4 ms·

> identifying integer powers of 2 Doesn't this just get you the octaves, which are by no means unique to the 12-tone scale? > group properties of C12 Can you

> identifying integer powers of 2 Doesn't this just get you the octaves, which are by no means unique to the 12-tone scale? > group properties of C12 Can you expand on this? "Group" in what sense? C12? I'm genuinely intrigued! In any case, I don't think I disagree per se. But from what I can tell, the 12 tone scale only "falls out" if we apply two constraints: (1) roughly 10 tones in an octave, (2) steps are roughly equidistant (I'm _not_ alluding to equal temperament tuning here). Besides those constraints, why does, say, 16:15 get a tone (m2) but not 7:6? So the question becomes: are those constraints justified? EDIT: Partially answered my own question by consulting this diagram [1]: 16:15 has a lower "prime limit" than 7:6. [1] https://en.m.wikipedia.org/wiki/File:Equal_Temper_w_limits.svg https://en.m.wikipedia.org/wiki/File:Equal_Temper_w_limits.s...

wyager 4y ago
> Doesn't this just get you the octaves, which are by no means unique to the 12-tone scale? Correct, I was just referring to the convention that you use the same letter for f and 2f. > Can you expand on this? "Group" in what sense? C12? The cyclic group with 12 elements https://en.m.wikipedia.org/wiki/Cyclic_group https://en.m.wikipedia.org/wiki/Cyclic_group Isomorphic to the integers mod 12 (Z/Z12) - this is just a set with 12 things that wraps around. Things like the "circle of fifths" falls naturally out of the behavior of this group (7 is a generator of C12, and adding 7 in this group corresponds to multiplying by 2^(7/12) \approx 3/2).
_Microft 4y ago
>> group properties of C12 > Can you expand on this? "Group" in what sense? C12? Have a look here: https://en.wikipedia.org/wiki/Group_theory https://en.wikipedia.org/wiki/Group_theory https://mathworld.wolfram.com/CyclicGroupC12.html https://mathworld.wolfram.com/CyclicGroupC12.html

wyager 4y ago

> Doesn't this just get you the octaves, which are by no means unique to the 12-tone scale? Correct, I was just referring to the convention that you use the same letter for f and 2f. > Can you expand on this? "Group" in what sense? C12? The cyclic group with 12 elements https://en.m.wikipedia.org/wiki/Cyclic_group https://en.m.wikipedia.org/wiki/Cyclic_group Isomorphic to the integers mod 12 (Z/Z12) - this is just a set with 12 things that wraps around. Things like the "circle of fifths" falls naturally out of the behavior of this group (7 is a generator of C12, and adding 7 in this group corresponds to multiplying by 2^(7/12) \approx 3/2).

_Microft 4y ago

>> group properties of C12 > Can you expand on this? "Group" in what sense? C12? Have a look here: https://en.wikipedia.org/wiki/Group_theory https://en.wikipedia.org/wiki/Group_theory https://mathworld.wolfram.com/CyclicGroupC12.html https://mathworld.wolfram.com/CyclicGroupC12.html