Recently, I launched a teaching unit in all of my math classes (Algebra 2, Pre-Calculus/Trig and Calculus) analyzing the ratios or factors used to develop various musical tuning systems. Generally, my students "hate fractions," and they're even more suspicious of irrational numbers. So I figured it might be interesting for them to listen to fractions and to hear the effects of multiplying by irrationals.
An introduction to the harmonic series and the construction of the 12-tone chromatic scale based on whole number ratios of the upper partials compared to a fundamental, folded within the space of an octave, came first. Then the concept of logarithmic constants used to exactly double a frequency through 12 multiplications was covered, looking at the use of the 12th root of 2 to construct a 12 tone equal tempered tonal palette. Some historical background was thrown in to try to give the students some sense of how seriously people used to take this stuff, particularly regarding religion, philosophy, psychology and spirituality. Finally, an introduction to the 43-tone 11-limit just tuned Harry Partch octave extended the topic, including a showing of _The Dreamer That Remains_, the 1972 documentary directed by Stephen Pouliot.
In researching to prepare the unit, one of the endlessly distracting toys I found was the midicode online synthesizer. There's a tuning drop-down menu that includes 12-tone equal temperament as well as a great many other tunings (Zarlino, Pythagorean, Ptolemaic, a variety of microtonal tunings, even "Wendy Carlos Super Just," which I have yet to analyze), including, lo and behold, Partch's 43-tone octave.
The whole experience has me thinking about the tonal musical palette in general. Experimental studies show that some highly percipient listeners can hear frequency differences on the order of a mere 5 cents. (The cent is a logarithmic measure of change in frequency, where 100 cents is equal to multiplying by the 12th root of 2, in other words, an equal tempered half step is 100 cents). For those who can hear 5 cents at a time, there's a universe of 20 distinct tones between, for example, A440 and A#. This of course would lead to a 240-tone octave.
I'm defining any octave that includes 240 tones as "panchromatic."
Out of this panchromatic scale (whether equal tempered or just tuned...I have to look into exactly how the ratios would look in whole numbers to get 240 tones into an octave....) one could construct all possible perceivable music. Tonally, anyway.
This panchromatic idea has me also thinking about whether a fairly simple General Theory of Harmony could be proposed. Removing all considerations of "good" and "bad" from the terms consonance and dissonance, perhaps these two fundamental principles (which, after all, turn out to be rooted in how we perceive frequency ratios) could be used as a foundation for such a General Theory. Composers would then be entirely liberated to write panchromatic music, drawing on all of the perceivable tones in the 240 note octave. Of course, composers would still be free to use only 12, or 7, or 5, or 43, or whatever number.
Perhaps problems in performance could at least be partly ameliorated with the aid of technology. But such an idea would shift music education from its current obsession with mechanical technique and facility, toward a highly refined tonal sensibility.
I have nothing against the 12-tone equal tempered scale. But the vast universe of perceivable tones available for our auditory perception has barely been explored, at least for the past 400 years or so.
