On Wednesday, 14 June 2017 20:22:11 UTC+2,

*Post by SteveGG*Interesting to speculate why 12 seems

to be the magic number in our music . . .

No need to speculate; it's because 2^19 is very

close to 3^12. Were it otherwise, none of it

would work out as neatly as it actually does.

One way to find more magic numbers is to look at

the continued fraction series of

x = log(3)/log(2) = 1.58496250072...

This gives us a sequence of _best approximations_ p/q

of x, "best" meaning there is no rational number with a

smaller denominator which is closer than p/q to x.

So, best approximations of log(3)/log(2) are

1/1, 2/1, 3/2, 8/5, 19/12, 65/41, 84/53, ...

Twelve (approximately) equal ratios will bring us

from 1 to its octave 2, with a stop at 3/2

along the way, near to 2^(19/12).

We can do something similar to create an (approximately)

equal-tempered pentatonic scale, five (approximately)

equal ratios bringing us from 1 to 2 with a stop at 3/2,

near to 2^(8/5).

Or similarly, a 53-tone scale.

There are other choices for the magic number, not many,

but a few. Of course, what humans find appealing or

even usable restricts those few magic numbers further.

(And why must a scale have approximately equal

steps, anyway?)