Hexanies are sooooo different!

OK, so everybody knows this, but I was SO surprised as to how different
one hexany tuning sounds from another. I guess this is to be expected
since there are only 6 pitches and the ratios resulting from the
factoring are quite different in each. I bet a person could devote a
WHOLE LIFETIME to just working with different hexanies. (Is that YOU,
Kraig Grady??) This is a MUCH different experience that working with
the various "flavors" of 19 per octave, exciting as that is... where the
tuning differences are subtle, at best (surprise, surprise.)

The point is, one cannot easily "substitute" one hexany tuning for
another and still have the same composition! It isn't just a different
"flavor," it's a WHOLE DIFFERENT PIECE!!

I have a question concerning "degenerate" hexanies, which are my
favorite ;-). Why are they called "degenerate?" Is it because the
ratios don't quite work out to small numbers. I'm thinking of the
example:

1.3.11.33 Hexany, degenerate pentatonic form
0: 1/1 0.000 unison, perfect prime
1: 4/3 498.045 perfect fourth
2: 11/8 551.318 undecimal semi-augmented fourth
3: 16/11 648.682 undecimal semi-diminished fifth
4: 3/2 701.955 perfect fifth
5: 2/1 1200.000 octave

I guess that would make sense, with the "semi-diminished fifth" and so
forth, the "traditional" pentatonic from the pythagorean having smaller
ratios... (that is except for the 27/16!)

Please explain "degeneracy" to me. This is something I need to know
about quickly.

Finally, (thank god you say) how are the larger 12-note "hexanies"
derived from the "regular" 6-note patterns??

Lets take Scala "hexanys.scl":

|
Hexanys 1 3 5 7 9
0: 1/1 0.000 unison, perfect prime
1: 35/32 155.140 septimal neutral second
2: 9/8 203.910 major whole tone
3: 5/4 386.314 major third
4: 21/16 470.781 narrow fourth
5: 45/32 590.224 tritone
6: 3/2 701.955 perfect fifth
7: 105/64 857.095 septimal neutral sixth
8: 27/16 905.865 Pythagorean major sixth
9: 7/4 968.826 harmonic seventh
10: 15/8 1088.269 classic major seventh
11: 63/32 1172.736 octave - septimal comma
12: 2/1 1200.000 octave

How are we (ahem, I mean some of you) getting this from the basic 6 note
patterning.

As usual, any help and the usual patience are greatly appreciated!

______________ _______ ____ ___ __ _
Joseph Pehrson

Joseph Pehrson wrote,

> I have a question concerning "degenerate" hexanies, which are my
> favorite ;-). Why are they called "degenerate?" Is it because the
> ratios don't quite work out to small numbers.

Nope . . .

> I'm thinking of the
> example:
>
> 1.3.11.33 Hexany, degenerate pentatonic form
> 0: 1/1 0.000 unison, perfect prime
> 1: 4/3 498.045 perfect fourth
> 2: 11/8 551.318 undecimal semi-augmented
fourth
> 3: 16/11 648.682 undecimal semi-diminished
fifth
> 4: 3/2 701.955 perfect fifth
> 5: 2/1 1200.000 octave
>
> I guess that would make sense, with the "semi-diminished fifth" and
so
> forth, the "traditional" pentatonic from the pythagorean having
smaller
> ratios... (that is except for the 27/16!)
>
> Please explain "degeneracy" to me. This is something I need to know
> about quickly.

It's degenerate because it doesn't have six notes -- it only has
five!
The reason is that of the six possible ways to take the factors two
at
a time, two are identical: 1*33 = 3*11.

The notes of the hexany are
1*3
1*11
1*33
3*11
3*33
11*33
Let's choose 3*11 as the 1/1. So we divide by 33 (and throw in powers
of two to get the scale within one octave) and get

16/11
4/3
1/1
1/1
3/2
11/8

So the "hexany" has two instances of 1/1 -- it's degenerate, and
really a 5-note scale.