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Message: 6725 - Contents - Hide Contents

Date: Fri, 04 Apr 2003 11:20:51

Subject: 45/44 & 49/48

From: Gene Ward Smith

Here are 11-limit scales for which both 45/44 and 49/48 are chromas
(which entails that (45/44)/(49/48) = 540/539 is a comma.) They all
have the property that the scale degree is greater than the 7-limit
Graham complexity, and in five cases, for Wizard[28], Octoid[56],
Contraschismic[53], Dreielf[66] and Heptadec[9] it is greater than the
11-limit Graham complexity--these are presumably the really fun cases.


Wizard[28] [225/224, 385/384, 4000/3993]
[12, -2, 20, -6, -31, -2, -51, 52, -7, -86] [[2, 1, 5, 2, 8], [0, 6,
-1, 10, -3]]

bad 3830.785828 comp 107.1605720 rms 1.584514314
graham 26 scale size 28


Octoid[56] [540/539, 1375/1372, 4000/3993]
[24, 32, 40, 24, -5, -4, -45, 3, -55, -71] [[8, 13, 19, 23, 28], [0,
-3, -4, -5, -3]]

bad 4139.348941 comp 173.2618570 rms .7687062821
graham 40 scale size 56


Magic[19] [100/99, 225/224, 245/243]
[5, 1, 12, -8, -10, 5, -30, 25, -22, -64] [[1, 0, 2, -1, 6], [0, 5, 1,
12, -8]]

bad 4474.854491 comp 61.02789552 rms 4.730404304
graham 20 scale size 19


Catakleismic[38] [225/224, 385/384, 4375/4374]
[6, 5, 22, -21, -6, 18, -54, 37, -66, -135] [[1, 0, 1, -3, 9], [0, 6,
5, 22, -21]]

bad 4805.476809 comp 117.8180381 rms 1.697136764
graham 43 scale size 38


Meanpop[19] [81/80, 126/125, 385/384]
[1, 4, 10, -13, 4, 13, -24, 12, -44, -71] [[1, 2, 4, 7, -2], [0, -1,
-4, -10, 13]]

bad 5420.225566 comp 61.58085622 rms 5.644270538
graham 23 scale size 19


Schismatic[29] [225/224, 385/384, 2200/2187]
[1, -8, -14, 23, -15, -25, 33, -10, 81, 113] [[1, 2, -1, -3, 13], [0,
-1, 8, 14, -23]]

bad 5478.852555 comp 102.3231433 rms 2.447558936
graham 37 scale size 29


Contraschismic[53] [540/539, 1375/1372, 5120/5103]
[1, 33, 27, -18, 50, 40, -32, -30, -156, -144] [[1, 2, 16, 14, -4],
[0, -1, -33, -27, 18]]

bad 6259.259444 comp 177.5735716 rms 1.115729896
graham 51 scale size 53


Dreielf[66] [540/539, 1375/1372, 8019/8000]
[18, 39, 42, 9, 20, 16, -48, -12, -114, -120] [[3, 2, 1, 2, 9], [0, 6,
13, 14, 3]]

bad 6297.038152 comp 184.8474423 rms 1.049817759
graham 42 scale size 66


Heptadec[9] [36/35, 56/55, 77/75]
[5, 3, 7, 4, -7, -3, -11, 8, -1, -13] [[1, 1, 2, 2, 3], [0, 5, 3, 7, 4]]

bad 6400.766110 comp 32.19555159 rms 19.64440328
graham 7 scale size 9


Fourththirds[5] [16/15, 28/27, 77/75]
[1, -1, 3, -4, -4, 2, -10, 10, -6, -22] [[1, 2, 2, 4, 2], [0, -1, 1,
-3, 4]]

bad 6476.838089 comp 20.25383770 rms 43.03787612
graham 7 scale size 5


Pajarous[10] [50/49, 55/54, 64/63]
[2, -4, -4, 10, -11, -12, 9, 2, 37, 42] [[2, 3, 5, 6, 6], [0, 1, -2,
-2, 5]]

bad 6667.906202 comp 43.76707564 rms 12.26714784
graham 14 scale size 10

Message: 6726 - Contents - Hide Contents

Date: Sat, 05 Apr 2003 08:11:06

Subject: Scales of Wizard

From: Gene Ward Smith

Here are scales for wizard. Since the main interest I presume is using
it for 72-et, I've reduced everything using 72-et, which means
including 243/242. A poptimal generator however is found in the
310-et, in case some one cares.

Scales of Wizard

Wizard commas [225/224, 243/242, 385/384, 4000/3993]
72 et commas [225/224, 243/242, 385/384, 4000/3993]
Wedgie [12, -2, 20, -6, -31, -2, -51, 52, -7, -86]


Wizard[28] 

Chromas 45/44, 49/48

72 et period [1, 3, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3]

Scale
[1, 33/32, 35/33, 15/14, 11/10, 25/22, 7/6, 33/28, 40/33, 5/4, 9/7,
33/25, 4/3, 11/8, 99/70, 16/11, 3/2, 50/33, 14/9, 8/5, 33/20, 5/3,
12/7, 44/25, 20/11, 15/8, 66/35, 35/18]

[[35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4, 14/9, 35/18, 40/33,
50/33, 66/35, 33/28], [3/2, 15/8, 7/6, 16/11, 20/11, 25/22, 99/70,
44/25, 11/10, 11/8, 12/7, 15/14, 4/3, 5/3]]

Circles of 5/4 generator

[1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
[1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
[1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
[1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
[1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
[1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
[1, 5/4, 3/2, 12/7] [1, 33/28, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
[1, 14/11, 3/2, 7/4] [1, 6/5, 3/2, 7/4]


[1, 44/35, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
[1, 44/35, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
[1, 9/7, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
[1, 9/7, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
[1, 9/7, 22/15, 44/25] [1, 7/6, 22/15, 5/3]
[1, 9/7, 22/15, 44/25] [1, 7/6, 22/15, 5/3]
[1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
[1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
[1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
[1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
[1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
[1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 56/33]


[1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
[1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
[1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
[1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
[1, 7/6, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
[1, 7/6, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
[1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
[1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
[1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
[1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
[1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
[1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
[1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]




Wizard[34]

Chroma 21/20

72 et period
[1, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2]

Scale
[1, 33/32, 25/24, 35/33, 15/14, 11/10, 25/22, 7/6, 33/28, 6/5, 40/33,
5/4, 9/7, 33/25, 4/3, 15/11, 11/8, 99/70, 16/11, 22/15, 3/2, 50/33,
14/9, 8/5, 33/20, 5/3, 56/33, 12/7, 44/25, 20/11, 15/8, 66/35, 27/14,
35/18]

[[15/11, 56/33, 35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4, 14/9,
35/18, 40/33, 50/33, 66/35, 33/28, 22/15], [27/14, 6/5, 3/2, 15/8,
7/6, 16/11, 20/11, 25/22, 99/70, 44/25, 11/10, 11/8, 12/7, 15/14, 4/3,
5/3, 25/24]]

Circles of 5/4 generator

[1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
[1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
[1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
[1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
[1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
[1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
[1, 5/4, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
[1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
[1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 9/5]


[1, 11/9, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
[1, 11/9, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
[1, 9/7, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
[1, 9/7, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
[1, 9/7, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
[1, 9/7, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
[1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
[1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
[1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
[1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
[1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
[1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
[1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]





Wizard[38]

Chroma 22/21

72-et period 
[1, 3, 3, 1, 2, 1, 3, 3, 1, 2, 1, 2, 1, 3, 3, 1, 2, 1, 2]
Scale
[1, 33/32, 25/24, 35/33, 15/14, 12/11, 11/10, 25/22, 7/6, 33/28, 6/5,
40/33, 5/4, 9/7, 35/27, 33/25, 4/3, 15/11, 11/8, 99/70, 16/11, 22/15,
3/2, 50/33, 54/35, 14/9, 8/5, 33/20, 5/3, 56/33, 12/7, 44/25, 20/11,
11/6, 15/8, 66/35, 27/14, 35/18]

[[12/11, 15/11, 56/33, 35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4,
14/9, 35/18, 40/33, 50/33, 66/35, 33/28, 22/15, 11/6], [54/35, 27/14,
6/5, 3/2, 15/8, 7/6, 16/11, 20/11, 25/22, 99/70, 44/25, 11/10, 11/8,
12/7, 15/14, 4/3, 5/3, 25/24, 35/27]]

Circles of 5/4 generator

[1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7] [1, 5/4, 11/7, 11/6] [1,
44/35, 11/7, 12/7]
[1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
[1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
[1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
[1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
[1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
[1, 5/4, 3/2, 11/6] [1, 44/35, 3/2, 12/7]
[1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
[1, 25/21, 3/2, 7/4] [1, 6/5, 3/2, 18/11]


[1, 27/20, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
[1, 27/20, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
[1, 9/7, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
[1, 9/7, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
[1, 9/7, 11/7, 66/35] [1, 7/6, 11/7, 5/3]
[1, 9/7, 11/7, 66/35] [1, 7/6, 11/7, 5/3]
[1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
[1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 35/22]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
[1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]


[1, 11/9, 11/7, 11/6] [1, 27/20, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 27/20, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
[1, 7/6, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
[1, 7/6, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
[1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
[1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]

Message: 6727 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 00:00:07

Subject: Circulating miracles

From: Gene Ward Smith

Here are Miracle[20], Miracle[21] and Miracle[22], this time 11-limit
reduced assuming 72 et. Miracle[22] seems well worth exploring for
anyone interested in 11-limit harmony.

I had before been restricting myself to where the Graham complexity
was less than the scale size; since 11/10 is a consonance of the
11-limit, this means it could never appear, whereas ratios of
consonances such as
15/14 = (5/4)/(7/6) could so long as we did not insist on getting
twice the Graham complextity less than the scale size. However, it
occurred to me this was unduly restrictive, given that 11/10 is well
outside of the range of 7-limit Graham complexity.

Miracle 

72 et commas [225/224, 243/242, 385/384, 4000/3993]

Miracle commas [225/224, 243/242, 385/384]

Wedgie [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]

Circulating miracles

Miracle[20]

Chroma 25/24

Scale

[1, 15/14, 12/11, 8/7, 7/6, 11/9, 5/4, 21/16, 4/3, 7/5, 10/7, 3/2,
32/21, 8/5, 18/11, 12/7, 7/4, 11/6, 15/8, 49/25]

[12/11, 7/6, 5/4, 4/3, 10/7, 32/21, 18/11, 7/4, 15/8, 1, 15/14, 8/7,
11/9, 21/16, 7/5, 3/2, 8/5, 12/7, 11/6, 49/25]

Chords in chains of secors 

[1, 6/5, 3/2, 27/16] [1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 27/16] [1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 6/5, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 16/9]
[1, 5/4, 3/2, 7/4] [1, 5/4, 3/2, 16/9]
[1, 5/4, 14/9, 7/4] [1, 5/4, 14/9, 16/9]
[1, 5/4, 14/9, 7/4] [1, 5/4, 14/9, 16/9]
[1, 5/4, 14/9, 7/4] [1, 5/4, 14/9, 16/9]
[1, 5/4, 14/9, 7/4] [1, 5/4, 14/9, 16/9]
[1, 5/4, 14/9, 7/4] [1, 5/4, 14/9, 16/9]
[1, 5/4, 14/9, 7/4] [1, 5/4, 14/9, 16/9]


[1, 9/7, 3/2, 9/5] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 27/16]
[1, 9/7, 3/2, 15/8] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 27/16]
[1, 9/7, 3/2, 15/8] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 7/4]
[1, 9/7, 3/2, 15/8] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 7/4]
[1, 9/7, 3/2, 15/8] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 7/4]
[1, 9/7, 3/2, 15/8] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 9/8, 3/2, 8/5] [1, 9/8, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 7/6, 3/2, 8/5] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 7/6, 3/2, 8/5] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 7/6, 3/2, 8/5] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 7/6, 3/2, 8/5] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 7/6, 3/2, 8/5] [1, 7/6, 3/2, 7/4]
[1, 4/3, 3/2, 15/8] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 4/3, 14/9, 15/8] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]
[1, 4/3, 14/9, 15/8] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]
[1, 4/3, 14/9, 15/8] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]
[1, 4/3, 14/9, 15/8] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]
[1, 4/3, 14/9, 15/8] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]
[1, 4/3, 14/9, 15/8] [1, 7/6, 14/9, 5/3] [1, 7/6, 14/9, 7/4]


[1, 11/9, 11/7, 11/6] [1, 11/9, 3/2, 12/7] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 3/2, 12/7] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 3/2, 12/7] [1, 9/7, 11/7, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 9/7, 18/11, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 9/7, 18/11, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 9/7, 18/11, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 4/3, 18/11, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 4/3, 18/11, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 4/3, 18/11, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 4/3, 18/11, 12/7]
[1, 11/9, 18/11, 11/6] [1, 11/9, 3/2, 12/7] [1, 4/3, 18/11, 12/7]
[1, 11/9, 18/11, 21/11] [1, 11/9, 3/2, 12/7] [1, 4/3, 18/11, 12/7]
[1, 11/9, 18/11, 21/11] [1, 11/9, 3/2, 16/9] [1, 4/3, 18/11, 16/9]
[1, 11/9, 18/11, 21/11] [1, 11/9, 3/2, 16/9] [1, 4/3, 18/11, 16/9]
[1, 11/9, 18/11, 21/11] [1, 11/9, 14/9, 16/9] [1, 4/3, 18/11, 16/9]
[1, 11/9, 18/11, 21/11] [1, 11/9, 14/9, 16/9] [1, 4/3, 18/11, 16/9]
[1, 11/9, 18/11, 21/11] [1, 11/9, 14/9, 16/9] [1, 4/3, 18/11, 16/9]
[1, 14/11, 18/11, 21/11] [1, 14/11, 14/9, 16/9] [1, 4/3, 18/11, 16/9]
[1, 14/11, 18/11, 21/11] [1, 14/11, 14/9, 16/9] [1, 4/3, 18/11, 16/9]
[1, 14/11, 18/11, 21/11] [1, 14/11, 14/9, 16/9] [1, 4/3, 18/11, 16/9]



Miracle[21] (Blackjack)

Chromas 33/32, 36/35

Scale

[1, 45/44, 15/14, 12/11, 8/7, 7/6, 11/9, 5/4, 21/16, 4/3, 7/5, 10/7,
3/2, 32/21, 8/5, 18/11, 12/7, 7/4, 11/6, 15/8, 49/25]

[45/44, 12/11, 7/6, 5/4, 4/3, 10/7, 32/21, 18/11, 7/4, 15/8, 1, 15/14,
8/7, 11/9, 21/16, 7/5, 3/2, 8/5, 12/7, 11/6, 49/25]

Chords in chains of secors 

[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 9/5] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 9/7, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 3/2, 7/4] [1, 7/6, 3/2, 5/3]
[1, 5/4, 16/11, 7/4] [1, 7/6, 16/11, 5/3]
[1, 5/4, 16/11, 7/4] [1, 7/6, 16/11, 5/3]
[1, 5/4, 16/11, 7/4] [1, 7/6, 16/11, 5/3]
[1, 5/4, 16/11, 7/4] [1, 7/6, 16/11, 5/3]
[1, 5/4, 16/11, 7/4] [1, 7/6, 16/11, 5/3]
[1, 5/4, 16/11, 7/4] [1, 7/6, 16/11, 5/3]


[1, 11/9, 11/7, 11/6] [1, 11/9, 8/5, 11/6] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 8/5, 11/6] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 8/5, 11/6] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 8/5, 11/6] [1, 9/7, 11/7, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 9/7, 32/21, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 9/7, 32/21, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 9/7, 32/21, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 5/4, 32/21, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 5/4, 32/21, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 5/4, 32/21, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 5/4, 32/21, 12/7]
[1, 11/9, 32/21, 11/6] [1, 11/9, 8/5, 11/6] [1, 5/4, 32/21, 12/7]
[1, 11/9, 32/21, 16/9] [1, 11/9, 8/5, 16/9] [1, 5/4, 32/21, 12/7]
[1, 11/9, 32/21, 16/9] [1, 11/9, 8/5, 16/9] [1, 5/4, 32/21, 5/3]
[1, 11/9, 32/21, 16/9] [1, 11/9, 14/9, 16/9] [1, 5/4, 32/21, 5/3]
[1, 11/9, 32/21, 16/9] [1, 11/9, 14/9, 16/9] [1, 5/4, 32/21, 5/3]
[1, 11/9, 32/21, 16/9] [1, 11/9, 14/9, 16/9] [1, 5/4, 32/21, 5/3]
[1, 11/9, 32/21, 16/9] [1, 11/9, 14/9, 16/9] [1, 5/4, 32/21, 5/3]
[1, 25/21, 32/21, 16/9] [1, 25/21, 14/9, 16/9] [1, 5/4, 32/21, 5/3]
[1, 25/21, 32/21, 16/9] [1, 25/21, 14/9, 16/9] [1, 5/4, 32/21, 5/3]
[1, 25/21, 32/21, 16/9] [1, 25/21, 14/9, 16/9] [1, 5/4, 32/21, 5/3]




Miracle[22]

Chroma 11/10

Scale

[1, 45/44, 21/20, 15/14, 12/11, 8/7, 7/6, 11/9, 5/4, 21/16, 4/3, 7/5,
10/7, 3/2, 32/21, 8/5, 18/11, 12/7, 7/4, 11/6, 15/8, 49/25]

[45/44, 12/11, 7/6, 5/4, 4/3, 10/7, 32/21, 18/11, 7/4, 15/8, 1, 15/14,
8/7, 11/9, 21/16, 7/5, 3/2, 8/5, 12/7, 11/6, 49/25, 21/20]

Chords in chains of secors 

[1, 11/8, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 27/14] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 11/8, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 12/7]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 14/9]
[1, 5/4, 3/2, 7/4] [1, 12/11, 3/2, 14/9]
[1, 5/4, 15/11, 7/4] [1, 12/11, 15/11, 14/9]
[1, 5/4, 15/11, 7/4] [1, 12/11, 15/11, 14/9]
[1, 5/4, 15/11, 7/4] [1, 12/11, 15/11, 14/9]
[1, 5/4, 15/11, 7/4] [1, 12/11, 15/11, 14/9]
[1, 5/4, 15/11, 7/4] [1, 12/11, 15/11, 14/9]
[1, 5/4, 15/11, 7/4] [1, 12/11, 15/11, 14/9]


[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 27/14]
[1, 9/7, 3/2, 9/5] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 7/4]
[1, 9/7, 3/2, 18/11] [1, 9/7, 3/2, 11/6] [1, 9/7, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 11/6] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 3/2, 18/11] [1, 7/6, 3/2, 5/3] [1, 7/6, 3/2, 7/4]
[1, 7/6, 15/11, 18/11] [1, 7/6, 15/11, 5/3] [1, 7/6, 15/11, 7/4]
[1, 7/6, 15/11, 18/11] [1, 7/6, 15/11, 5/3] [1, 7/6, 15/11, 7/4]
[1, 7/6, 15/11, 18/11] [1, 7/6, 15/11, 5/3] [1, 7/6, 15/11, 7/4]
[1, 7/6, 15/11, 18/11] [1, 7/6, 15/11, 5/3] [1, 7/6, 15/11, 7/4]
[1, 7/6, 15/11, 18/11] [1, 7/6, 15/11, 5/3] [1, 7/6, 15/11, 7/4]
[1, 7/6, 15/11, 18/11] [1, 7/6, 15/11, 5/3] [1, 7/6, 15/11, 7/4]


[1, 11/9, 11/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 11/7, 12/7]
[1, 11/9, 11/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 11/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 10/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 10/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 9/7, 10/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 7/6, 10/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 7/6, 10/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 7/6, 10/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 49/25] [1, 7/6, 10/7, 12/7]
[1, 11/9, 10/7, 11/6] [1, 11/9, 12/7, 16/9] [1, 7/6, 10/7, 12/7]
[1, 11/9, 10/7, 5/3] [1, 11/9, 12/7, 16/9] [1, 7/6, 10/7, 12/7]
[1, 11/9, 10/7, 5/3] [1, 11/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]
[1, 11/9, 10/7, 5/3] [1, 11/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]
[1, 11/9, 10/7, 5/3] [1, 11/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]
[1, 11/9, 10/7, 5/3] [1, 11/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]
[1, 11/9, 10/7, 5/3] [1, 11/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]
[1, 10/9, 10/7, 5/3] [1, 10/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]
[1, 10/9, 10/7, 5/3] [1, 10/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]
[1, 10/9, 10/7, 5/3] [1, 10/9, 14/9, 16/9] [1, 7/6, 10/7, 14/9]

Message: 6728 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 12:16:25

Subject: Re: The canonical homomorphism

From: Carl Lumma

>Since the ratio between TH(a^n)TH(b^n) and TH((ab)^n) is bounded, when
>we take the nth root they tend towards the same limit, and so TH is a
>homomorphic mapping--that is to say TH(1) = 1, TH(a)TH(b) = TH(ab).
>
>This is as slick as goose grease, and whether or not we want to use
>this exact tuning is not to me the real question. What puts a smile on
>my face is that in this way we can in a theoretical sense define the
>temperament as being the mapping. The definition has nothing whatever
>to do with octaves or octave equivalence, and in no way depends on the
>definition or choice of consonances. I propose to call this the
>canonical homomorphism.


Wow, sometimes you do get what you've always wanted.  Can you show
how this works with an example?

-Carl

Message: 6729 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 00:21:32

Subject: Re: Scales of Wizard

From: Joseph Pehrson

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 

Yahoo groups: /tuning-math/message/6168 * [with cont.] 

wrote:

> Here are scales for wizard. Since the main interest I presume is 
using
> it for 72-et, I've reduced everything using 72-et, which means
> including 243/242. A poptimal generator however is found in the
> 310-et, in case some one cares.
> 
> Scales of Wizard
> 
> Wizard commas [225/224, 243/242, 385/384, 4000/3993]
> 72 et commas [225/224, 243/242, 385/384, 4000/3993]
> Wedgie [12, -2, 20, -6, -31, -2, -51, 52, -7, -86]
> 
> 
> Wizard[28] 
> 
> Chromas 45/44, 49/48
> 
> 72 et period [1, 3, 3, 3, 1, 3, 3, 3, 3, 1, 3, 3, 3, 3]
> 
> Scale
> [1, 33/32, 35/33, 15/14, 11/10, 25/22, 7/6, 33/28, 40/33, 5/4, 9/7,
> 33/25, 4/3, 11/8, 99/70, 16/11, 3/2, 50/33, 14/9, 8/5, 33/20, 5/3,
> 12/7, 44/25, 20/11, 15/8, 66/35, 35/18]
> 
> [[35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4, 14/9, 35/18, 40/33,
> 50/33, 66/35, 33/28], [3/2, 15/8, 7/6, 16/11, 20/11, 25/22, 99/70,
> 44/25, 11/10, 11/8, 12/7, 15/14, 4/3, 5/3]]
> 
> Circles of 5/4 generator
> 
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 22/15, 12/7] [1, 33/28, 22/15, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 33/28, 3/2, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 12/7] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> [1, 14/11, 3/2, 7/4] [1, 6/5, 3/2, 7/4]
> 
> 
> [1, 44/35, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 44/35, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 8/7, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 7/6, 22/15, 5/3]
> [1, 9/7, 22/15, 44/25] [1, 7/6, 22/15, 5/3]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 44/25] [1, 7/6, 3/2, 56/33]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 56/33]
> 
> 
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 8/7, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 7/6, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 7/6, 22/15, 12/7] [1, 11/8, 22/15, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 12/7] [1, 11/8, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
> [1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
> [1, 7/6, 3/2, 7/4] [1, 7/5, 3/2, 7/4]
> 
> 
> 
> 
> Wizard[34]
> 
> Chroma 21/20
> 
> 72 et period
> [1, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2, 1, 3, 3, 3, 1, 2]
> 
> Scale
> [1, 33/32, 25/24, 35/33, 15/14, 11/10, 25/22, 7/6, 33/28, 6/5, 
40/33,
> 5/4, 9/7, 33/25, 4/3, 15/11, 11/8, 99/70, 16/11, 22/15, 3/2, 50/33,
> 14/9, 8/5, 33/20, 5/3, 56/33, 12/7, 44/25, 20/11, 15/8, 66/35, 
27/14,
> 35/18]
> 
> [[15/11, 56/33, 35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4, 14/9,
> 35/18, 40/33, 50/33, 66/35, 33/28, 22/15], [27/14, 6/5, 3/2, 15/8,
> 7/6, 16/11, 20/11, 25/22, 99/70, 44/25, 11/10, 11/8, 12/7, 15/14, 
4/3,
> 5/3, 25/24]]
> 
> Circles of 5/4 generator
> 
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 10/7, 5/3] [1, 8/7, 10/7, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 8/7, 3/2, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 5/3] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> [1, 21/16, 3/2, 7/4] [1, 6/5, 3/2, 9/5]
> 
> 
> [1, 11/9, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 11/9, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 10/9, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 10/7, 12/7] [1, 7/6, 10/7, 5/3] [1, 11/8, 10/7, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 5/3] [1, 11/8, 3/2, 5/3]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 12/7] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 11/8, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 7/4] [1, 36/25, 3/2, 7/4]
> 
> 
> 
> 
> 
> Wizard[38]
> 
> Chroma 22/21
> 
> 72-et period 
> [1, 3, 3, 1, 2, 1, 3, 3, 1, 2, 1, 2, 1, 3, 3, 1, 2, 1, 2]
> Scale
> [1, 33/32, 25/24, 35/33, 15/14, 12/11, 11/10, 25/22, 7/6, 33/28, 
6/5,
> 40/33, 5/4, 9/7, 35/27, 33/25, 4/3, 15/11, 11/8, 99/70, 16/11, 
22/15,
> 3/2, 50/33, 54/35, 14/9, 8/5, 33/20, 5/3, 56/33, 12/7, 44/25, 20/11,
> 11/6, 15/8, 66/35, 27/14, 35/18]
> 
> [[12/11, 15/11, 56/33, 35/33, 33/25, 33/20, 33/32, 9/7, 8/5, 1, 5/4,
> 14/9, 35/18, 40/33, 50/33, 66/35, 33/28, 22/15, 11/6], [54/35, 
27/14,
> 6/5, 3/2, 15/8, 7/6, 16/11, 20/11, 25/22, 99/70, 44/25, 11/10, 11/8,
> 12/7, 15/14, 4/3, 5/3, 25/24, 35/27]]
> 
> Circles of 5/4 generator
> 
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7] [1, 5/4, 11/7, 11/6] [1,
> 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 11/7, 11/6] [1, 44/35, 11/7, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 44/35, 3/2, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 11/6] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 12/7]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> [1, 5/4, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> [1, 25/21, 3/2, 7/4] [1, 6/5, 3/2, 18/11]
> 
> 
> [1, 27/20, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 27/20, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 11/9, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 7/6, 11/7, 5/3]
> [1, 9/7, 11/7, 66/35] [1, 7/6, 11/7, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 5/3]
> [1, 9/7, 3/2, 66/35] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> [1, 9/7, 3/2, 9/5] [1, 7/6, 3/2, 35/22]
> 
> 
> [1, 11/9, 11/7, 11/6] [1, 27/20, 11/7, 12/7]
> [1, 11/9, 11/7, 11/6] [1, 27/20, 11/7, 12/7]
> [1, 11/9, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 11/9, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 7/6, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 7/6, 11/7, 11/6] [1, 9/7, 11/7, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 11/6] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 12/7]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]
> [1, 7/6, 3/2, 7/4] [1, 9/7, 3/2, 18/11]


***Thanks, Gene... This looks promising...

J. Pehrson

Message: 6730 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 05:33:47

Subject: The limit generator

From: Gene Ward Smith

I've thought of a new approach to the question of defining a canonical
generator for a given linear temperament.

If T is a linear temperament, and r is a rational representative of
the generator (eg, the TM reduced generator), and if TM is the
Tenny-Minkowski reduction function for T, then we define the limit
generator for T as

g = lim n-->infinity (TM(r^n))^(1/n)

in terms of cents, this would be

g = lim n--> infinity cents(TM(r^n))/n

The limit does exist, so this definition defines a unique generator,
without, alas, telling us a good way to compute it. The limit secor in
the 7-limit is about 116.785 cents, which we may compare to the rms
value of 116.573 cents and the minimax value of 116.588 cents. Here
are the ratios of successive terms for the TM reduced sequence for
miracle:

15/14, 16/15, 343/320, 15/14, 16/15, 15/14, 16/15, 15/14, 16/15, 
343/320, 15/14, 15/14, 16/15, 15/14, 16/15, 15/14, 343/320, 15/14,
16/15, 15/14, 16/15, 15/14, 343/320, 15/14, 16/15, 15/14, 16/15,
15/14, 343/320, 16/15, 15/14, 16/15, 15/14, 15/14, 2401/2250, 15/14,
15/14, 16/15, 15/14, 15/14, 16/15,343/320, 15/14, 16/15, 15/14, 15/14,
16/15, 343/320, 15/14, 16/15, 15/14, 16/15, 15/14, 15/14, 2401/2250,
15/14, 16/15, 15/14, 15/14, 15/14, 2401/2250, 15/14, 16/15, 15/14,
15/14, 15/14, 2401/2250, 15/14, 16/15, 15/14, 15/14, 16/15,
343/320, 15/14, 16/15, 15/14, 16/15, 15/14, 15/14, 2401/2250, 15/14,
16/15, 15/14, 15/14, 15/14, 2401/2250, 15/14, 16/15, 15/14, 15/14,
15/14, 2401/2250, 15/14, 16/15, 15/14, 15/14, 16/15, 343/320, 15/14...

The limit secor is the limit of geometric averages of this sequence,
(or C(1) summation applied to the logs of the sequence, if you speak
that language.)

I'm uploading a graph of the secors approaching their limit for values
from 300 to 999, and another of the Borel summation method applied to
the first graph.

Message: 6731 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 05:40:23

Subject: Re: The limit generator

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:


> I'm uploading a graph of the secors approaching their limit for values
> from 300 to 999, and another of the Borel summation method applied to
> the first graph.


You'll find them in "photos", where there is, incidentally, lots of
free space.

Message: 6732 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 05:53:33

Subject: Limit generator for 5-limit meantone

From: Gene Ward Smith

This turns out to be 1/4-comma meantone; the sequence of fifths falls
into a regular pattern of three pure fifts followed by a 40/27.

Message: 6733 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 06:43:18

Subject: Re: Limit generator for 5-limit meantone

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:


> This turns out to be 1/4-comma meantone; the sequence of fifths falls
> into a regular pattern of three pure fifts followed by a 40/27.


I should add for this and the miracle example that the limit octave is
2. This is not always the case; Paul may or may not like to know that
for Pajara, for instance, it is distinctly flat. Carl I hope is taking
note of this, as it seems to be along the lines he has been agitating for.

Message: 6734 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 07:54:28

Subject: Re: The limit generator

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> I've thought of a new approach to the question of defining a 
canonical
> generator for a given linear temperament.

cool!

> If T is a linear temperament, and r is a rational representative of
> the generator (eg, the TM reduced generator), and if TM is the
> Tenny-Minkowski reduction function for T, then we define the limit
> generator for T as
> 
> g = lim n-->infinity (TM(r^n))^(1/n)
> 
> in terms of cents, this would be
> 
> g = lim n--> infinity cents(TM(r^n))/n
> 
> The limit does exist, so this definition defines a unique 
generator,
> without, alas, telling us a good way to compute it. The limit secor 
in
> the 7-limit is about 116.785 cents, which we may compare to the rms
> value of 116.573 cents and the minimax value of 116.588 cents. Here
> are the ratios of successive terms for the TM reduced sequence for
> miracle:
> 
> 15/14, 16/15, 343/320, 15/14, 16/15, 15/14, 16/15, 15/14, 16/15, 
> 343/320, 15/14, 15/14, 16/15, 15/14, 16/15, 15/14, 343/320, 15/14,
> 16/15, 15/14, 16/15, 15/14, 343/320, 15/14, 16/15, 15/14, 16/15,
> 15/14, 343/320, 16/15, 15/14, 16/15, 15/14, 15/14, 2401/2250, 
15/14,
> 15/14, 16/15, 15/14, 15/14, 16/15,343/320, 15/14, 16/15, 15/14, 
15/14,
> 16/15, 343/320, 15/14, 16/15, 15/14, 16/15, 15/14, 15/14, 
2401/2250,
> 15/14, 16/15, 15/14, 15/14, 15/14, 2401/2250, 15/14, 16/15, 15/14,
> 15/14, 15/14, 2401/2250, 15/14, 16/15, 15/14, 15/14, 16/15,
> 343/320, 15/14, 16/15, 15/14, 16/15, 15/14, 15/14, 2401/2250, 
15/14,
> 16/15, 15/14, 15/14, 15/14, 2401/2250, 15/14, 16/15, 15/14, 15/14,
> 15/14, 2401/2250, 15/14, 16/15, 15/14, 15/14, 16/15, 343/320, 
15/14...
> 
> The limit secor is the limit of geometric averages of this 
sequence,
> (or C(1) summation applied to the logs of the sequence, if you 
speak
> that language.)


so it picks the mode of the distribution? or something?

Message: 6735 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 07:56:56

Subject: Re: Limit generator for 5-limit meantone

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...>
> wrote:
> 

>> This turns out to be 1/4-comma meantone; the sequence of fifths 
falls
>> into a regular pattern of three pure fifts followed by a 40/27.
> 

> I should add for this and the miracle example that the limit octave 
is
> 2. This is not always the case; Paul may or may not like to know 
that
> for Pajara, for instance, it is distinctly flat. Carl I hope is 
taking
> note of this, as it seems to be along the lines he has been 
agitating for.


can you explain, step by step, all the details of this calculation?

Message: 6736 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 08:46:04

Subject: Re: The limit generator

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:


> so it picks the mode of the distribution? or something?


I'm trying to figure out what it does, but the mode has nothing to do
with it. It averages the generator along a line of generators, and I
see no reason to think that it is independent of generator-pair
choice, so I suppose its canonicity is open to question.

Message: 6737 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 08:49:24

Subject: Re: Limit generator for 5-limit meantone

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
> <gwsmith@s...>
>> wrote:
>> 

>>> This turns out to be 1/4-comma meantone; the sequence of fifths 
> falls

>>> into a regular pattern of three pure fifts followed by a 40/27.
>> 

>> I should add for this and the miracle example that the limit octave 
> is

>> 2. This is not always the case; Paul may or may not like to know 
> that

>> for Pajara, for instance, it is distinctly flat. Carl I hope is 
> taking

>> note of this, as it seems to be along the lines he has been 
> agitating for.
> 

> can you explain, step by step, all the details of this calculation?


Which one? In general, you show something is going to be the best
possible choice given the commas you have--for instance 2^n is often
the most reduced for any n, but for some temperaments, such as pajara
with its 64/63 in the mix, it isn't (64 reduces to 63.)

Message: 6738 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 09:00:43

Subject: Re: Limit generator for 5-limit meantone

From: wallyesterpaulrus

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:

>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
<gwsmith@s...> 
>> wrote:

>>> --- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" 
>> >> wrote:
>>> 

>>>> This turns out to be 1/4-comma meantone; the sequence of 
fifths 
>> falls

>>>> into a regular pattern of three pure fifts followed by a 
40/27.
>>> 

>>> I should add for this and the miracle example that the limit 
octave 
>> is

>>> 2. This is not always the case; Paul may or may not like to 
know 
>> that

>>> for Pajara, for instance, it is distinctly flat. Carl I hope is 
>> taking

>>> note of this, as it seems to be along the lines he has been 
>> agitating for.
>> 

>> can you explain, step by step, all the details of this 
calculation?
> 

> Which one? In general, you show something is going to be the best
> possible choice given the commas you have--for instance 2^n is often
> the most reduced for any n, but for some temperaments, such as 
pajara
> with its 64/63 in the mix, it isn't (64 reduces to 63.)


right, show all the details of *this* calculation.

Message: 6739 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 09:34:58

Subject: Re: Limit generator for 5-limit meantone

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:


>> Which one? In general, you show something is going to be the best
>> possible choice given the commas you have--for instance 2^n is often
>> the most reduced for any n, but for some temperaments, such as 
> pajara

>> with its 64/63 in the mix, it isn't (64 reduces to 63.)
> 

> right, show all the details of *this* calculation.


I'm confused. Obviously, 63 is less than 64, so 2^6 reduces to 63.
What is there to show? On the other hand, for 5-limit meantone, if we
have 5^n then multiplying it by 80/81 obviously increases its height,
whereas attempting to lower the power of 5 by multiplying by 81/80
gives 81*5^(n-1) / 16, with height 2^4 * 3^4 * 5^(n-1) > 5^n. Hence,
5^n is always reduced, and so 5 must be exactly generated by g^4 where
g is the generator, so that g = 5^(1/4). If we look at the TM
reduction of 1, 3/2, (3/2)^2, (3/2)^3, (3/2)^4 we get 1, 3/2, 9/4,
10/3, 5, ...giving ratios of 3/2, 3/2, 40/27, 3/2, ...

Unfortunately it seems to get a lot more complicated in the 7-limit
and beyond.

Message: 6740 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 10:15:50

Subject: Re: Limit generator for 5-limit meantone

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...>
wrote:


> Unfortunately it seems to get a lot more complicated in the 7-limit
> and beyond.


"A lot" is exaggerated; I find these can be done. The 7-limit meantone
case is easy--exactly the same argument applies to 126/125 as to
81/80, and the limit generator is 1/4-comma meantone again. Miracle is
more difficult, but you can show that (27/5)^n cannot be further
reduced by either 225/224 or 1029/1024; since 27/5 is represented by
25 generator steps, the limit secor is (27/5)^(1/25), or 116.782 cents.

Now the question is how to turn it into an algorithm.

Message: 6741 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 04:12:10

Subject: Re: Limit generator for 5-limit meantone

From: Carl Lumma

>This turns out to be 1/4-comma meantone; the sequence of fifths falls
>into a regular pattern of three pure fifts followed by a 40/27.


It's not clear to me why big n should be special.  Just because it
causes convergence?  Why should I optimize my generator for whatever
a chain of 500 generators approximates, over what a chain of 5 of
them approximates?


>I should add for this and the miracle example that the limit octave is
>2. This is not always the case; Paul may or may not like to know that
>for Pajara, for instance, it is distinctly flat. Carl I hope is taking
>note of this, as it seems to be along the lines he has been agitating
>for.

Indeed.  -C.

Message: 6742 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 04:17:33

Subject: Re: The limit generator

From: Carl Lumma

>I'm trying to figure out what it does, but the mode has nothing to do
>with it. It averages the generator along a line of generators, and I
>see no reason to think that it is independent of generator-pair
>choice, so I suppose its canonicity is open to question.


If a "generator-pair" is what I think it is, doesn't this method claim
to give the optimum size(s) *given* a generator-pair?  Would we expect
two different gen-pair choices for a temperament to result in the same
tuning if carried out forever at their respective optimum sizes?

-Carl

Message: 6743 - Contents - Hide Contents

Date: Sun, 06 Apr 2003 13:56:17

Subject: The canonical homomorphism

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> If a "generator-pair" is what I think it is, doesn't this method claim
> to give the optimum size(s) *given* a generator-pair?  Would we expect
> two different gen-pair choices for a temperament to result in the same
> tuning if carried out forever at their respective optimum sizes?


The very question I've been pondering. The answer is no! 

This method gives something much nicer than a canonical generator,
namely a canonical homomorphism. For T a regular temperament
(therefore now no longer necessarily linear) and q a rational number,
define TR(q) to be the Tenney-Minkowski reduction function for T, and

TH(q) = lim n--> infinity TR(q^n)^(1/n)

TH turns out to be a canonically defined homomorphism, which can be
thought of as the temperament mapping. To show this, note that if
a and b are rational numbers, then

TH(a)TH(b) = lim TR(a^n)^(1/n) lim TR(b^n)^(1/n) = 
lim (TR(a^n)*TR(b^n))^(1/n)

Since the ratio between TH(a^n)TH(b^n) and TH((ab)^n) is bounded, when
we take the nth root they tend towards the same limit, and so TH is a
homomorphic mapping--that is to say TH(1) = 1, TH(a)TH(b) = TH(ab).

This is as slick as goose grease, and whether or not we want to use
this exact tuning is not to me the real question. What puts a smile on
my face is that in this way we can in a theoretical sense define the
temperament as being the mapping. The definition has nothing whatever
to do with octaves or octave equivalence, and in no way depends on the
definition or choice of consonances. I propose to call this the
canonical homomorphism.

Message: 6744 - Contents - Hide Contents

Date: Mon, 07 Apr 2003 01:19:17

Subject: Re: The canonical homomorphism

From: Gene Ward Smith

--- In tuning-math@xxxxxxxxxxx.xxxx Carl Lumma <ekin@l...> wrote:


> Wow, sometimes you do get what you've always wanted.  Can you show
> how this works with an example?


First I want to fix what I've written, which is defective, but not, I
think, fatally so.

Message: 6745 - Contents - Hide Contents

Date: Mon, 07 Apr 2003 06:46:51

Subject: Re: Limit generator for 5-limit meantone

From: wallyesterpaulrus

i have no idea what you're talking about. :(

--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith" <gwsmith@s...> 
wrote:

> --- In tuning-math@xxxxxxxxxxx.xxxx "wallyesterpaulrus"
> <wallyesterpaulrus@y...> wrote:
> 

>>> Which one? In general, you show something is going to be the 
best
>>> possible choice given the commas you have--for instance 2^n is 
often
>>> the most reduced for any n, but for some temperaments, such as 
>> pajara

>>> with its 64/63 in the mix, it isn't (64 reduces to 63.)
>> 

>> right, show all the details of *this* calculation.
> 

> I'm confused. Obviously, 63 is less than 64, so 2^6 reduces to 63.
> What is there to show? On the other hand, for 5-limit meantone, if 
we
> have 5^n then multiplying it by 80/81 obviously increases its 
height,
> whereas attempting to lower the power of 5 by multiplying by 81/80
> gives 81*5^(n-1) / 16, with height 2^4 * 3^4 * 5^(n-1) > 5^n. 
Hence,
> 5^n is always reduced, and so 5 must be exactly generated by g^4 
where
> g is the generator, so that g = 5^(1/4). If we look at the TM
> reduction of 1, 3/2, (3/2)^2, (3/2)^3, (3/2)^4 we get 1, 3/2, 9/4,
> 10/3, 5, ...giving ratios of 3/2, 3/2, 40/27, 3/2, ...
> 
> Unfortunately it seems to get a lot more complicated in the 7-limit
> and beyond.

Message: 6747 - Contents - Hide Contents

Date: Sun, 13 Apr 2003 04:35:21

Subject: Canonical homomorphisms revisted

From: Gene Ward Smith

The problem with my first definition is that it works beautifully when
it works, but it doesn't always work (1/2 time in the 5-limit, 1/4 of
the time in the 7-limit, and so forth.) Taking it as a source of
inspiration, here is a definition which works in general.
Unfortunately it no longer is as slick as goose grease.

If T is a temperament, call q a "subgroup comma" for T if q>1 is not a
power of anything else, and if only three primes are involved in its
factorization (these commas are easily found from the wedgie.) Call a
prime "reducible" if it appears by itself in either numerator or
denominator for the factorization of a subgroup comma. If T is
p-limit, let P be the set of primes <= p, and let R be the set of
reducible primes. Then N = P\R is the set of non-reducible primes. If
card(N) = g, where g is the number of generators (2 for a linear
temperament, 3 for a planar temperament, and so forth) then set G = N.
If card(N)<g, then fill it out by adding the smallest of the remaining
primes, and set the result to G; if card(N)>g, then reduce it by
removing the largest of the primes in N, and call that G. We end with
a set G of primes such that card(G)=g, and we may use G as a set of
generators for the temperament T.

Strange as it may seem, this definition actually corresponds to my
previous one in those cases where the previous one gives a
homomorphism. It also seems to give us reasonable results, or at least
so it seems to me, YMMV. Here is what we get for meantone and miracle;
I give the mapping of the generators (2 and 3/2 in the case of
meantone, 2 and 16/15 in the case of miracle), and the mapping applied
to primes:

5-limit meantone 81/80
Generators [2, 5^(1/4)]
Prime mapping [2, 2*5^(1/4), 5]

7-limit meantone [1, 4, 10, 4, 13, 12]
Generators [2, 2^(3/10)*7^(1/10)]
Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7]

11-limit meantone [1, 4, 10, 18, 4, 13, 25, 12, 28, 16]
Generators [2, 2^(3/10)*7^(1/10)]
Prime mapping [2, 2*2^(3/10)*7^(1/10), 2*2^(1/5)*7^(2/5), 7,
7/4*2^(2/5)*7^(4/5)]

11-limit meanpop [1, 4, 10, -13, 4, 13, -24, 12, -44, -71]
Generators [7^(13/71)*11^(10/71), 7^(11/71)*11^(3/71)]
Prime mapping [7^(13/71)*11^(10/71), 7^(24/71)*11^(13/71),
7^(44/71)*11^(12/71), 7, 11]

5-limit miracle 34171875/33554432
Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
Prime mapping [3^(7/25)*5^(6/25), 3, 5]

7-limit miracle [6, -7, -2, -25, -20, 15]
Generators [3^(7/25)*5^(6/25), 1/5*3^(3/25)*5^(24/25)]
Prime mapping [3^(7/25)*5^(6/25), 3, 5, 3^(3/5)*5^(4/5)]

11-limit miracle [6, -7, -2, 15, -25, -20, 3, 15, 59, 49]
Generators [5^(15/59)*11^(7/59), 1/5*5^(57/59)*11^(3/59)]
Prime mapping [5^(15/59)*11^(7/59), 5^(3/59)*11^(25/59), 5,
5^(49/59)*11^(15/59), 11]

Message: 6748 - Contents - Hide Contents

Date: Sun, 13 Apr 2003 10:17:58

Subject: Doing 12-equal within 133-et

From: Gene Ward Smith

Since 133 = 11*12+1, it might seem I am one off here. However, let's
consider the dominant sevenths temperament, which more or less
characterizes Western common-practice music in the 17-19th centuries. 

Dominant sevenths is defined by the wedgie [1, 4, -2, 4, -6, -16] or
the comma basis [36/35, 64/63]. The canonical map is given by
[245^(1/8), 6125^(1/8), 5, 7]
or in terms of generators 2 and 3/2, by
[245^(1/8), 5^(1/4)].

This, of course, has pure 5's and 7's, with flat 2's and 3's. We can
get something related to this by finding the least squares x for
(28*x - cents(5))^2 + (34*x - cents(7))^2, which turns out to be
99.25611588 cents. This is reasonably close to 2^(11/133), which is 
99.24812030 cents. The mapping to primes using this step value is
11 * [12, 19, 28, 34] = [132, 209, 308, 374]; here we take these as
133-equal values. In comparison the standard val for 133-equal would
be [133, 211, 309, 373].

Retuning any 12-et piece into this tuning is a straightforward task,
and I hope to get around to actually doing it soon. (I've either got
to boot up DOS, which I seldom do now, or get my Linux up to speed.)

Message: 6749 - Contents - Hide Contents

Date: Mon, 14 Apr 2003 00:11:01

Subject: Another strange property of 72

From: Gene Ward Smith

I went looking for vals which could be used for approximating the
7-limit miracle canonical map, stepping through divisions of 3 rather
than of 2, and seeing how well the canonical map was approximated. It
turns out that not only does [72, 114, 167, 202] work well, it works
extremely well--vastly better than anything else.

If we do a least squares for (114*x-cents(3))^2+(167*x-cents(5))^2, we
get x=16.68429155 as the size of a step; [7*x, 72*x] gives us

[116.7900408, 1201.268992]

and the difference in cents from the canonical map is inaudible:

[-.6300e-2, -.54236e-1, .37024e-1, -.2922e-2]

We can do the same calculation and get a least squares optimum for
all four images of the primes under the canonical map, with
essentially the same result: x=16.68427953 cents, with generators

[116.7899567, 1201.268126]

and difference from the canonical map of

[-.5434e-2, -.52865e-1, .39031e-1, -.494e-3]

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