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Square scales (also known as Carthesian cross sets or Carthesian
products) are of the form:
[1 3 5] x [1 3 5] = 1/1 9/8 5/4 3/2 25/16 15/8
Diamonds (as in Partch tonality diamond) are of the form:
[1 3 5] x [1/1 1/3 1/5] = 1 6/5 5/4 4/3 3/2 8/5 5/3
Squares may be written:
[1 3 5]^2
Diamonds may be written:
[1 3 5] x u[1 3 5]
The number of notes in these scales is given by:
square(f) = (pi(f) * (pi(f)+1)) / 2
diamond(f) = (pi(f) * (pi(f)-1)) + 1
...where pi(n) is the number of elements in a list of mutually
prime factors f. If the factors in f are not mutually prime,
some number of doubled notes must be subtracted out to give the
number of unique notes.
As pi(f) gets big, the number of notes in diamonds becomes much
larger than the number of notes in squares. Diamonds provide
more consonant chords per note than squares, though, as long as
utonal (subharmonic) chords are considered consonant.
Note: In Scala, the command to create a diamond is called
"square", and the command to create a square is called "product".
Squares and diamonds are types of cross sets. Cross sets can be
made from two different sets...
[1 3] x [5 7] = 1/1 5/4 21/16 3/2 7/4 15/8
...two sets of different sizes...
[1 3] x [1 3 5] = 1/1 9/8 5/4 3/2 15/8
...or more than two sets...
[1 3 5] x [3 5 7] x u[5 7 9] = . . .
Have fun!