OK. So I am still a bit confused, but:
Finally something that uses the anthropic principle to explain why sound is the way it is: because this is the simplest construction next to octaves.
Rule 1: Given a frequency f, the simplest way to get another note is to double the frequency: 2 * f. BUT, that gives you the same note, so that’s not good. But that is good enough for us to create a rule (Rule 1) – if a note exists with frequency f, then there is also a note with frequency 2 *f.
Then the second simplest way to get another note is to triple the frequency: 3 * f. So now we have three notes:
2. f * 2 (by rule 1)
3. f * 3
but, between f and f * 3, there is another note: f * 3 / 2 (by rule 1 again). And so we can repeat the process. So really, the notes we have are:
f, f*3, f*3*3, f*3*3*3, and so forth. So all notes can be formed by f*(3^i) where i is an integer value. So then we want the process to stop eventually, which means that f * (3 ^ i ) = f * (2 ^ k) for some i and k being integer value.
f * (3 ^ i ) = f * (2 ^ k) =>
(3 ^ i ) = (2 ^ k) =>
k = log base 2 (3 ^ i ) =>
k = log (3 ^ i ) / log 2
Now, in order for things to work we need to solve this so that i and k are whole numbers (or as close to them as possible). And if we choose i to be 12 (as in that article above), then k is 19.01955, which is really close to a whole number. For 24, it is 38.0391
put in different numbers for i instead of 12. They are all singificantly farther from a whole number. Moreover, perhaps when compensating for this, I wonder what the average sensitivity of the ear is. Also, note that if you compensate in each note, you get an error of:
0.01955 / 12 = 0.00162916667 and
0.0391 / 24 = 0.00162916667
which is really small when compared to, for example, if we decided to use 4 notes (i = 4), then k = 0.339850003 and k / 4 = 0.0849625007.
Now, of course, my next question is why do we like music that resonates well with other notes? Why don’t we instead like that which clashes and does not carry redundant information in a chord (resonating frequencies)? Psychological? Nature of beauty and simplicity – being able to recognize structure and speak the language (i.e. why we find a circle more pleasant to the eye than a square). So then it might all go back to the notion of languages and ability to recognize information, but that’s for some other time…