12edo: Difference between revisions
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| Line 57: | Line 57: | ||
! Monzo | ! Monzo | ||
! Cents | ! Cents | ||
![[Color notation/Temperament Names|Color Name]] | |||
! Name 1 | ! Name 1 | ||
! Name 2 | ! Name 2 | ||
| Line 64: | Line 65: | ||
| {{Monzo| 3 4 -4 }} | | {{Monzo| 3 4 -4 }} | ||
| style="text-align:right;" | 62.57 | | style="text-align:right;" | 62.57 | ||
|Quadgu | |||
| style="text-align:center;" | Major Diesis | | style="text-align:center;" | Major Diesis | ||
| style="text-align:center;" | Diminished Comma | | style="text-align:center;" | Diminished Comma | ||
| Line 71: | Line 73: | ||
| {{Monzo| 2 2 -1 -1 }} | | {{Monzo| 2 2 -1 -1 }} | ||
| style="text-align:right;" | 48.77 | | style="text-align:right;" | 48.77 | ||
|Rugu | |||
| style="text-align:center;" | Septimal Quarter Tone | | style="text-align:center;" | Septimal Quarter Tone | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 78: | Line 81: | ||
| {{Monzo| 7 0 -3 }} | | {{Monzo| 7 0 -3 }} | ||
| style="text-align:right;" | 41.06 | | style="text-align:right;" | 41.06 | ||
|Trigu | |||
| style="text-align:center;" | Diesis | | style="text-align:center;" | Diesis | ||
| style="text-align:center;" | Augmented Comma | | style="text-align:center;" | Augmented Comma | ||
| Line 85: | Line 89: | ||
| {{Monzo| 1 0 2 -2 }} | | {{Monzo| 1 0 2 -2 }} | ||
| style="text-align:right;" | 34.98 | | style="text-align:right;" | 34.98 | ||
|Biruyo | |||
| style="text-align:center;" | Tritonic Diesis | | style="text-align:center;" | Tritonic Diesis | ||
| style="text-align:center;" | Jubilisma | | style="text-align:center;" | Jubilisma | ||
| Line 92: | Line 97: | ||
| {{Monzo| 6 -2 0 -1 }} | | {{Monzo| 6 -2 0 -1 }} | ||
| style="text-align:right;" | 27.26 | | style="text-align:right;" | 27.26 | ||
|Ru | |||
| style="text-align:center;" | Septimal Comma | | style="text-align:center;" | Septimal Comma | ||
| style="text-align:center;" | Archytas' Comma | | style="text-align:center;" | Archytas' Comma | ||
| Line 99: | Line 105: | ||
| {{Monzo| -19 12 }} | | {{Monzo| -19 12 }} | ||
| style="text-align:right;" | 23.46 | | style="text-align:right;" | 23.46 | ||
|Lalawa | |||
| style="text-align:center;" | Pythagorean Comma | | style="text-align:center;" | Pythagorean Comma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 106: | Line 113: | ||
| {{Monzo| -4 4 -1 }} | | {{Monzo| -4 4 -1 }} | ||
| style="text-align:right;" | 21.51 | | style="text-align:right;" | 21.51 | ||
|Gu | |||
| style="text-align:center;" | Syntonic Comma | | style="text-align:center;" | Syntonic Comma | ||
| style="text-align:center;" | Didymos Comma | | style="text-align:center;" | Didymos Comma | ||
| Line 113: | Line 121: | ||
| {{Monzo| 0 -2 5 -3 }} | | {{Monzo| 0 -2 5 -3 }} | ||
| style="text-align:right;" | 21.18 | | style="text-align:right;" | 21.18 | ||
|Triru-aquinyo | |||
| style="text-align:center;" | Gariboh | | style="text-align:center;" | Gariboh | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 120: | Line 129: | ||
| {{Monzo| 11 -4 -2 }} | | {{Monzo| 11 -4 -2 }} | ||
| style="text-align:right;" | 19.55 | | style="text-align:right;" | 19.55 | ||
|Sagugu | |||
| style="text-align:center;" | Diaschisma | | style="text-align:center;" | Diaschisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 127: | Line 137: | ||
| {{Monzo| -1 -2 -1 1 0 1 }} | | {{Monzo| -1 -2 -1 1 0 1 }} | ||
| style="text-align:right;" | 19.13 | | style="text-align:right;" | 19.13 | ||
|Thozogu | |||
| style="text-align:center;" | Superleap | | style="text-align:center;" | Superleap | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 134: | Line 145: | ||
| {{Monzo| 26 -12 -3 }} | | {{Monzo| 26 -12 -3 }} | ||
| style="text-align:right;" | 17.60 | | style="text-align:right;" | 17.60 | ||
|Sasa-trigu | |||
| style="text-align:center;" | Misty Comma | | style="text-align:center;" | Misty Comma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 141: | Line 153: | ||
| {{Monzo| -1 2 0 -2 1 }} | | {{Monzo| -1 2 0 -2 1 }} | ||
| style="text-align:right;" | 17.58 | | style="text-align:right;" | 17.58 | ||
|Loruru | |||
| style="text-align:center;" | Mothwellsma | | style="text-align:center;" | Mothwellsma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 148: | Line 161: | ||
| {{Monzo| 2 -2 2 0 -1 }} | | {{Monzo| 2 -2 2 0 -1 }} | ||
| style="text-align:right;" | 17.40 | | style="text-align:right;" | 17.40 | ||
|Luyoyo | |||
| style="text-align:center;" | Ptolemisma | | style="text-align:center;" | Ptolemisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 155: | Line 169: | ||
| {{Monzo| 1 2 -3 1 }} | | {{Monzo| 1 2 -3 1 }} | ||
| style="text-align:right;" | 13.79 | | style="text-align:right;" | 13.79 | ||
|Zotrigu | |||
| style="text-align:center;" | Septimal Semicomma | | style="text-align:center;" | Septimal Semicomma | ||
| style="text-align:center;" | Starling Comma | | style="text-align:center;" | Starling Comma | ||
| Line 162: | Line 177: | ||
| {{Monzo| 5 -4 3 -2 }} | | {{Monzo| 5 -4 3 -2 }} | ||
| style="text-align:right;" | 13.47 | | style="text-align:right;" | 13.47 | ||
|Rurutriyo | |||
| style="text-align:center;" | Octagar | | style="text-align:center;" | Octagar | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 169: | Line 185: | ||
| {{Monzo| 4 0 -2 -1 1 }} | | {{Monzo| 4 0 -2 -1 1 }} | ||
| style="text-align:right;" | 9.86 | | style="text-align:right;" | 9.86 | ||
|Lorugugu | |||
| style="text-align:center;" | Valinorsma | | style="text-align:center;" | Valinorsma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 176: | Line 193: | ||
| {{Monzo| 7 -4 0 1 -1 }} | | {{Monzo| 7 -4 0 1 -1 }} | ||
| style="text-align:right;" | 9.69 | | style="text-align:right;" | 9.69 | ||
|Saluzo | |||
| style="text-align:center;" | Pentacircle | | style="text-align:center;" | Pentacircle | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 183: | Line 201: | ||
| {{Monzo| -9 8 -4 2 }} | | {{Monzo| -9 8 -4 2 }} | ||
| style="text-align:right;" | 8.04 | | style="text-align:right;" | 8.04 | ||
|Labizogugu | |||
| style="text-align:center;" | Varunisma | | style="text-align:center;" | Varunisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 190: | Line 209: | ||
| {{Monzo| -5 2 2 -1 }} | | {{Monzo| -5 2 2 -1 }} | ||
| style="text-align:right;" | 7.71 | | style="text-align:right;" | 7.71 | ||
|Ruyoyo | |||
| style="text-align:center;" | Septimal Kleisma | | style="text-align:center;" | Septimal Kleisma | ||
| style="text-align:center;" | Marvel Comma | | style="text-align:center;" | Marvel Comma | ||
| Line 197: | Line 217: | ||
| {{Monzo| 6 0 -5 2 }} | | {{Monzo| 6 0 -5 2 }} | ||
| style="text-align:right;" | 6.08 | | style="text-align:right;" | 6.08 | ||
|Zozoquingu | |||
| style="text-align:center;" | Hemimean | | style="text-align:center;" | Hemimean | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 204: | Line 225: | ||
| {{Monzo| 10 -6 1 -1 }} | | {{Monzo| 10 -6 1 -1 }} | ||
| style="text-align:right;" | 5.76 | | style="text-align:right;" | 5.76 | ||
|Saruyo | |||
| style="text-align:center;" | Hemifamity | | style="text-align:center;" | Hemifamity | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 211: | Line 233: | ||
| {{Monzo| -3 2 -1 2 -1 }} | | {{Monzo| -3 2 -1 2 -1 }} | ||
| style="text-align:right;" | 3.93 | | style="text-align:right;" | 3.93 | ||
|Luzozogu | |||
| style="text-align:center;" | Werckisma | | style="text-align:center;" | Werckisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 218: | Line 241: | ||
| {{Monzo| 25 -14 0 -1 }} | | {{Monzo| 25 -14 0 -1 }} | ||
| style="text-align:right;" | 3.80 | | style="text-align:right;" | 3.80 | ||
|Sasaru | |||
| style="text-align:center;" | Garischisma | | style="text-align:center;" | Garischisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 225: | Line 249: | ||
| {{Monzo| -15 8 1 }} | | {{Monzo| -15 8 1 }} | ||
| style="text-align:right;" | 1.95 | | style="text-align:right;" | 1.95 | ||
|Layo | |||
| style="text-align:center;" | Schisma | | style="text-align:center;" | Schisma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 232: | Line 257: | ||
| {{Monzo| -11 2 7 -3 }} | | {{Monzo| -11 2 7 -3 }} | ||
| style="text-align:right;" | 1.63 | | style="text-align:right;" | 1.63 | ||
|Latriru-asepyo | |||
| style="text-align:center;" | Meter | | style="text-align:center;" | Meter | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 239: | Line 265: | ||
| {{Monzo| -4 6 -6 3 }} | | {{Monzo| -4 6 -6 3 }} | ||
| style="text-align:right;" | 0.33 | | style="text-align:right;" | 0.33 | ||
|Trizogugu | |||
| style="text-align:center;" | Landscape Comma | | style="text-align:center;" | Landscape Comma | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| Line 246: | Line 273: | ||
| {{Monzo| -3 4 -2 -2 2 }} | | {{Monzo| -3 4 -2 -2 2 }} | ||
| style="text-align:right;" | 0.18 | | style="text-align:right;" | 0.18 | ||
|Bilorugu | |||
| style="text-align:center;" | Kalisma | | style="text-align:center;" | Kalisma | ||
| style="text-align:center;" | Gauss' Comma | | style="text-align:center;" | Gauss' Comma | ||
| Line 253: | Line 281: | ||
| {{Monzo| 161 -84 -12 }} | | {{Monzo| 161 -84 -12 }} | ||
| style="text-align:right;" | 0.02 | | style="text-align:right;" | 0.02 | ||
|Sepbisa-quadtrigu | |||
| style="text-align:center;" | Atom | | style="text-align:center;" | Atom | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
Revision as of 07:50, 13 December 2019
12edo, perhaps better known as 12et since it really is a temperament, is the predominating tuning system in the world today. It achieved that position because it is the smallest equal division of the octave (EDO) which can seriously claim to represent 5-limit harmony, and because as 1/12 Pythagorean comma (approximately 1/11 syntonic comma or full schisma) meantone, it represents meantone.
It divides the octave into twelve equal parts, each of exactly 100 cents each unless octave shrinking or stretching is employed. Its has a fifth which is quite good at two cents flat. It has a major third which is 13+2/3 cents sharp, which works well enough for some styles of music and is not really adequate for others, and a minor third which is flat by even more, 15+2/3 cents. It is probably not an accident that as tuning in European music became increasingly close to 12et, the style of the music changed so that the defects of 12et appeared less evident, though it should be borne in mind that in actual performance these are often reduced by the tuning adaptations of the performers.
The seventh partial (7/4) is "represented" by an interval which is sharp by over 31 cents, and stands out distinctly from the rest of the chord in a tetrad. Such tetrads are often being used as dominant seventh chords in functional harmony, for which the 5-limit JI version would be 1/1 - 5/4 - 3/2 - 16/9, and while 12et officially supports septimal meantone via the val <12 19 28 34|, its credentials in the 7-limit department are distinctly cheesy. It cannot be said to represent 11 or 13 at all, though it does a quite credible 17 and an even better 19. Nevertheless its relative tuning accuracy is quite high, and 12edo is the fourth zeta integral edo.
In terms of the kernel, which is to say the commas it tempers out, it tempers out the Pythagorean comma, 3^12/2^19, the Didymus comma, 81/80, the diesis, 128/125, the diaschisma, 2048/2025, the Archytas comma, 64/63, the septimal quartertone, 36/35, the jubilisma, 50/49, the septimal semicomma, 126/125, and the septimal kleisma, 225/224. Each of these affects the structure of 12et in specific ways, and tuning systems which share the comma in question will be similar to 12et in precisely those ways.
Rank two temperaments
- List of 12et rank two temperaments by badness
- List of 12et rank two temperaments by complexity
- List of edo-distinct 12f rank two temperaments
| Periods per octave |
Generator | Temperaments |
|---|---|---|
| 1 | 1\12 | Ripple |
| 1 | 5\12 | Meantone/dominant |
| 2 | 1\12 | Srutal/pajara/injera |
| 3 | 1\12 | Augmented |
| 4 | 1\12 | Diminished |
| 6 | 1\12 | Hexe |
Commas
12 EDO tempers out the following commas. (Note: This assumes val < 12 19 28 34 42 44 |.)
| Rational | Monzo | Cents | Color Name | Name 1 | Name 2 | Name 3 |
|---|---|---|---|---|---|---|
| 648/625 | [3 4 -4⟩ | 62.57 | Quadgu | Major Diesis | Diminished Comma | |
| 36/35 | [2 2 -1 -1⟩ | 48.77 | Rugu | Septimal Quarter Tone | ||
| 128/125 | [7 0 -3⟩ | 41.06 | Trigu | Diesis | Augmented Comma | |
| 50/49 | [1 0 2 -2⟩ | 34.98 | Biruyo | Tritonic Diesis | Jubilisma | |
| 64/63 | [6 -2 0 -1⟩ | 27.26 | Ru | Septimal Comma | Archytas' Comma | Leipziger Komma |
| 531441/524288 | [-19 12⟩ | 23.46 | Lalawa | Pythagorean Comma | ||
| 81/80 | [-4 4 -1⟩ | 21.51 | Gu | Syntonic Comma | Didymos Comma | Meantone Comma |
| 3125/3087 | [0 -2 5 -3⟩ | 21.18 | Triru-aquinyo | Gariboh | ||
| 2048/2025 | [11 -4 -2⟩ | 19.55 | Sagugu | Diaschisma | ||
| 91/90 | [-1 -2 -1 1 0 1⟩ | 19.13 | Thozogu | Superleap | ||
| 5201701/5149091 | [26 -12 -3⟩ | 17.60 | Sasa-trigu | Misty Comma | ||
| 99/98 | [-1 2 0 -2 1⟩ | 17.58 | Loruru | Mothwellsma | ||
| 100/99 | [2 -2 2 0 -1⟩ | 17.40 | Luyoyo | Ptolemisma | ||
| 126/125 | [1 2 -3 1⟩ | 13.79 | Zotrigu | Septimal Semicomma | Starling Comma | |
| 4000/3969 | [5 -4 3 -2⟩ | 13.47 | Rurutriyo | Octagar | ||
| 176/175 | [4 0 -2 -1 1⟩ | 9.86 | Lorugugu | Valinorsma | ||
| 896/891 | [7 -4 0 1 -1⟩ | 9.69 | Saluzo | Pentacircle | ||
| 321489/320000 | [-9 8 -4 2⟩ | 8.04 | Labizogugu | Varunisma | ||
| 225/224 | [-5 2 2 -1⟩ | 7.71 | Ruyoyo | Septimal Kleisma | Marvel Comma | |
| 3136/3125 | [6 0 -5 2⟩ | 6.08 | Zozoquingu | Hemimean | ||
| 5120/5103 | [10 -6 1 -1⟩ | 5.76 | Saruyo | Hemifamity | ||
| 441/440 | [-3 2 -1 2 -1⟩ | 3.93 | Luzozogu | Werckisma | ||
| 33554432/33480783 | [25 -14 0 -1⟩ | 3.80 | Sasaru | Garischisma | ||
| 32805/32768 | [-15 8 1⟩ | 1.95 | Layo | Schisma | ||
| 703125/702464 | [-11 2 7 -3⟩ | 1.63 | Latriru-asepyo | Meter | ||
| 250047/250000 | [-4 6 -6 3⟩ | 0.33 | Trizogugu | Landscape Comma | ||
| 9801/9800 | [-3 4 -2 -2 2⟩ | 0.18 | Bilorugu | Kalisma | Gauss' Comma | |
| [161 -84 -12⟩ | 0.02 | Sepbisa-quadtrigu | Atom |
Scales
Intervals
An expanded version of the above, including some higher-limit intervals:
Selected just intervals by error
The following table shows how some prominent just intervals are represented in 12edo (ordered by absolute error).
| Interval, complement | Error (abs., in cents) |
|---|---|
| 4/3, 3/2 | 1.955 |
| 15/13, 26/15 | 2.259 |
| 9/8, 16/9 | 3.910 |
| 13/11, 22/13 | 10.790 |
| 16/15, 15/8 | 11.731 |
| 5/4, 8/5 | 13.686 |
| 6/5, 5/3 | 15.641 |
| 7/5, 10/7 | 17.488 |
| 14/11, 11/7 | 17.508 |
| 10/9, 9/5 | 17.596 |
| 15/14, 28/15 | 19.443 |
| 14/13, 13/7 | 28.298 |
| 8/7, 7/4 | 31.174 |
| 7/6, 12/7 | 33.129 |
| 11/10, 20/11 | 34.996 |
| 9/7, 14/9 | 35.084 |
| 18/13, 13/9 | 36.618 |
| 15/11, 22/15 | 36.951 |
| 13/12, 24/13 | 38.573 |
| 16/13, 13/8 | 40.528 |
| 13/10, 20/13 | 45.786 |
| 11/9, 18/11 | 47.408 |
| 11/8, 16/11 | 48.682 |
| 12/11, 11/6 | 49.323 |