Diatonic Hexachord Pairs

Paul E. called it to my attention that the term "double diatonic" is
already in use (Injera). So here are my results for comparing pairs
of hexachords taken from the diatonic collection. It should be pretty
self-explanatory. There are 28 combinations (7+6+5+4+3+2+1) based
on which note of the scale you omit (no-C would be D,E,F,G,A,and B
for example). This reduces to 16 when you consider mirror symmetry
(noC&noD is the mirror of noD&noE for example). Anyway, here it
is:

noC&noC 6 2 8 6 4 8 2
noC&noD 5 3 8 5 5 8 2
noC&noE 5 2 8 6 5 8 2
noC&noF 5 2 8 6 4 10 1
noC&noG 5 3 7 6 4 9 2
noC&noA 5 3 7 7 4 8 2
noC&noB 5 3 8 6 4 9 1
noD&noD 6 4 6 4 6 8 2
noD&noF 5 3 7 6 5 9 1
noD&noG 5 4 6 5 5 9 2
noF&noF 6 2 8 6 4 10 0
noF&noG 5 3 8 6 4 9 1
noF&noA 5 3 7 6 5 9 1
noF&noB 5 2 8 6 4 10 1
noG&noG 6 4 6 6 4 8 2
noG&noA 5 4 7 6 4 8 2

Looks like [noC&noF and noF&noB] have the same interval vector and
so do [noD&noF and noF&noA] as well as [noC&noB and noF&noG]. These
are the Double-Z relations (Z2). This reduces the interval vector
count to 13, 13 unique energy states between diatonic hexachords.

--Paul Hj

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> Paul E. called it to my attention that the term "double diatonic" is
> already in use (Injera). So here are my results for comparing pairs
> of hexachords taken from the diatonic collection. It should be
pretty
> self-explanatory. There are 28 combinations (7+6+5+4+3+2+1) based
> on which note of the scale you omit (no-C would be D,E,F,G,A,and B
> for example). This reduces to 16 when you consider mirror symmetry
> (noC&noD is the mirror of noD&noE for example). Anyway, here it
> is:
>
> noC&noC 6 2 8 6 4 8 2
> noC&noD 5 3 8 5 5 8 2
> noC&noE 5 2 8 6 5 8 2
> noC&noF 5 2 8 6 4 10 1
> noC&noG 5 3 7 6 4 9 2
> noC&noA 5 3 7 7 4 8 2
> noC&noB 5 3 8 6 4 9 1
> noD&noD 6 4 6 4 6 8 2
> noD&noF 5 3 7 6 5 9 1
> noD&noG 5 4 6 5 5 9 2
> noF&noF 6 2 8 6 4 10 0
> noF&noG 5 3 8 6 4 9 1
> noF&noA 5 3 7 6 5 9 1
> noF&noB 5 2 8 6 4 10 1
> noG&noG 6 4 6 6 4 8 2
> noG&noA 5 4 7 6 4 8 2
>
> Looks like [noC&noF and noF&noB] have the same interval vector and
> so do [noD&noF and noF&noA] as well as [noC&noB and noF&noG]. These
> are the Double-Z relations (Z2). This reduces the interval vector
> count to 13, 13 unique energy states between diatonic hexachords.
>
> --Paul Hj

A little tidbit to add as a P.S. the noA set is C,D,E,F,G,B which
is G7&C together. Its complement is the blues scale on Eb:
Eb,Gb,Ab,A,Bb,Db,Eb. Since we are dealing with a 2n set, the
measurements for complementary sets are the same. So this could
also be seen to measure energy states between the blues scale
and comp(noG), comp(noF), comp(noC) which are merely (bk=blackkeys)
bluescale=bk+A and bk+G, bk+F and bk+C. The remaining comp scales
are bk+B, bk+D, bk+E. All of which are used in jazz improvisation,
where your melody is the pentatonic scale plus a passing-tone or
leading-tone. (I might take this over to tuning group!)

PHj