The Western system quantizes notes with 12 bins per octave, so the 12 note up doubles the pitch (frequency). Sure enough, as I note, the guitar strings are exactly cut in half at the 12th fret. When are they cut by 3/4? At the fifth fret, nearly exactly, within 4% (a quarter inch).
Why do we care?
Because side frequencies are multiples of two from the center frequency, so the Yaw,or Twang sound as two notes mix is undetectable.
Yaw is phase aligned exactly every four or two cycles. So it never 'carries' forward,m it gets reset back to zero. Mix a halt string and a third string, on the guitar, it is is at least six cycles before phase is repeated. Mix a 1/5 (not Fifth), a 1/3 and a 1/2 string (that is make the wavelengths have these fundamental ratios (2,3,5), and it takes 15 cycles before a phase repeats. The Yaw sound is very detectable, it carries forward five times as far, in time. The pressure cycles in our cavity resonator reach steady state.
These are the kind of insights we get when working with a single tone spectrum, like the spread sheet composer.
Why not adopt a higher quantization accuracy, say a 16 level tone set? Because the instruments need to be acoustic and portable. Hence we have physical limitations. And, more importantly, we do not have 4 bit precision with our fingers, we could never manage squeeing in another four frets on the guitar neck. If we did, the high notes down the neck would have tiny separations between the frets, tinier than our finger widths. Inter spacing between frets gets way smaller because we separate note tones by ratio. So as the pitch gets higher, the string gets shorter, and the fret distance is a fixed proportion of the string length.
Making more notes in an octave means making the note ratio get closer to one, meaning many more bins (or frets) are needed to count up the pitches.
Consider the major triads. They are the first, the middle and the perfect fifth. So, the first and fifth resolce quickly, but the tird will have a stand out yaw. It is the difference between ther first and fifth, ised with the third that gives the chords the major or minor feel. The instrument can lengthen or shorten the yaw by just a semi-tone, when switching between minor and major chords. We notice, the standout note has either a weak or strong frequency, its a slight mix of the note frequency and the beat frequency, chords do a frequency divide for us..
Let's do the numbers. Speech is about 440 hertz, i typical banwidth. A tird p from 440 is about 10-20% (magic ratio to the third power). The yaw rate will be about 80, in pitch. But if the middle note only goes up by a step, the yaw rate goes to 20, close to a 1/16 brat, and it interferes with speech. So, human seech has reached a compromise between our ability to speak in chords (wider spectrum) and our ability to speak fast.
What else? The portable instruments will be about the same size as our resonating cavity, a violin or guitar is abnout the size of our chest. Instrument engineers creating robot versions of our chest cavity and vocal chords. Hence, clearly we listen by silently humming or silently speaking in rhythm.
I did an undergraduate research project on the topic, ablating the part of the rat brain that inhibits muscular thought. fter ablation, the rat could not maintain a rhythm with the environment, it had no ability to predict environmental actions and was always startled.
This is how we speak. Our resonator is the entire respiratory cavity, and we can shpe it. As we learn, we find matches between the shapes we can contort, and the Yaw, or twang we make. And those Yaws and Twangs are the standout mis-matched frequencies, and they make words. They divide down so the Yaw and syllable rate start to overlap. It is part of the theory of everything, everything has a slight overlap as we rate divide the world. The individuals in the aggregate remain slightly defocused, the central limit point is always an imperfect creation.
It is the rate overlap that allows nature to detect and remove redundant transactions.
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