Re Hexanies, CPS, etc.

I was inspired by Carl Lumma's recent discussion of the hexany and eikosany to look for me notes from the 1970's when I was studying the CPS with Erv Wilson. Erv noticed that all hexanies have the melodic sequence c b a b c d, where the intervals a, b c and d are not necessarily different. A consequence of this finding is that hexanies can be defined in ET without concern of the possible JI interpretation of the generating tetrad. BTW, the generators can be tetrachords or even a set or irrational numbers -- e.pi.phi.sqr(2) is a perfectly good hexany generator.

Fortunately I was able to locate some of my notes and re-coded my hexanies in ETs program from Fortran to Basic and ran it. Here's the output for 12-tet. The hard part is figuring out how to get a non-redundant list, but the number here agrees with my previous work and Erv's list (the algorithm is ultimately from Erv, of course).

Getting the generating tetrad is not as hard as I thought. If the generalized hexany has the interval sequence c b a b c d, the generator tetrad is 0 b c + b a + 2*b + c. Take sums two at a time to get b c + b a + 2*b + c c + 2*b a + 3*b + c a + 3*b + 2*c, sort in ascending order and subtract the smallest term from all so that the hexany starts on 0.

The list:

# HEXANY INTERVALS GENERATING HEXAD HEXANY ET= 12
1 1 1 1 1 1 7 0 1 2 4 0 1 2 3 4 5
2 1 1 2 1 1 6 0 1 2 5 0 1 2 4 5 6
3 1 1 3 1 1 5 0 1 2 6 0 1 2 5 6 7
4 1 1 4 1 1 4 0 1 2 7 0 1 2 6 7 8
5 1 2 1 2 1 5 0 2 3 6 0 1 3 4 6 7
6 1 2 2 2 1 4 0 2 3 7 0 1 3 5 7 8
7 1 2 3 2 1 3 0 2 3 8 0 1 3 6 8 9
8 2 1 1 1 2 5 0 1 3 5 0 2 3 4 5 7
9 2 1 2 1 2 4 0 1 3 6 0 2 3 5 6 8
10 2 1 3 1 2 3 0 1 3 7 0 2 3 6 7 9
11 2 2 1 2 2 3 0 2 4 7 0 2 4 5 7 9
12 2 2 2 2 2 2 0 2 4 8 0 2 4 6 8 10
13 3 1 1 1 3 3 0 1 4 6 0 3 4 5 6 9
14 3 1 2 1 3 2 0 1 4 7 0 3 4 6 7 10
15 3 2 1 2 3 1 0 2 5 8 0 3 5 6 8 11
16 4 1 1 1 4 1 0 1 5 7 0 4 5 6 7 11

The inversion of a hexany-generating tetrad produces the same hexany as the prime form. However, the 12-tet complement of a hexany is not always a hexany. In the above table, hexanies 6, 9 and 14 have complements that are not hexanies. I'm not a dodecaphonic theorist, but one might look at Allen Forte's treatment of these hexanies as hexads. IIRC, some of the tetrad generators are Babbitt's "All-Combinatorial Source Sets," but I don't recall which.

There are a lot of things one can do with hexanies---e.g., joining them by common triads to create 12-tone scales is a fruitful approach. Erv called the resulting scales "flankers." Another approach is to construct the 1.3.5.9 hexany and transpose it by 3/1 or 9/3 to generate a major scale between the two conjoined hexanies, then harmonize it according to hexany rules with 1.3.5, 1.3.9, 1.5.9 and 3.5.9 triads and their inversions in 12-tet. The result sounds rather modal and fresh compared to the usual treatment.

Erv conceived hexanies both as scales and as frameworks for modulation. In fact, he discovered them and the other CPS while pondering what pitch bases (aka keys) I should use for my JI tables back in 1968. So, one could modulate a 7 limit scale around the pitches of the 1.3.5.7 or other hexany. Erv diagrammed the result of modulating the Partch gamut around the 1.3.5.7.9.11 20-any.

I haven't reinvestigated the tempered 20-anies , as I haven't located all my notes from the late 70's. As I recall, the procedure was analogous to the hexany case, but avoiding redundancy was harder as was the problem of extracting the hexad generator, but I do recall that hexanies or any hexatonic scale can be used as 20-any generators, though the resulting 20-any may be degenerate with fewer than 20 different notes. This is not necessarily a problem as some notes have dual functions and be interpreted as different notes in different 20-any tetrads. This is not too different from double emploi in traditional harmony. In fact, Erv tuned a set of tubulongs to a redundant 80-any with only 56 tones. (I think it was the 1.3.5.7.9.11.13.15, 4 at a time 80-any).

--john

If one makes the next homolog of a melodic hexany -- d c b a b c d e,
with 8 tones, the result is an Euler's genus. I was calling these
Octonies, until Erv noted that they are 8 note Euler's genera. These
can be draped over the vertices of a cube and naturally partition into
faces which are tetrads, each of which is also an EG.

Of course, they can be done in JI too such as the 3.5.7 EG. 1/1 35/32
5/4 21/16 3/2 105/64 7/4 15/8 2/1 with face chords like 1/1 5/4 3/2 15/8
and corner chords such as 1/1 5/4 3/2 7/4 and their inversions. One can also
play face-diagonal chords ( the corner noe and the note at the opposite vertex
of each of the 3 square faces meeting each corner)

I find 10 8-anies in 12:

# OCTONY INTERVALS OCTONY ET= 12

1 1 1 1 1 1 1 1 5 0 1 2 3 4 5 6 7

2 1 1 1 2 1 1 1 4 0 1 2 3 5 6 7 8

3 1 1 1 3 1 1 1 3 0 1 2 3 6 7 8 9

4 1 1 2 1 2 1 1 3 0 1 2 4 5 7 8 9

5 1 1 2 2 2 1 1 2 0 1 2 4 6 8 9 10

6 1 2 1 1 1 2 1 3 0 1 3 4 5 6 8 9

7 1 2 1 2 1 2 1 2 0 1 3 4 6 7 9 10 8 1 2 2 1 2 2 1 1 0 1 3 5 6 8 10 11

9 2 1 1 1 1 1 2 3 0 2 3 4 5 6 7 9

10 3 1 1 1 1 1 3 1 0 3 4 5 6 7 8 11

Note that #7 is the octatonic scale.

The next homolog would be a dekany, not sure that this is the same as
the half-20any dekany, but probably is.

Anyway, I'm not sure I want to promote more 12-tet atonality, but the
hexany-EG system and homologs offers an approach which may be only
implicit in the standard literature.

--john

--- In tuning@yahoogroups.com, "John H. Chalmers" <JHCHALMERS@...> wrote:

> There are a lot of things one can do with hexanies---e.g., joining them
> by common triads to create 12-tone scales is a fruitful approach.

Here's an interesting scale I call the bihexany, which shows up as a hole in the 11-limit lattice of octave-equivalent intervals, just as the 1.3.5.7 lattice is the deep hole in the 7-limit lattice.

! bihexany.scl
Hole around [0, 1/2, 1/2, 1/2]
12
!
35/33
7/6
5/4
14/11
15/11
3/2
35/22
5/3
7/4
20/11
21/11
2

Tempering it, for instance by tuning it to 600et, makes some sense.

As I recall, the procedure was
> analogous to the hexany case, but avoiding redundancy was harder as was
> the problem of extracting the hexad generator, but I do recall that
> hexanies or any hexatonic scale can be used as 20-any generators, though
> the resulting 20-any may be degenerate with fewer than 20 different
> notes. This is not necessarily a problem as some notes have dual
> functions and be interpreted as different notes in different 20-any
> tetrads.

If you consider the sets you start out with and end up with to actually be multisets, this becomes straightforward. Apply the cps construction to a multiset, and get another multiset, and iterate that as you choose. The final result can be converted to an ordinary set if it isn't already one.

The diamond and genus constructions can likewise be applied to multisets and iterated.