# © 1999 by Paul Erlich

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Onelist Tuning Digest 463
Message: 21
From:	Paul H. Erlich
Sent:	Wednesday, December 29, 1999 5:59 PM
To:	'tuning@onelist.com'
Subject:	RE: [tuning] RE: 11-limit, 31 tones, 9 hexads within 2.7c of  just

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 [Erv] 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. . . .

Onelist Tuning Digest 464
Message: 11
Date: Thu, 30 Dec 1999 01:54:25 -0500
From: "Paul H. Erlich"
Subject: Partch scale as periodicity block?

100:99 = [22 *] 3-2 * 52 * 70 * 11-1

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

81:80
-----  = 243:242 or 35 * 5 0 * 70 * 11-2
121:120

55:54
-----  = 245:243 or 3-5 * 51 * 72 * 110
99:98

45:44
-----  = 225:224 or 32 * 52 * 7-1 * 110
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.

Onelist Tuning Digest 465
Message: 8
From:	Paul H. Erlich
Sent:	Thursday, December 30, 1999 5:08 AM
To:	'tuning@onelist.com'
Subject:	RE: [tuning] Partch scale as periodicity block?

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

Onelist Tuning Digest 464
Message: 12
Date: Thu, 30 Dec 1999 02:56:26 -0500
From: "Paul H. Erlich"
Subject: RE: Partch scale as periodicity block?

... Since translating a note of a periodicity block by one or two
unison vectors does not change its important properties, Partch's
scale with Wilson's two equivalencies is a periodicity block.

```

updated:

2002.10.10 -- reformatted for better viewing
2000.1.20

 For many more diagrams and explanations of historical tunings, see my book. If you don't understand my theory or the terms I've used, start here or try some definitions.