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 the syntonic comma (81/80) is being divided and n is a fraction between -1 and 1 with a denominator 22 or smaller.
Scope of table
Characteristics included
Some of the characteristics this table mentions for each temperament include:
- Whether it saw historical (pre-1950) use
- Whether it is close to (i.e. within 1/2 a degree of closing of) an edo smaller than 100
- Whether it is the closest on the table to the optimal CTE or POTE tuning of meantone or superpyth in any JI limit
- Whether it approximates a very simple n-Pythagorean comma meantone
- Whether it is about equally sharp of 3/2 as some other listed temperament is flat
- Whether it is close to exactly one just-noticeable difference away from 3/2
Occasional other comments may be included as well.
Characteristics omitted
Dozens of tunings on this table are significant to negative harmony temperament theory, enough that labelling them all individually would clutter the table.
Every tuning on this table is close to some arbitrarily large edo, but labeling them beyond 100edo would clutter the table.
Special cases included
A small number of additional temperaments are included. Not too many, to avoid clutter, just the bare minimum:
- 7, 12, 17 and 5 edos (to delineate small MOS shapes and boundaries of diamond monotone)
- any tunings listed under "historical temperaments" (e.g. 4/25-comma), but only the ones of the form "n-comma".
Cautions
Preservation of meantone behavior
Temperaments that fall outside of the "historically-defined meantone" range will not possess most of the musical properties that meantone usually possesses, but they are included for completeness.
Temperaments that fall outside of the "diamond monotone" range preserve even fewer meantone properties, but they are also included for completeness.
The table
Flatter than flattest historically-defined meantone
| Meantone Temperament | Generator (cents) | Comments |
|---|---|---|
| 1/1-comma | 680.449 | Close to 30edo |
| 21/22-comma | 681.426 | Close to 37edo. |
| 20/21-comma | 681.473 | |
| 19/20-comma | 681.524 | |
| 18/19-comma | 681.581 | |
| 17/18-comma | 681.644 | |
| 16/17-comma | 681.713 | |
| 15/16-comma | 681.793 | Close to 44edo. |
| 14/15-comma | 681.883 | |
| 13/14-comma | 681.985 | |
| 12/13-comma | 682.103 | |
| 11/12-comma | 682.241 | |
| 10/11-comma | 682.404 | Close to 51edo. |
| 19/21-comma | 682.497 | |
| 9/10-comma | 682.599 | |
| 17/19-comma | 682.713 | |
| 8/9-comma | 682.838 | Close to 58edo. |
| 15/17-comma | 682.979 | |
| 7/8-comma | 683.137 | Close to 65edo. |
| 13/15-comma | 683.316 | Close to 72edo. |
| 19/22-comma | 683.381 | |
| 6/7-comma | 683.521 | Close to 79edo. |
| 17/20-comma | 683.675 | |
| 11/13-comma | 683.757 | Close to 86edo. |
| 16/19-comma | 683.844 | |
| 21/25-comma | 683.890 | Close to 93edo |
| 5/6-comma | 684.033 | Close to 100edo. |
| 14/17-comma | 684.244 | |
| 9/11-comma | 684.359 | |
| 13/16-comma | 684.481 | |
| 17/21-comma | 684.545 | |
| 4/5-comma | 684.750 | |
| 15/19-comma | 684.976 | |
| 11/14-comma | 685.057 | |
| 7/9-comma | 685.228 | |
| 17/22-comma | 685.337 | |
| 10/13-comma | 685.412 | |
| 13/17-comma | 685.509 | |
| 16/21-comma | 685.569 | 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 | 685.825 | Everything from this point onwards generates 12 and 19 tone MOS scales. |
| 14/19-comma | 686.108 | |
| 11/15-comma | 686.184 | |
| 19/26-comma | 686.239 | |
| 8/11-comma | 686.314 | |
| 13/18-comma | 686.423 | |
| 5/7-comma | 686.593 | |
| 17/24-comma | 686.721 | |
| 12/17-comma | 686.774 | |
| 7/10-comma | 686.901 | |
| 9/13-comma | 687.066 | |
| 11/16-comma | 687.169 | |
| 13/19-comma | 687.240 | |
| 15/22-comma | 687.292 | |
| 17/25-comma | 687.331 | |
| 19/28-comma | 687.361 | |
| 2/3-comma | 687.617 | Close to 89edo. |
| 17/26-comma | 687.893 | Close to 82edo. |
| 15/23-comma | 687.929 | |
| 13/20-comma | 687.976 | |
| 11/17-comma | 688.039 | Close to 75edo |
| 9/14-comma | 688.129 | |
| 7/11-comma | 688.269 | Close to 68edo. |
| 12/19-comma | 688.372 | |
| 5/8-comma | 688.514 | Close to 61edo and 43/32. |
| 13/21-comma | 688.641 | |
| 1/ϕ-comma | 688.663 | |
| 8/13-comma | 688.720 | |
| 11/18-comma | 688.812 | Close to 54edo. |
| 14/23-comma | 688.864 | |
| 3/5-comma | 689.051 | |
| 13/22-comma | 689.247 | |
| 10/17-comma | 689.304 | |
| 7/12-comma | 689.410 | Close to 47edo. |
| 11/19-comma | 689.504 | |
| 4/7-comma | 689.666 | Close to 87edo. |
| 9/16-comma | 689.858 | |
| 5/9-comma | 690.007 | Close to 40edo. |
| 11/20-comma | 690.127 | |
| 6/11-comma | 690.224 | |
| 7/13-comma | 690.375 | Close to 73edo. |
| 8/15-comma | 690.485 | |
| 9/17-comma | 690.569 | |
| 10/19-comma | 690.636 | |
| 11/21-comma | 690.690 | Close to 33edo |
| 1/2-comma | 691.202 | Close to 92edo, 59edo. Historically significant (see historical temperaments). Everything up to this point does not have a whole tone between 10/9 and 9/8. |
Historically-defined meantone
| Temperament | Generator (cents) | Comments |
|---|---|---|
| 10/21-comma | 691.714 | |
| 9/19-comma | 691.768 | Close to 85edo. |
| 8/17-comma | 691.834 | |
| 7/15-comma | 691.919 | |
| 6/13-comma | 692.029 | |
| 5/11-comma | 692.179 | |
| 9/20-comma | 692.277 | Close to 26edo. |
| 4/9-comma | 692.397 | |
| 7/16-comma | 692.546 | |
| 3/7-comma | 692.738 | |
| 8/19-comma | 692.899 | |
| 5/12-comma | 692.994 | Close to 71edo. |
| 7/17-comma | 693.099 | |
| 9/22-comma | 693.157 | |
| 2/5-comma | 693.352 | Close to 45edo. |
| 9/23-comma | 693.539 | |
| 7/18-comma | 693.591 | |
| 5/13-comma | 693.683 | |
| 1/(ϕ+1)-comma | 693.740 | Close to 64edo. |
| 8/21-comma | 693.762 | |
| 3/8-comma | 693.890 | Close to 83edo. |
| 7/19-comma | 694.032 | |
| 4/11-comma | 694.134 | Almost exactly 1/3-Pythagorean comma meantone. |
| 5/14-comma | 694.274 | |
| 6/17-comma | 694.365 | |
| 7/20-comma | 694.428 | |
| 8/23-comma | 694.475 | |
| 9/26-comma | 694.511 | |
| 1/3-comma | 694.786 | Close to 19edo. Historically significant (see historical temperaments). |
| 9/28-comma | 695.042 | |
| 8/25-comma | 695.073 | |
| 7/22-comma | 695.112 | |
| 6/19-comma | 695.164 | |
| 5/16-comma | 695.234 | |
| 4/13-comma | 695.338 | |
| 3/10-comma | 695.503 | Close to 88edo and Lucy tuning. Historically significant (see historical temperaments). |
| 5/17-comma | 695.630 | Close to 69edo. |
| 7/24-comma | 695.682 | |
| 2/7-comma | 695.810 | Historically significant (see historical temperaments). |
| 5/18-comma | 695.981 | Close to 50edo. |
| 3/11-comma | 696.090 | |
| 7/26-comma | 696.165 | Close to golden meantone. Historically significant (see historical temperaments). |
| 4/15-comma | 696.220 | Close to 5-limit meantone POTE tuning. |
| 5/19-comma | 696.295 | Close to 81edo. |
| 1/4-comma | 696.578 | Close to septimal and tridecimal meantone POTE tunings. Historically significant (see historical temperaments). |
| 5/21-comma | 696.834 | Close to 31edo. |
| 4/17-comma | 696.895 | |
| 3/13-comma | 696.992 | Close to septimal & tridecimal meantone CTE tunings. Close to undecimal meantone POTE tuning. |
| 5/22-comma | 697.067 | |
| 2/9-comma | 697.176 | Close to 5-limit and undecimal meantone CTE tunings. Historically significant (see historical temperaments). |
| 3/14-comma | 697.346 | Close to 74edo. Historically significant (see historical temperaments). |
| 4/19-comma | 697.427 | |
| 1/5-comma | 697.654 | Close to 43edo. Historically significant (see historical temperaments). |
| 4/21-comma | 697.859 | |
| 3/16-comma | 697.923 | |
| 2/11-comma | 698.045 | Close to 55edo. Historically significant (see historical temperaments). |
| 3/17-comma | 698.159 | |
| 4/23-comma | 698.215 | |
| 1/6-comma | 698.371 | Historically significant (see historical temperaments). Everything up to this point has a fifth which is flat of Pythagorean tuning by at least the just-noticeable difference. |
| 4/25-comma | 698.514 | Close to 67edo. |
| 3/19-comma | 698.559 | |
| 2/13-comma | 698.646 | Close to 79edo. |
| 3/20-comma | 698.729 | |
| 1/7-comma | 698.883 | Close to 91edo. Historically significant (see historical temperaments). |
| 3/22-comma | 699.022 | |
| 2/15-comma | 699.088 | |
| 1/8-comma | 699.267 | |
| 2/17-comma | 699.425 | |
| 1/9-comma | 699.565 | |
| 2/19-comma | 699.691 | |
| 1/10-comma | 699.804 | |
| 2/21-comma | 699.907 | |
| 1/11-comma | 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 | 700.163 | Everything from this point onwards generates 12 and 17 tone MOS scales. |
| 1/13-comma | 700.301 | |
| 1/14-comma | 700.419 | |
| 1/15-comma | 700.521 | |
| 1/16-comma | 700.611 | |
| 1/17-comma | 700.690 | |
| 1/18-comma | 700.760 | |
| 1/19-comma | 700.823 | |
| 1/20-comma | 700.879 | |
| 1/21-comma | 700.931 | |
| 1/22-comma | 700.977 |
Negative harmony theory-defined meantone (most often approached as superpyth)
| Meantone Temperament | Generator (cents) | Comments |
|---|---|---|
| Pythagorean tuning | 701.955 | Historically significant (see historical temperaments.) Everything from this point onwards does not have a whole tone between 10/9 and 9/8. |
| -1/22-comma | 702.933 | |
| -1/21-comma | 702.979 | |
| -1/20-comma | 703.030 | |
| -1/19-comma | 703.087 | |
| -1/18-comma | 703.150 | |
| -1/17-comma | 703.220 | |
| -1/16-comma | 703.299 | |
| -1/15-comma | 703.389 | Close to 11/13 third-kleisma temperament. |
| -1/14-comma | 703.491 | Close to 29edo. |
| -1/13-comma | 703.609 | |
| -1/12-comma | 703.747 | |
| -1/11-comma | 703.910 | About as sharp of Pythagorean tuning as 12edo is flat. |
| -2/21-comma | 704.003 | Close to 75edo. |
| -1/10-comma | 704.105 | |
| -2/19-comma | 704.219 | |
| -1/9-comma | 704.344 | Close to 46edo, 11/7 quarter-kleisma temperament. |
| -2/17-comma | 704.483 | |
| -1/8-comma | 704.643 | |
| -2/15-comma | 704.823 | Close to 63edo. |
| -3/22-comma | 704.888 | |
| -1/7-comma | 705.027 | Close to 80edo. |
| -3/20-comma | 705.181 | |
| -2/13-comma | 705.350 | |
| -3/19-comma | 705.350 | |
| -4/25-comma | 705.396 | |
| -1/6-comma | 705.538 | Everything from this point onwards has a fifth which is sharp of Pythagorean tuning by at least the just-noticeable difference. |
| -4/23-comma | 705.695 | |
| -3/17-comma | 705.750 | About as sharp of Pythagorean tuning as 55edo is flat. |
| -2/11-comma | 705.865 | Everything up to this point generates 17 and 29 tone MOS scales. |
| 17edo | 705.882 | The largest MOS scale this can generate is 17 tone. Vaguely resembles Middle Eastern neutral third scales. |
| -3/16-comma | 705.987 | Everything from this point onwards generates 17 and 22 tone MOS scales. |
| -4/21-comma | 706.051 | |
| -1/5-comma | 706.256 | About as sharp of Pythagorean tuning as 43edo is flat. |
| -4/19 comma | 706.483 | |
| -3/14-comma | 706.563 | About as sharp of Pythagorean tuning as 74edo is flat. |
| -2/9-comma | 706.734 | |
| -5/22-comma | 706.843 | |
| -3/13-comma | 706.918 | Close to 39edo. |
| -4/17-comma | 707.015 | |
| -5/21-comma | 707.076 | About as sharp of Pythagorean tuning as 31edo is flat. |
| -1/4-comma | 707.332 | |
| -5/19-comma | 707.615 | |
| -4/15-comma | 707.690 | About as sharp of Pythagorean tuning as golden meantone is flat. |
| -7/26-comma | 707.745 | |
| -3/11-comma | 707.820 | Almost exactly -1/4-Pythagorean comma meantone |
| -5/18-comma | 707.930 | About as sharp of Pythagorean tuning as 50edo is flat. Close to 100edo. |
| -2/7-comma | 708.100 | |
| -7/24-comma | 708.227 | |
| -5/17-comma | 708.280 | |
| -3/10-comma | 708.407 | Nearly as sharp of Pythagorean tuning as Lucy tuning is flat. Nearly as sharp of Pythagorean tuning as 88edo is flat. |
| -4/13-comma | 708.572 | |
| -5/16-comma | 708.675 | |
| -6/19-comma | 708.746 | |
| -7/22-comma | 708.800 | |
| -8/25-comma | 708.837 | |
| -9/28-comma | 708.867 | |
| -1/3-comma | 709.124 | Close to 22edo. About as sharp of Pythagorean tuning as 19edo is flat. |
| -9/26-comma | 709.399 | Close to 2.3.7-limit superpyth POTE tuning. |
| -8/23-comma | 709.435 | |
| -7/20-comma | 709.482 | |
| -6/17-comma | 709.545 | Close to 11-limit superpyth CTE tuning. |
| -5/14-comma | 709.636 | Close to 93edo. Close to 2.3.7-limit and 7-limit superpyth CTE tunings. |
| -4/11-comma | 709.775 | Almost exactly -1/3-Pythagorean comma meantone. |
| -7/19-comma | 709.878 | Close to 13-limit superpyth CTE tuning. |
| -3/8-comma | 710.019 | |
| -8/21-comma | 710.148 | |
| -1/(ϕ+1)-comma | 710.170 | Close to 11-limit superpyth POTE tuning. |
| -5/13-comma | 710.227 | Close to 49edo. Close to 7-limit superpyth POTE tuning. |
| -7/18-comma | 710.319 | |
| -9/23-comma | 710.371 | |
| -2/5-comma | 710.558 | Close to 13-limit superpyth POTE tuning. |
| -9/22-comma | 710.753 | |
| -7/17-comma | 710.810 | |
| -5/12-comma | 710.915 | |
| -8/19-comma | 711.010 | |
| -3/7-comma | 711.172 | Close to 27edo. |
| -7/16-comma | 711.364 | |
| -4/9-comma | 711.513 | |
| -9/20-comma | 711.633 | |
| -5/11-comma | 711.731 | |
| -6/13-comma | 711.880 | Close to 59edo. |
| -7/15-comma | 711.991 | |
| -8/17-comma | 712.075 | |
| -9/19-comma | 712.142 | |
| -10/21-comma | 712.196 |
Sharper than sharpest negative harmonic-defined meantone
| Meantone Temperament | Generator (cents) | Comments |
|---|---|---|
| -1/2-comma | 712.708 | Close to 32edo. Everything from this point onwards does not have a whole tone being between 9/8 and 729/640. |
| -11/21-comma | 713.220 | |
| -10/19-comma | 713.274 | |
| -9/17-comma | 713.340 | |
| -8/15-comma | 713.425 | |
| -7/13-comma | 713.535 | Close to 37edo. |
| -6/11-comma | 713.686 | |
| -11/20-comma | 713.783 | |
| -5/9-comma | 713.903 | |
| -9/16-comma | 714.052 | |
| -4/7-comma | 714.244 | Close to 42edo. |
| -11/19-comma | 714.406 | |
| -7/12-comma | 714.500 | |
| -10/17-comma | 714.606 | |
| -13/22-comma | 714.663 | |
| -3/5-comma | 714.859 | Close to 47edo. |
| -14/23-comma | 715.046 | |
| -11/18-comma | 715.098 | |
| -8/13-comma | 715.190 | |
| -1/ϕ-comma | 715.247 | |
| -13/21-comma | 715.268 | |
| -5/8-comma | 715.396 | Close to 52edo and 387/256. |
| -12/19-comma | 715.538 | |
| -7/11-comma | 715.641 | |
| -9/14-comma | 715.780 | Close to 57edo. |
| -11/17-comma | 715.871 | |
| -13/20-comma | 715.934 | |
| -2/3-comma | 716.293 | Close to 62edo. |
| -15/22 comma | 716.618 | Close to 67edo. |
| -13/19 comma | 716.669 | Close to 72edo. |
| -11/16-comma | 716.741 | |
| -9/13-comma | 716.844 | Close to 77edo. |
| -7/10-comma | 717.009 | |
| -12/17-comma | 717.136 | Close to 82edo. |
| -17/24-comma | 717.188 | Close to 87edo. |
| -5/7-comma | 717.317 | Close to 92edo. |
| -13/18-comma | 717.487 | Close to 97edo. |
| -8/11-comma | 717.596 | |
| -19/26-comma | 717.671 | |
| -11/15-comma | 717.726 | |
| -14/19-comma | 717.802 | |
| -3/4-comma | 718.085 | About as sharp of Pythagorean tuning as 7edo is flat. |
| -21/26-comma | 718.325 | |
| -16/21-comma | 718.341 | |
| -13/17-comma | 718.401 | |
| -10/13-comma | 718.498 | |
| -17/22-comma | 718.574 | |
| -7/9-comma | 718.682 | |
| -11/14-comma | 718.853 | |
| -15/19-comma | 718.934 | |
| -4/5-comma | 719.160 | |
| -17/21-comma | 719.365 | |
| -13/16-comma | 719.429 | |
| -9/11-comma | 719.551 | |
| -14/17-comma | 719.666 | |
| -5/6-comma | 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 | 720.020 | Everything from this point onwards generates 13 and 18 tone MOS scales. |
| -16/19-comma | 720.066 | |
| -11/13-comma | 720.153 | |
| -17/20-comma | 720.235 | |
| -6/7-comma | 720.399 | |
| -19/22-comma | 720.529 | |
| -13/15-comma | 720.594 | |
| -7/8-comma | 720.773 | |
| -15/17-comma | 720.931 | |
| -8/9-comma | 721.017 | |
| -17/19-comma | 721.197 | |
| -9/10-comma | 721.311 | |
| -19/21-comma | 721.413 | |
| -10/11-comma | 721.506 | |
| -11/12-comma | 721.669 | |
| -12/13-comma | 721.807 | |
| -13/14-comma | 721.925 | |
| -14/15-comma | 722.028 | |
| -15/16-comma | 722.117 | |
| -16/17-comma | 722.196 | |
| -17/18-comma | 722.266 | |
| -18/19-comma | 722.329 | |
| -19/20-comma | 722.386 | |
| -20/21-comma | 722.437 | |
| -21/22-comma | 722.484 | |
| -1/1-comma | 723.461 | Close to 68edo. |