-c scales

Gene Ward Smith wrote :

> If you want to give a formal definition, I'd suggest calling the set
> in question S, and saying if f1<f2 are elements of S, and if
> for some integer n, 2^n (f2-f1) is an element of S, then the interval
> f1:f2 is S-coherent.

perfect.

> Then you could define what a coherent scale is--for instance, is it
> a set S of intervals 1<=s<2 such that 1:s is S-coherent?

= differential coherence with a given 1/1 - that's an option I call "modal -c"

Other options are having all the seconds, or thirds, or fourths of a scale -c.
Examples :
24 26 32 36 39 48
2 6 4 3 9
is a "2-c" scale = difference tones of consecutive tones belong to the scale.

32 36 44 48 52 59 64 72
12 12 8 11 12 13
is a "3-c" scale = differentials of intervals between 3 consecutive tones belong
to the scale - (etc.)

Some scales can also combine several attributes :

29 32 37 40 42 48 52 58 64
3 5 3 2 6 4 6 6
8 8 5 8 10 10 12
is a "2+3-c" scale = differentials of intervals between 2 or 3 consecutive tones
belong to the scale.

Here's a differentially (almost) coherent Bohlen-Pierce scale:
The difference tone's interval class is 2.

0: 1/1 1.000000
1: 140.704 cents 1.084668
2: 293.449 cents 1.184715
3: 434.263 cents 1.285104
4: 586.825 cents 1.403492
5: 727.820 cents 1.522578
6: 879.964 cents 1.662441
7: 1021.373 cents 1.803930
8: 1172.944 cents 1.968986
9: 1314.918 cents 2.137264
10: 1464.799 cents 2.330542
11: 1608.449 cents 2.532169
12: 1756.919 cents 2.758911
13: 3/1 3.000000

>For what we were talking about, if you start with an equal division of the
octave
>that provides a good approximation of the interval you want to be -c,
there will
>be less to refine and you will be able to have at once a maximum of them.

I've also added code to the EQUALTEMP/DATA command to find the best
interval
class to make coherent for a given equal division.

Manuel