RE: JI pentatonic, "diatonic", and decatonic scales as periodicity blocks

>In the 3-limit pentatonic case, modulating the scale by a single ratio of 3
>simply moves one note by 256:243.
<>
>Interestingly, as I was writing this, Carl posted something about 1D
>periodicity blocks, to which this may relate.

You bet! Of course any chain-of-fifths tuning can be viewed as a
periodicity block whose unison vector is the chromatic interval, not just
the pentatonic scale, and not just the MOS's.

>In the 5-limit diatonic case, 81:80 is already assumed as a unison vector,
>and the chromatic interval by which one note moves when modulating by a
>ratio of 3 is 25:24.

Hmmm. What are you transposing by 3:2 to get one 25:24 move? When I
traspose the first major lattice you give up a 3:2, I get...

27/16 - 5/3 = 81/80
45/32 - 4/3 = 135/128

-C.

Okay, I've made the following chart...

Tones MOS U. Vector Sm. 2nd Chromatic
----- --- --------- ------- ---------
3 Y |+3| 293 205 205
4 |+4| 410 205 205
5 Y |+5| 88 205 88
6 .... 585 88 88
7 Y 117 88 88
8 380 88 88
9 322 88 88
10 176 88 88
11 527 88 88
12 Y 29 88 29
13 468 29 29

...showing values in cents for chains of 24 steps of 41tET. There should
be errors in the "chromatic" column, because it was made simply by
comparing the (non-octave reduced, if that matters) unison vector with the
_nearest_ scale step. However, the nearest one isn't always the one that
changes, at least I don't think it is. For example, in the 4-tone case,
the change actually jumps a scale degree (the chromatic step should be
81/64, or 410 cents in 41tET). Can anybody fix the chart? Or better yet,
explain how to transpose a scale in Scala without confusing the hell out of
yourself?

-C.