Before going into the realms of harmonics, which have been referred to a few times in the various quotes, I want to interject with another perspective, related to the above subject, which will require a rather large quote from the book, Turbulent Mirror, by John Briggs and David Peatt. They are talking about research on Solitons and Equipartition. To clarify, a Soliton is an unexpected, solitary (hence Soliton) wave: – whereas a normal wave tends to break up over time and lose energy, a Soliton is a wave that becomes reinforced and bound by other waves joining together and can pass through other waves without losing its integrity. As Peatt and Briggs put it:
A soliton is born on the edge. If too much energy is involved in the initial interaction, the wave breaks up into turbulence. If too little energy, the wave dissipates…nonlinear interactions at critical values don’t produce chaos, they produce spontaneous self-organizing forms. (B&P, p.120)
This becomes relevant in a series of experiments three scientists performed in 1955. They were researching Equipartition of energy. Equipartition is the assumption, or the action of newly introduced energy into a system, to even itself out over the range of that system, as opposed to gathering in one place. Physicist Enrico Fermi and mathematicians Stanislav Ulam and J. Pasta were exploring the movement of vibrations through metal.
The internal structure of metal contains a stable pattern, called a lattice, of atoms. When energy, in the form of heat, is given to the metal it causes the atoms to vibrate. But because these atoms are all bound together in the lattice they vibrate in a collective way, producing a single “note”. In fact, there are many notes, many different modes of vibration within the lattice, and each of these is associated with a characteristic energy.
According to the principle of equipartition, if all the heat energy were to be given to a certain note – that is, to a particular vibration of the lattice – then pretty soon that energy would spread out and distribute itself to all the other “notes” of the lattice. This was the great assumption of thermodynamics and, since no one could actually get inside a lattice to see what was happening, it had never been observed directly. But with the coming of the computer the lattice could be looked at indirectly, through a mathematical model. To observe the way energy was shared between all the vibrational notes in the lattice, Fermi, Pasta, and Ulam set up a model containing five notes or modes. The plan was to feed one mode with energy and watch how this energy obeyed the strictures of thermodynamics by distributing itself through the other modes. In order to mathematically represent this sharing of energy it was necessary to add a tiny extra term – a nonlinear term – corresponding to the interaction between modes. If it was not added there was no way that “energy” in the model could pass from one note to another. As it turned out, this tiny additional term dominated the whole system and transformed it from a linear, well-behaved lattice into an arena for solitons.
In the 1950s when the Fermi-Ulam-Pasta calculation was carried out no one was seriously thinking about solitons, so the three scientists were quite confident that once the system had settled down from its initial burst of energy, the energy would soon be parceled out among all the other vibrational modes.
As expected, after a few hundred cycles of the calculation, mode 1 began to fall rapidly in energy and modes 2,3,4 and 5 began to gain. And after 2,500 iterations of the equation everything was still going according to plan. Then something wonderlandish occurred. While vibrational mode 1 continued to lose energy, mode 4 began to gain at the expense of all the other modes. By 3,500 cycles mode 4 had peaked and now mode 3 was beginning to gather energy. To the complete surprise of the scientists, energy was not being shared out equally but was bunching itself together in one or another of the modes. By the end of 30,000 cycles, energy was not equipartitioned at all but had returned and gathered itself again into the first mode!
The result was especially shocking because it was found that this concentration of energy doesn’t depend on the strength of the nonlinear interaction; even a very weak coupling of feedback will cause the system to bunch.The computer calculation indicated that the nonlinear lattice had a sort of “memory” not possessed by its linear counterpart. Given sufficient time, the system would return again and again to the state it was in when it first received its burst of energy…Analysis of the Fermi-Pasta-Ulam model shows that the phenomenon involved formation of a soliton – not of water or air but of energy – which moves through the lattice in a coherent wave.
The model is illuminating because it shows that the nonlinear world is holistic; it’s a world where everything is interconnected, so there must always be a subtle order present. Even what appears on the surface as disorder contains a high degree of implicit correlation. Sometimes this below-the-surface correlation can be triggered and emerges to shape the system. Soliton behavior is, therefore, a mirror of chaos. On one side of the mirror, the orderly system falls victim to an attracting chaos; on the other, the chaotic system discovers the potentiality in its interactions for an attracting order. On one side, a simple regular system reveals its implicit complexity. On the other, complexity reveals its implicit coherence. (B&P p.127)
I am particularly interested in the ‘lattice’ idea, since it is one that I came across in Hans Jenny’s book Cymatics. As he puts it when referring to liquids:
The wave lattice generated in a liquid by the action of sound…imposes a spatial pattern on a diffusion process occurring there. Into the vibrating liquid we drip some of the same liquid which has been colored with a marker dye, expecting that we shall see it mix intimately with the outspread film. However, instead of diffusing uniformly [as expected in the previous experiment: equipartition], the colored liquid first shoots, as it were, in jets, through the meshes of the lattice. If we greatly intensify the process, say by turning up the volume of the tone (increase of amplitude), we see how the colored liquid jets forth but always in a particular direction. If we examine one of these jets more closely… we can see that the liquid moves through the lattice in a complicated manner… observation is guided forward step by step by the phenomenon itself. We might note that structural patterns generated in a particular medium by vibration do in actual fact exercise a spatially directive function” (p.13)
This last part of the description is remarkably similar to the previous description of a Soliton and I find myself wondering how the presence of a Soliton accelerates, impacts or influences the movement of tone, or energy, within a system any more than its constituent parts i.e. If waves of energy or tone were generated and this Soliton effect did not happen, does ‘healing’ and resonance still occur, for instance, with individually resonating waves? In other words, is a Soliton a by-product or a necessity to sonic and vibrational influences in, and on, a system, with particular emphasis on its health and well-being?
Finally, we have reached the explication of harmonics. In music, when one goes to a concert or hears someone sing, moment by moment, we are hearing a fundamental tone; otherwise known as the first partial, the root, the tonic. The reason it is pointed out, so to speak, is because it is not the only tone that is occurring, even if, for some of us, it is the only one that we think we are hearing, or are in fact hearing. The way I perceive it is like a ladder; the first note, the ‘visible’ and obvious note is sounded, for example ‘middle C’ (256 Hz). Once that note is sounded out, whether struck, blown, bowed or sung, an immediate rainbow of other notes is revealed, embedded within the outward movement of the selected note. There is an order to this apparently secret set of notes and this order is something that can be found throughout the Universe. It is an order of ratios based on mathematical principles that is constant, i.e. the harmonic ‘scale’ is itself consistent. What differentiates the sound (timbre/character) of one instrument or persons voice from another is the shape and the material of each body (as well as its temperature and humidity and various other organic as well as personal factors) as these two main elements dictate which of the infinite number of harmonics get automatically accentuated. There is no effort involved here, it is involuntary and revealing, especially on the part of a human’s voice.
One thing I will say at this point though, is that I am very aware of how we, as individuals, impact the timbre of our voice, through our relationship to spirit, to our own selves – our psychology – and even in relation to our thinking. How we hold ourselves and with how much awareness, to each of these systems, impacts the timbre of our voice. i.e. We reveal ourselves in the way we speak (or sing), not only via the content of our words but by their voice’s textual quality. As a voice teacher, this is something that I find particularly exciting within a person since their ability to transform themselves becomes immediately apparent in their presence, particularly evident in their voice. In other words, self- transformation is real and it affects our relationship to a cosmic textual and aesthetic architecture, if you will, as different notes, perhaps even different fundamentals get selected as we grow and change.
Without wishing to go into too much detail on harmonics (aka ‘the overtone series’) at this point, I will outline their form, taking ‘A’ @ 220Hz (found an octave + m3 below middle C) as the fundamental in this example. The ascending tones are all considered in relation to the fundamental. The rules are simple: each consecutive harmonic is a whole multiple of the fundamental. And the intervalic ratio between each ascending harmonic and its predecessor diminishes (i.e. Fundamental, 8ve, P5, P4, M3, minor 3 etc.). The ratios between the partials (harmonics) thus look like this 2:1, 3:1, 4:1, 5:1 etc. – which corresponds to the length of a string on a monochord in terms of whether the string is being held down in the middle, in thirds, in four places etc.- Whilst the mathematical relationship between the harmonics and the fundamental are a direct I/Thou relationship, not in relation to each of the other harmonics. So, if the fundamental note is ‘A’ at 220Hz, the first harmonic (at 2:1) is 2×220= 440Hz, the 2nd harmonic (at 3:1) is 3×220=660Hz, the third harmonic (at 4:1) is 4×220=880Hz and the fourth harmonic (at 5:1) is 5×220=1100Hz. These particular note names are therefore A (fundamental), A (8ve above), E (P5 above), A (P4 above), C# (M3 above). [Pse. Refer to p.52 Sound at back for full description w/diagram]
There is one other point to make here, and that is the existence of so many different types of scales and musical systems. This harmonic system is based on what is commonly known as the ‘just intonation’ system. This is a reference to pure tone, i.e. tone that has not been manipulated or adjusted. There is also the fact of the microtonal and quarter tone system commonly used in China, India and the Middle East, whereas here, in the West, we have Bach to thank for the development of the piano which, in turn, introduced the tempered system (it was the only way the instrument could be made that would work for cross-platform i.e. Group playing, with the ability to modulate easily and still ‘make sense’ musically). The tempered system is an adjustment of the natural pure tuning of the ‘just intonation’ system, and is made up of intervals of steps and half steps (i.e. c, c#, d, d#).
When we refer to ‘A’ @ 440Hz or 220Hz, we are referring to the ‘equal tempered’ system. In this system, ‘middle c’ is actually 261.6Hz, whereas in the ‘just intonation’ system (pure tunings) it is, in fact, 256Hz. It seems that this system was introduced by ‘squishing down’ (!) an octave so that a ‘perfect octave’ (i.e. perfectly predictable) could exist. Rather like chaos theory, if this wasn’t done, the incremental changes in increasing and decreasing octaves would become so large that they would render the idea of an exact octave with exact and predictable intervals impossible and therefore playing in large groups would be disastrous without an exceptionally high level of musicality.
Hans Jenny, in Cymatics, points to the wonderful link between harmonics and form in Nature:
The resultants of harmonic vibrations are at all times so strictly law-ordered that it is possible to draw up a systematology of morphogenesis. What one must bear in mind is that under this or that quite specific set of conditions Nature produces this form only and no other. Nothing here is diffuse and indeterminate; everything presents itself in a precisely defined form.
The more one studies these things, the more one realizes that sound is the creative principle. It must be regarded as primordial…Tone and sound are, so to speak, the entelechies which are active here. (p.100)
Part 6 is coming!