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.
The comma being divided here is the syntonic comma (81/80).
Temperaments that fall outside of the "historically considered meantone" range will not provide most of the properties that meantone usually provides, but they are included for completeness.
Scope
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 MOS shapes)
- any tunings listed under "historical temperaments" (e.g. 4/25-comma), but only the ones of the form "n-comma".
Dozens of tunings on this table are significant to negative harmony temperament theory, enough that labelling them all individually would clutter the table.
All tunings on this table are close to some arbitrarily large EDO, but labelling them beyond 100edo would clutter the table.
Flatter than flattest allowed historical meantone
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 |