Notes regarding music, mathematics and relating the unrelated

Summary

Find here some unedited notes to a text developing the boundary of connections between abstract music and likewise mathematics. The main inspiration originate from an idea to uncover 'the mathematics' in music, not through acoustics or by telling musicians about pythagoreans and trigonometry and so on, but by observation and non-intrusive description connecting musicians language about music as is, to mathematicians language about mathematics. What can be reasoned about, what will we find?

I am Danish so please by all means correct my grammer :) — it would be very helpful! The text and whatever come out of it is a contribution to oeis.org. See license agreement for oeis.org and contributors below.

Please note also that the loose sentence structures are deliberate: ... something and this etcetera ... , indicate a 'line of thought'.

Sketches

...took note of that the linguists have suggested that the origin of mathematics is the Proto-Indo-European root *mendh-... meaning: “That what has been learned“.

That what has been learned? It make a lot of sense to me; take Euclid and the geometry, that what at the time had been learned and then collected? Analyzed and written down in a more general and precise form. From what had been learned by carpenters, artisans, artists to accountants; The ordinary men and women as well as the Leonardo Da Vinci's and Pacioli's of the time so to speak.

...music...it has nothing to do with mathematics and yet in some way is mathematical reasoning in its own right....

However more often than seldom the talk become awkward when attempting to demonstrate how music and mathematics is connected. That an octave is twice the unison in frequency or 2 to 1, 1/2 if considered the other way I suppose, is not the reason a certain piece of music is to be liked. The technicalities of acoustics are not what makes listening to musical phenomena pleasurable; neither is popularity or anything else apart from the evident: that music is music to us the moment we perceive it as such. And otherwise just a sounding distraction.

...music as mathematics in its own right. It is interesting to think about this; say we look at what a musician do and think about in which way this is similar to abstract mathematics? Suppose an improviser explains a little and we understand that the struggle is to find the way to render a line which is more true than other; and once understood it become part of the improvisers repertoire. Then it has a strong reminiscence of that what the mathematicians explains to me about the mathematical proof and the beauty of a particular ingenious one. ... could this line of thinking lead us to discoveries about music and mathematics? Maybe?

Notes about the sequences and ways to interpret numbers

...listening to integer sequences most often take the point of departure in a direct translation of the integer number to some scale of a typical equal tempered tuning, say C-major or the chromatic scale...

This is not the only way.

Apart from that the structure behind a particular sequence could possess some interesting properties which could inspire to music, then there is an obvious other line of investigation which could be attempted.

Sound is essentially pressure waves in air (or other mediums). So it make just as much sense to look at the integers one wish to ponder on for their sounding qualities as the number of vibrations per unit of time, or frequency. We do not even have to begin to reach out for a physics book as even though a “1” corresponding to one vibration per second is a ridiculously low frequency then we can just take them to be the i'th overtone of a fundamental of unity. The particular reference unity can then be chosen as one pleases...

The curious thing about this point of attack is that every sequence based on some adding and multiplication modulo something will naturally have quite a lot of resonance build into it as it is the nature of vibrating instruments to be behave that way as well...

...note on the transformed and relative relationships: the perception of melodic identity (here used in the broadest sense) is different from a purely analytical identity; a transposed melody within a uniform scala retain its perceived familiarity to the ear, but already a transposition within a typical major or minor scale do not; and when the note sequence is reversed or mirrored it loose most of its relationship with the origin apart from its timing — is timing changed as well then there seem not to be anything distinguishing the derived melody from a random development. Say a sequence a(n) is 1, 3, 5, 4, 2 then a(n)+1 would sound very close to identical to a(n) in a uniform scale such as the chromatic scale; but clearly different, though strongly related, in a C-major scale. Further transforms would change it so much that while sounding more or less related we could not tell if this was developed from one another or purely coincidental; unless the composer decide to 'bend it in neon' by repeating the melody and its transformations a few times...

...what can the capabilities of the human ear and mind to perceive, or not perceive, melodic similarity, tell us when applied to an integer sequence? For one thing what seem unrelated compared as sequences could be related under some simple melodic, or acoustic, transformation. An obvious example would be that some sequence read as scale steps in a natural scale could be identical to another sequence read as frequencies. But we could also contemplate sequences having repeated temporal patterns hidden within them as well as sequences which look interesting but sound chaotic or dull. It can be contemplated what human cognition is trying to do in such scenarios...to which extent the 'behavior of our ear' could inspire to mathematical findings of asymmetries under certain constraints; are they there or are they effects of auditory illusions...