang that started the universe, and the big crunch that will happen when the universe goes into reverse and collapses back into itself, is only a part of many cycles of universes, just like ours, appearing and disappearing, just like the subatomic particles in our world.
Ervin Laszlo states that information can be transferred from one cycle to the next, which explains how the precise numbers for gravity, electromagnetism etc. come to be so exact when there has not been enough time for these to have formed randomly. Those numbers are transferred from previous universes.
The Akashic Field, being the background to the subatomic worlds, also flows through the other realms of stars, galaxies and human life, and is an activating force in all those realms. It is the force moving the stars and galaxies and the spark that gives life to bunches of molecules, driving the power of evolution and giving us the ability to develop our consciousness and experience the unity of the universe.
Ervin Laszlo’s theory says we are therefore linked to all people who have ever lived, and we can get access to them by accessing the Akashic field. This can explain life after death, because the past has never gone away, telling us that the past is ever present in everything we do. It clearly points to a universe where all is one and everything is linked. And if we tune ourselves into the Akashic field, we can access abilities that appear to be supernatural, but are in fact completely natural. Activities such as meditation can help us plug into the Akashic field and become much more than we are at present. Science and spirituality do not need to be set against each other as we have tended to think for so long.
Ervin Laszlo links the world of science with spiritual traditions, presenting it in a seemingly clear and logical way, incorporating all the latest research in so many fields of scientific research and tying it all together into a wonderful cohesive theory that makes sense of so many strange and contradictory parts of the universe. It explains what is usually called the supernatural in natural terms, giving strong evidence for the truth of such things as clairvoyance, reincarnation and spiritual healing.
Information In the universe, states Ervin Laszlo, information is entirely basic. In the latest conception the universe doesn’t consist of matter and space, it consists of energy and information. Energy exists in the form of wave-patterns and wave-propagations in the quantum vacuum that fills space; in its various forms, energy is the “hardware” of the universe. The software is information. The universe is not an assemblage of bits of inert matter moving passively in empty space, it is a dynamic and coherent whole. The energy that constitutes its hardware is always and everywhere in-formed. It is in-formed by what David Bohm called the implicate order and what physicists now regard as the quantum vacuum or zero-point field (also called physical spacetime, universal field, or nuether). This is the in-formation that structures the physical world, the information we grasp as the laws of nature. Without information the energy waves and patterns of the universe would be as random and unstructured as the behavior of a computer without its software. But the universe is not random and unstructured; it is precisely in-formed. Would it be any less precisely informed, complex systems could not have emerged in it, and we would not be here to ask how this on first sight highly improbable development could have come about.
The answer science has to the ‘what’ question refers to an entangled, holographic, non-local connecting in-formation field in the cosmos. In his books – in greatest detail in Science and the Akashic Field – he discusses the evidence for this field and notes that the Hindu seers referred to it as Akasha, the fundamental element of the cosmos. In recognition of this feat of insight, he is calling the information field of the universe the Akashic Field.
But how does the scientific answer to the question regarding the fundamental significance of the spiritual experience relate to the answer given by religion?
For the world’s religions, the larger and deeper reality to which the spiritual experience connects us is a numinous, divine reality. It is either a spirit or consciousness that infuses the natural world (the immanentist view), or a spirit or consciousness that is above and beyond it (the transcendentalist claim). Traditional polytheistic religions leaned toward the former, while the Abrahamic monotheistic religions (with some exceptions) embraced the latter.
The difference between a divine intelligence immanent in the world and one that transcends it is not negligible, but it is still just a difference in interpretation. The raw data for both positions is the same: it is the spiritual experience, a quantum communion with universal oneness. In the Western religious perspective this is communion with the spirit that infuses the cosmos, identified as God. Deepak Chopra writes, “Spirituality is the experience of that domain of awareness where we experience our universality. This domain of awareness is a core consciousness that is beyond our mind, intellect, and ego. In religious traditions this core consciousness is referred to as the soul which is part of a collective soul or collective consciousness, which in turn is part of a more universal domain of consciousness referred to in religions as God.”
Our experience of the core consciousness of the world is ultimately an experience of the universal domain of consciousness Western religions call God. The experience itself, if not its interpretation, is the same in all religions, and in all religions it inspires a sense of oneness and belonging. Michael Beckwith affirms that “when you strip away the culture, history, and dogma of every religion, the teachers of those religions were teaching very similar principles and practices that led to a sense of oneness, that ended a sense of separation from the Whole.”
Science’s answer to the question of what the spiritual experience connects us to is immanentist. The information that underlies the universe, the Akashic Field, is part of the universe. This doesn’t mean that the immanentist position necessarily states the ultimate truth; it only means that science can only take an immanentist position. Scientists are limited to speaking about the natural world; they must leave speculation about transcendent realities to poets, philosophers, and spiritual masters.
It’s time to conclude. If the substance of the spiritual experience is always and everywhere the same, differences in its expression and interpretation are secondary and not a valid cause for conflict and intolerance.
The world to which our quantum brain connects us is fundamentally one, whether its oneness is due to an information field within the natural world or the work of a divine transcendent intelligence. To enter into communion with this oneness has been the quest of all the great teachers and spiritual masters. And to understand the nature of this oneness has been, and is, the ultimate quest of all great scientists.
Still today, physicists seek the one equation that would anchor their famous “Theory of Everything,” the theory that would account for all the laws of nature and explain everything that ever happened in our integrally whole universe. Einstein said that knowing this equation would be reading the mind of God.
About the Author
(image) David Storoy is a deputy head of a Norwegian interest organization in mental health care called White Eagle. His main work is in the community of Bergen as a consultant in the archive of building projects.
His main passion is practicing Vedanta teachings. Vedanta is called Science of Consciousness and he stopped chasing and searching for experience and now he is doing self-inquiry (reflections, contemplation, analyzing, logical thinking and systematizing) as a means of knowledge: Self-Knowledge of Vedanta teachings. James Swartz is his teacher and he has 40 years experience with Vedanta teachings. He has been influenced by Swami Chinmayananda and Swami Dayananda Saraswati. They follow and are influenced by a traditional Vedanta lineage.
Vedanta means end of knowledge and the source is mainly Upanishads. He is also grateful to Science and Nonduality and Deepak Chopra for the influence of following Vedanta teachings.
Source: The Consciousness Revolution
The Consciousness Revolution
Mar 6, 2022
by Ervin Laszlo The Laszlo Institute of New Paradigm Research
There is not only a revolution in the way our consciousness works, see my blog You can change your mindset, there is also a revolution in our very understanding of the nature of consciousness. There is a new concept emerging at the cutting edge of science and philosophy, and this concept is very different from the old established concept.
Consciousness is at the same time the most familiar and the most mysterious element of our life. Consciousness is mysterious because it is not clear what it is and where it comes from. Is the flow of sensations that makes up our consciousness generated in, and confined to, our brain? Or does it extend in some way beyond our body and brain? The new concept opts for the latter. And if the new concept is true, we are not what we thought we were, and the world is not what we thought it was. Consciousness in the new conception is more than a plaything of our imagination—it is the very substance of our beings. That of course is not the old concept, but /the heart of the new one.
Here I shall suggest the basic features of the new concept, but first I outline the old idea, so as to see the differences.
The old idea of consciousness
Until a few years ago, nobody other than deeply spiritual or religious people would have subscribed to the proposition that consciousness is more than a product of the workings of the brain. The accepted concept of consciousness was consistent with the physics of Newton. In the Newtonian universe, there is no place for consciousness. In the last count, all that exists in the universe are bits of matter moving in passive space and equitably flowing time. Consciousness is an epiphenomenon: something generated by real phenomena but is not real in itself. Consciousness is like the electricity generated by a stream of electrons in a turbine. The electrons are real, the turbine is real, but the electricity generated by them is a secondary phenomenon. It disappears when the electrons cease to move in the turbine. The existence of electricity is contingent on the working of the turbine, just as the existence of consciousness is contingent on the working of the brain. After all, consciousness can no more exists in a dead brain than electric charge could exists in a stationary turbine.
We do not see, hear, or taste electricity; we know it only by the effect it produces. This is said to be the same with consciousness. We experience the flow of sensations, feelings, and intuitions we call consciousness, but we do not perceive consciousness itself. No amount of scrutiny of the brain will disclose anything we could call consciousness. We only find gray matter with networks of neurons firing in sequence, creating the flow of electrons that generates the sensations we experience. When the brain is damaged, consciousness is distorted, and when the brain stops working, consciousness ceases.
For the classical concept there is nothing mysterious, about the presence of consciousness in the universe. Human consciousness is the product of the workings of the human brain.
The new concept of consciousness
The turbine concept of consciousness is a hypothesis and, as other hypotheses, it can be upheld if the predictions generated by it are confirmed by observations. In this instance, the relevant prediction is that when the brain stops working, consciousness will vanish This is confirmed by observation. People who are brain-dead do not possess consciousness.
The above claim does not admit of exceptions. We can no more account for the presence of consciousness in a dead brain than we could account for the presence of electric charge in a stationary turbine. Evidence to the contrary would place in question the basic tenet of the old concept. But evidence to the contrary does exist. It surfaces in rigorously protocolled experiments. There is real and credible evidence today that in some cases consciousness does not cease when brain function does.
The most widely known evidence is furnished by people who have reached the portals of death but returned to the ranks of the living. In some cases, their consciousness persists even when their brain functions are “flat.” Many temporarily brain-dead people report having had conscious experiences during their near-death episode. NDEs—near-death experiences—are surprisingly widespread: in some cases they are reported by up to 25 percent of the people who experienced a condition near physical death.
There are indications that conscious experience persists not only during the temporary cessation of brain function, but also in its permanent absence: when the individual is fully and irreversibly dead. These surprising experiences became known as ADEs: after-death experiences. The evidence for them is offered by mediums in deeply altered states of consciousness. In these trance-states they appear able to communicate with deceased persons. They “hear” the deceased recount their experiences after they have died and in some cases experience visual contact with them as well.
Reports of ADEs have been subjected to systematic scrutiny, exploring the possibility that the mediums would have invented the messages, or picked them up from living persons through some form of extrasensory perception. In a non-negligible number of cases, the theory that they were invented by the mediums or received by them in some nonordinary way could be ruled out: the messages contained surprisingly accurate information the mediums were unlikely to have accessed or invented themselves.
Given the mounting stream of evidence, we are logically obliged to accept that consciousness does not always and necessarily cease with the death of the brain that produced it. But, perhaps, the brain did not actually produce it? The new concept claims that consciousness is more than a product of brain function. “Our” consciousness is a local and temporary manifestation of a consciousness that is an element of the real world. More and more consciousness researchers, brain scientists, psychologists, and psychiatrists uphold this concept. Consciousness could and does exist beyond the brain.
In the new concept of consciousness, the flow of sensations we call consciousness is as real as energy, frequency, amplitude, phase, and information, and more real than “matter.” The brain is not a material turbine that generates consciousness, and consciousness is not its by-product. Consciousness is a real-world phenomenon. The brain is not its generator, only its receiver and transmitter.
Consciousness exists as a real phenomenon in the universe, and this phenomenon is universal; it is “one.” Famed quantum physicist Erwin Schrödinger said that consciousness in the world does not exist in the plural: the overall number of minds in the world is one. In his last years, Carl Jung came to a similar conclusion. The psyche is not a product of the brain and is not located within the skull; it is part of the one-universe: of the unus mundus. In David Bohm’s quantum physics, the roots of consciousness are traced to the deep reality of the cosmos: the implicate order. A number of contemporary scientists, such as Henry Stapp, elaborate this concept. Consciousness, they say, is nonlocal: it is present throughout the universe.
The quantum scientists revive an ancient wisdom: We are connected through our participation in the world’s one-consciousness. This is a very different condition from being a separate entity with a separate brain producing a separate consciousness.
The new concept of consciousness is more than a theory of consciousness: it is a revolution in our understanding of being.
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
Source: Ervin Laszlo’s Akashic Field and The Dilemmas of Modern Consciousness Research
My Related Posts
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The Great Chain of Being
Indira’s Net: On Interconnectedness
Geometry of Consciousness
Charles Sanders Peirce’s Continuum
On Synchronicity
On Holons and Holarchy
Hua Yan Buddhism : Reflecting Mirrors of Reality
What is Yogacara Buddhism (Consciousness Only School)?
Mind, Consciousness, and Quantum Entanglement
Consciousness of Cosmos: A Fractal, Recursive, Holographic Universe
Law of Dependent Origination
Five Types of System Philosophy
Systems View of Life: A Synthesis by Fritjof Capra
Key Sources of Research
Science and the Akashic Field: An Integral Theory of Everything
This book offers an original hypothesis capable of unifying evolution in the physical universe with evolution in biology; herewith it lays the conceptual foundations of “transdisciplinary unified theory”. The rationale for the hypothesis is presented first; then the theoretical framework is outlined, and thereafter it is explored in regard to quantum physics, physical cosmology, micro– and macro–biology, and the cognitive sciences (neurophysiology, psychology, with attention to anomalous phenomena as well). The book closes with a variety of studies, both by the author and his collaborators, sketching out the implications of the hypothesis in regard to brain dynamics, cosmology, the concept of space, phenomena of creativity, and the prospects for the elaboration of a mature transdisciplinary unified theory. The Foreword is written by philosopher of science Arne Naess, and the Afterword is contributed by neuroscientist Karl Pribram.
The Systems View of the World: The Natural Philosophy the New Developments in the Sciences
The Age of Bifurcation: Understanding the Changing World
Ervin Laszlo Gordon and Breach, 1991 – Bifurcation theory – 126 pages
The Connectivity Hypothesis: Foundations of an Integral Science of Quantum, Cosmos, Life, and Consciousness
Author Ervin Laszlo Contributor Ralph H. Abraham Edition illustrated Publisher State University of New York Press, 2003 ISBN 0791457850, 9780791457856 Length 147 pages
The Evolution of Cognitive Maps: New Paradigms for the Twenty-first Century
Volume 5 of World futures general evolution studies, ISSN 1043-9331 Editor Ervin Laszlo Edition illustrated, reprint Publisher Psychology Press, 1993 ISBN 2881245595, 9782881245596 Length 292 pages
System, Structure, and Experience: Toward a Scientific Theory of Mind
Human Values and Natural Science: Proceedings, Volume 3
Volume 4 of Current topics of contemporary thought Human Values and Natural Science: Proceedings, State University of New York College, Geneseo
Editors Ervin Laszlo, James Benjamin Wilbur Contributor State University of New York College, Geneseo Publisher Gordon and Beach, 1970 Original from University of Minnesota Digitized Jan 19, 2010 ISBN 0677139608, 9780677139609 Length 292 pages
Introduction to Systems Philosophy: Toward a New Paradigm of Contemporary Thought
Harper Torchbooks: Philosophy
Author Ervin Laszlo Edition illustrated Publisher Gordon and Breach, 1972 ISBN 067703850X, 9780677038506 Length 328 pages
Studies on the Conceptual Foundations: The Original Background Papers for Goals for Mankind
We Are in the Midst of a Global Transformation (pt. 2 of 2)
Prolific author and philosopher Ervin Laszlo discusses his most recent books, in which he outlines how the latest discoveries in science converge with spiritual insights and point to the ways in which society might evolve in ways that will help overcome contemporary crises.
Consciousness in the Universe is Tuned by a Musical Master Code, Part 3: A Hydrodynamic Superfluid Quantum Space Guides a Conformal Mental Attribute of Reality
In its original form the Titius-Bode law appeared under Bonnet’s name in 1766. In 1772 Titius identified himself as the author of the law, but in the same year Bode borrowed under his own name Titius’ formulation of the law. Titius attributed the law, wholly arbitrarily, to Bonnet, Lambert, and Wolff. From 1772 until 1787 Bode was practically alone among astronomers to mention the law in its primitive, sequential form, in his various writings. The algebraic, functional form of the law was given by Wurm in 1787. The distance of Uranus, discovered in 1781, fitted well into the law which inspired the search for the missing planet between Mars and Jupiter. The discovery of Ceres in 1801 was a triumph of the law only until the discovery of Pallas in 1802, which produced the opposite effect. Most leading astronomers of the period considered the law as a mere game with numbers.
Source: The Original Formulation of the Titius-Bode Law
Source: Stamping Through Astronomy
Source: Stamping Through Astronomy
Source: Jovicentricity in the Solar System: The history of a discovery
The Titius-Bode law is simple mathematical relation describing the distances of planets from the sun. The relation comes from starting with a simple arithmetic progression of numbers:
0, 3, 6, 12, 24, 48, 96, 192, 384.
Note that each number is twice the previous. Then, by adding 4 to each number and dividing the result by 10, this yields a sequence of numbers that roughly corresponds to the spacing of planets in our solar system out to Uranus (in AU):
Mercury
Venus
Earth
Mars
Asteroid Belt
Jupiter
Saturn
Uranus
Neptune
Pluto
Predicted:
0.4
0.7
1
1.6
2.8
5.2
10
19.6
38.8
77.2
Actual:
0.39
0.72
1
1.52
2.7
5.2
9.54
19.19
30.1
39.5
When it was initially published, it was found that this law correctly predicts the distances of all known planets from Mercury to Saturn. It also correctly predicted the (then unknown) locations of the asteroid belt and Uranus, but not for Neptune or Pluto (Fig. 1). The Titius-Bode relation has been the subject of much speculation, but the so-called “law” is now largely thought to be a mathematical coincidence rather than an actual physical law since it is not well physically motivated and fails to apply to the outermost planets in our solar system. Even so, there have been suggestions that this relation is a mathematical result of orbital resonances and gravitational interactions within multi-body planetary systems.
It is difficult to say whether the Titius-Bode relation has any deeper significance just from looking in our own solar system. However, the plethora of exoplanet discoveries over the past several years allows for a larger sample of planetary systems in which we can examine this relation. In the four years that the Keplermission has been active, over 3000 extrasolar planetary systems have been discovered. About 1/5 of these planet-hosting stars are believed to host multiple planets.
In this paper, the authors use Kepler data to see if a generalized Titius-Bode relation holds for extrasolar multi-planet systems. This analysis is based on a previous paper by Bovaird & Lineweaver (hereafter BL13), in which the authors attempt to test the Titius-Bode relation on known extrasolar planetary systems. BL13 predicts that these extrasolar systems should follow a Titius-Bode relation (one that is modified and generalized from the relation that applies to the solar system), and that there may be undetected planets that fit into this relation. Specifically, the paper predicted the existence of 141 additional exoplanets in 68 multiple-exoplanet systems.
The analysis in this paper focuses on Kepler data taken over a 100 day time span. From examining the light curves from 56 exoplanet systems, the authors only managed to detect 5 of the predicted planets. That is, a majority of the planets predicted from the modified Titius-Bode relation were not found.
It is also possible that there are observational biases that prevent these “missing” planets from being detected. For example, it is assumed that most of the planets in a planetary system will lie roughly in the same orbital plane. This is not necessarily true, and any strong deviations in orbital inclination angle will reduce the number of expected observable transits. Additionally, planets could also avoid detection due to their small size and lack of observed signal in their light curves. After taking these factors into consideration, the authors predict that they should find roughly 15 planets that obey a Titius-Bode relation.
The authors ultimately only detect only 5 of the 141 predicted planets. Even after correcting for observational biases, this number is significantly smaller than expected. The authors conclude that it is questionable that a Titius-Bode relation will hold for all extrasolar planetary systems. Even if the Titius-Bode relation turns out to be a mathematical oddity, it is still insightful to see if our own solar system shares any common characteristics with any extrasolar counterparts.
About the Author
About Anson Lam
I am a graduate student at UCLA, where I am working with Steve Furlanetto on models of galaxy clustering and their applications to the reionization era. My main interests involve high redshift cosmology, dark matter, and structure formation. Previously, I was an undergraduate at Caltech, where I did my BS in astrophysics. When I’m not doing astronomy, I enjoy engaging in some linear combination of swimming/biking/running.
Source: The complex planetary synchronization structure of the solar system
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: Solar System Dynamics
Source: The Titius-Bode Law of Planetary Distances Its History and Theory
Source: The Titius-Bode Law of Planetary Distances Its History and Theory
Source: The Titius-Bode Law of Planetary Distances Its History and Theory
Source: Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
Source: Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
Source: Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
Source: Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
Source: Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
Source: Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
Source: Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
The HARPS search for southern extra-solar planets⋆
XXVII. Up to seven planets orbiting HD 10180: probing the architecture of low-mass planetary systems
C. Lovis1, D. Se ́gransan1, M. Mayor1, S. Udry1, W. Benz2, J.-L. Bertaux3, F. Bouchy4, A. C. M. Correia5, J. Laskar6, G. Lo Curto7, C. Mordasini8,2, F. Pepe1, D. Queloz1, and N. C. Santos9,1
Astronomy & Astrophysics manuscript no. HD10180 August 13, 2010
Scaling, Mirror Symmetries and Musical Consonances Among the Distances of the Planets of the Solar System
Michael J. Bank1†
Nicola Scafetta2*†
1Danbury Music Centre, Danbury, CT, United States 2Department of Earth Sciences, Environment and Georesources, University of Naples Federico II, Complesso Universitario di Monte S. Angelo, Naples, Italy
Exoplanet Predictions Based on Harmonic Orbit Resonances
by Markus J. Aschwanden 1 and Felix Scholkmann 2,*
1 Lockheed Martin, Solar and Astrophysics Laboratory, Org. A021S, Bldg. 252, 3251 Hanover St., Palo Alto, CA 94304, USA 2 Research Office for Complex Physical and Biological Systems, Mutschellenstr. 179, 8038 Zürich, Switzerland *Author to whom correspondence should be addressed.
The Planetary Theory of Solar Activity Variability: A Review
Nicola Scafetta1*
Antonio Bianchini2* 1Department of Earth Sciences, Environment and Georesources, Complesso Universitario di Monte S. Angelo, University of Naples Federico II, Naples, Italy 2INAF, Astronomical Observatory of Padua, Padua, Italy
Overview of the Spectral Coherence between Planetary Resonances and Solar and Climate Oscillations
by Nicola Scafetta 1,* and Antonio Bianchini 2
1 Department of Earth Sciences, Environment and Georesources, University of Naples Federico II, Complesso Universitario di Monte S. Angelo, via Cinthia, 21, 80126 Napoli, Italy 2 INAF, Astronomical Observatory of Padua, Vicolo Osservatorio 5, 35122 Padova, Italy * Author to whom correspondence should be addressed.
Climate 2023, 11(4), 77; https://doi.org/10.3390/cli11040077 Received: 8 March 2023 / Revised: 23 March 2023 / Accepted: 25 March 2023 / Published: 27 March 2023 (This article belongs to the Special Issue Natural Drivers of Climate Change: Emerging Research)
Agol, E., Dorn, C., Grimm, S. L., Turbet, M., Ducrot, E., Delrez, L., et al. (2021). Refining the Transit-Timing and Photometric Analysis of TRAPPIST-1: Masses, Radii, Densities, Dynamics, and Ephemerides. Planet. Sci. J. 2, 1. doi:10.3847/psj/abd022
Aschwanden, M. J. (2018). Self-organizing Systems in Planetary Physics: Harmonic Resonances of Planet and Moon Orbits. New Astron. 58, 107–123. doi:10.1016/j.newast.2017.08.002
Beer, J., Tobias, S. M., and Weiss, N. O. (2018). On Long-Term Modulation of the Sun’s Magnetic Cycle. Monthly Notices R. Astronomical Soc. 473, 1596–1602. doi:10.1093/mnras/stx2337
Bones, O., Hopkins, K., Krishnan, A., and Plack, C. J. (2014). Phase Locked Neural Activity in the Human Brainstem Predicts Preference for Musical Consonance. Neuropsychologia 58, 23–32. doi:10.1016/j.neuropsychologia.2014.03.011
Cartwright, J. H. E., González, D. L., and Piro, O. (2021). Dynamical Systems, Celestial Mechanics, and Music: Pythagoras Revisited. Math. Intelligencer 43, 25–39. doi:10.1007/s00283-020-10025-x
Charvàtovà, I. (1997). Solar-terrestrial and Climatic Phenomena in Relation to Solar Inertial Motion. Surv. Geophys. 18, 131–146. doi:10.1023/a:1006527724221
Gillon, M., Triaud, A. H. M. J., Demory, B.-O., Jehin, E., Agol, E., Deck, K. M., et al. (2017). Seven Temperate Terrestrial Planets Around the Nearby Ultracool dwarf star TRAPPIST-1. Nature 542 (7642), 456–460. doi:10.1038/nature21360
Kepler, J. (1997). “Harmonices Mundi (1619),” in The Harmony of the World, Translated by. Editors E. J. Aiton, A. M. Duncan, and J. V. Field (Philadelphia: American Philosophical Society).
McDermott, J. H., Lehr, A. J., and Oxenham, A. J. (2010). Individual Differences Reveal the Basis of Consonance. Curr. Biol. 20 (11), 1035–1041. doi:10.1016/j.cub.2010.04.019
Moons, M., Morbidelli, A., and Migliorini, F. (1998). Dynamical Structure of the 2/1 Commensurability with Jupiter and the Origin of the Resonant Asteroids. Icarus 135, 458–468. doi:10.1006/icar.1998.5963
Moons, M., and Morbidelli, A. (1995). Secular Resonances in Mean Motion Commensurabilities: The 4/1, 3/1, 5/2, and 7/3 Cases. Icarus 114, 33–50. doi:10.1006/icar.1995.1041
Scafetta, N. (2014b). Discussion on the Spectral Coherence between Planetary, Solar and Climate Oscillations: a Reply to Some Critiques. Astrophys Space Sci. 354, 275–299. doi:10.1007/s10509-014-2111-8
Scafetta, N., Milani, F., Bianchini, A., and Ortolani, S. (2016). On the Astronomical Origin of the Hallstatt Oscillation Found in Radiocarbon and Climate Records throughout the Holocene. Earth-Science Rev. 162, 24–43. doi:10.1016/j.earscirev.2016.09.004
Scafetta, N. (2020). Solar Oscillations and the Orbital Invariant Inequalities of the Solar System. Sol. Phys. 295 (2), 33. doi:10.1007/s11207-020-01599-y
Stefani, F., Stepanov, R., and Weier, T. (2021). Shaken and Stirred: When Bond Meets Suess-De Vries and Gnevyshev-Ohl. Sol. Phys. 296, 88. doi:10.1007/s11207-021-01822-4
“There is geometry in the humming of the strings, there is music in the spacing of the spheres.” — Pythagoras
Counting, rhythm, scales, intervals, patterns, symbols, harmonies, time signatures, overtones, tone, pitch. The notations of composers and sounds made by musicians are connected to mathematics. The next time you hear or play classical, rock, folk, religious, ceremonial, jazz, opera, pop, or contemporary types of music, think of what mathematics and music have in common and how mathematics is used to create the music you enjoy.
With apologies to the musical Grease, mathematics and music go together like rama lama lama ke ding a de dinga a dong. You need to look no further than Plusto see how the links between the two have fascinated researchers for centuries — see all Plus articles tagged with the mathematics and music tag.
Their article Geometrical Music Theory, published in the April 18 edition of Science, outlines their theory that musical operations, such as transpositions, can be expressed as symmetries of n-dimensional space.
They categorise sequences such as chords, rhythms and scales into mathematical “families”. The families can be represented by points on the complex plane, and different types of categorisation produce different geometrical spaces. The authors argue that through this method researchers will be able to analyse music more deeply and understand how music has changed over time. They also say that the theory will allow the comparison of many kinds of Western music, although only some non-Western styles. This is because the theory is based on concepts such as the “chord”, which are present in nearly all Western styles, but not all non-Western styles.
The basis of geometrical music theory is that it provides a unified mathematical framework for musical events that are described differently depending on the scenario, but are fundamentally the same. For example, a “C” followed by the “E” and “G” above it may be described as a “C major chord,” “an ascending C major arpeggio,” “a major chord” and the list goes on. The authors describe five different methods of categorising such collections of notes. These are called “OPTIC symmetries,” with each letter of OPTIC representing a different categorisation method that may, for example, look at what octaves the notes are in, their order of play, or how many times each note is played. Each categorisation focuses on one aspect of the music, ignoring the others. The five symmetries can then be combined together to produce different musical concepts, some familiar, some new. Three-note chords end up on a triangular doughnut while other chord types live on the surface of a cone.
Tymoczko believes that their theory can be used to investigate the differences between musical styles. “Our methods are not so great at distinguishing Aerosmith from The Rolling Stones,” he said. “But they might allow you to visualise some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music.”
The authors even hope that through their work, new musical instruments may be developed. “You could create new kinds of musical instruments or new kinds of toys,” said Tymoczko. He also envisaged new visual shows that could accompany the music. “Imagine going to a classical music concert where the music was being translated visually.”
So next time you go see a visually spectacular Madonna or Kylie show, look at the big screens — you might learn some maths!
The new shape of music: Music has its own geometry, researchers find
The figure shows how geometrical music theory represents four-note chord-types — the collections of notes form a tetrahedron, with the colors indicating the spacing between the individual notes in a sequence. In the blue spheres, the notes are clustered, in the warmer colors, they are farther apart. The red ball at the top of the pyramid is the diminished seventh chord, a popular 19th-century chord. Near it are all the most familiar chords of Western music. Credit: Dmitri Tymoczko, Princeton University
The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.
And the so-called musica universalis or “music of the spheres” emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies — the sun, moon and planets — could be viewed as a form of music, inaudible but perfectly harmonious.
Now, three music professors – Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University — have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.
Writing in the April 18 issue of Science, the trio has outlined a method called “geometrical music theory” that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into “families.” They have found a way to assign mathematical structure to these families, so they can then be represented by points in complex geometrical spaces, much the way “x” and “y” coordinates, in the simpler system of high school algebra, correspond to points on a two-dimensional plane.
Different types of categorization produce different geometrical spaces, and reflect the different ways in which musicians over the centuries have understood music. This achievement, they expect, will allow researchers to analyze and understand music in much deeper and more satisfying ways.
The work represents a significant departure from other attempts to quantify music, according to Rachel Wells Hall of the Department of Mathematics and Computer Science at St. Joseph’s University in Philadelphia. In an accompanying essay, she writes that their effort, “stands out both for the breadth of its musical implications and the depth of its mathematical content.”
The method, according to its authors, allows them to analyze and compare many kinds of Western (and perhaps some non-Western) music. (The method focuses on Western-style music because concepts like “chord” are not universal in all styles.) It also incorporates many past schemes by music theorists to render music into mathematical form.
“The music of the spheres isn’t really a metaphor — some musical spaces really are spheres,” said Tymoczko, an assistant professor of music at Princeton. “The whole point of making these geometric spaces is that, at the end of the day, it helps you understand music better. Having a powerful set of tools for conceptualizing music allows you to do all sorts of things you hadn’t done before.”
Like what?
“You could create new kinds of musical instruments or new kinds of toys,” he said. “You could create new kinds of visualization tools — imagine going to a classical music concert where the music was being translated visually. We could change the way we educate musicians. There are lots of practical consequences that could follow from these ideas.”
“But to me,” Tymoczko added, “the most satisfying aspect of this research is that we can now see that there is a logical structure linking many, many different musical concepts. To some extent, we can represent the history of music as a long process of exploring different symmetries and different geometries.”
Understanding music, the authors write, is a process of discarding information. For instance, suppose a musician plays middle “C” on a piano, followed by the note “E” above that and the note “G” above that. Musicians have many different terms to describe this sequence of events, such as “an ascending C major arpeggio,” “a C major chord,” or “a major chord.” The authors provide a unified mathematical framework for relating these different descriptions of the same musical event.
The trio describes five different ways of categorizing collections of notes that are similar, but not identical. They refer to these musical resemblances as the “OPTIC symmetries,” with each letter of the word “OPTIC” representing a different way of ignoring musical information — for instance, what octave the notes are in, their order, or how many times each note is repeated. The authors show that five symmetries can be combined with each other to produce a cornucopia of different musical concepts, some of which are familiar and some of which are novel.
In this way, the musicians are able to reduce musical works to their mathematical essence.
Once notes are translated into numbers and then translated again into the language of geometry the result is a rich menagerie of geometrical spaces, each inhabited by a different species of geometrical object. After all the mathematics is done, three-note chords end up on a triangular donut while chord types perch on the surface of a cone.
The broad effort follows upon earlier work by Tymoczko in which he developed geometric models for selected musical objects.
The method could help answer whether there are new scales and chords that exist but have yet to be discovered.
“Have Western composers already discovered the essential and most important musical objects?” Tymoczko asked. “If so, then Western music is more than just an arbitrary set of conventions. It may be that the basic objects of Western music are fantastically special, in which case it would be quite difficult to find alternatives to broadly traditional methods of musical organization.”
The tools for analysis also offer the exciting possibility of investigating the differences between musical styles.
“Our methods are not so great at distinguishing Aerosmith from the Rolling Stones,” Tymoczko said. “But they might allow you to visualize some of the differences between John Lennon and Paul McCartney. And they certainly help you understand more deeply how classical music relates to rock or is different from atonal music.”
The deep connection between music and mathematics was recognized at least as early as the time of Pythagoras. Now, Ian Quinn, an assistant professor in Yale’s music department and its cognitive science program, and his colleagues have devised a new mathematical means of understanding music. This “geometrical music theory” can translate the language of music theory into that of contemporary geometry and create visual representations of music’s underlying mathematical structure.
In the April 18 issue of Science, they describe five ways (“symmetries”) of categorizing groups of notes that are similar but not identical: the same note in different octaves, or the same group of notes in a different order. Then they show how these symmetries can be combined to map musical works in coordinate space where, for instance, two-note chords take the shape of a Mobius strip, three-note chord types take the shape of a three-dimensional cone, and four-note chord types somewhat resemble a pyramid.
“We can put any music into the model,” Quinn says, “and visualize the structure behind similarities and differences among musical styles—why Chopin, for instance, sounds different from Mozart.” Or Lennon from McCartney.
The translation of music theoretical terms into precise geometrical language provides a framework for investigating contemporary music-theoretical topics, Quinn says. It can also be useful in analysis, composition, pedagogy, and even the design of new kinds of instruments. Adds Quinn, “My students have used the models to write in the styles of various composers. Somewhat to my surprise these complex topics are fairly easily taught.”
Geometry shapes sound of music, FSU professor says
Through the ages, the sound of music in myriad incarnations has captivated human beings and made them sing along, and as scholars have suspected for centuries, the mysterious force that shapes the melodies that catch the ear and lead the voice is none other than math.
The space of three-note chord types is a cone. Numbers refer to pitch classes, with 0 = C, 1 = C#, etc. Points represent equivalence classes of transpositionally related chords. Thus, (C, D, E) and (D, E, F#) are both instances of 024. (Image made with Dmitri Tymoczko’s “ChordGeometries” program available here.)
It’s geometry, to be more precise, and now, a trio of 21st-century music professors from Florida State University, Yale University and Princeton University have analyzed and categorized in brand-new ways the mathematics intrinsic to musical harmony. Their cutting-edge collaboration has produced a powerful tool they call “geometrical music theory,” which translates the language of music theory into that of contemporary geometry.
The research is described in the April 18 issue of the journal Science, where the publication of work by music theorists and composers is rare if not unprecedented, said Clifton Callender, an assistant professor of composition in FSU’s College of Music. Callender is co-author of the paper “Generalized Voice-Leading Spaces” with Ian Quinn of Yale and Dmitri Tymoczko of Princeton.
“Our research offers a variety of tools for understanding and exploring music by drawing upon contemporary mathematics in natural and musically relevant ways,” Callender said. “It also provides a way to compare chords, and represents all possible combinations of pitches, including those found in non-Western music and avant-garde works that don’t conform to the traditional scales of Western music.”
As a result, composers could explore all sorts of uncharted musical possibilities; musicians may well be trained differently; new types of toys and musical instruments might be created; and music could be manifested visually (and geometry manifested aurally) in previously unimagined ways.
Geometrical music theory represents a culminating moment in the longstanding marriage of music and math. That marriage began when Pythagoras described pleasing musical intervals with simple mathematical ratios more than 2,600 years ago and further evolved during the Middle Ages when deep thinkers used those same ratios to model the “music of the spheres”—what many at that time believed to be the literally harmonious movements of the sun, moon and planets.
Understanding and interpreting music, say the authors of the study, is a process of discarding information—which in turn is the key to discovering its underlying mathematical structure.
A chord, for example, may be variously described as “the opening chord of Bach’s G minor Sonata for Unaccompanied Violin,” “G minor triad,” “minor triad” or simply “triad.”
“Each of these terms can refer to the same musical object at different levels of abstraction,” Callender said.
“We also experience a sense of distance when moving from one chord to another,” he said. “Changing one note just a little feels like a small motion between similar chords, while changing many notes by large amounts feels like a large motion between dissimilar chords.
“So, building on my own research and that of my Princeton colleague, our research modeled these spatial intuitions about chords at various levels of abstraction geometrically, using what mathematicians call ‘quotient spaces.’ Most of those spaces are warped and twisted such that they contain multiple ‘straight’ paths connecting any pair of points,” Callender said.
“Imagine being near the peak of a mountain and needing to get to the immediately opposite location,” he said. “You could proceed clockwise around the peak, counter-clockwise, or directly over the peak. These same three paths represent unique types of motions between major and minor triads in the space of three-note chord types, which is a cone. In fact, these motions and chords have been ubiquitous in Western music since medieval times to the present day.”
At each level of abstraction, musical objects are grouped into families of chords or melodies. Mathematical structure is assigned to the “families” so that they can be represented as points within complex geometrical spaces in much the same way that “x” and “y” coordinates correspond to points on a two-dimensional plane in simple high school algebra. The different families produce an exotic maze of diverse geometrical spaces such as twisted triangular donuts and pinched cones—and even some spaces that mathematicians haven’t dreamed up names for yet.
“My fellow researchers and I have found it thrilling to discover unexplored areas of mathematics in the course of solving musical problems,” Callender said.
“Professor Callender and his colleagues at Yale and Princeton are working at the forefront in this rarified area of music theory,” said Don Gibson, dean of the FSU College of Music. “Their research—and its publication in Science—represents a signal achievement in the discipline.”
A few years ago, Princeton University music theorist and composer Dmitri Tymoczko was sitting in the living room of his home playing with a piece of paper. Printed on the sheet were rows and columns of dots representing all the two-note chords that can be played on a piano — AA, AB b , AB, and so on for the rows; AA, B b A, BA, and so on for the columns. It was a simple drawing, something a child could make, yet Tymoczko felt that the piece of paper was trying to show him something that no one ever had seen before.
Suddenly Tymoczko (pronounced tim-OSS-ko) realized that if he cut two triangles from the piece of paper, turned one of the triangles upside down, and reconnected the two triangles where the chords overlapped, the two-note chords on one edge of the resulting strip of paper would be the reversed versions of those on the opposite edge. If he then twisted the paper and attached the two edges, the chords would line up. “That’s when I got a tingly feeling in my fingers,” he says.
Tymoczko had discovered the fundamental geometric shape of two-note chords. They occupy the space of a Möbius strip, a two-dimensional surface embedded in a three-dimensional space. Music is not just something that can be heard, he realized. It has a shape.
He soon saw that he could transform more complex chords the same way. Three-note chords occupy a twisted three-dimensional space, and four-note chords live in a corresponding but impossible-to-visualize four-dimensional space. In fact, it worked for any number of notes — each chord inhabit ed a multidimensional space that twisted back on itself in unusual ways — a non-Euclidean space that does not adhere to the classical rules of geometry. A physicist friend told him that these odd multidimensional spaces were called orbifolds — a name chosen by the graduate students of Princeton mathematician William Thurston, who first described them in the 1970s. In the 1980s, physicists found a few applications for orbifolds in arcane areas of string theory. Now Tymoczko had discovered that music exists in a universe of orbifolds.
Tymoczko’s insight, made possible through a research collaboration with Clifton Callender from Florida State University and Ian Quinn from Yale University, has created “quite a buzz in Anglo-American music-theory circles,” says Scott Burnham, the Scheide Professor of Music History at Princeton. His work has “physicalized” music. It provides a way to convert melodies and harmonies into movements in higher dimensional spaces. It has given composers new tools to write music, has revealed new ways to teach music students, and has revealed surprising musical connections between composers as distant as Palestrina — the Italian Renaissance composer — and Paul McCartney.
In a book to be published in March by Oxford University Press, A Geometry of Music: Harmony and Counterpoint in the Extended Common Practice, Tymoczko uses the connection between music and geometry to analyze the music of the last millennium and position modern composers in a new landscape. He rejects the idea that music can be divided into distinct genres. As Tymoczko sees it, medieval polyphony, the high classical music of Beethoven and Mozart, the chromatic romanticism of Wagner and Debussy, the jazz improvisations of Bill Evans, and the Beatles’ Sgt. Pepper’s Lonely Hearts Club Band all are built on the same handful of principles. Tymoczko writes in the preface: “It would make me happy to think that these ideas will be helpful to some young musician, brimming with excitement over the world of musical possibilities, eager to understand how classical music, jazz, and rock all fit together — and raring to make some new contribution to musical culture.”
The link between geometry and music has deep roots. Sometime between 530 and 500 B.C., in the town of Kroton on the rocky southern coast of Italy, Pythagoras and his followers made one of the most consequential discoveries in the history of science. If the string of a harp is shortened by half, it creates a tone one octave above that of the unshortened string. If the original string is shortened by two-thirds, the resulting tone is separated from the octave tone by a euphonious interval we know today as a fifth. Further experimentation showed that dividing the string into four parts produces intervals now known as fourths, with fur ther divisions of the string producing the familiar 12-note chromatic scale that the Greeks bequeathed to history.
Pythagoras and his followers thought big. The rational division of the musical scale was not just beautiful or pleasing — it was a sign that the universe was constructed on a rational basis and could be understood. “It was the first consistent realization that there is a mathematical rationality in the universe and that the human mind can make sense of that rationality,” says Kitty Ferguson, the author of The Music of Pythagoras: How an Ancient Brotherhood Cracked the Code of the Universe and Lit the Path from Antiquity to Outer Space.
Two-and-a-half millennia later, fifths and fourths are still the basis not just of three-chord rock-and-roll, but of much basic music theory. Students learn how to recognize intervals and relate those intervals to different kinds of scales, like major and minor scales. They practice transposing, inverting, and modulating melodies and chords. They absorb, perhaps without fully realizing it, the mathematician Gottfried Leibniz’s injunction that music is the “unknowing exercise of our mathematical faculties.”
Tymoczko falls squarely into the mathematical tradition in music. His father, Thomas, was a well-known philosopher of mathematics at Smith College who was fascinated by the use of computers in mathematics. His sister, Julianna, is a mathematician specializing in algebraic geometry at the University of Iowa.
But Tymoczko, growing up in the 1980s in Northhampton, Mass., spent more time listening to the Talking Heads, John Coltrane, and Brian Eno than solving equations. He swapped his piano lessons for guitar lessons and began playing in bands. He entered Harvard intending to study music, but the abstract and atonal music his professors preferred left him cold, and he switched from composition to philosophy. After studying philosophy at Oxford on a Rhodes scholarship, he kicked around Harvard for a few years as a teaching assistant, composing on the side and dabbling in journalism. Finally he decided to become serious about music again and enrolled in music graduate school at the University of California, Berkeley.
Tymoczko got interested in the mathematics of music for what he calls a “selfish reason” — he wanted his music to sound better. His music is distinctly modern but also “tonal” — a term that he defines in his new book as music that adheres to several basic principles. In compositions such as “Four Dreams” and “The Agony of Modern Music” (termed “brilliant” by music critic Alex Ross), he combines the free-flowing vibe of jazz, the strangeness of 20th-century music, and the raw exuberance of the rock ‘n’ roll he listened to growing up. Combining such different kinds of music is a delicate balancing act, and to do it well, Tymoczko decided that he needed to understand music at a more fundamental level.
Early on, Tymoczko recognized that tonal music has two major distinguishing features. First, melodies tend to move short distances from note to note — a characteristic known in music theory as “efficient voice leading.” Think of the tunes you can hum off the top of your head. They probably have notes that are fairly close to each other, rather than strings of notes that jump wildly up and down. Melodies with close-together notes are easier to sing and to play on most instruments, and they’re easier to listen to.
A second important feature of tonal music is that it uses harmonies that sound good rather than bad. For reasons that aren’t entirely clear, humans (including infants) tend to prefer certain kinds of chords to other kinds of chords. Echoing the findings of Pythagoras, we tend to like chords that divide the octave almost, but not completely, evenly. Triads, seventh chords, ninths, elevenths — the archetypal chords of everything from early polyphony to jazz — divide the octave into ever-smaller but approximately equal-sized segments, much as the vertical lines on a ruler divide feet and inches into equal-sized intervals. In contrast, notes that are next to each other sound harsh and dissonant when played together, like a child banging his fists on a piano.
These two properties may seem to be unrelated. But the “amazing and mysterious” thing about music, Tymoczko says, is that each requires the other. Three singers can go from a pleasing C-major chord to the complementary and more plaintive A-minor chord by moving just one note: changing from CEG to CEA. Someone playing “Hey Jude” on the piano can move his or her fingers very little while moving from one sonorous chord to another. “Miraculously, the chords that sound good together and the ones that produce efficient voice leading are the same,” Tymoczko says.
Tymoczko and other music theorists knew that these obser vations must have a mathematical representation, and previ ous theorists had captured some of these properties using geo metric ideas. He developed an intense e-mail relationship with Quinn, Callender, and several other theorists who were working on the same general problem: How can music be captured mathematically in the most powerful possible way? “People had been coming up with ad hoc solutions,” says Yale’s Quinn. “We wanted to do something general.”
Initially, Tymoczko took an algebraic approach, representing chord progressions by equations. But a critical contribution came from Callender. In a paper published in Music Theory Online, he explained how you could arrange two-note chords on a two-dimensional surface, so that changes from one chord to another could be represented by movements of a point on the surface. However, this surface had some very odd properties. For example, if a point moving on the surface hit an edge it would essentially bounce off, like a billiard ball careening off a bumper. In mathematical terms, the surface had a discontinuity or “singularity.”
Callender, Quinn, and Tymoczko all knew that the unusual properties of the space meant that it somehow harbored additional, non-Euclidean dimensions. But how were those dimensions configured? This is where Tymoczko’s checkered academic history came in. From his time at Harvard and Berkeley, he had lots of mathematician and physicist friends. One suggested making a physical representation of the space and playing with it. That’s the piece of paper he printed out, stared at in his living room, and used to discover the Möbius strip representation of two-note chords. Tymoczko’s scientist friends also told him that when they make a big discovery they submit the work to Science. A 2006 paper by Tymoczko and a 2008 paper by all three collaborators became the first on music theory in Science’s 129-year history.
Tymoczko’s discovery of the orbifold structure of musical space immediately had an important consequence: It explained his earlier observations about efficient voice leading and euphonious chords. When orbifolds are used to represent musical sounds, the chords that most evenly divide the octave reside in the central regions of the space. For example, in the Möbius strip representation of all two-note chords, fourths and fifths occupy the central area of the strip, while dissonant chords of closely spaced notes huddle near the edge of the strip. Composers can move from one euphonious chord to another while moving short distances in the central region of a musical space. Movements of short distances correspond to notes that are close together, producing singable melodies.
The implications for composers are momentous, says Tymoczko. “Imagine that you’re a blind man in a city who knows how to get from your home and to a coffee shop. Suddenly your vision is restored. You realize that not only are the post office and coffee shop very close to each other, but that you can get there many other ways, and many of them are better than the way you knew.”
The geometric representation of music also provides a powerful way to analyze past compositions. In his new book, Tymoczko uses geometric concepts to “retell the history of Western music.” For example, on the Möbius strip representation of two-note chords, medieval composers of two-voice polyphony tended to remain near the center of the space, with occasional leaps to its edge whenever the two singers were to voice the same note. By the high classical period of Bach, Mozart, and Beethoven, composers had become experts at crafting harmonious pieces that fully exploited these central regions of musical orbifolds. Later composers then began to branch out, exploring new regions of musical space. By the 20th century, minimalist composers like Steve Reich were relying on burbling arpeggios of closely spaced notes right on the edges of orbifolds, while jazz composers like Miles Davis and Bill Evans were further exploring the chromatic spaces between the centers and edges of orbifolds pioneered by Wagner and Debussy.
This way of visualizing music provides new insights into how composers wrote some of the world’s most beautiful music. In his living room, Tymoczko plays Chopin’s “Prelude in E” through the speakers of his laptop as the computer mon itor displays a three-dimensional projection of a four-dimensional orbifold. With each new chord, a ball moves through a latticework of points on the screen. Repeatedly the prelude returns to a particular point in the lattice — representing a diminished seventh chord — from which it branches first to a chord on the immediate right and then to a chord on the immediate left. But at other points in the prelude, the ball moves freely along the lattice, a kind of improvisation more commonly associated with modern music. “Composers in the 19th century had an intuitive understanding of the bizarre geometry of musical chord space,” Tymoczko says. “In fact, they had a better feel for non-Euclidean, higher-dimensional spaces than did their mathematical contemporaries.”
Tymoczko’s approach also makes it possible to compare very different kinds of music. At various times in his new book he compares Schumann’s “Chopin” movement in Carna val to Nirvana’s “Heart-Shaped Box,” Shostakovich’s “G-Minor Piano Quintet” to the Black Sabbath song “Sabbath, Bloody Sabbath,” and Philip Glass’ opera Einstein on the Beach to the opening of the TV show Battlestar Galactica. “Music that superficially seems quite different, like Renaissance music and jazz, make remarkably similar use of musical space,” he says.
In fact, Tymoczko argues, the music of the past millennium in the West, and much of non-Western music as well, constitutes an “extended common practice” characterized by the continued broadening of usable musical space. That history has, of course, been shaped by the creativity and idiosyncrasies of individual composers. But it has followed certain broad paths because of how musical orbifolds are configured. It’s like a mountaineer ascending a rock face, he says. In principle, the climber is free to move in any direction. But the structure of the rock provides certain natural routes, offering handholds and footholds along the way. “The trick for the historian is to make room for historical contingency while also capturing the way in which music history sometimes follows the path of least resistance, like a climber ascending a cliff by way of a particularly inviting chute.”
Tymoczko’s work falls into the category of basic research — it doesn’t have a guaranteed payoff. Yet already it’s finding lots of applications. Among the first enthusiasts were composers who use computers to produce musical ideas — and in some cases, complete compositions. Composer Michael Gogins, for example, uses orbifolds to add harmonic and melodic structure to notes generated randomly by the computer. “It gives you more power,” he says.
Some music theorists have begun using orbifolds to teach composition and theory to undergraduates, a trend Tymoczko hopes his new book will encourage. The value of the idea, says Quinn at Yale, is that a geometric approach demonstrates both the possibilities and constraints of music. “Once composers said, ‘Let’s get out of the box,’ that’s when they began wrapping themselves around the singularities and other weird parts of the space.”
Other applications of orbifolds are still on drawing boards. Tymoczko envisions orbifold-inspired children’s toys that could teach chord structure through play. A dancer could move through a space wired to generate the chords associated with each location. Composers could write music by moving through orbifolds rather than writing notes on staves.
Has the discovery of music’s geometry achieved Tymoczko’s original purpose — to make his music sound better? It helps, he says, but it can’t replace inspiration. Composition has both a mechanical phase and a mysterious phase. The use of geometric ideas can suggest melodies and harmonies — it frees a composer “from repeating the formulas of the past.” But when he’s writing music, says Tymoczko, who is working on an album of jazz, funk, and classical fusion pieces, the mechanical part must be followed by the mysterious part, when a composer “learns what a piece is really about.”
The saxophone genius Charlie Parker put it a bit differently: Learn all the theory you can, he said, and then forget it when you play.
Steve Olson, based in Seattle, has written about genetics, race, evolution, climate change, talent, and punk-rock music, among other things. His most recent book, Anarchy Evolution: Faith, Science, and Bad Religion in a World Without God, co-written by Greg Graffin, was published in September by itbooks.
How is the Beatles’ “Help!” similar to Stravinsky’s “Dance of the Adolescents?” How does Radiohead’s “Just” relate to the improvisations of Bill Evans? And how do Chopin’s works exploit the non-Euclidean geometry of musical chords?
In this groundbreaking work, author Dmitri Tymoczko describes a new framework for thinking about music that emphasizes the commonalities among styles from medieval polyphony to contemporary rock. Tymoczko identifies five basic musical features that jointly contribute to the sense of tonality, and shows how these features recur throughout the history of Western music. In the process he sheds new light on an age-old question: what makes music sound good?
A Geometry of Music provides an accessible introduction to Tymoczko’s revolutionary geometrical approach to music theory. The book shows how to construct simple diagrams representing relationships among familiar chords and scales, giving readers the tools to translate between the musical and visual realms and revealing surprising degrees of structure in otherwise hard-to-understand pieces.
Tymoczko uses this theoretical foundation to retell the history of Western music from the eleventh century to the present day. Arguing that traditional histories focus too narrowly on the “common practice” period from 1680-1850, he proposes instead that Western music comprises an extended common practice stretching from the late middle ages to the present. He discusses a host of familiar pieces by a wide range of composers, from Bach to the Beatles, Mozart to Miles Davis, and many in between.
A Geometry of Music is accessible to a range of readers, from undergraduate music majors to scientists and mathematicians with an interest in music. Defining its terms along the way, it presupposes no special mathematical background and only a basic familiarity with Western music theory. The book also contains exercises designed to reinforce and extend readers’ understanding, along with a series of appendices that explore the technical details of this exciting new theory.
Table of Contents
PREFACE
PART I. Theory
CHAPTER 1. Five Components of Tonality 1.1 The five features. 1.2. Perception and the five features. 1.3 Four Claims. A. Harmony and counterpoint constrain each other. B. Scale, macroharmony, and centricity are independent. C. Modulation involves voice leading. D. Music can be understood geometrically. 1.4 Music, magic, and language. 1.5 Outline of the book, and a suggestion for impatient readers.
CHAPTER 2. Harmony and Voice Leading 2.1 Linear pitch space. 2.2 Circular pitch-class space. 2.3 Transposition and inversion as distance-preserving functions. 2.4 Musical objects. 2.5 Voice leadings and chord progressions. 2.6 Comparing voice leadings. 2.7Voice-leading size. 2.8 Near identity. 2.9 Harmony and counterpoint revisited. 2.10 Acoustic consonance and near-evenness
CHAPTER 3. The Geometry of Chords 3.1 Ordered pitch space. 3.2 The Parable of the Ant. 3.3 Two-note chord space. 3.4 Chord progressions and voice leadings in two-note chord space. 3.5 Geometry in analysis. 3.6 Harmonic consistency and efficient voice leading. 3.7 Pure parallel and pure contrary motion. 3.8 Three-dimensional chord space. 3.9 Higher-dimensional chord spaces. 3.10 Voice leading lattices. 3.11 Triads are from Mars, seventh chords are from Venus. 3.12 Two musical geometries. 3.13 Study guide.
CHAPTER 4. Scales 4.1 A scale is a ruler. 4.2 Scale degrees, scalartransposition, scalar inversion. 4.3 Evenness and scalar transposition. 4.4 Constructing common scales. 4.5 Modulation and voice leading. 4.6 Voice leading between common scales . 4.7 Two examples. 4.8 Scalar and interscalar transposition. 4.9 Interscalar transposition and voice leading. 4.10 Combining interscalar and chromatic transpositions.
CHAPTER 5. Macroharmony and Centricity 5.1 Macroharmony. 5.2 Small-gap macroharmony. 5.3 Pitch-class circulation. 5.4 Modulating the rate of pitch-class circulation. 5.5 Macroharmonic consistency. 5.6 Centricity. 5.7 Where does centricity come from? 5.8 Beyond “tonal” and “atonal.”
PART II. History and Analysis
CHAPTER 6. The Extended Common Practice 6.1Disclaimers. 6.2 Two-voice medieval counterpoint. 6.3 Triads and the Renaissance. 6.4 Functional harmony. 6.5 Schumann’s Chopin. 6.6 Chromaticism. 6.7 Twentieth-century scalar music. 6.8 The extended common practice.
CHAPTER 7. Functional Harmony 7.1 The thirds-based grammar of elementary tonal harmony. 7.2 Voice leading in functional harmony. 7.3 Sequences. 7.4 Modulation and key distance. 7.5 The two lattices. 7.6 A challenge from Schenker. CHAPTER 8. Chromaticism 8.1 Decorative chromaticism. 8.2 Generalized augmented sixths. 8.3 Brahms and Schoenberg. 8.4 Schubert and the major-third system. 8.5 Chopin’s tesseract. 8.6 The Tristan Prelude. 8.7 Alternative approaches. 8.8Conclusion
CHAPTER 9. Scales in Twentieth-Century Music 9.1 Three scalar techniques. 9.2 Chord-first composition. A. Grieg’s “Drömmesyn,” (Vision), Op. 62 no. 5 (1895). B. Debussy’s “Fetes” (1899). C. Michael Nyman’s “The Mood That Passes Through You” (1993). 9.3 Scale-first composition. A. Debussy’s “Des pas sur la neige” (1910). B. Janácek’s “On an Overgrown Path,” Series II, no. 1 (1908). C. Shostakovich’s Fs minor Prelude and Fugue, Op. 87 (1950). D. Reich’s “New York Counterpoint” (1985). E. Reich’s “The Desert Music,” movement 1 (1984). F. The Who’s “Can’t Explain” (1965) and Bob Seger’s “Turn the Page” (1973). 9.4 The Subset Technique. A. Grieg’s “Klokkeklang,” (Bell Ringing), Op. 54 no. 6 (1891). B. “PetitAirs,” from Stravinsky’s Histoire du Soldat (1918). C. Reich’s “City Life” (1995). D. Stravinsky’s “Dance of the Adolescents” (1913). E. The Miles Davis Group’s “Freedom Jazz Dance” (1966). 9.5 Conclusion.
CHAPTER 10. Jazz. 10.1 Basic jazz voicings. 10.2 From thirds to fourths. 10.3 Tritone substitution. 10.4 Altered chords and scales. 10.5 Bass and upper-voice tritone substitutions. 10.6 Polytonality, sidestepping, and “playing out.” 10.7 Bill Evans’s “Oleo.” 10.8 Jazz as modernist synthesis.
CONCLUSION
APPENDIX A. Measuring voice-leading size APPENDIX B.Chord geometry: a technical look. APPENDIX C.Discrete voice leading lattices. APPENDIX D. The interscalar interval matrix. APPENDIX E.Scale, macroharmony, and Lerdahl’s “basic space.” APPENDIX F. Some study questions, problems, and activities.
Prelude: transposition along a collection 2. Rock logic 1. A melodic principle 2. A harmonic principle 3. A first chord-loop family 4. Two more families 5. Shepard-tone passacaglias 6. Minor triads and other trichords 7. A fourth family 8. Other modalities 9. Function and retrofunction 10. Continuity or reinvention?
Prelude: the Tinctoris transform 3. Line and configuration 1. The imperfect system 2. Voice exchanges 3. Other intervals 4. The circle of diatonic triads 5. Voice exchanges and multiple chord types 6. Four-voice triadic counterpoint 7. Counterpoint within the chord 8. Seventh chords 9. Harmony and counterpoint
Prelude: sequence and function 4. Repetition 1. Repetition reimagined 2. Repeating contrapuntal patterns 3. The geometry of two-voice sequences 4. Three voices and the circle of triads 5. Three voices arranged 2+1 6. Four voices 7. Contrary-motion sequences 8. Melodic sequences and near sequences 9. Near sequences 10. Sequences as reductional targets
Prelude: three varieties of analytical reduction 5. Nonharmonic tones 1. The first practice and the SNAP system 2. Schoenberg’s critique 3. Monteverdi’s “Ohimè” 4. The standardized second practice 5. A loophole 6. After nonharmonicity
Prelude: functional and scale-degree analysis 6. The origins of functional harmony 1. The logical structure of protofunctionality 2. Similarities and differences 3. Origin and meaning 4. Harmony and polyphony 5. The Pope Marcellus Kyrie 6. A broader perspective 7. “I Cannot Follow”
Prelude: could the Martians understand our music? 7. Functional progressions 1. A theory of harmonic cycles 2. A more principled view 3. Rameau and Bach 4. Functional melody, functional harmony 5. Fauxbourdon and linear idioms 6. Sequences 7. Bach the dualist
Prelude: chromatic or diatonic? 8. Modulation 1. Two models of key distance 2. Enharmonicism and loops in scale space 3. Minor keys 4. Modulatory schemas 5. Up and down the ladder 6. Modal homogenization and scalar voice leading 7. Generalized set theory
Prelude: hearing and hearing-as 9. Melodic strategies 1. Strategy and reduction 2. Two models of the phrase 3. Chopin and the Prime Directive 4. An expanded vocabulary of melodic templates 5. Simple harmonic hierarchy 6. The four-part phrase 7. Grouping, melody, harmony 8. Beyond the phrase: hierarchy at the level of the piece
Prelude: why Beethoven? 10. Beethoven theorist 1. Meet the Ludwig 2. From schema to flow 3. The Tempest 4. The Fifth Symphony 5. The “Pastorale” sonata, op. 28 6. Schubert’s Quartettsatz 7. The prelude to Lohengrin
From musical chords to twin primes / Jack Douthett, David Clampitt & Norman Carey
Hypercubes and the generalized Cohn cycle / Jack Douthett, Peter Steinbach & Rick Hermann
Associahedra, combinatorial block designs and related structures / Franck Jedrezejewski
Rhythmic and melodic l-canons / Jeremy Kastine
The Fibonacci sequence es metric suspension in Luigi Nono’s Il canto sospeso / Jon Kochavi
Note samba : navigating notes and their meanings within modes and exo-modes / Thomas Noll
Difference sets and all-directed-interval chords / Robert W. Peck
Harmonious opposition / Richard Plotkin
Orbifold path models for voice leading : dealing with doubling / James R. Hughes
Remarks on the geometry of chords / Thomas A. Ivey
Theoretical physics and category theory as tools for analysis of musical performance and composition / Maria Mannone
Intuitive musical homotopy / Aditya Sivakumar & Dmitri Tymoczko
Geometric generalizations of the Tonnetz and their relation to Fourier phases space / Jason Yust
Deterministic geometries : a technique for the systematic generation of musical elements in composition / Brent A. Milam
Flamenco music and its computational study / Francisco Gómez
Examining fixed and relative similarity metrics through jazz melodies / David J. Baker & Daniel Shanahan
In search of arcs of prototypicality / Daniel Shanahan.
Publisher’s summary Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music. remove. (source: Nielsen Book Data)
Questions about variation, similarity, enumeration, and classification of musical structures have long intrigued both musicians and mathematicians. Mathematical models can be found from theoretical analysis to actual composition or sound production. Increasingly in the last few decades, musical scholarship has incorporated modern mathematical content. One example is the application of methods from Algebraic Combinatorics, or Topology and Graph Theory, to the classification of different musical objects. However, these applications of mathematics in the understanding of music have also led to interesting open problems in mathematics itself.
The reach and depth of the contributions on mathematical music theory presented in this volume is significant. Each contribution is in a section within these subjects: (i) Algebraic and Combinatorial Approaches; (ii) Geometric, Topological, and Graph-Theoretical Approaches; and (iii) Distance and Similarity Measures in Music.
remove Contents:
Section I:
From Musical Chords to Twin Primes (Jack Douthett, David Clampitt and Norman Carey)
Hypercubes and the Generalized Cohn Cycle (Jack Douthett, Peter Steinbach and Richard Hermann)
Associahedra, Combinatorial Block Designs and Related Structures (Franck Jedrzejewski)
Rhythmic and Melodic L-canons (Jeremy Kastine)
The Fibonacci Sequence as Metric Suspension in Luigi Nono’s II Canto Sospeso (Jon Kochavi)
One Note Samba: Navigating Notes and Their Meanings Within Modes and Exo-modes (Thomas Noll)
Difference Sets and All-Directed-Interval Chords (Robert W Peck)
Harmonious Opposition (Richard Plotkin)
Section II:
Orbifold Path Models for Voice Leading: Dealing with Doubling (James R Hughes)
Reflections on the Geometry of Chords (Thomas A Ivey)
Theoretical Physics and Category Theory as Tools for Analysis of Musical Performance and Composition (Maria Mannone)
Intuitive Musical Homotopy (Aditya Sivakumar and Dmitri Tymoczko)
Geometric Generalizations of the Tonnetz and Their Relation to Fourier Phases Spaces (Jason Yust)
Deterministic Geometries: A Technique for the Systematic Generation of Musical Elements in Composition (Brent A Milam)
Section III:
Flamenco Music and Its Computational Study (Francisco Gómez)
Examining Fixed and Relative Similarity Metrics Through Jazz Melodies (David J Baker and Daniel Shanahan)
In Search of Arcs of Prototypicality (Daniel Shanahan)
Readership: Students and researchers in Mathematical Music Theory.Mathematics and Music;Algebra;Geometry;Topology;Graph Theory;Combinatorics;Distance and Similarity Measures;Discrete Fourier Transform0 Key Features:
It includes the most prominent authors in the field
It gathers a gamut of the most recent work in the field, which is something very difficult to find in one volume
It will appeal to mathematicians, music theorists, and computer scientists. Within mathematics, it offers a variety of areas and techniques related to musical phenomena that cannot be found together in other volumes
1 Department of Mathematics and Computer Science, Saint Joseph’s University, Philadelphia, PA 19131, rhall@sju.edu
Music theorists have frequently invoked geometry in modeling musical objects such as chords, rhythms, and scales; however, no unified geometric perspective has hitherto emerged. On page XXX of this issue, Callender, Quinn, and Tymoczko (1) demonstrate that many musical terms can be understood as expressing symmetries of n -dimensional space. Identifying— “gluing together”—points related by these symmetries produces a family of non-Euclidean quotient spaces that subsume a large number of geometric models proposed in the literature. The use of mathematics to describe, analyze, and create music goes back millennia. Questions have previously emerged in music theory that are appealing, nontrivial, and, in several cases, connected to other scientific fields (2,3,4). Math inspired composers such as Schoenberg, Messaien, and Xenakis; musical investigations have even motivated mathematical discoveries (5). However, the “geometrical music theory” proposed by Callender, Quinn, and Tymoczko stands out both for the breadth of its musical implications and the depth of its mathematical content. It suggests a wealth of new techniques for studying music theoretical topics, including chord similarity and melodic contour, and provides attractive visualizations of harmonic relationships. Although the authors’ use of contemporary geometry departs from the discrete mathematics normally used to describe music, they make a convincing argument for the validity of their models. Musicians think in degrees of abstraction: While “middle C” is a particular pitch (frequency measured on a logarithmic scale), the letter name (or pitch class) “C” refers to any pitch that is a whole number of octaves away from it. Any collection of C, E, and G notes is a “C major chord.” A “major chord” is any transposition of a C major chord; a “consonant triad” is any three-note chord containing a major third, a minor third, and a perfect fifth. Music theorists invoke the mathematical concept of equivalence class —a set of objects that are “the same” if we ignore certain information—to describe these musical structures. Callender, Quinn, and Tymoczko show that many musical terms describe equivalence classes under combinations of five basic “OPTIC” relations: Octave shifts, permutation (reordering), transposition (the relation between pitches sharing the same succession of intervals, regardless of their starting note), inversion (turning a sequence “upside down”), and cardinality equivalence (ignoring repetitions). The authors go beyond traditional music theory by showing that there are two distinct ways in which these symmetries can apply to progressions of musical objects: “uniformly,” where the same symmetry applies to each object in a progression, and “individually,” where distinct symmetry operations apply to the harmonies in a progression. This distinction allows them to formalize the relationships among a large number of musical terms, as in table S1 of their paper. Following Tymoczko (6), they develop geometrical models of these relationships. Pitches correspond to real numbers; each point in the Euclidean n -dimensional “configuration” space represents a sequence of n pitches. Points near each other differ by microtones (a C major chord
The Geometry of Music
A composer has taken equations from string theory to explain why Bach and bebop aren’t so different
When you first hear them, a Gregorian chant, a Debussy prelude and a John Coltrane improvisation might seem to have almost nothing in common–except that they all include chord progressions and something you could plausibly call a melody. But music theorists have long known that there’s something else that ties these disparate musical forms together. The composers of these and virtually every other style of Western music over the past millennium tend to draw from a tiny fraction of the set of all possible chords. And their chord progressions tend to be efficient, changing as few notes, by as little as possible, from one chord to the next.
Exactly how one style relates to another, however, has remained a mystery–except over one brief stretch of musical history. That, says Princeton University composer Dmitri Tymoczko, “is why, no matter where you go to school, you learn almost exclusively about classical music from about 1700 to 1900. It’s kind of ridiculous.”
But Tymoczko may have changed all that. Borrowing some of the mathematics that string theorists invented to plumb the secrets of the physical universe, he has found a way to represent the universe of all possible musical chords in graphic form. “He’s not the first to try,” says Yale music theorist Richard Cohn. “But he’s the first to come up with a compelling answer.”
Tymoczko’s answer, which led last summer to the first paper on music theory ever published in the journal Science, is that the cosmos of chords consists of weird, multidimensional spaces, known as orbifolds, that turn back on themselves with a twist, like the Möbius strips math teachers love to trot out to prove to students that a two-dimensional figure can have only one side. Indeed, the simplest chords, which consist of just two notes, live on an actual Möbius strip. Three-note chords reside in spaces that look like prisms–except that opposing faces connect to each other. And more complex chords inhabit spaces that are as hard to visualize as the multidimensional universes of string theory.
But if you go to Tymoczko’s website music.princeton.edu/~dmitri) you can see exactly what he’s getting at by looking at movies he has created to represent tunes by Chopin and, of all things, Deep Purple. In both cases, as the music progresses, one chord after another lights up in patterns that occupy a surprisingly small stretch of musical real estate. According to Tymoczko, most pieces of chord-based music tend to do the same, although they may live in a different part of the orbifold space. Indeed, any conceivable chord lies somewhere in that space, although most of them would sound screechingly harsh to human ears.
The discovery is useful for at least a couple of reasons, says Tymoczko. “One is that composers have been exploring the geometrical structure of these maps since the beginning of Western music without really knowing what they were doing.” It’s as though you figured out your way around a city like Boston, for example, without realizing that some of your routes intersect. “If someone then showed you a map,” he says, “you might say, ‘Wow, I didn’t realize the Safeway was close to the disco.’ We can now go back and look at hundreds of years of this intuitive musical pathmaking and realize that there are some very simple principles that describe the process.”
Computational geometric aspects of rhythm, melody, and voice-leading
Godfried Toussaint 1 School of Computer Science and Center for Interdisciplinary Research in Music Media and Technology, McGill University, Montréal, Québec, Canada
Many problems concerning the theory and technology of rhythm, melody, and voice-leading are fundamentally geometric in nature. It is therefore not surprising that the field of computational geometry can contribute greatly to these problems. The interaction between computational geometry and music yields new insights into the theories of rhythm, melody, and voice-leading, as well as new problems for research in several areas, ranging from mathematics and computer science to music theory, music perception, and musicology. Recent results on the geometric and computational aspects of rhythm, melody, and voice-leading are reviewed, connections to established areas of computer science, mathematics, statistics, computational biology, and crystallography are pointed out, and new open problems are proposed.
Exploring Musical Spaces: A Synthesis of Mathematical Approaches
By Julian Hook
Oxford, 2022
Exploring Musical Spaces is a comprehensive synthesis of mathematical techniques in music theory, written with the aim of making these techniques accessible to music scholars without extensive prior training in mathematics. The book adopts a visual orientation, introducing from the outset a number of simple geometric models―the first examples of the musical spaces of the book’s title―depicting relationships among musical entities of various kinds such as notes, chords, scales, or rhythmic values. These spaces take many forms and become a unifying thread in initiating readers into several areas of active recent scholarship, including transformation theory, neo-Riemannian theory, geometric music theory, diatonic theory, and scale theory.
Concepts and techniques from mathematical set theory, graph theory, group theory, geometry, and topology are introduced as needed to address musical questions. Musical examples ranging from Bach to the late twentieth century keep the underlying musical motivations close at hand. The book includes hundreds of figures to aid in visualizing the structure of the spaces, as well as exercises offering readers hands-on practice with a diverse assortment of concepts and techniques.
A guided tour of the mathematical principles inherent in music.
Taking a “music first” approach, Gareth E. Roberts’s From Music to Mathematics will inspire students to learn important, interesting, and at times advanced mathematics. Ranging from a discussion of the geometric sequences and series found in the rhythmic structure of music to the phase-shifting techniques of composer Steve Reich, the musical concepts and examples in the book motivate a deeper study of mathematics.
Comprehensive and clearly written, From Music to Mathematics is designed to appeal to readers without specialized knowledge of mathematics or music. Students are taught the relevant concepts from music theory (notation, scales, intervals, the circle of fifths, tonality, etc.), with the pertinent mathematics developed alongside the related musical topic. The mathematics advances in level of difficulty from calculating with fractions, to manipulating trigonometric formulas, to constructing group multiplication tables and proving a number is irrational.
Topics discussed in the book include
• Rhythm • Introductory music theory • The science of sound • Tuning and temperament • Symmetry in music • The Bartók controversy • Change ringing • Twelve-tone music • Mathematical modern music • The Hemachandra–Fibonacci numbers and the golden ratio • Magic squares • Phase shifting
Featuring numerous musical excerpts, including several from jazz and popular music, each topic is presented in a clear and in-depth fashion. Sample problems are included as part of the exposition, with carefully written solutions provided to assist the reader. The book also contains more than 200 exercises designed to help develop students’ analytical skills and reinforce the material in the text. From the first chapter through the last, readers eager to learn more about the connections between mathematics and music will find a comprehensive textbook designed to satisfy their natural curiosity.
physics and philosophy, but the most important is music, for in the patterns of the arts are the keys to all learning.
Plato, c. 428-347 BC
The first concrete argument for a fundamental link between mathematics and music was perhaps made by the early philosopher and mathematician Pythagoras (569-475 BC), often referred to as the “father of numbers.” He can also be considered the “father of harmony,” given that his discovery of the overtone series and analyses of the acoustics and ratios involved in music have served as the foundation of harmony in western-hemisphere music composition ever since. The Pythagorean, Quadrivium and Platonic classifications of mathematics were based on hierarchical dimensions, starting with arithmetic, then geometry, astronomy and finally music.
Reportedly, Pythagoras experimented with the tones produced when plucking strings of different lengths. He found that some specific ratios of string lengths created pleasing combinations (“harmonies”) and others did not. Based on his careful observations, Pythagoras identified the physics of intervals, or distances between notes, that form the primary harmonic system which is still used today (Parker, 2009, pp. 3-5).
Music is based on proportional relationships. The mathematical structure of harmonic sound begins with a single naturally occurring tone, which contains within it a series of additional frequencies above its fundamental frequency (“overtones”), of which we are normally unaware on a conscious level. Within this harmonic or overtone series, there is a mathematical relationship between the frequencies – they are specific integer multiples of each other. For example, if the slowest frequency (the “fundamental”) were 100 Hz, then the overtones would be 2 x 100 (200 Hz), 3 x 100 (300 Hz) and so forth. (The overtone series is often referred to as harmonics.)
Pythagoras observed several ratios of sound wave frequencies and the corresponding intervals between them, including 4:3 (known to musicians as the interval of a perfect fourth, or two pitches that are five semitones apart from each other) and 3:2 (a perfect fifth, seven semitones apart). Note that pitch is the frequency or rate of vibration of a physical source such as a plucked string.
The most prominent interval that Pythagoras observed highlights the universality of his findings. The ratio of 2:1 is known as the octave (8 tones apart within a musical scale).
When the frequency of one tone is twice the rate of another, the first tone is said to be an octave higher than the second tone, yet interestingly the tones are often perceived as being almost identical. For example, a woman’s voice may fluctuate around 220 Hertz while a man’s voice is around 110 Hertz, approximately half the frequency of the woman’s. However, if they sing together, it may sound as though they are singing the same melody together in unison, even though they are actually an octave apart. This 2:1 ratio is so elemental to what humans consider to be music, that the octave is the basis of all musical systems that have been documented – despite the diversity of musical cultures around the world. Moreover, this physical phenomenon is so fundamental that even non-human species such as monkeys and cats recognize it (Levitin, 2008, p. 31).
The inherent properties of physics and mathematics within music, perceived so long ago by Pythagoras, may help to explain why many physicists and mathematicians are also musicians. This point is illustrated by a quotation from Einstein: “The theory of relativity occurred to me by intuition, and music was the driving force behind that intuition….My new discovery was the result of musical perception” (Suzuki, 1969, 90).
*Portions of this article are adapted from Edel Sander’s chapter in Musik i forskola och tidiga skolar (2015), a Swedish textbook for music educators.
10 – Music theory and mathematics
from PART II – SPECULATIVE TRADITIONS
Published online by Cambridge University Press: 28 March 2008
In Chapter 6 of The Manual of Harmonics (early second century CE), Nicomachus of Gerasa narrates the legendary story of Pythagoras passing by the blacksmith’s shop, during which in an epiphany of sonorous revelation, he discovered the correlation of sounding intervals and their numerical ratios. According to Nicomachus, Pythagoras perceived from the striking of the hammers on the anvils the consonant intervals of the octave, fifth, and fourth, and the dissonant interval of the whole tone separating the fifth and fourth. Experimenting in the smithy with various factors that might have influenced the interval differences he heard (force of the hammer blows, shape of the hammer, material being cast), he concluded that it was the relative weight of the hammers that engendered the differences in the sounding intervals, and he attempted to verify his conclusion by comparing the sounds of plucked strings of equal tension and lengths, proportionally weighted according to the ratios of the intervals.
Physical and logical incongruities or misrepresentations in Nicomachus’s narrative aside, the parable became a fixture of neo-Pythagorean discourse because of its metaphoric resonance: it encapsulated the essence of Pythagorean understanding of number as material or corporeal, and it venerated Pythagoras as the discoverer of the mathematical ratios underlying the science of harmonics. The parable also established a frame of reference in music-theoretical thought in the association between music and number, or more accurately, music theory and mathematical models, since it is not through number alone but through the more fundamental notions of universality and truth embedded in Pythagorean and Platonic mathematics and philosophy that one can best begin to apprehend the broad range of interrelationships between music theory and mathematics.
A new way of visualizing Western music theory could inspire innovations in everything from musical instrument design to music composition techniques—even to new kinds of toys.
Western Music Moves in Three and Even Four (!) Dimensional Spaces: How the Pioneering Research of Princeton Theorist Dmitri Tymoczko Helps Us Visualize Music in Radical, New Ways
Supplementary Sets and regular complementary unending canons,
Dan Tudor Vuza,
Perspectives of New Music, Numbers 29(2), 30(1), 30(2), and 31(1). Journal of Mathematics and Music, Special Issue on Tiling Problems in Music, Vol. 3, No. 2
Computational Geometric Aspects of Rhythm, Melody, and Voice-Leading
Godfried Toussaint∗
School of Computer Science
and Center for Interdiisciplinary Research in Music Media and Technology McGill University Montr ́eal, Qu ́ebec, Canada
To appear in: Computational Geometry: Theory and Applications, 2009. doi:10.1016/j.comgeo.2007.01.003
Source: Awakening of Geometrical Thought in Early Culture
Source: Awakening of Geometrical Thought in Early Culture
Source: The ritual origin of geometry
Let us sum up the history of geometry from its beginnings in peg-and-cord constructions for circles and squares.
The circle and square were sacred figures and were studied by the priests for the same reason they studied the stars, namely, to know their gods better. The observation that the square on the diagonal of a rectangle was the sum of the squares on the sides found an immediate ritual application. Its elaboration in the sacrificial ritual gave it a dominant position in ancient thought and ensured its conservation for thousands of years. This initial elaboration took place well before 2000 B.C. By 2000 B.C., it was already old and had diffused parts of itself into Egypt and Babylonia (unless, indeed, one of these places was the homeland of the elaboration). These parts became the basis of a new development in these centers. The new, big development was the solution of the quadratic. A thousand years and more later, Greece inherited algebra from Babylonia, but its geometry has more of an Indian than a Babylonian look. It inherited geometric algebra, the problem of squaring the circle, the problem of expressing √2 rationally, and some notions of proof.
Source: Mathematics in India / Part 1: Geometry in Vedic and Sūtra literature
Source: The ritual origin of geometry
Source: The ritual origin of counting
Source: On the volume of a sphere
Source: The Ritual Origin of the Circle and Square
The penultimate version of the chapter to appear in: Sriraman, Bharat (ed) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham, 2022.
Philosophical Problems of Mathematics in the Light of Evolutionary Epistemology
YEHUDA RAV
Séminaire de Philosophie et Mathématiques, 1988, fascicule 6 « Philosophical problems of mathematics in the light of evolutionary epistemology », , p. 1-30
Variations of the Vedic Unit “Prakrama” (Step) Applied for Measuring Length at Ritual Sites (With Excursus: “Prakramas in the Kerala Vedic Tradition”)
TESHIMA Hideki Manuscript for contributing “Sciences of the Ancient World Matter 2” (ed. Karine CHEMLA, Agathe KELLER, et al., in preparation for publishing)
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For more than twenty years, I have been studying the image generating properties of reflective spheres stacked in 52 degree angle pyramids. The 52 (51.827) degree angle slope of the sides of The Great Pyramid in Cairo, Egypt embodies the Golden Mean which is the ratio that is used in Nature to generate growth patterns in space. Sacred Geometry studies such primal systems which reveal the unity of the cosmos by representing the relationships between numbers geometrically. The Vesica Piscis is one of the most fundamental geometrical forms of this ancient discipline and it reveals the relationship between the The Great Pyramid and the 2 dimensional expansion of a circle of one unit radius R as shown in Figure 1. This relationship is more completely described in The New View Over Atlantis by John Michell published by Thames and Hudson.
Figure 1 Vesica Piscis in 2 Dimensions
In the early 1970s, I became very interested in the three dimensional representation of this geometry and I visualized this as a three dimensional pyramid inside two intersecting spheres shown in Figure 2.
Figure 2 Vesica Piscis in 3 Dimensions
In an effort to visualize these 3D relationships in yet another way, I stacked reflective Christmas Tree balls in an inverted pyramid shell. I discovered that the patterns of multiple reflections created on the interior surfaces of reflective spheres stacked at this angle produce images that relate to the human form as photographed and shown in Figure 3 and Figure 4.
I made many pyramidal configurations of reflective spheres with different light sources and I photographed the patterns on the interior surfaces from many points of view. Another is shown here in Figure 5.
Then in 1977, I discovered one stacking structure and viewpoint that produced a very convincing image of an archetypal human face. This structure of 10 spheres (2 5-ball pyramids) forming a cluster is shown in Figure 6.
The face image is shown in the lower third of the pattern on one sphere inside this 10 sphere cluster and is shown in Figure 7.
Figure 7 When I made this photograph, the structure was enclosed in a mirrored pyramid.
Later, I realized that the most natural structure for enclosing would be another 10 sphere, 2 pyramid structure that would totally enclose a smaller but similar cluster. I worked out the math to find that by multiplying the inner sphere’s diameters by Pi gives the dimension for the outer sphere’s diameters as shown in Figure 8.
Figure 8 One sphere is removed from the outer cluster to reveal the inner cluster. However, the inner cluster must be upside down with respect to the outer cluster to fit inside.
The expansion by Pi reinforced my suspicion that this 10 sphere cluster is a fundamental unit that is linked to the properties of three dimensional space.
Close-packed reflective spheres clustered in this concentric shell structure produce an optical distribution network that links the Golden Mean and Pi. The Golden Mean is expressed in the 52 degree angle pyramid structure and Pi is expressed in the ratio of the diameter to the circumference of each sphere of course; but it is also expressed in the ratio of the sizes of spheres in the 10 spheres within 10 spheres concentric shell structure that I discovered. This concentric shell structure could continue to expand with many shells and still retain the same ratio between shells. It was not until 1991 that I was able to build and photograph the images inside a ten-within-ten (2 shell) structure. I used 10 – 6″ diameter spheres and 10 – 19″ diameter hemispheres. The structure is shown here in Figure 9.
The photographs that I made from this 10-within-10 sphere structure revealed the inherent limitations of photography for this work. The final images were not what I was seeing with my eyes while directly viewing the interior of the structure. But this approach did reveal a more complete face form and I also realized that I would need at least 10 more spheres (about 60″ in diameter at $1500 each) to complete the enclosure and remove the remaining gaps in the images. Also, the lighting system was limited to the exterior and it was very difficult to control the positioning, color and brightness. So, in 1992, I purchased a computer to model these structures with ray-tracing software which enabled me to investigate more thoroughly the relationship between this cluster geometry and the archetypal images generated therein. During this tour into cyberspace, I could take a camera into the sacred chamber central to the concentric shells of reflective spheres which are simulated by a computer program (Real 3D by Realsoft Oy, Finland) that most accurately renders the effects of real-world light sources and records the patterns generated by multiple reflections on metal surfaces. With this method of investigation, I am able to more conveniently control the many variables which led to these discoveries and conclusions:
1.) At least 40 spheres (4-10 ball shells) are necessary to enclose the central area and fill in the gaps in the images.
2.) Most of the lights should be point sources placed at the points of contact between the innermost spheres of the structure; although additional symmetrically-paired, point-lights are necessary in the central area.
3.) Some of the innermost spheres are reduced in size and they can float within certain areas in the central space. Figure 12 shows typical positions and sizes.
4.) The camera position and field of view as shown in Figure 12 produces the most convincing image of the human form.
5.) The image of this artificial anthropoid that is produced in these structures can be animated when the positions and sizes of certain spheres are modulated as shown in the three animations listed below.
animation 1 .8 MB … estimated download time at 56k … 3.5 min.
animation 2 2.4 MB … estimated download time at 56k … 10 min.
Figure 13 Human Form From Sacred Geometry
The computer image in Figure 13 was made from the camera position and field of view shown in Figure 12.
Figure 14 is from the same camera position (shown in Figure 12) but the lens set at a very wide angle.
Figure 16 is a stereo image and it shows the interior of the cluster in 3D. Stare through the images with your eyes focused at a distant point and the two images will turn into three images and the center image will appear in 3D.
In 1996, I produced an animation Sacred Spaces (6 minutes) which has been screened in many national and international film festivals and it has won some awards (see resume). I have also produced Flesh Tones (5.5 Minutes) another animation completed in 1997 and I have produced many high resolution images for prints/slides (some examples are shown in Figures 13 through 36) which I have presented with the video animations at lecture presentations. These images are stills from the animations and they demonstrate the variety of image generating that is inherent in this system. I have concentrated on exhibiting the videotape documentation of my work thus far because it is the most portable presentation format that describes this research most completely.
This interdisciplinary research has taken me into many related areas of study. Geometry, Optics, Ancient History of Art and Religion, Computer Imaging, Photography, Animation, Graphic Art and of course Sculpture are the major connecting disciplines that have contributed to this work.
I feel that I have rediscovered some of what was a highly developed understanding of Mankind’s relationship to the Universe and this knowledge was utilized and documented in the geometry of ancient structures. Sacred Geometry, the study of the unity of the cosmos, demonstrates relationships between Number and Space and the Human Form. It was used in the construction of ancient glyphs and monuments thereby preserving the knowledge of these principles of Natural Law for future generations. This construction of reflective spheres may embody the technology that produced the animated images of the deities in the temples of antiquity. The Tree of Life which is a graphic representation of the interaction between cosmic forces is shown in Figure 22. It is found in many ancient texts of the Kabbalah.
The construction of this structure of clustered metal reflective spheres (offering bowls) is well within the capability of many ancient cultures and with the addition of a few glass lenses, these images could be projected onto walls or into smoke. Perhaps there is some Truth behind the smoke and mirrors of Ancient Religion … perhaps it is geometry … Sacred Geometry.
This research which is documented in four computer animations Sacred Spaces, Flesh Tones, Our Mothers and Sacred Spaces 2, in color computer prints, and in color slides has given me new insight into the motives that may have inspired the construction of The Great Pyramid.
1998 – 2000 update
The images shown in Figures 24 – 35 were made with 50 spheres and 144 point-light sources. Each of these images was made with unique brightness, color and value settings for various groupings of lights.
InFigure32, the camera is aimed at the sphere opposite the face shown inFigure 31.Figure 12 shows the camera position for Figure 31. The image in Figure 32 was formed when the camera position was rotated 180 degrees around the vertical axis shown in Figurre 12 and zoomed in. The face image (a child?) in Figure 32 is much smaller than the face (mother?) in Figure 31.
Figure35This is a stereo 3D image that requires shutter glasses to view and your monitor must be set to interlace mode.
Figure 36 Stereo Image for Cross-Eyed Viewing
With your monitor at arm’s length away, focus on a point 6 inches in front of your nose (put your index finger 6 inches in front of your nose and focus on it). You will see a third image in 3D between the two images on the monitor (at the tip of your finger). Shift your attention from your finger to this third/middle image which will appear in 3D.
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The cross-eyed viewing method is perhaps the most effective way to put the viewer inside the cluster to see the human image as it would exist in 3D from the cameras position as seen in Figure 12. There are many more identifiable images in this clustering geometry viewed from this position and from other camera positions and even more images with other color settings for the point lights.
Figure 37 Rods connecting centers of nearest neighbors in 3 shells
I’m now very curious about the relationship between Sacred Geometry and Sacred Music and the Human Form. Number relates to all that science measures by virtue of the way that 3D Space is defined. Number is also used to measure Time … and, as Pythagoras observed, Music is a manifestation of Number in Time. The distribution of sounds i.e. amplitude and frequency, may well find an idealized model in the 10 within 10 sphere, space-filling, close-packing geometric system. The representation of this geometry with sticks or strings or rods as shown in Figure 37 (in which the centers of nearest neighbor spheres in three shells are joined) may represent the ideal space-filling matrix of linear oscillating elements. It may also be used to define spatially distributed, hierarchical, cellular arrays.
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The camera positions on the symmetric planes within the cluster produce a bilateral symmetry that we identify with animal and human form. Naturally, we would expect this symmetry in the idealized images of higher life forms. The multiplicity of idealized beings in this cluster of reflective spheres suggests the presence of The Company of the Gods as described in many ancient Egyptian texts. It seems that the face images occur on a vertical plane linking the centers of spheres. There seem to be faces facing faces and faces within faces throughout this reflective environment. My guess is that the sacred part of what I have discovered is a result of the way the deity put the higher life forms in three dimensions.
This cluster geometry may have other properties that would be useful for spatial organization. The nesting of 10 reflective spheres within 10 reflective spheres geometry produces a distribution system that could be used for processing of optical information between the interior to the exterior of the structure. Because of the spatial distribution of the reflective spheres in two concentric shells of ten each, optical information must be reflected and diverged in order to enter or exit the system with the exception of a few radially arrayed directions. The system becomes a more selective filter of optical information as more shells are added to the structure. This inside-to-outside transformation/translation should have many practical applications in pattern recognition tasks. For example, any point source of coherent light (laser light) anywhere outside the two-shell cluster will produce a unique light distribution pattern on the inside as viewed from the center area of the cluster. This pattern could be recorded in a holographic medium and the exterior point source could then be reconstructed using conventional holographic means.
Is Number (Geometry and Time) the link between Art, Science and Religious experience? The language of number is perhaps the most convincing form of expression between humans and between humans and the Gods. We think and imagine in visual forms. Einstein constructed his mathematics based on mental images. He said that he would first try to visualize a space/time image and then mathematize it. We use mental images to construct possible scenarios of the future so that we don’t have to live out each one in “reality”. Words and pictures and mathematical formulas are ways to document, test, realize, and communicate these visions. Although there is seldom a need to mathematize images, we sense that it would be possible. We know that we could count and number the grains of sand on the beach. The geometry of our visions is what makes them real to us and it allows us to mentally work on them and to integrate them convincingly into our life here in 3D.
I’m not quite sure how the physical human form fits into the grand scheme of things but it does seem to be a result of the space-filling, spherical, close-packing geometric system that I’ve discovered and it is indeed “Sacred Geometry” by virtue of the definition God gave to three dimensions. I don’t think this geometric system is the matrix for all life systems but I think it can serve as a model for the interactions between the various dimensional realities in which we are immersed. It may also guide us in our attempt to develop new sciences and technologies that utilize the forces that operate in the regions that we now call consider paranormal.
According to contemporary Superstring Theory as described by Dr. Michio Kaku in his many recent writings, the mathematics that most appropriately describes the forces of nature requires an expression in ten dimensions. Einstein tried to describe the forces of nature in the mathematics of 3D and Time and found that the formulas were not broad enough to include all of the forces. We can perceive 3D and Time. The other 6D in Superstrings are hidden from our normal senses due to their incredibly small size according to Dr. Kaku. Our instincts inform us that there are more than 3 Dimensions and Time in the universe and the possiblity that something else exists mathematically beyond our perceptual horizon drives my curiosities about our possible links to these worlds. The new science of parapsychology has discovered many ways that humans can perceive by means beyond the physical senses and it has found that there are some people that are more capable of extrasensory perception than others. It may well be that our only contacts with this duality of nature are through numbers and mathematics on one hand or through dreams, remote viewing, telepathy, Ouija boards and tarot cards on the other. This would certainly confirm the existence of a God with a sense of humor
I am interested in any information relating this technology to ancient religious traditions. Any references that you send will certainly be appreciated and I will certainly credit any references in future publications.
A more detailed version of this research is now available on CD-ROM and it includes the 10 minute animation Sacred Spaces 2 in streaming format.
I am also making available unlimited editions of selected images on this website which will be printed in very high resolution on archival paper with archival inks.
For more specific information about the availability, formats, sizes and pricing of the prints and CD relating to this work you may contact me at:
Interaction / Intersection of Menger Sponge and Apollonian Sphere in cosmic geometry produces pythagorean triples / triads.
Seven plus one spheres to triads of three.
Apollonian Gasket, Circle and Sphere Packing and Cosmic Geometry
Source: PRECISE CALCULATION OF HAUSDORFF DIMENSION OF APOLLONIAN GASKET
A transfer operator method is proposed to calculate 𝑑𝐻, the Hausdorff dimension of the Apollonian gasket. Compared with previous operator-based methods, we make two improvements in this paper. We adopt an infinite set of contractive Möbius transformations (rather than a finite set of parabolic ones) to generate the Apollonian gasket. We also apply an efficient finite matrix approximation of an infinite sum of infinite-dimensional operators. By using this method, a high precision estimate of 𝑑𝐻 is obtained:
𝑑𝐻=1.305 686 728 049 877 184 645 986 206 851 0….
Source: THE FRACTAL DIMENSION OF THE APOLLONIAN SPHERE PACKING
The fractal dimension of the Apollonian sphere packing has been computed numerically up to six trusty decimal digits. Based on the 31 944 875 541 924 spheres of radius greater than 2−19 contained in the Apollonian packing of the unit sphere, we obtained an estimate of 2.4739465, where the last digit is questionable. Two fundamentally different algorithms have been employed. Outlines of both algorithms are given.
Source: THE FRACTAL DIMENSION OF THE APOLLONIAN SPHERE PACKING
Source: What Type of Apollonian Circle Packing Will Appear?
Source: Self-similar space-filling sphere packings in three and four dimensions
Source: Self-similar space-filling sphere packings in three and four dimensions
Sapta Matrikas
Brahmi
Maheswari
Kumari
Vaishnavi
Varahi
Indrani
Chamunda
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
In Varahapurana the Devi, Vaishnavi in account of the creation of the matrika, is doing asceticism on mount Mandara. At one point she losses her concentration. From her disgraced mind, several beautiful attendants were created. They later became Devi’s helpmates on the battlefield when she fights the demon. Although the Matrikas are described as lovely in this account, it is important to note that they are born when Devi losses control of her concentration. This suggests that the matrikas are essentially of uncontrolled natures. Born from lack of mental control, they lack control themselves. Varahapurana relates them to vices or inauspicious emotions; Brahmi of Mada, Maheswari of Krodha, Kumari of Moha, Vaishnavi of Lobha, Indrani of Matsarya, Varahi of Asuya and Chamunda of Paisunya.
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
The follower of Tantrasara has an esoteric interpretation of the seven matrikas. According to them, Brahmi represents the primordial Nada, the energy in which even the first throb has not yet appeared. This is the manifest sound, the origin of all creation. It is the same substance or energy represented by the pranava. When Brahmi creates the universe, the power of Vaishnavi gives definite shape. The symmetry, beauty, organization and order in the universe are the work of Vaishnavi. Maheswari stands for the power that gives individuality to the created beings. She resides in the hearts of all and makes them play, as dolls mounted on a machine. Kumari represents the ever present force of aspiration of the evolving soul. She is ‘Guruguha’, the Guru in guha (the cave of the heart, the intellect). Varahi is the all-consuming power of assimilation and enjoyment. Because of her, all living beings get their food and enjoyments. Indrani symbolizes the terrible power that destroys all that opposes the cosmic law. Chamunda is the force of concentrated awareness, the spiritual awakening in the heart that devours that ceaseless activity of the immature mind and uplifts it to the highest level (Harshananda 1981.95-99).
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
Source: Saptamatrikas in Kerala: Iconography and Distribution Pattern
Source: Matrikas/Siddha Pedia
Source: The Seven Ancient Mothers
The Pythagoreans considered the figure seven as the image and model of the divine order and harmony in nature. As the harmony of cosmic sound takes place on the space between the seven planets, the harmony of audible sound takes place on a smaller plane within the musical scale of the seven tones. Therefore, the syrinx of the nature god Pan consists of seven pipes, and the lyre of Apollo (the god of music) consists of seven strings. As the number seven is a union between the number three (the symbol of the divine triad) and of four (the symbol of the cosmic forces or elements), the number seven points out symbolically to the union of the divine with the universe.
Pierre Auclair∗ Cosmology, Universe and Relativity at Louvain (CURL), Institute of Mathematics and Physics, University of Louvain, 2 Chemin du Cyclotron, 1348 Louvain-la-Neuve, Belgium (Dated: November 15, 2022)
Estimate for the fractal dimension of the Apollonian gasket in d dimensions
R. S. Farr* Unilever R&D, Olivier van Noortlaan 120, AT3133 Vlaardingen, The Netherlands and The London Institute for Mathematical Sciences, 22 S. Audley Street, Mayfair, London, United Kingdom
E. Griffiths
297 Sandy Bay Road, Sandy Bay, Tasmania, Australia
A Study of the Sapta Matrikas’ Origins and Evolution: From the Perspectives of the Art and Literature of Western India Dating from B.C. 1400 to 500 A.D.
Rebecca Sholes 1982
Mother Goddess in Central India
Om Prakash Misra Agam Kala Prakashan, 1985
The Little Goddesses (mātrikās)
K. C. Aryan Rekhā, 1980
The Iconography of the Saptamatrikas: Seven Hindu Goddesses of Spiritual Transformation
Katherine Anne Harper E. Mellen Press, 1989
ISBNS 9780889460614, 0889460612 OCLC ocm19921123
Saptamātr̥kā Worship and Sculptures: An Iconological Interpretation of Conflicts and Resolutions in the Storied Brāhmanical Icons
Self-similar space-filling sphere packings in three and four dimensions *
D. V. St ̈ager 1 , ∗ and H. J. Herrmann 1, 2 , † 1 Computational Physics for Engineering Materials, IfB, ETH Zurich, Wolfgang-Pauli-Strasse 27, CH-8093 Zurich, Switzerland 2 Departamento de F ́ısica, Universidade Federal do Cear ́a, 60451-970 Fortaleza, Ceara ́, Brazil
Indira’s Pearls: Apollonian Gasket, Circle and Sphere Packing
Source: Matrikas
Key Terms
Circle Packing
Disk Packing
Sphere Packing
Indira’s Net
Indira’s Pearls
Apollonian Gasket
Fractals
Square and Circle
Circular Temple
Earth and Heaven
64 Yogini Hindu Shakti Temples
64 Bhairav
64 Yogini
7 Matrikas
Mandala
Curvilinear Sierpinski Gasket
Poincare Disc
Jagannath Puri, Odisha
Bhuvneswar, Odisha
Konark, Odisha
Non Euclidian Geometry
Euclidian Geometry
Fractal Geometry
Sacred Geometry
Hyperbolic Geometry
1, 2, 3 where is the 4th
1 2 3 4
Fifth and Fourth in Music Theory
Fractals, Circle, and Sphere Packing
Source: Circle Packing Explorations
Source: Circle Packing Explorations
Source: On a Diophantine Equation That Generates AllIntegral Apollonian Gaskets
Source:
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: Recursive Apollonian Gasket
Source: On the cover: Apollonian packing
What can you do with this space? So asks Andrew Stacey. ‘Fill it’ is the prompt reply, but fill it with what? Maybe like Andrew you want to use a single curve, but I want to use circles. If you do this in the way shown above in blue, the result is called an Apollonian packing, a variant of which can be seen on the cover of this issue.
Here we shall explore the history of this entrancing object, which spans over 2000 years, and percolates into a surprising variety of mathematical disciplines. Starting in the familiar world of Euclidean geometry, Apollonian packings extend into fractal geometry and measure theory; Möbius transformations and the hyperbolic plane; and then on into the distant reaches of geometric group theory, number theory, orbital mechanics, and even ship navigation.
At times we may wonder off into thickets of more obscure mathematics, so those readers who get lost should feel free to skip ahead to the next section.
Apollonius of Perga
Apollonius (c 230 BC) was a Hellenistic mathematician, considered one of the greatest after Euclid and Archimedes. Perhaps his most important work was his eight book treaties Κωνικα on conic sections—once lost to European civilisation, but fortuitously preserved by the more enlightened Middle Eastern scholars and later reintroduced by Edmund Halley in 1710. The same unfortunately cannot be said of Έπαφαι (De Tractionibus or Tangencies). Although now lost, we have accounts of the work from other ancient authors, particularly in the writings of Pappus of Alexandria. In it, Apollonius posed and solved the following problem.
Problem: Given three geometric objects in the plane (points, lines, and/or circles), find all circles which meet all three simultaneously (ie which pass through any points, and are tangent to any lines or circles).
So for example, given three points which don’t lie on the same line, there is exactly one circle which passes through all three. The case which interests us at present is when we are given three circles, each of which is tangent to the other two. In the very special case that all three are tangent at the same point there are infinitely many circles tangent to all three. Usually, however, the circles will be pairwise tangent at three distinct points, in which case there are exactly two other circles tangent to all three simultaneously.
Given three mutually tangent circles (black) there are usually exactly two others (blue) tangent to all three.
This is as far as Apollonius went; the next step would not be taken until 1643, when René Descartes discovered a formula for the size of the two tangent circles, which he wrote in a letter to Princess Elizabeth of the Palatinate. The same formula was later rediscovered by Frederick Soddy and published as a poem in Nature in 1936.
The size of a circle is determined by its radius rr. If rr is small, the circle will be small, but it will also be very curved. We can define the curvature of the circle to be k=1/rk=1/r. Descartes showed that if three given circles are mutually tangent at three distinct points, and have curvatures k1k1, k2k2, and k3k3, then a fourth circle which is tangent to all three has curvature k4k4 satisfying
For technical algebraic reasons, sometimes this equation gives negative values for the curvature k4k4, which we can interpret as corresponding to a circle with curvature |k||k|which contains the other circles in its interior. Notice that this equation is quadratic in the variable k4k4, so there are two solutions; these will correspond to the two possibilities for the fourth circle found by Apollonius.
Apollonian packings
So far we have constructed at most 5 mutually tangent circles. The step to infinity may seem obvious, but took another 63 years and some 1900 years after Apollonius. The earliest description seems to appear in a letter from Leibniz to des Bosses (11 March 1706):
Imagine a circle; in it draw three other circles that are the same size and as large as possible, and in any new circle and in the space between circles again draw the three largest circles of the same size that are possible. Imagine proceeding to infinity in this way…
A finite iteration of a nested Apollonian packing similar to the one described by Leibniz. Image: adapted from Todd Stedl, CC BY-SA 4.0
What Leibniz is describing is in fact a nested Apollonian packing, since at each step he fills in every circle as well as the gaps between circles. This early description makes the nested Apollonian packing one of the first fractals, although it wasn’t studied properly until mathematicians like Cantor, Weierstrass, von Koch, and Sierpinski started discovering other fractals in the late nineteenth and early twentieth centuries. This may be because Leibniz was not interested in the mathematical construction, but rather was trying to draw an analogy to argue against the existence in infinitesimals in nature.
Henceforth we shall only consider the un-nested Apollonian packing. As a fractal, it has a number of interesting properties: it is a set of measure 0, which means that if you tried to make it by starting with a disc of metal, and then drilled out infinitely many ever smaller holes (and if you ignore that metal is made out of atoms), then you would finish up with a single piece of metal (you haven’t removed everything), but nevertheless with exactly 0 mass. It has fractal dimension approximately 1.30568, which means that mathematically it lives somewhere between a 1D curve and a 2D area. Finally, if you look at just the portion of an Apollonian packing which lies in the triangular region between three tangent circles, this is homeomorphic to the Sierpinski triangle, which means that one can be bent and stretched to look like the other.
A portion of an Apollonian packing is homeomorphic to the Sierpinski triangle—just squash all of the circles to make them triangular. Images: adapted from Todd Stedl, CC BY-SA 4.0 and Beojan Stanislaus, CC BY-SA 3.0.
There is a curious combinatorial consequence of Descartes’ formula for Apollonian packings. If we start with three mutually tangent circles with curvatures k1k1, k2k2, and k3k3, we can solve (1)(1) to find that the curvatures k+4k4+ and k−4k4− of the other two circles are
The integral Apollonian packing starting with curvatures -10, 18, 23, and 27. Image: adapted from Todd Stedl, CC BY-SA 4.0.
Now suppose we start constructing an Apollonian packing by drawing four mutually tangent circles whose curvatures k1k1, k2k2, k3k3, and k+4k4+ are all integers. From equation (2)(2) it follows that 2√k1k2+k2k3+k3k12k1k2+k2k3+k3k1 must be an integer since k+4k4+ is an integer, and so k−4k4− is also an integer. Now we can build the packing by filling in a fifth circle wherever we see four mutually tangent circles. By the observation above, if the four circles have integer curvatures, the fifth circle will also have integer curvature. Inductively therefore we will end up with an Apollonian packing consisting of infinitely many tangent circles, all of which have integer curvatures.
Hyperbolic geometry
All these scorpions have the same hyperbolic size.
If you have some familiarity with non-Euclidean geometry, Apollonian packings may remind you of the Poincaré model of the hyperbolic plane. The hyperbolic plane H2H2 is a 2D surface on which we can do geometry just like we can on the flat Euclidean plane. Whereas a sphere has constant positive curvature (it curves the same way in all directions), and the Euclidean plane has constant zero curvature (it’s flat), H2H2 is an infinite surface which has constant negative curvature, which means that at every point it curves in the same way as a Pringle. This negative curvature makes the surface crinkle up on itself more and more as you move out towards infinity, which is inconvenient when we try to work with it. Usually then we represent it on a flat surface so we can draw pictures of it in magazines and the like. One way to do this is with the Poincaré model. This views the hyperbolic plane as a disc. In order to fit the whole infinity of H2H2 into a finite disc, we have to shrink distances as we move out towards the edge of the disc. Using this skewed way of measuring distances, the circular edge of the disc is infinitely far away from its centre.
We can think of an Apollonian packing as living in the Poincaré disc, with the outermost circle of the packing as the boundary circle of H2H2. Then the circles in the packing which are not tangent to this boundary are also circles in the strange hyperbolic way of measuring distance, that is, all points are equidistant from some other point in the plane—the circle’s hyperbolic centre. Circles in the packing which are tangent to the boundary are called horocycles (in Greek this literally means border circle), which are circles with infinite radius in the hyperbolic metric. Horocycles have no analogue in the Euclidean plane.
Something interesting happens when we see what an Apollonian packing looks like in the upper half-plane (UHP) model for H2H2. This model is similar to the Poincaré model, but instead of using a disc, we use the half-plane above the xx-axis {(x,y)∈R2:y>0}{(x,y)∈R2:y>0}, where the xx-axis behaves like the boundary circle and should be thought of as at infinity. There is a problem, in that in the Poincaré disc, the boundary of H2H2 was a circle, and so it closed up on itself. In the UHP, the boundary is a line which doesn’t close up on itself, but these are supposed to be models for the same thing. To fix this, we imagine there is a point at infinity ∞∞ which joins up the two ends of the boundary to form an infinite diameter circle.
If we start with any Apollonian packing living in the Poincaré disc, there is a map from the disc to the UHP preserving hyperbolic distances, under which the outer circle of the packing becomes the xx-axis (together with the point at infinity), and exactly one of the horocycles (one of the circles tangent to the outer circle in the packing) becomes the horizontal line y=1y=1. All other circles and horocycles in the packing are sent to circles which are tangent to each other as before, but are now sandwiched between the lines y=0y=0 and y=1y=1.
If we focus on just those circles which meet the xx-axis we get what are called Ford circles. Remarkably each of these circles is tangent to the xx-axis at a rational number p/qp/q, and has radius 1/2q21/2q2. Moreover every rational number is the point of tangency of one of the circles (see below). Now some magic happens: suppose the Ford circles at a/ba/b and c/dc/d are tangent to each other, then there is a unique circle sandwiched between these two circles and the xx-axis. The rational point at which this circle meets the xx-axis is given by the Farey sum of a/ba/b and c/dc/d
ab⊕cd=a+bc+dab⊕cd=a+bc+d
Note that for this to be well-defined, a/ba/b and c/dc/d must be written in their simplest form. This Farey sum, and the associated Farey sequences FnFn you get by looking at all rational numbers between 0 and 1 which can be written as a fraction with denominator at most nn, turn up in several places across number theory. These include rational approximation of irrational numbers and the Riemann Hypothesis.
Möbius transformations
If you haven’t seen hyperbolic geometry before, you may wonder how we can map the Poincaré disc model to the UHP model, and in such a way that the strange distance measure in the two models is preserved—for a start one is a finite region while the other is an infinite half-plane. The answer is to view both models as living inside the complex plane CC (or more accurately the extended complex plane ˆC=C∪{∞}C^=C∪{∞}): the Poincaré disc is the unit disc {z∈C:|z|<1}{z∈C:|z|<1}, and the UHP is the region above the real axis {z∈C:{z∈C: Im(z)>0}(z)>0}. Then a function like
z↦−iz+1z−1=−iz−iz−1(3)(3)z↦−iz+1z−1=−iz−iz−1
will do the trick. This function is an example of a Möbius transformation, which in general is a complex function of the form
z↦az+bbz+dz↦az+bbz+d
were we require ad−bc≠0ad−bc≠0 so that this function is invertible. The function (3)(3) sends the unit disc to the UHP, but it is not the only Möbius transformation which does this. In fact there are infinitely many such functions, all of which preserve the hyperbolic metric. In the previous section I claimed that starting with any Apollonian packing, we could choose one of these Möbius transformations such that the image had a very specific form, sandwiched between the lines Im(z)=0(z)=0 and Im(z)=1(z)=1.
An exercise: If you have seen Möbius transformations before, you may wish to try and prove that the purported mapping exists yourself. (Hint: remember that Möbius transformations send circles and lines to circles and lines, and are completely determined by their image on 3 distinct points.)
The upshot of this is that all Apollonian packings are the same in the hyperbolic plane, because they can all be mapped to the same packing by (invertible) functions which preserve hyperbolic distance. Once we have started thinking about the Apollonian packing living in the complex plane, the whole world of complex functions is open to us, and we can start to do crazy things. If we don’t restrict ourselves to just Möbius transformations, but see what happens when we apply holomorphic or anti-holomorphic functions to the packing (these are complex functions with a good notion of derivative in the sense of calculus, which in particular have the property that they preserve angles between intersecting curves), we can get some very pretty designs. We need not even require (anti-)holomorphicity. The patterns featured on the front and back covers were drawn in this way.
Beyond the packing
Let us return to Apollonius of Perga. Remember that his treaties Έπαφαι, where he stated and solved the problem of finding tangent circles, is lost to history—how then do we know what he proved and how? The answer is that we don’t. The only record we have appears in the writings of Pappus of Alexandria, who lived some 400 years after Apollonius, but who references many of Apollonius’ works, including six which are no longer extant. All he says of Tangencies is the general problem which Apollonius was interested in, and that he solved it by solving many simple special cases and working up from there.
The first person to reprove Apollonius’ results in ‘modern’ times was Adriaan van Roomen in 1596. His solution, however, does not use ruler and compass constructions, so cannot have been the one Apollonius used. The result was later proved using methods available to Apollonius, and in the way described by Pappus, by van Roomen’s friend François Viéte.
A ship’s location determined by its distance from three points.
The method of Viéte was later reworked and simplified by several mathematicians, including Isaac Newton in his Principia. Newton related the position of the centre of the fourth circle to its distance from the centres of the three circles to which it is supposed to be tangent. This is called hyperbolic positioning or trilateration. Newton used this viewpoint to describe the orbits of planets in the solar system, but it can also be used to help navigate ships, and to locate the source of a signal based on the different times the signal is received at three different locations. In the first world war this was used to locate artillery based on when shots were heard. This is also how modern GPS works (not by triangulation as is commonly believed).
So this 2000-year-old problem in abstract geometry turned out to have extremely useful applications in the real world. The Apollonian packing also shows up in lots of different areas of mathematics. For example, Ford circles inspired the Hardy–Littlewood circle method, an important tool in analytic number theory which was used to solve Waring’s Problem: for an integer kk, can every integer be written as a sum of at most nn kkth powers for some value of nn? This is true: for example, every integer is the sum of 4 squares, 9 cubes, 19 fourth powers, and so on. In 2013, Harald Helfgott used the circle method to prove the weak Goldbach conjecture: every odd number greater than 5 is the sum of 3 primes.
To infinity
As a final application, I am a geometric group theorist, and I cannot help but talk about one place the Apollonian packing shows up in my field. Be warned: there is definitely some advanced maths coming up, but if you don’t mind skipping over some of the details, there are some very pretty pictures to make it worthwhile.
It turns out that the extended complex plane ˆCC^ can be thought of as the boundary of 3 dimensional hyperbolic space H3H3. If we model H3H3 as the upper half-space {(x,y,z)∈R3∣z≥0}∪{∞}{(x,y,z)∈R3∣z≥0}∪{∞} then ˆCC^ is identified with the plane {(x,y,z)∈R3∣z=0}∪{∞}{(x,y,z)∈R3∣z=0}∪{∞}. When Möbius transformations act on ˆCC^, they also act on the whole of H3H3, and preserve hyperbolic distance. If we start by choosing just a few Möbius transformations, these generate a group which acts on H3H3. In doing so, the group creates a pattern on the complex plane called its limit set. This is a picture of how the group acts ‘at infinity’. Choosing the Möbius transformations carefully gives a group whose limit set is precisely the Apollonian packing.
A valid arrangement of circles, with their interiors shaded. Note that the straight line is an infinite radius circle through ∞∞, so it still counts.
Let’s be a bit more precise; pick a point p∈ˆCp∈C^ and choose gg pairs of circles (C+i,C−i)gi=1(Ci+,Ci−)i=1g, each of which doesn’t intersect pp. Each circle cuts ˆCC^ into two regions, call the region containing pp the exterior of that circle, and the complementary region the circle’s interior. We also want to arrange things so that no two circles have overlapping interiors (although two circles are allowed to be tangent). Next, for each pair of circles (C+i,C−i)(Ci+,Ci−) choose a Möbius transformation mimiwhich maps C+iCi+ to C−iCi− and which sends the interior of C+iCi+ to the exterior of C−iCi−. The group G=⟨m1,…,mg⟩G=⟨m1,…,mg⟩ generated by these transformations is called a (classical) Schottky group and it acts as a subgroup of the group of isometries of H3H3. Since we chose the circles to have non-overlapping interiors, we can use the delightfully named ‘Ping-Pong Lemma’ to prove that GG is abstractly isomorphic to the free group on gggenerators.
A packing with the four starting circles emphasised in black, and the Schottky group generators shown in pink and blue.
So how do we get a Schottky group whose limit set is the Apollonian packing? We can cheat slightly by working backwards; starting off with the picture we want to create, then we will choose the pairs of circles in the right way. Remember that one way we thought about constructing the Apollonian packing was to start off with four mutually tangent circles and then inductively draw the fifth circle wherever we can. Our strategy will be to choose Möbius transformations which do the same thing. We are helped by the following curious fact which you may want to try and prove yourself (again using Möbius transformations): given any three mutually tangent circles, there is a unique circle (possibly through ∞∞) which passes through all three circles at right angles. Given the four initial circles, there are (43)=4(43)=4 triples of mutually tangent circles, so we let C±1C1± and C±2C2± be the four circles orthogonal to each of these triples, as shown on the left. The corresponding Möbius transformations are :
The limit set of G=⟨m1,m2⟩G=⟨m1,m2⟩ is indeed the Apollonian packing we started with. If we perturb the starting Möbius transformations just slightly by varying the matrix entries (while being careful to ensure that the resulting group acts nicely on H3H3), we get a group whose limit set is a twisted Apollonian packing.
The limit sets of a one parameter family of groups which contains the Schottky group GG. Click the image to view a higher quality version.
Even though some of these perturbed limit sets look like they are still made up more or less of circles, they are in fact made up of a single continuous closed curve which is fractal, and does not intersect itself anywhere. They are examples of Jordan curves and illustrate why the Jordan Curve Theorem is so difficult to prove despite being ‘obvious’. Playing around more with different choices of Möbius generators we can produce even more beautiful examples of fractal limit sets; below are just a few to finish off. If you want to learn more about Schottky groups, their limit sets, and how to draw these pictures, I highly recommend the book Indra’s pearls: the vision of Felix Klein. It is the basis of this final section of this article, and gives details on exactly how you can draw these and many other pictures yourself.
David Sheard
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Source: The Dual Language of Geometry in Gothic Architecture: The Symbolic Message of Euclidian Geometry versus the Visual Dialogue of Fractal Geometry
Chausath Jogini Mandir (64 Joginis Temple) is situated in a hamlet called Hirapur, 20 km outside Bhubaneswar, the capital of Odisha state of Eastern India.
The temple is believed to be built by the Queen Hiradevi of Bramha dynasty during 9th century.
It’s built in a circular fashion, completely put together with blocks of sand stone. The inside of the circular wall has cavities, each housing the statue of a Goddess. There are around 56 such idols, made of black granite, inscribed within the wall cavities, centring on the main idol which is the Goddess Kali, who stands on a human head representing the triumph of the heart over the mind. The temple houses a central altar (Chandi Mandapa) which has the remaining 8 Goddess idols on all 4 sides. Some historians believe that an idol of Maha Bhairava was worshiped in the Chandi Mandapa.
64 Joginis Temple is a tantric temple, with hypaethral architecture as tantric prayer rituals involve worshiping the bhumandala (environment consisting all the 5 elements of nature – fire, water, earth, sky and ether).
The legend behind the temple according to local priests is of the Goddess Durga taking the form of 64 demi-goddesses in order to defeat a demon. After the fight the 64 goddesses (Joginis) asked Durga to commemorate them in the form of a temple structure.
The Jogini idols are generally representing a female figurine standing on an animal, a demon or a human head depicting the victory of Shakti (Feminine power). The idols express everything from rage, sadness, pleasure, joy, desire and happiness.
Such temples are also seen at Ranipur-Jharial site of the Balangir district in Odisha and seven other places in India.
The number 64 finds its reference in Hindu mythology in various forms viz Kala for time, Kala for performing arts etc. The temple complex is maintained by Archaeological Survey of India.
Source: 64 Yogini Temple in Hirapur, Odisha
Source: Chausathi Yogini Temple
Source: Chausathi Yogini Temple
Chaunsath Yogini ( 64 योगिनी )
Jul 26 2019
In very ancient days, eight great Female Goddesses Shaktis emerged from the cosmic soul of the Principle Dieties and formed Kali Durga, the Universal Shakti Power, These were the grand Mothers (Ashta Matrikas) of all subsequent Yoginis. According to Kaula Tantra, these eight manifested each in turn into eight Divine Shaktis, thus resulting in the 64 Tantric Yoginis. These 64 powerful Goddesses have unique personas and powers to fulfill your desires, drive away negativity and fear, prevent misfortunes, and give you knowledge, peace, all-around prosperity, good progeny, and auspiciousness of all kinds.
An examination of the ancient Tantric tradition reveals a particular sanctity assigned to the number eight. The eight mother faculties (tatvas) of the manifested universe, the eight directions with four cardinal and four intermediate points (digbandahs), the eight miraculous yogic powers (siddhis), eight “limbs” of Yoga (astanga) eight forms of the Divine Mother (matrikas) and eight primary mystic symbols (mudras) are just a few examples. The square of eight, or sixty-four, occupies an even more profound position in the field of Tantra which, from the point of view of the practitioner, first and foremost identifies the sixty-four Tantric Yoginis.
Legends about the 64 Yoginis
The Lalitha Sahasranama and Vishnu Bhagavatha Purana reverently call the Divine Mother Shakti as Maha Yogini (Great Ascetic) and Kula Yogini. The Yogini Sahasranamaand Kaula Kularnava Tantra refers to the Supreme Devi as Maha Yogini. In ancient days, 8 great Female Shaktis emerged from the Universal Shakthi Energy called Parashakthi – these were the Divine Grand Mothers (called Ashta Matrikas) of all subsequent Yoginis. These 8 Matrikas manifested each in turn into Eight Sacred Shaktis, thus resulting in the 64 Tantric Yoginis.
The Yoginis are also popularly known and worshipped in Hinduism, Jainism and Buddhism in several countries as Bhairavis, Dhakinis, Shakinis, Sakinis, Sakthis, etc. The 64 Yoginis are known to be capable to manifest in physical form to give dharshan even in this Kali Yuga.
References about the 64 Yoginis and their powers can be found in the ancient scriptures like : Brahmananda Purana, Agni Purana, Skanda Purana, Kalika Purana, Jnanarnava Tantra, Brihad Nandikeswara Purana, Chandi Purana of Sarala Das, Brihndla Tantra, Bata Avakasa of Balaram Das, etc. Historical romances and semi-historical literature like Somadevasuri’s Yasastilaka of AD 959, Kalhana’s Rajatarangini of c.1150 and Somadeva’s Kathasarit Sagara of c.1070 contain legendary stories about the all-powerful Yoginis.
Each form of the Divine Mother Kali is a Mahavidya. Dasa Mahavidyas are:
Kālī
Tārā
Ṣodaśī
Bhuvaneśvarī
Chinnamastā
Bhairavī
Dhūmāvatī
Bagalāmukhī
Mātangī
Kamala
Shodashi (Sixteen) Nityas of Lalita
Kamesvari
Bhagamalini
Nityaklinna
Bherunda
Vahnivasini
Mahavajresvari
Sivaduti
Tvarita
Kulasundari
Nitya
Nilapataka
Vijaya
Sarvamangala
Jvalamalini
Citra
Mahanitya
Shodashi (Sixteen) names of Devi and 16 days worship Rituals
Source: The Millennium old 16-day Durga Puja in Odisha
Source: The sixteen names of Durgā and their explanations
Chausath Yogini Temples in India
Source: When was Chousath Yogini Temple built?
There are 17 Chausath Yogini temples found across India out of which two are in Odisha, five in Madhya Pradesh, three in Uttar Pradesh, and one in Tamil Nadu. The most prominent ones are Hirapur, Ranipur Jharial, Khajuraho, Bhedaghat, Mitauli, Dudhai, and badoh. Several yogini temples have been lost, whether through deliberate destruction or plundering for building materials. Yogini images have been discovered from Shahdol, Hinglajgadh, Lokhari, Nareshwar, Rikhiyan, shahdol, Kanchipuram, greater Bengal, Varanasi, and Delhi.
Chausath Yogini Temple of Hirapur, also called Mahamaya Temple, is 20 km outside Bhubaneswar, the capital of Odisha state of Eastern India. It is devoted to the worship of the yoginis, auspicious goddess-like figures. The temple is believed to have been built by Queen Hiradevi of the Bramha dynasty during the 9th century. The temple is small and circular, only 25 feet in diameter. It is hypaethral and built of blocks of sandstone. The inside of the circular wall has niches, each housing the statue of a Goddess. 56 of the 64 idols, made of black stone, survive. They surround the main image at the center of the temple, the Goddess Kali
Chausath Yogini Temple of Ranipur Jharial in Balangir District, Odisha, The Chausath Yogini Temple, built in the 9th or 10th century in Ranipur-Jharial, in an isolated position some miles from the towns of Titilagarh and Kantabanjhi in Balangir district, Odisha, is a circular, hypaethral, 64-yogini temple made of sandstone, some 50 feet in diameter. 62 of the yogini images survive. The site, on an outcrop of rock, must have been important, given the presence of a large temple built of brick and several small temples of stone. The primary entrance is an opening in the circular wall towards the east; unlike at the Hirapur yogini temple, there was once a further opening towards the south, now filled in.
Chausath Yogini temple in Khajuraho town of Madhya Pradesh, India. Dated to the late 9th century, it is the oldest surviving temple at Khajuraho. According to an inscription dated to 1323 CE (Vikram Samvat 1383), the temple was built by the Kachchhapaghata king Devapala (1055 – 1075CE). It is said that the temple was the venue for providing education in astrology and mathematics based on the transit of the Sun. It is the earliest extant temple in the Chandela capital, Khajuraho. The temple is among the Western group of temples on a 5.4 m high platform. It has a rectangular plan measuring 31.4 m x 18.3 m. It is one of the historic Yogini temples across India; many of the others have a circular plan, though those at Rikhiyan and Badoh are also rectangular, so there was at least a local tradition of building them in this shape. Like all Yogini temples, the Khajuraho temple is hypaethral, open to the air. The temple is made of large, coarse granite blocks, with an open courtyard at the center. The centeryard was originally surrounded by 65 shrine cells: 10 on the front (north) wall, 11 on the back wall, and 22 on each side. Only 35 of these 65 cells now survive; each has a small doorway made of two squared granite pillars, a lintel stone, and a curvilinear tower roof. Above the lintel of the best-surviving cells is a triangular pediment. There is no surviving trace of a central shrine, whether to Shiva or the Goddess, as found in other Yogini temples.
Chausath Yogini Temple, Bhedaghat, also called the Golaki Math, is one of India’s yogini temples, but exceptionally it has shrines for 81 rather than the usual 64 yoginis. The temple is the largest of the circular yogini temples, some 125 feet in diameter. The scholar Shaman Hatley calls it the “most imposing and perhaps best known of the yogini temples”. It has a covered walkway with 81 cells for yoginis around the inside of its circular wall; three niches, two to the west, and one to the southeast remain open as entrances. There is a later shrine in the center of the courtyard; the temple was adapted as a Gauri-Shankar temple with the construction of the building in the south-centre of the circle in 1155 AD, at which time the central deities were moved. The temple was built early in the 11th Century AD by King Yuvaraja II, of the dynasty of the Kalachuris of Tripuri; he lived around 975-1025 AD
Chausath Yogini Temple, Mitaoli, also known as Ekattarso Mahadeva Temple, is an 11th-century temple in the Morena district in the Indian state of Madhya Pradesh. It is one of the few well-preserved Yogini temples in India. The temple is formed by a circular wall with 65 chambers, apparently for 64 yoginis and the goddess Devi, and an open mandapa in the center of a circular courtyard, sacred to Shiva. According to an inscription dated to 1323 CE (Vikram Samvat 1383), the temple was built by the Kachchhapaghata king Devapala (1055 – 1075CE). It is said that the temple was the venue for providing education in astrology and mathematics based on the transit of the Sun. The temple is on a hill about 100 feet (30 m) in height; there are 100 steps to climb up to the entrance. It is circular with a radius of 170 feet (52 m), while inside it has 65 small chambers, each with a mandapa which is open and a facia of pilasters and pillars. The roof of the ring of shrines is flat, as is that of the central shrine to Shiva; the circular courtyard is hypaethral, open to the sky, with an open porch as its entrance. The parliament building of India is said to have been based on this temple. The temple is in the Seismic Zone III region and has survived several earthquakes, seemingly without any serious damage. This fact was cited when the issue of safety from the earthquake effect of the circular Parliament House, its design supposedly based on the Mitaoli temple, was debated in the Indian Parliament.
Chausath Yogini Temple, Dudhai, Lalitpur, Uttarpradesh, details of temples are not available. The temple locally called as Akhada /Akhara, is situated in a forest of Buri Dudhai. Temple is believed to be built in the 10th century CE by Chandela kings. The temple is on a circular plan and has a hypaethral elevation. The temple is about 50 feet in diameter. The temple has 12 flat-roofed cells on the north, and five flat-roofed cells on the south, and the eastern and western portions are completely lost.
Chausath Yogini Temple, Badoh, Some 30 miles from Dudahi, at Badoh in Vidisha district, Madhya Pradesh is the Gadarmal temple of the Mothers, another 42-niche yogini temple, and one of the few that are rectangular. 18 broken images of the goddesses that once fitted into grooves in the temple platform are preserved from the waist down. It is composed of a rectangular shrine and a tall and massive Shikhara, adjacent to some Jain temples. Vidya Dehejia writes that the yogini temple must once have been hypaethral. Gadarmal Devi temple dates back to the 9th century. The architecture of this yogini temple is a fusion of Pratihara and Parmara styles.
LokhariThere appears to have been an early 10th-century yogini temple on a hilltop at Lokhari, Banda District, and Uttar Pradesh. A set of twenty images, nearly all theriomorphic, the figures having the heads of animals such as horse, cow, rabbit, snake, buffalo, goat, bear, and deer, has been recorded. Dehejia describes these as striking rather than specially artistic.
Nareshwar Another set of twenty 10th-century images, with careless later inscriptions from the 12th century, was rescued from Nareshwar (also called Naleshvar and Naresar) in Madhya Pradesh, a site which still has some twenty small Shaivite temples, to the Gwalior Museum, some fifteen miles away.
Hinglajgarh The site of Hinglajgarh, on the border of Madhya Pradesh and Rajasthan, was cleared of statuary for the building of the Gandhi Sagar Dam. The rescued statues contain enough fragments of yogini images for Dehejia to state that there was once a yogini temple at Hinglajgarh
Rikhiyan, Some 150 miles north of Khajuraho on the south bank of the River Yamuna, in the Banda District, Uttar Pradesh, are the fragmentary remains of what seems to have been a rectangular 64-Yogini temple in the Rikhiyan valley. This is part of a complex of other temples, unlike the solitary Yogini temples such as Mitaoli. When the site was photographed in 1909, ten four-Yogini slabs were present. Dehejia states that the multiples of 4 suggest a 64-Yogini total, while the straightness of the slabs implies a rectangular plan (as at Khajuraho). Seven were stolen on various occasions, and the last 3 of the slabs were moved to Gadhwa fort nearby for their safety. The slabs portray the Yoginis on a plain background without the usual attendant figures. They sit in the ceremonial pose of Lalitasana, one leg resting on their animal vehicle. They have “heavy breasts, broad waist[s] and large stomach[s]”. One has the head of a horse and holds a corpse, a severed head, a club, and a bell, and so maybe Hayanana, “The Horse-headed”. This and other Yoginis shown with corpses link the temple to a corpse ritual. Also photographed in 1909 were three three-Matrika slabs; Dehejia suggests that these formed part of a rectangular shrine to the Eight Matrikas accompanied by Ganesh. A well-preserved four-Yogini slab from Rikhiyan is held in the Denver Art Museum
Shahdol, Yogini images from Shahdol district (anciently Sahasa-dollaka) in Madhya Pradesh have been taken to the Dhubela Museum near Khajuraho, the Indian Museum at Calcutta, and the village temples of Antara and Panchgaon in Shahdol district. The yoginis are seated in the ceremonial Lalitasana pose, and they have haloes flanked by flying figures behind their heads
Kanchipuram or Kaveripakkam Dehejia publishes and discusses yogini images of the Chola period, around 900 AD, recovered from northern Tamil Nadu. These include one now in the British Museum, others in the Madras Museum, the Brooklyn Museum, the Minneapolis Institute of Arts, the Detroit Institute of Arts, and the Royal Ontario Museum. The British Museum yogini is ascribed to Kanchipuram; the collection site is not known, but many sculptures of the same style were recovered from a large “tank” (artificial lake) at Kaveripakkam, seemingly derived from nearby temples. The image formed part of a large set of yoginis.
Varanasi, 12th-century texts including the Varanasimahatmya of the Bhairavapradurbhava suggest that there was a circular hypaethral yogini temple in Varanasi (also called Benares and Kashi) in the 11th century. Several yogini-related sites have been identified in the city. Just above the Chaumsathi Ghat cremation ground is Chaumsathi Devi temple; it is not mentioned in the scriptures but is where modern-day devotees gather, especially at Holi, as prescribed in the Kashikhanda.
Delhi, Legend has it that a yogini temple was built in the south Delhi district of Mehrauli; tradition places this as the Yogmaya Temple there, without reliable evidence. The region outside the imperial city of Indraprastha, described in the Mahabharata, was called Yoginipura, the yogini city. Indraprastha has been identified with Delhi.
Source of Picture: Google
Chausath Yogini Temple, Bhedha Ghat, Jabalpur, MP, India
Source: Chausath Yogini Temple – Complete Inventory of Goddesses and Gods
Source: Chausath Yogini Temple – Complete Inventory of Goddesses and Gods
Dupuis, Stella (2008). The Yoginī Temples of India : in the pursuit of a mystery, travel notes. Varanasi: Pilgrims Publishing. ISBN978-81-7769-665-3. OCLC298129207.
Dehejia, V. 1986. Yoginī Cult and Temples: A Tantric Tradition. New Delhi: National Museum.
Donaldson, Thomas E. 1985. Hindu Temple Art of Orissa 3 vols. Leiden: Brill.
Gadon, Elinor. ‘Probing the Mysteries of the Hirapur Yoginis’. In ReVision Vol. 25, no. 1 (2002): 33-41.
Shaw, Miranda. 1994. Passionate Enlightenment: Women in Tantric Buddhism. New Jersey: Princeton University Press.
Mahapatra, K. N. 1953. ‘A Note on the Hypaethral Temple of Sixty-four Yoginis at Hirapur,’ Orissa Historical Research Journal II: 23–40; reprinted in H. K. Mahtab, ed., Orissa Historical Research Journal, Special Volume, 1982.
Hatley, Shaman. 2007. ‘The Brahmayamalatantra and Early Saiva Cult of Yoginis’. Unpublished PhD. dissertation, University of Pennsylvania.
Stietencron, Henrich von. 2013. ‘Cosmographic buildings of India: The circles of the yogini,’ Yogini in South Asia: Interdisciplinary Approaches(Routledge Studies in Asian Religion and Philosophy), ed. István Keul, pp. 70-83. London: Routledge.
Further readings
Hatley, Shaman. ‘Matr to Yogini: Continuity and Transformations in the cult of the Mother Goddesses,’ in Transformations and Transfer of Tantra in Asia and Beyond, edited by István Keul, pp. 99–129, Berlin: Walter de Gruyter, 2012.
Mishra, P.B. Orissa under the Bhauma Kings. Calcutta: Vishwamitra Press, 1934.
Panigrahi, K. C. Archaeological Remains at Bhubaneswar. Bombay: Orient Longmans, 1961.
Sharma, Rajkumar. The Temple of Chaunsatha-yogini at Bheraghat. Delhi: Agam Kala Prakashan, 1978.
Source: A Unifying Model of the Arts: The Narration/ Coordination Model
The Narration/Coordination model is presented as a unifying model of the arts with regard to psychological processing and social functions. The model proposes a classification of the arts into the two broad categories of the narrative arts and the coordinative arts. The narrative arts function to tell stories, often to promote social learning through the modeling of prosocial behaviors. The coordinative arts function to stimulate group participation through synchronized action, thereby serving as a reinforcer of group affiliation and a promoter of social cooperation. These two categories vary with regard to a number of psychological and social features related to personal engagement, role playing, cognitive structure, and performance. The arts are evolutionarily adaptive because they promote social cooperation through two distinct routes: the simulation of prosocial behaviors via the narrative arts, and the stimulation of group synchronization and cohesion via the coordinative arts.
Narrative and Coordinative Arts
Source: A UNIFYING MODEL OF THE ARTS: THE NARRATION/ COORDINATION MODEL
Narration/Coordination Model of the Arts
Source: A UNIFYING MODEL OF THE ARTS: THE NARRATION/ COORDINATION MODEL
Features of Narrative and Coordinative Arts
Source: A UNIFYING MODEL OF THE ARTS: THE NARRATION/ COORDINATION MODEL
Classification of Arts
Source: TOWARD A UNIFICATION OF THE ARTS
Interaction among the Arts
Source: TOWARD A UNIFICATION OF THE ARTS
Modular Aspects of Performance Arts
Source: TOWARD A UNIFICATION OF THE ARTS
Connections Between the arts: an Indian Perspective
Source: ART AND COSMOLOGY IN INDIA
The view that the arts belong to the domain of the sacred and that there is a connection between them is given most clearly in a famous passage in the Vishnudharmottara Purana in which the sage Markandeya instructs the king Vajra in the art of sculpture, teaching that to learn it one must first learn painting, dance, and music:
Vajra: How should I make the forms of gods so that the image may always manifest the deity?
Markandeya: He who does not know the canon of painting (citrasutram) can never know the canon of image-making (pratima lakshanam).
Vajra: Explain to me the canon of painting as one who knows the canon of painting knows the canon of image-making.
Markandeya: It is very difficult to know the canon of painting without the canon of dance (nritta shastra), for in both the world is to be represented.
Vajra: Explain to me the canon of dance and then you will speak about the canon of painting, for one who knows the practice of the canon of dance knows painting.
Markandeya: Dance is difficult to understand by one who is not acquainted with instrumental music (atodya).
Vajra: Speak about instrumental music and then you will speak about the canon of dance, because when the instrumental music is properly understood, one understands dance.
Markandeya: Without vocal music (gita) it is not possible to know instrumental music.
Vajra: Explain to me the canon of vocal music, because he, who knows the canon of vocal music, is the best of men who knows everything.
Markandeya: Vocal music is to be understood as subject to recitation that may be done in two ways, prose (gadya) and verse (padya). Verse is in many meters.
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Viewpoints which Matter 
