MODMOS's for fractional period MOS's

MODMOS's have so far posed a bit of a conceptual problem where
fractional periods are concerned. This turns out to not be a problem
at all, and can be easily represented by extending the periodicity
block notation I've posted so far to two dimensions.

Previously I represented the diatonic scale as an unmapped rank-2
periodicity block, which you can see here:

.--------------------.--------------------.--------------------.
|Fb Cb Gb Db Ab Eb Bb|F C G D A E B |F# C# G# D# A# E# B#|
'--------------------'--------------------'--------------------'

This representation ignores the octave entirely because it assumes
octave-equivalence. This is precisely what leads to all of the
problems in working out fractional periods. The actual 2d
representation should be:

.--------------------.--------------------.--------------------.
|Fb Cb Gb Db Ab Eb Bb|F C G D A E B |F# C# G# D# A# E# B#|
!--------------------!--------------------!--------------------!
|Fb Cb Gb Db Ab Eb Bb|F C G D A E B |F# C# G# D# A# E# B#|
!--------------------!--------------------!--------------------!
|Fb Cb Gb Db Ab Eb Bb|F C G D A E B |F# C# G# D# A# E# B#|
'--------------------'--------------------'--------------------'

I haven't put octave numbers here because there are a number of ways
to line things out. We could assume here that the horizontal axis is
mapped to either the fifth or the twelfth, or we could assume that the
horizontal axis contains scales that are modulo'd to within an octave,
and the vertical axis specifies which octave that is. Take your pick.
None of this changes the basic procedure.

The periodicity block for 3L3s, then, is going to look something like
this (using 12-tet notation for simplicity):
.-----.-----.-----.
|Ab Eb|Bb F |C G |
| | | |
|E B |F# C#|G# D#|
| | | |
|C G |D A |E B |
!-----!-----!-----!
|G# D#|Bb F |C G |
| | | |
|E B |F# C#|G# D#|
| | | |
|C G |D A |E B |
!-----!-----!-----!
|G# D#|Bb F |C G |
| | | |
|E B |F# C#|G# D#|
| | | |
|C G |D A |E B |
'-----'-----'-----'

There are, again, a million ways to do this, but in this case I said
that the generator was going to be the ~3/2. Now let's say we want to
take the scale C D# E G Ab B C and modify it such that we sharpen the
E by the appropriate chromatic vector. The chromatic vector here is
L-s = 200 cents, so the resulting scale will be something like C D# F#
G Ab B C. Here's what we're left with:

.-----.-----.-----.
|Ab Eb|Bb F |C G |
'--. '--. '--. '--.
|B F#|C# G#|D# A#|
.--' .--' .--' .--'
|C G |D A |E B |
!-----!-----!-----!
|Ab Eb|Bb F |C G |
'--. '--. '--. '--.
|B F#|C# G#|D# A#|
.--' .--' .--' .--'
|C G |D A |E B |
!-----!-----!-----!
|Ab Eb|Bb F |C G |
'--. '--. '--. '--.
|B F#|C# G#|D# A#|
.--' .--' .--' .--'
|C G |D A |E B |
'-----'-----'-----'

What if, however, we had arranged things out such that we viewed the
generator as a minor third, not a perfect fifth? Then we'd have the
following periodicity block:

.-----.-----.-----.
|Ab B |D F |Ab B |
| | | |
|E G |Bb C#|E G |
| | | |
|C Eb|Gb A |C Eb|
!-----!-----!-----!
|Ab B |D F |Ab B |
| | | |
|E G |Bb C#|E G |
| | | |
|C Eb|Gb A |C Eb|
!-----!-----!-----!
|Ab B |D F |Ab B |
| | | |
|E G |Bb C#|E G |
| | | |
|C Eb|Gb A |C Eb|
'-----'-----'-----'

Fifths now appear as diagonal lines going upward and to the right.
Note that while four steps to the right is some octave-equivalent
unison vector on this chart, this is only because I'm using 12-tet
notation to simplify things, and this wouldn't be the case in
15-equal, where the unison vector would instead be 5 steps up and to
the right. However, three steps up or down is a unison vector no
matter what tuning we use, since this is 3L3s. So the chromatic vector
now appears most simply as two steps right and one step down.

So the corresponding MODMOS view of sharpening C D# F# G Ab B C would now be

.-----.
|Ab B |
'--. |-----.
|G |Bb C#|
.--' '--. |-----.
|C Eb F#|A |C Eb|
'-----.--' '--. |
|Ab B |D F Ab|B |
'--. |-----.--' '--.
|G |Bb C#|E G Bb|
.--' '--. |-----.--'
|C Eb F#|A |C Eb|
'-----.--' '--. |
|Ab B |D F Ab|B |
'--. |-----.--' '--.
|G |Bb C#|E G Bb|
.--' '--. |-----.--'
|C Eb F#|A |C Eb|
'-----.--' '--. |
|D F Ab|B |
'-----.--' '--.
|E G Bb|
'--------'

etc. The same also applies to MODMOS's of meantone that repeat every
two octaves instead of every one, say take two octaves of the diatonic
scale:

C D E F G A B C D E F G A B C

Let's say we want to just sharpen the C every other octave to make the
following two-octave scale:

C D E F G A B C# D E F G A B C

The resulting plot will look similar to those above.

-Mike