________________________________________________________________________

Date: Mon, 4 Oct 1999.
Subject: JI pentatonic, "diatonic", and decatonic scales as ... blocks


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 1-D
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 at:

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 . . .