Difference diamonds

A planar, or perfect, difference set is a set of residues a[i] mod n
such that every nonzero mod n residue is uniquely defined as a
difference a[i]-a[j]. If v is a p-limit n-equal val, a difference
diamond for v may be defined as a tonality diamond based on a set of
p-limit rational numbers s[i] such that {v(s[i])} is a perfect
difference set. Another way to say it is that it is a

http://mathworld.wolfram.com/PerfectDifferenceSet.html

http://mathworld.wolfram.com/DifferenceSet.html

An example would be {9/8, 5/4, 3/2}, which 7-et maps to {1, 2, 4}, a
difference set mod 7. The ratios give 1-10/9-6/5-4/3-3/2-5/3-9/5,
which Scala tells us is Ptolemy's diatonic and also Al Farabi's
diatonic #2. Because of the way it is contructed, it is epimorphic.

After the {1,2,4} perfect difference set, the next most complex is the
mod 13 perfect difference set. {10/9, 7/6, 4/3, 10/7} map, using the
standard 13 val, to {1, 2, 5, 7}; taking all the ratios gives us the
13-note epimorphic scale

[1, 21/20, 15/14, 8/7, 6/5, 60/49, 9/7, 14/9, 49/30, 5/3,
7/4, 28/15, 40/21]

7 = 2^2+2+1 gives the smallest size for a difference diamond. Next is
13 = 3^2+3+1, and then 21=4^2+4+1.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:

> A planar, or perfect, difference set is a set of residues a[i] mod n
> such that every nonzero mod n residue is uniquely defined as a
> difference a[i]-a[j].

Another use for these perfect difference sets is that certain MOS can
be arranged as a tonality diamond. We can, for instance, arrange
Meantone[7], the diatonic scale, using {0,1,3} as a difference set,
leading to

0 1 3
6 0 2
4 5 0

as the diamond, or in letter terms

C D F
B C E
G A C

MOS of size q^2+q+1, where q=p^n is a prime power, can similarly be so
arranged; examples are Orwell[13], Miracle[21] (Blackjack) and
Miracle[31] (Canasta.) The diamond would then be a (q+1)x(q+1) sized
square, so 4x4 for 13 notes, 5x5 for 21 notes, and 6x6 for 31 notes.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > A planar, or perfect, difference set is a set of residues a[i] mod n
> > such that every nonzero mod n residue is uniquely defined as a
> > difference a[i]-a[j].
>
> Another use for these perfect difference sets is that certain MOS can
> be arranged as a tonality diamond. We can, for instance, arrange
> Meantone[7], the diatonic scale, using {0,1,3} as a difference set,
> leading to
>
> 0 1 3
> 6 0 2
> 4 5 0

It might make it clearer to do this without reducing mod 7, so that
you'd get

0 1 3
-1 0 2
-3 -2 0

though no doubt Yahoo is going to mess that up for me. We can arrange
things in terms of the fifth generator if we like, getting

C G A
F C D
Eb Bb C