Slowly recognising a mathematical order in sound and human aesthetics
Pythagoras – proportionality
The awareness developed slowly of a geometric and mathematical order to the natural world and our place as humanity within it. Pythagoras developed music theory around 500 BCE, noticing the connection between the weights of a blacksmith’s hammers and the pitch of the note that rang out when metal struck metal.
“A hammer weighing half as much as another sounded a note twice as high, an octave, with a ratio of 2:1. A pair weighing a ratio of 3:2 sounded beautiful, a fifth apart. [A pair in the ratio of 5:4 sounds a major third apart.] Simple ratios made appealing sounds.” [Quoting Anthony Ashton. Harmonograph: A Visual Guide to the Mathematics of Music. Wooden Books Ltd., Glastonbury. Revised Edition 2005.]
He went on to discover that all simple instruments, whether they are struck, plucked or blown, work in much the same way. The same proportions of scale when paired are pleasant to the ear. He identified a number series, equivalent to ‘notes’, the pairing of which produced harmonious sound – 16, 12, 9, 8, 6 and 4. From this, Pythagoras developed a theory of numbers. He started to see the structure of nature as harmony, much as physicists now believe that movement in nature conforms to laws expressed structurally in mathematical or algebraic formulae.
Harmony is, of course, a psychological appreciation of the relatedness of two or more tonal pitches of sound. (Pitch is the highness or lowness of your voice, tone is the quality or mood of your voice, and volume is the loudness or softness of your voice. By varying these elements, people can create contrast, emphasis, and interest in their speech.) Any note produced in nature, including human speech, has a tonal quality produced by ‘overtones’ that fill out the basic pitch. There is thus a nuanced relationship between the psychology of harmony, and the resonant order of nature, as we shall go on to see.
Resonance is how one vibration at a certain frequency can induce a remote vibration of tone and more complex overtones elsewhere in nature. This close and subtly interrelated feature of movement in the physical structure of nature and human embodiment will become important later, when we consider how the mathematically rationalised scientific theory of music slightly mismatches the wider spirituality of embodied human relatedness. By spirituality, I mean living our dynamic connection into the local and the nonlocal movements of shared relational life, which requires our acceptance that not everything that is happens locally can be integrated into a rationalised Big Picture. There has to be some element of mystery about truly nonlocal movement, and some interpret this as spirituality.
Harmony is the synergising of movement as our lived experience aligns with the ‘musicality’ of relational nature. Musicality is that rhythmic co-creation of mutuality in expression and response creating aesthetic joy in its beauty (and a lack of beauty in aesthetic disharmony). Whereas resonance is spatially located, harmony is the timing and synchronicity of nonlocal relationships. Harmony is a present state of being that shapes ongoing events by our inclusion in the mutuality of a process. Our interpretations of memorised events and our hopes also influence that present state of being. This shapes ‘meaning’ into our experiences of nature and its ongoing creation, which may explain why, for some people, harmony is a spiritual or sacred experience of relatedness beyond our spatial framework of thinking and living and moving. Harmony is in the making of that framework, as we shall go on to see.
Harmony stabilises resonant movement by the proportionality of feedback that Pythagoras noticed. In our Third Millennium CE, technology has advanced so that auditory vibrational tones can be seen. These are not only in oscilloscope traces on a screen, but even in 1787, Ernst Chladni found that by scattering fine dust on a square plate (and on other shaped surfaces), and bowing the edge or otherwise vibrating it, he could produce patterns from harmonies. Figure 1 (xx) shows Chladni patterns of harmonious sounds. Disharmonies produce a chaotic mess on the plate. These harmonic patterns shape up slowly, progressively refining into patterns that form, and transform as synergies dispersed throughout ‘the whole’ harmonically change.
It is worthy of note at this stage that recent astronomical measurements into the depths of space have shown that the distribution of known ‘visible’ matter within and between galaxies aligns into patterns of variable density much like Chladni patterns. This suggests there is a vibrational wholeness to the entire cosmos, which is replicated microcosmically at the local scales where human lives appreciate beauty. The vibrational wholeness on the cosmological scale is measurable as gravitational waves, electromagnetic (EM) waves, and thermodynamic vibrations (temperature). At the local scale, the equivalent vibrational wholeness is experienced as social harmonies among groups of people who are appreciating the same proportional relatedness in the aesthetic enjoyment of all of our senses.
The psychological appreciation of beauty connects musical vibration with visual arts, with the spoken word and the kineasthetics of embodied movement in dance, in the texture of movement in touch or intimacy, in the biochemical movements of aroma and taste, in the mathematical beauty of proportion in formulas emerging as they seem to do into nature’s forms and processes, and even in the human love of wisdom, for example aspiring to Ideals behind the messiness and complexity of reality’s ongoing processes. (Is beauty is in the eye of the beholder, or in some higher spiritual realm, or in the proportional relationship of both?). The common feature of all these features of life is proportional movement.
Euclid – The Golden Ratio in depth and replication as octaves
Surprisingly the early Pythagoreans were not particularly interested in circles. They were particularly strong on triangular proportions, the story behind the hypotenuse. It was Euclid in about 300 BCE who fitted triangles, squares, pentangles and more inside the circumference of circles. He thus discovered a feature that, since the Renaissance, is now called the Golden Ratio. An explanation of the Golden Ratio will be fruitful, and we shall return to it later. It is everywhere, including the musical scales that we propose are vibrationally significant for the material, psychological and spiritual unity that stabilises the continuous creation of life. Those same vibrational principles would shape from within the healing and restoration of separated ‘parts’ into a dynamic wholeness when life has become disharmonious or even dislocated and broken.
Euclid first discovered the Golden Ratio in the five sided geometric shape called a pentagon. By taking five points equidistant on the circumference of a circle, and connecting them with straight lines that criss-cross, he drew a pentangle star (one internal line of which in this diagram is omitted). Its outer borders can be joined to make the geometric pentagon. He realised that the ratio of the length of line AC to AB, where AB is bisected at C by line DE, is proportionally the same as that found to be aesthetically harmonious in the musical and visual arts.
That harmonious ratio is now mathematically defined as 1.61803… (or 1 / 1.618 = 0.618…), an irrational (unending) number that since the Renaissance has been called the Golden Ratio, phi φ. It so turns out that line AE, the side of the regular pentagon, is the same length as AC (ACE making an isosceles triangle). The same phi φ ratio is therefore easier to see as the length of the longest internal line of the five-pointed star (AB) divided by the length of its pentagon side (AE).
This Golden Ratio proportion is identifiable in a remarkably wide range of physical structures in nature, and also heard in the musical proportions within the octaves. Examples in nature include the proportions of the forms of many animal limbs, fish bodies, plants, and even the proportions of embodied human beings as seen by Leonardo da Vinci. Amazingly it is also now identified in the proportional ratio of the turns of DNA’s double helix in relation to its molecular width, and even in the proportions of rotating galaxies. This suggests that a stabilising proportional balance of movement must be innate to the microstructure of all matter.
To leap ahead and give a hint of where this article is going, there is an ancient Celtic icon of the unending movement behind all that seems constant, dating back to the fourth century BCE. It is now called a triquetra (tri- = three; quetra = corners), meaning a three-cornered shape that is not a triangle. Following the unending line along gives a feel of integration to constant movement even among the diversity of its angled turns, especially as connecting into networks, such as the liganded double triquetra. These internal reflective turns can open out into ligands of movement, making triquetra a distinct geometric (topological) shape from a trefoil, which is only an internally curling mobius strip.
The Golden Ratio is visible in the triquetra. One line loops behind and then over the other two, creating a helicoid waveform. Where it crosses the second line is approximately 2/3 to 1/3 along its own length. This waveform allows the flexibility of internal movement iconically represented here to stabilize into a Golden Ratio that integrates wider patterns.
A most striking geometric feature of the Golden Ratio, however, is how this ratio of lengths allows the pentagon to scale further into the centre of its expanding circle.
Depth scaling brings the potential for resonance and overtones across different octaves of movement. This is a scalable quality of connectedness among ‘substates’ of a greater wholeness, bringing what is now called a fractal replication of patterning into life’s movements and growth, replicating image and likeness. It is the scalable features of this geometry that leads some people to describe the Golden Ratio as a sacred balance – a balance that proportions life in all its diversity of growing forms – which thus becomes the source of life’s potential for continuous renewal and restoration.
Archimedes and Ptolemy – circles and pi (π)
Archimedes in 250 BC first calculated the circumference of a circle using polygons. He fitted a regular hexagon inside a circle (which has a true diameter, unlike a pentagon) then made a calculation adding its side lengths. He then increased the detail of this method up to a 96 sided polygon, such that the diameter of the circle multiplied by his version of π yields its circumference. His measure was further refined by Ptolemy in 150 BC as π = 3.1416. In our present days computers can bring greater specificity, to millions of decimal points, of this unendingly imprecise ‘irrational number’. The important implication to bring into sharp focus here is that all mathematical calculations of curved movement are approximations only. Drawing a circle is an analogue curve of movement. Drawing a curve by hand is not a static digital process adding steps of whole numbers of sides of whole polygons, even if this model could be extended to an infinite ‘number’ of polygonal sides. To draw or analyse a curve mathematically requires the proportions of Pythagorean geometry (the sine and cosine of right-angled triangles), which also connect the curve to its ‘centre’.
This important point of connection and imprecision in life will reappear later. It derives from this key observation and fact that proportionality (not number) is foundational to cyclical movement.
Fibonacci – scaling and fractals
Moving on to the early 13th Century in Italy, a brilliant mathematician and number theorist, Leonardo Pisano Fibonacci, introduced the Indo-Arabic numerical system to Europe. He had shown how adding ‘zero’ as a number (representing a state of stasis) was superior to the Roman numerical system for representing the movements of nature. The Roman system is ‘positivist’, in that all numbers refer to defined or boundaried actualities. It allows no scope for potential or probability of change or growth, whereas in life and nature all is in changing movement, mostly cyclical or stabilised in the familiar scale of our planet Earth. Even the rocks have cyclical movement ongoing in the minutiae of their cooled, wave-particle elements, packed tightly together in ways that can loosen under the pressure of gravity in the earth’s molten core.
Fibonacci’s number theory enabled him to quantify the scaling of the harmonic Golden Ratio. The Fibonacci sequence of numbers is 0, 1, 1, 2, 3, 5, 8, 13, 21, 34… Each number is obtained by simply adding together the previous two. This expanding sequence of proportions can identify the Golden Ratio everywhere in nature, appreciated as harmonious beauty. The ammonite or nautilus shell is famously demonstrating Fibonacci scaling.
Each square of the Fibonacci ‘vector diagram’ shown here gains in size (or reduces) with the Golden Ratio proportion. The Nautilus shell shows infinite gradual progression of scaling from φ in nature. Phi is the stabilising proportion within the cumulative movements that constitute growth, which is a slow and minutely detailed process in which every connection has to be made and stabilised for its overall form to be replicated or scaled.
Renaissance Golden Ratio – aesthetics with phi (φ) shaping material forms
The Renaissance (Re-birth) started in the 15th Century CE as a rediscovery of Classical beauty after a period of intellectual, political and religious decline in the late Middle Ages. The printing press liberated learning and questioning, resulting in the authority of religious traditions being challenged, and a Protestant revolution shaking established cultures across the world. Some historians have attributed the growth of humanism in this three century era to the psychological impact of the 1531 earthquake and tsunami that destroyed most of Lisbon causing 30,000 deaths and the decline of Portugal’s global influence. A profound questioning of divine providence accompanied the cultural grief. A resurgence of classical order was aspired to, perhaps in the maxim of Protagoras, that ‘Man is the measure of all things’. Three centuries of exploratory development followed in every area of human endeavour. The ‘Renaissance Man’ was widely read and imaginative (although it was to be another four centuries before the Renaissance Woman could be equally valued). Polymaths of great accomplishment arose, perhaps most exemplified in Leonardo da Vinci, for whom art expressed truth and spiritual depth in complete unity with scientific and technological advancements. This profusion of creativity in art brought to light the aesthetic of Golden Ratio proportionality. It captured there a mystical intuition of order underlying all diverse things. The truth of proportionality took on sacred and mystical associations, which for some became protected as ‘occult knowledge’.
The quest for mystical depth of lived experience outside religion thus also led to alchemical and gnostic diversification during this time, as people tried to capture a deeper meaning to life and its beauty in ways other than art and music. It was not until the following era, known as The Enlightenment during the 18th to 20th Centuries CE, that a scientific method of observation developed rather than participation in this shared life. Out of the mental separation required for dispassionate observation arose the dualist worldview of materialism.
Materialism became a paradigm of thought and education in the ‘Enlightened’ Westernised world. It empowered human control over nature by bringing a mental separation of physics and chemistry from a view of the depths of life as spiritual connectedness. In the Renaissance era, these two perspectives on the underlying unity of life had been fully integrated as people explored how to develop new ways of living together. In this 21st Century CE this paradigm of materialism is being questioned again. Quantum physics is shaking its roots by questioning materialism’s view of internally static and separated particles. Simultaneously, people’s human need to account for the mysterious depths of their lived experience is challenging again their former education into a view of life where there is a separation of random material from the connecting spiritual. A further cultural revolution is in progress.
The reintegration of worldviews that potentially could occur over the next two or three decades or centuries could unfold around understanding how the Golden Ratio stabilizes a vibrational view of matter.
In this latest century around the turn of the Millennium, the exponential growth of the New Science of adaptive systems and living systems in biochemistry, ecology and cosmology has affirmed the widespread presence of Golden Ratio proportionality in nature. The proposal being framed in this article is that the entire cosmos is an open system of movement influenced in its probabilities for change (but not ‘controlled’) by stabilising all dynamic processes in a tendency towards a Golden Ratio of harmonic balance. The balance nevertheless is bursting with potential for life forms to grow and self-replicate in suitable environments.
It is worth restating that the known universe is not only expanding or growing. It is expanding at an ever increasing rate. This suggests there must be a process everywhere that leads to expansive movement, with reflective synergy and infilling by patterning of movement that can stabilize where conditions allow. The triquetra is an icon of movement that intrinsically displays Golden Ratio proportionality even at the Planck length. It may give people a visual… not a building block, but better… a visual icon of a bundle of spin that builds from within – and may continue to expand, reflect, and infill.
Part 2 – Coming soon
From static circles to helicoid movement in waves
The cultural setting in which to consider the importance of proportional harmony
The Schrödinger Equation – probabilities of change in transient processes
The Pythagorean Comma – divergence of maths and Golden Ratio proportionality
Resonance and Harmony – linear maths mismatches the Fibonacci curled cochlea
Social Human Musicality – the stabilisation of movement in harmonic patterns
The Triquetral Golden Ratio in Movement – the creativity of sung or spoken Word
