Yahoo Tuning Groups Ultimate Backup tuning Partch scale as periodicity block?

I wrote,

>Anyway, now that we're dealing with 4-d (11-prime limit) ones, a natural
>question would be, can we represent Partch's 43-tone scale with a
>periodicity block? Following Wilson, we'll identify 11/10 and 10/9 as an
>equivalent pair (and their inversions as another) so we are dealing with a
>41-tone periodicity block, one of whose unison vectors is 100:99. Unlike
the
>19- and 31-tone periodicity blocks found so far, this 41-tone one will not
>use 81:80 as a unison vector, as several 81:80 pairs appear in Partch's
>scale. . . .

100:99 = [2^2*]3^(-2)*5^2*7^0*11^(-1)

What unison vectors can we find? By dividing pairs of step sizes, some
simple candidates are:

81:80
----- = 243:242 or 3^5*5^0*7^0*11^(-2)
121:120

55:54
----- = 245:243 or 3^(-5)*5^1*7^2*11^0
99:98

45:44
----- = 225:224 or 3^2*5^2*7^(-1)*11^0
56:55

And lo and behold, the determinant of

-2 2 0 -1
5 0 0 -2
-5 1 2 0
2 2 -1 0

is -41! Next, we'll compare the Fokker hyperparallelopiped(?) with Partch's
scale.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

Define Fokker1 as the notes in the hyperparallelopiped centered around 1/1
with edge vectors

( -2 2 0 -1 ),
( 5 0 0 -2 ),
( -5 1 2 0 ),
( 2 2 -1 0 ).

Here we go:

Fokker1 Partch Fokker1/Partch
1 1 1
45/44 81/80 100/99
28/27 33/32 896/891 = 100/99*224/225
81/77 21/20 540/539 = 100/99*243/245
15/14 16/15 225/224
12/11 12/11 1
10/9 11/10 100/99
" 10/9 1
9/8 9/8 1
154/135 8/7 539/540 = 245/243*99/100
81/70 7/6 243/245
33/28 32/27 891/896 = 225/224*99/100
6/5 6/5 1
11/9 11/9 1
5/4 5/4 1
14/11 14/11 1
9/7 9/7 1
405/308 21/16 540/539 = 100/99*243/245
4/3 4/3 1
15/11 27/20 100/99
25/18 11/8 100/99
140/99 7/5 100/99
99/70 10/7 99/100
36/25 16/11 99/100
22/15 40/27 99/100
3/2 3/2 1
616/405 32/21 539/540 = 245/243*99/100
14/9 14/9 1
11/7 11/7 1
8/5 8/5 1
18/11 18/11 1
5/3 5/3 1
56/33 27/16 896/891 = 100/99*224/225
140/81 12/7 245/243
135/77 7/4 540/539 = 100/99*243/245
16/9 16/9 1
9/5 9/5 1
11/6 11/6 1
28/15 15/8 224/225
154/81 40/21 539/540 = 245/243*99/100
27/14 64/33 891/896 = 225/224*99/100
88/45 160/81 99/100

Since translating a note of a periodicity block by one or two unison vectors
does not change it's important properties, Partch's scale with Wilson's two
equivalencies is a periodicity block. I think we can find a better
parallelopiped, though . . .

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

I wrote,

>I think we can find a better parallelopiped, though . . .

I have.

Define Fokker2 as the notes in the hyperparallelopiped centered around 1/1
with edge vectors

( 4 0 -1 1 )
( 2 -1 2 -1 )
( -5 1 2 0 )
( -2 2 0 -1 )

in other words, unison vectors

896/891
441/440
245/243
100/99

Observe:

Fokker2 Partch Fokker2/Partch
1 1 1
81/80 81/80 1
28/27 33/32 896/891
21/20 21/20 1
297/280 16/15 891/896
12/11 12/11 1
10/9 11/10 100/99
" 10/9 1
9/8 9/8 1
8/7 8/7 1
7/6 7/6 1
33/28 32/27 891/896
6/5 6/5 1
11/9 11/9 1
5/4 5/4 1
14/11 14/11 1
9/7 9/7 1
21/16 21/16 1
4/3 4/3 1
27/20 27/20 1
11/8 11/8 1
7/5 7/5 1
10/7 10/7 1
16/11 16/11 1
40/27 40/27 1
3/2 3/2 1
32/21 32/21 1
14/9 14/9 1
11/7 11/7 1
8/5 8/5 1
18/11 18/11 1
5/3 5/3 1
56/33 27/16 896/891
12/7 12/7 1
7/4 7/4 1
16/9 16/9 1
9/5 9/5 1
" 20/11 99/100
11/6 11/6 1
560/297 15/8 896/891
40/21 40/21 1
27/14 64/33 891/896
160/81 160/81 1

Notice that I forgot to include 20/11 last time. Anyway, this is a pretty
good agreement -- probably as good as a hyperparallelopiped will get.

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

🔗David C Keenan <d.keenan@xx.xxx.xxx>

"Paul H. Erlich" <PErlich@Acadian-Asset.com> wrote in TD465.9:
>As a final insult to Partch purists, I ask Dave Keenan to figure out,
>what is the best way to distribute some of the commas (probably 441:440
>(3.9�) and 896:891 (9.7�)) in the tuning so as to increase the number of
>hexads while keeping them within an insect's poop of JI?

How could I resist such a challenge. I tried to do it algebraicly and
failed. I haven't figured out simultaneous multiple comma distributions
yet. But at least doing the two commas separately gave me the region in
which to look. I was forced to do it numerically by successive approximation.

The answer is (drum roll please):

Ratio Error Name
11:12 -4.06 narrow neutral second
10:11 0.60 neutral second
9:10 2.22 narrow major second
8:9 1.89 major second
7:8 0.60 supermajor second
6:7 -1.84 subminor third
5:6 -3.46 minor third
9:11 2.82 neutral third
4:5 4.11 major third
7:9 2.49 supermajor third
3:4 -1.24 perfect fourth
8:11 4.71 super fourth
5:7 -5.31 augmented fourth
7:10 4.71 diminished fifth
2:3 0.65 perfect fifth
7:11 5.31 subminor sixth
5:8 -4.71 minor sixth
3:5 2.86 major sixth
7:12 1.24 supermajor sixth
4:7 -1.20 subminor seventh
5:9 -2.82 wide minor seventh
6:11 3.46 neutral seventh
1:2 -0.60 perfect octave
5:11 0.00 neutral ninth
4:9 1.29 major ninth
3:7 -2.44 subminor tenth
5:12 -4.06 minor tenth
2:5 3.51 major tenth
3:8 -1.84 perfect eleventh
4:11 4.11 super eleventh
1:3 0.05 perfect twelfth
3:10 2.26 major thirteenth
2:7 -1.80 subminor fourteenth
3:11 2.86 neutral fourteenth
1:4 -1.20 perfect double octave

Unfortunately, as you can see, the worst error is 5.3 c in the 5:7 and
7:11. The only strictly just interval is the 5:11. Not really better than
1/4 comma meantone, and certainly not better than my recent 31-toner.

The two commas involved, while quite small, pull too much in opposite
directions with regard to the 7 versus the 5 and 11.

However, I have no idea how many hexads it gives you in 41 notes. Someone
else wanna draw the lattice? How many does Partch's 43 give?

Note the shameless plug for the interval naming scheme that Graham Breed
and I have been evolving (with help from Paul Erlich and Manuel Op de Coul).

Regards,

-- Dave Keenan
http://dkeenan.com