User:BudjarnLambeth/Table of n-comma meantone generators

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Here are all meantone tunings that can be written in the form "n-comma meantone", where n is a fraction between -1 and 1 with a denominator 18 or smaller.

Also included are 7, 12, 17 and 5 edos (to delineate MOS shapes), as well as the other tunings listed under "historical temperaments" (e.g. 4/25-comma), but only the ones of the form "n-comma". All of these tunings are closet to a large enough edo, but labelling them beyond 100edo would clutter the table.

The comma being divided here is the syntonic comma (81/80).

Temperaments that fall outside of the diamond monotone range will not provide most of the advantages that meantone usually provides, but they are included for completeness.

Dozens of tunings on the table are significant to negative harmony temperament theory, enough that labelling them all individually would clutter the table.

Flatter than flattest allowed historical meantone

Temperament	Generator (cents)	Comments
1/1-comma meantone	680.449	Close to 30edo
17/18-comma meantone	681.644	Close to 37edo.
16/17-comma meantone	681.713
15/16-comma meantone	681.793	Close to 44edo
14/15-comma meantone	681.883
13/14-comma meantone	681.985
12/13-comma meantone	682.103
11/12-comma meantone	682.241
10/11-comma meantone	682.404	Close to 51edo
9/10-comma meantone	682.599
8/9-comma meantone	682.838	Close to 58edo
15/17-comma meantone	682.979
7/8-comma meantone	683.137	Close to 65edo
13/15-comma meantone	683.316	Close to 72edo
6/7-comma meantone	683.521	Close to 79edo
11/13-comma meantone	683.757	Close to 86edo
16/19-comma meantone	683.844
21/25-comma meantone	683.890	Close to 93edo
5/6-comma meantone	684.033	Close to 100edo
14/17-comma meantone	684.244
9/11-comma meantone	684.359
13/16-comma meantone	684.481
4/5-comma meantone	684.75
11/14-comma meantone	685.057
7/9-comma meantone	685.228
10/13-comma meantone	685.412
13/17-comma meantone	685.509	Everything up to this point generates 9 and 16 tone MOS scales.
7edo	685.714	The largest MOS scale this can generate is 7 tone. Lower boundary of 5-limit diamond monotone.
3/4-comma meantone	685.825	Everything from this point onwards generates 12 and 19 tone MOS scales.
11/15-comma meantone	686.184
19/26-comma meantone	686.239
8/11-comma meantone	686.314
13/18-comma meantone	686.423
5/7-comma meantone	686.593
17/24-comma meantone	686.721
12/17-comma meantone	686.774
7/10-comma meantone	686.901
9/13-comma meantone	687.066
11/16-comma meantone	687.169
2/3-comma meantone	687.617	Close to 89edo
17/26-comma meantone	687.893	Close to 82edo.
15/23-comma meantone	687.929
13/20-comma meantone	687.976
11/17-comma meantone	688.039	Close to 75edo
9/14-comma meantone	688.129
7/11-comma meantone	688.269	Close to 68edo
5/8-comma meantone	688.514	Close to 61edo
Golden ratio comma meantone	688.663
8/13-comma meantone	688.720
11/18-comma meantone	688.812	Close to 54edo
3/5-comma meantone	689.051
10/17-comma meantone	689.304
7/12-comma meantone	689.410	Close to 47edo
4/7-comma meantone	689.666	Close to 87edo
9/16-comma meantone	689.858
5/9-comma meantone	690.007	Close to 40edo
6/11-comma meantone	690.224
7/13-comma meantone	690.375	Close to 73edo
8/15-comma meantone	690.485
9/17-comma meantone	690.569	Close to 33edo

Historically considered meantone

Temperament	Generator (cents)	Comments
1/2-comma meantone	691.202	Close to 92edo, 59edo. Historically significant (see historical temperaments). Everything up to this point does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
8/17-comma meantone	691.834	Close to 85edo
7/15-comma meantone	691.919
6/13-comma meantone	692.029
5/11-comma meantone	692.179
4/9-comma meantone	692.397	Close to 26edo
7/16-comma meantone	692.546
3/7-comma meantone	692.738
5/12-comma meantone	692.994	Close to 71edo
7/17-comma meantone	693.099
2/5-comma meantone	693.352	Close to 45edo
7/18-comma meantone	693.591
5/13-comma meantone	693.683
Split Golden ratio comma meantone	693.740	Close to 64edo
3/8-comma meantone	693.890	Close to 83edo
4/11-comma meantone	694.134	Almost exactly 1/3-Pythagorean comma meantone
5/14-comma meantone	694.274
6/17-comma meantone	694.365
7/20-comma meantone	694.428
8/23-comma meantone	694.475
9/26-comma meantone	694.511
1/3-comma meantone	694.786	Close to 19edo. Historically significant (see historical temperaments).
5/16-comma meantone	695.234
4/13-comma meantone	695.338
3/10-comma meantone	695.503	Close to 88edo
5/17-comma meantone	695.630	Close to 69edo.
7/24-comma meantone	695.682
2/7-comma meantone	695.810	Historically significant (see historical temperaments).
5/18-comma meantone	695.981	Close to 50edo. Historically significant (see historical temperaments).
3/11-comma meantone	696.090
7/26-comma meantone	696.165	Close to golden meantone. Historically significant (see historical temperaments).
4/15-comma meantone	696.220	Close to 81edo, close to golden meantone
1/4-comma meantone	696.578	Close to 31edo. Historically significant (see historical temperaments).
4/17-comma meantone	696.895
3/13-comma meantone	696.992	Close to septimal & tridecimal meantone CTE tunings.
2/9-comma meantone	697.176	Close to 5-limit and undecimal meantone CTE tunings. Historically significant (see historical temperaments).
3/14-comma meantone	697.346	Close to 74edo. Historically significant (see historical temperaments).
1/5-comma meantone	697.654	Close to 43edo. Historically significant (see historical temperaments).
3/16-comma meantone	697.923
2/11-comma meantone	698.045	Close to 55edo. Historically significant (see historical temperaments).
3/17-comma meantone	699.425
1/6-comma meantone	698.371	Historically significant (see historical temperaments).
4/25-comma meantone	698.514	Close to 67edo.
3/19-comma meantone	698.559
2/13-comma meantone	698.646	Close to 79edo.
1/7-comma meantone	698.883	Close to 91edo. Historically significant (see historical temperaments).
2/15-comma meantone	699.088
1/8-comma meantone	699.267
2/17-comma meantone	699.425
1/9-comma meantone	699.565
1/10-comma meantone	699.804
1/11-comma meantone	700.000	Everything up to this point generates 12 and 19 tone MOS scales.
12edo	700.000	The largest MOS scale this can generate is 12 tone. Historically significant (see historical temperaments.)
1/12-comma meantone	700.163	Everything from this point onwards generates 12 and 17 tone MOS scales.
1/13-comma meantone	700.301
1/14-comma meantone	700.419
1/15-comma meantone	700.521
1/16-comma meantone	700.611
1/17-comma meantone	700.690
1/18-comma meantone	700.760

Negative-harmonic-defined meantone

Temperament	Generator (cents)	Comments
0/1-comma meantone	701.955	Pythagorean tuning. Historically significant (see historical temperaments.) Everything from this point onwards does not meet the original historical definition of meantone (the whole tone being between 10/9 and 9/8).
-1/18-comma meantone	703.150
-1/17-comma meantone	703.220
-1/16-comma meantone	703.299
-1/15-comma meantone	703.389	Close to 11/13 third-kleisma temperament.
-1/14-comma meantone	703.491	Close to 29edo.
-1/13-comma meantone	703.609
-1/12-comma meantone	703.747
-1/11-comma meantone	703.910	About as sharp of Pythagorean tuning as 12edo is flat.
-1/10-comma meantone	704.105
-1/9-comma meantone	704.344	Close to 46edo, 11/7 quarter-kleisma temperament.
-2/17-comma meantone	704.483
-1/8-comma meantone	704.643
-2/15-comma meantone	704.823	Close to 63edo.
-1/7-comma meantone	705.027	Close to 80edo.
-2/13-comma meantone	705.350
-3/19-comma meantone	705.350
-4/25-comma meantone	705.396
-1/6-comma meantone	705.538
-3/17-comma meantone	705.750	About as sharp of Pythagorean tuning as 55edo is flat.
-2/11-comma meantone	705.865	Everything up to this point generates 17 and 29 tone MOS scales.
17edo	705.882	Vaguely resembles Middle Eastern neutral third scales.
-3/16-comma meantone	705.987	Everything from this point onwards generates 17 and 22 tone MOS scales.
-1/5-comma meantone	706.256	About as sharp of Pythagorean tuning as 43edo is flat.
-3/14-comma meantone	706.563
-2/9-comma meantone	706.734
-3/13-comma meantone	706.918
-4/17-comma meantone	707.015	About as sharp of Pythagorean tuning as 74edo is flat.
Negative Quarter-comma meantone	707.332
-4/15-comma meantone	707.690	About as sharp of Pythagorean tuning as Golden meantone is flat.
-7/26-comma meantone	707.745
-3/11-comma meantone	707.820	Almost exactly -1/4-Pythagorean comma meantone
-5/18-comma meantone	707.930	About as sharp of Pythagorean tuning as 50edo is flat. Close to 100edo
-2/7-comma meantone	708.100
-7/24-comma meantone	708.227
-5/17-comma meantone	708.280
-3/10-comma meantone	708.407	About as sharp of Pythagorean tuning as Lucy tuning is flat. Nearly as sharp of Pythagorean tuning as 88edo is flat.
-4/13-comma meantone	708.572
-5/16-comma meantone	708.675
-1/3-comma meantone	709.124	Close to 22edo. About as sharp of Pythagorean tuning as 19edo is flat.
-9/26-comma meantone	709.399
-8/23-comma meantone	709.435
-7/20-comma meanetone	709.482
-6/17-comma meantone	709.545
-5/14-comma meantone	709.636	Close to 93edo
-4/11-comma meantone	709.775	Almost exactly -1/3-Pythagorean comma meantone.
-3/8-comma meantone	710.019
Negative Split Golden ratio comma meantone	710.170
-5/13-comma meantone	710.227	Close to 49edo.
-7/18-comma meantone	710.319
-2/5-comma meantone	710.558
-7/17-comma meantone	710.810
-5/12-comma meantone	710.915
-3/7-comma meantone	711.172	Close to 27edo.
-7/16-comma meantone	711.364
-4/9-comma meantone	711.513
-5/11-comma meantone	711.731
-6/13-comma meantone	711.880	Close to 59edo.
-7/15-comma meantone	711.991
-8/17-comma meantone	712.075

Sharper than sharpest negative-harmonic-defined meantone

Temperament	Generator (cents)	Comments
-1/2-comma meantone	712.708	Close to 32edo. Everything from this point onwards does not meet the negative harmonic definition of meantone (the whole tone being between 9/8 and 729/640).
-9/17-comma meantone	713.340
-8/15-comma meantone	713.425
-7/13-comma meantone	713.535	Close to 37edo.
-6/11-comma meantone	713.686
-5/9-comma meantone	713.903
-9/16-comma meantone	714.052
-4/7-comma meantone	714.244	Close to 42edo.
-7/12-comma meantone	714.500
-10/17-comma meantone	714.606
-3/5-comma meantone	714.859	Close to 47edo.
-11/18-comma meantone	715.098
-8/13-comma meantone	715.190
Negative Golden ratio comma meantone
-5/8-comma meantone	715.396	Close to 52edo.
-7/11-comma meantone	715.641
-9/14-comma meantone	715.780	Close to 57edo.
-11/17-comma meantone	715.871
-2/3-comma meantone	716.293	Close to 62edo.
-15/22 comma meantone	716.618	Close to 67edo.
-13/19 comma meantone	716.669	Close to 72edo.
-11/16-comma meantone	716.741
-9/13-comma meantone	716.844	Close to 77edo.
-7/10-comma meantone	717.009
-12/17-comma meantone	717.136	Close to 82edo.
-17/24-comma meantone	717.188	Close to 87edo.
-5/7-comma meantone	717.317	Close to 92edo.
-13/18-comma meantone	717.487	Close to 97edo.
-8/11-comma meantone	717.596
-19/26-comma meantone	717.671
-11/15-comma meantone	717.726
-3/4-comma meantone	718.085	About as sharp of Pythagorean tuning as 7edo is flat.
-13/17-comma meantone	718.401
-10/13-comma meantone	718.498
-7/9-comma meantone	718.682
-11/14-comma meantone	718.853
-4/5-comma meantone	719.160
-13/16-comma meantone	719.429
-9/11-comma meantone	719.551
-14/17-comma meantone	719.666
-5/6-comma meantone	719.877	Everything up to this point generates 12 and 17 tone MOS scales.
5edo	720.000	The largest MOS scale this can generate is 5 tone. Upper boundary of 5-limit diamond monotone.
-21/25-comma meantone	720.020	Everything from this point onwards generates 13 and 18 tone MOS scales.
-16/19-comma meantone	720.066
-11/13-comma meantone	720.153
-6/7-comma meantone	720.399
-13/15-comma meantone	720.594
-7/8-comma meantone	720.773
-15/17-comma meantone	720.931
-8/9-comma meantone	721.017
-9/10-comma meantone	721.311
-10/11-comma meantone	721.506
-11/12-comma meantone	721.669
-12/13-comma meantone	721.807
-13/14-comma meantone	721.925
-14/15-comma meantone	722.028
-15/16-comma meantone	722.117
-16/17-comma meantone	722.196
-17/18-comma meantone	722.266
-1/1-comma meantone	723.461	Close to 68edo