JI pentatonic, "diatonic", and decatonic scales as periodicity bl ocks

These three basic scales, identified in my paper as the melodic bases for
3-, 5-, and 7-limit harmony, respectively, have JI representations that come
out as Fokker periodicity blocks when the typical "chromatic" interval
implied by the scales is used as a unison vector.

Interestingly, all these periodicity blocks are of the "most natural" type
catalogued by Kees van Prooijen (he skipped the 3-limit ones but they're
trivial -- apparantly not enough so to satisfy Carl?)

In the 3-limit pentatonic case, modulating the scale by a single ratio of 3
simply moves one note by 256:243. Using this as the unison vector (only one
is needed since octave-equivalent 3-limit space is one-dimensional), we get
the following periodicity block:

ratios for major

1/1-------3/2-------9/8------27/16-----81/64

ratios for minor

32/27-----16/9-------4/3-------1/1-------3/2

Interestingly, as I was writing this, Carl posted something about 1D
periodicity blocks, to which this may relate.

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. Using these as the unison vectors, the resulting
periodicity block is:

ratios for major

5/3-------5/4------15/8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
4/3-------1/1-------3/2-------9/8

or

10/9-------5/3-------5/4------15/8
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
4/3-------1/1-------3/2

or

100/81-----50/27
\ / \
\ / \
\ / \
\ / \
40/27-----10/9-------5/3
\ / \
\ / \
\ / \
\ / \
4/3--------1/1

ratios for minor

1/1-------3/2-------9/8
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
/ \ / \ / \
8/5-------6/5-------9/5------27/20

or

4/3-------1/1-------3/2-------9/8
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
\ / \ / \ /
8/5-------6/5-------9/5

or

40/27-----10/9
\ / \
\ / \
\ / \
\ / \
16/9-------4/3-------1/1
\ / \
\ / \
\ / \
\ / \
8/5--------6/5

In the 7-limit decatonic case, 64:63 and 50:49 are already assumed as unison
vectors. If you are unfamiliar with decatonic scales, see my paper
http://www-math.cudenver.edu/~jstarret/22ALL.pdf. When modulating by a
3-limit ratio, two notes move by a 48:49. Using these as the unison vectors,
the resulting periodicity block is:

ratios for symmetrical major

5/4------15/8 7/4------21/16
,'/:\`. ,'/:\`. ,'/:\`. ,'/:\`.
10/7-/-:-\15/14/-:-\45/28 or 1/1-/-:-\-3/2-/-:-\-9/8
: / 7/4------21/16\ : : /49/40----147/80\ :
:/,' `.\:/,' `.\: :/,' `.\:/,' `.\:
1/1-------3/2-------9/8 7/5------21/20-----63/40

or

16/9-------4/3-------1/1 80/63-----40/21-----10/7
:\`. ,'/:\`. ,'/: :\`. ,'/:\`. ,'/:
: \32/21/-:-\-8/7 / : : 160/147-:-\80/49/ :
56/45-----28/15------7/5 or 16/9-------4/3-------1/1
`.\:/,' `.\:/,' `.\:/,' `.\:/,'
16/15------8/5 32/21------8/7

or

5/4 7/4
.'/:\`. .'/:\`.
40/21-----10/7-/-:-\15/14 4/3-------1/1-/---\-3/2
:\`. ,'/: / 7/4 \ : :\`. ,'/: /49/40\ :
: \80/49/ :/,' `.\: or : \ 8/7 / :/.' `.\:
4/3-------1/1-------3/2 28/15------7/5------21/20
`.\:/,' `.\:/,'
8/7 8/5

ratios for symmetrical minor

4/3-------1/1 40/21-----10/7
,'/:\`. ,'/:\`. ,'/:\`. ,'/:\`.
32/21/-:-\-8/7-/-:-\12/7 or 160/147/-:-\80/49/-:-\60/49
: /28/15------7/5 \ : : / 4/3-----/-1/1 \ :
:/,' `.\:/,' `.\: :/,' `.\:/,' `.\:
16/15------8/5-------6/5 32/21------8/7------12/7

or

40/21-----10/7------15/14 4/3-------1/1-------3/2
:\`. ,'/:\`. ,'/: :\`. ,'/:\`. ,'/:
: \80/49/-:-\60/49/ : : \ 8/7-/-:-\12/7 / :
4/3-------1/1-------3/2 or 28/15------7/5------21/20
`.\:/,' `.\:/,' `.\:/,' `.\:/,'
8/7------12/7 8/5-------6/5

or

21/16 15/8
.'/:\`. .'/:\`.
1/1-------3/2-/-:-\-9/8 10/7------15/14/---\45/28
:\`. ,'/: 147/80\ : :\`. ,'/: /21/16\ :
: \12/7 / :/,' `.\: or : \60/49/ :/.' `.\:
7/5------21/20-----63/40 1/1-------3/2-------9/8
`.\:/,' `.\:/,'
6/5 12/7

I don't think the pentachordal decatonic scales can be thought of as
periodicity blocks, but I could be wrong . . .