12edo: Difference between revisions
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<span style="display: block; text-align: right;"> | <span style="display: block; text-align: right;">[[12平均律|日本語]]</span> | ||
12[[ | 12[[EDO]], 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 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. | 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 ([[ | 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 [[Vals and Tuning Space|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 [[The Riemann Zeta Function and Tuning#Zeta EDO lists|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. | 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== | ==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]] | ||
[[ | |||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! Periods <br/> per octave | ||
! Generator | |||
per octave | ! Temperaments | ||
! | |||
! | |||
|- | |- | ||
| 1 | |||
| 1\12 | |||
| [[Ripple|Ripple]] | |||
|- | |- | ||
| 1 | |||
| 5\12 | |||
| [[Meantone|Meantone]]/[[dominant|dominant]] | |||
|- | |- | ||
| 2 | |||
| 1\12 | |||
| [[Srutal|Srutal]]/[[pajara|pajara]]/[[Injera|injera]] | |||
|- | |- | ||
| 3 | |||
| 1\12 | |||
| [[augmented|Augmented]] | |||
|- | |- | ||
| 4 | |||
| 1\12 | |||
| [[Diminished|Diminished]] | |||
|- | |- | ||
| 6 | |||
| 1\12 | |||
| [[Hexe|Hexe]] | |||
|} | |} | ||
| Line 54: | Line 50: | ||
{| class="wikitable" | {| class="wikitable" | ||
|- | |- | ||
! | ! Rational | ||
! | ! Monzo | ||
! | ! Size (Cents) | ||
! | ! Name 1 | ||
! | ! Name 2 | ||
! | ! Name 3 | ||
|- | |- | ||
| style="text-align:center;" | 531441/524288 | | style="text-align:center;" | 531441/524288 | ||
| | | {{Monzo| -19 12 }} | ||
| style="text-align:right;" | 23.46 | | style="text-align:right;" | 23.46 | ||
| style="text-align:center;" | Pythagorean Comma | | style="text-align:center;" | Pythagorean Comma | ||
| Line 69: | Line 65: | ||
|- | |- | ||
| style="text-align:center;" | 648/625 | | style="text-align:center;" | 648/625 | ||
| | | {{Monzo| 3 4 -4 }} | ||
| style="text-align:right;" | 62.57 | | style="text-align:right;" | 62.57 | ||
| style="text-align:center;" | Major Diesis | | style="text-align:center;" | Major Diesis | ||
| Line 76: | Line 72: | ||
|- | |- | ||
| style="text-align:center;" | 128/125 | | style="text-align:center;" | 128/125 | ||
| | | {{Monzo| 7 0 -3 }} | ||
| style="text-align:right;" | 41.06 | | style="text-align:right;" | 41.06 | ||
| style="text-align:center;" | Diesis | | style="text-align:center;" | Diesis | ||
| Line 83: | Line 79: | ||
|- | |- | ||
| style="text-align:center;" | 81/80 | | style="text-align:center;" | 81/80 | ||
| | | {{Monzo| -4 4 -1 }} | ||
| style="text-align:right;" | 21.51 | | style="text-align:right;" | 21.51 | ||
| style="text-align:center;" | Syntonic Comma | | style="text-align:center;" | Syntonic Comma | ||
| Line 90: | Line 86: | ||
|- | |- | ||
| style="text-align:center;" | 2048/2025 | | style="text-align:center;" | 2048/2025 | ||
| | | {{Monzo| 11 -4 -2 }} | ||
| style="text-align:right;" | 19.55 | | style="text-align:right;" | 19.55 | ||
| style="text-align:center;" | Diaschisma | | style="text-align:center;" | Diaschisma | ||
| Line 97: | Line 93: | ||
|- | |- | ||
| style="text-align:center;" | 5201701/5149091 | | style="text-align:center;" | 5201701/5149091 | ||
| | | {{Monzo| 26 -12 -3 }} | ||
| style="text-align:right;" | 17.60 | | style="text-align:right;" | 17.60 | ||
| style="text-align:center;" | Misty Comma | | style="text-align:center;" | Misty Comma | ||
| Line 104: | Line 100: | ||
|- | |- | ||
| style="text-align:center;" | 32805/32768 | | style="text-align:center;" | 32805/32768 | ||
| | | {{Monzo| -15 8 1 }} | ||
| style="text-align:right;" | 1.95 | | style="text-align:right;" | 1.95 | ||
| style="text-align:center;" | Schisma | | style="text-align:center;" | Schisma | ||
| Line 111: | Line 107: | ||
|- | |- | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
| | | {{Monzo| 161 -84 -12 }} | ||
| style="text-align:right;" | 0.02 | | style="text-align:right;" | 0.02 | ||
| style="text-align:center;" | Atom | | style="text-align:center;" | Atom | ||
| Line 118: | Line 114: | ||
|- | |- | ||
| style="text-align:center;" | 36/35 | | style="text-align:center;" | 36/35 | ||
| | | {{Monzo| 2 2 -1 -1 }} | ||
| style="text-align:right;" | 48.77 | | style="text-align:right;" | 48.77 | ||
| style="text-align:center;" | Septimal Quarter Tone | | style="text-align:center;" | Septimal Quarter Tone | ||
| Line 125: | Line 121: | ||
|- | |- | ||
| style="text-align:center;" | 50/49 | | style="text-align:center;" | 50/49 | ||
| | | {{Monzo| 1 0 2 -2 }} | ||
| style="text-align:right;" | 34.98 | | style="text-align:right;" | 34.98 | ||
| style="text-align:center;" | Tritonic Diesis | | style="text-align:center;" | Tritonic Diesis | ||
| Line 132: | Line 128: | ||
|- | |- | ||
| style="text-align:center;" | 64/63 | | style="text-align:center;" | 64/63 | ||
| | | {{Monzo| 6 -2 0 -1 }} | ||
| style="text-align:right;" | 27.26 | | style="text-align:right;" | 27.26 | ||
| style="text-align:center;" | Septimal Comma | | style="text-align:center;" | Septimal Comma | ||
| Line 139: | Line 135: | ||
|- | |- | ||
| style="text-align:center;" | 3125/3087 | | style="text-align:center;" | 3125/3087 | ||
| | | {{Monzo| 0 -2 5 -3 }} | ||
| style="text-align:right;" | 21.18 | | style="text-align:right;" | 21.18 | ||
| style="text-align:center;" | Gariboh | | style="text-align:center;" | Gariboh | ||
| Line 146: | Line 142: | ||
|- | |- | ||
| style="text-align:center;" | 126/125 | | style="text-align:center;" | 126/125 | ||
| | | {{Monzo| 1 2 -3 1 }} | ||
| style="text-align:right;" | 13.79 | | style="text-align:right;" | 13.79 | ||
| style="text-align:center;" | Septimal Semicomma | | style="text-align:center;" | Septimal Semicomma | ||
| Line 153: | Line 149: | ||
|- | |- | ||
| style="text-align:center;" | 4000/3969 | | style="text-align:center;" | 4000/3969 | ||
| | | {{Monzo| 5 -4 3 -2 }} | ||
| style="text-align:right;" | 13.47 | | style="text-align:right;" | 13.47 | ||
| style="text-align:center;" | Octagar | | style="text-align:center;" | Octagar | ||
| Line 160: | Line 156: | ||
|- | |- | ||
| style="text-align:center;" | 321489/320000 | | style="text-align:center;" | 321489/320000 | ||
| | | {{Monzo| -9 8 -4 2 }} | ||
| style="text-align:right;" | 8.04 | | style="text-align:right;" | 8.04 | ||
| style="text-align:center;" | Varunisma | | style="text-align:center;" | Varunisma | ||
| Line 167: | Line 163: | ||
|- | |- | ||
| style="text-align:center;" | 225/224 | | style="text-align:center;" | 225/224 | ||
| | | {{Monzo| -5 2 2 -1 }} | ||
| style="text-align:right;" | 7.71 | | style="text-align:right;" | 7.71 | ||
| style="text-align:center;" | Septimal Kleisma | | style="text-align:center;" | Septimal Kleisma | ||
| Line 174: | Line 170: | ||
|- | |- | ||
| style="text-align:center;" | 3136/3125 | | style="text-align:center;" | 3136/3125 | ||
| | | {{Monzo| 6 0 -5 2 }} | ||
| style="text-align:right;" | 6.08 | | style="text-align:right;" | 6.08 | ||
| style="text-align:center;" | Hemimean | | style="text-align:center;" | Hemimean | ||
| Line 181: | Line 177: | ||
|- | |- | ||
| style="text-align:center;" | 5120/5103 | | style="text-align:center;" | 5120/5103 | ||
| | | {{Monzo| 10 -6 1 -1 }} | ||
| style="text-align:right;" | 5.76 | | style="text-align:right;" | 5.76 | ||
| style="text-align:center;" | Hemifamity | | style="text-align:center;" | Hemifamity | ||
| Line 188: | Line 184: | ||
|- | |- | ||
| style="text-align:center;" | 33554432/33480783 | | style="text-align:center;" | 33554432/33480783 | ||
| | | {{Monzo| 25 -14 0 -1 }} | ||
| style="text-align:right;" | 3.80 | | style="text-align:right;" | 3.80 | ||
| style="text-align:center;" | Garischisma | | style="text-align:center;" | Garischisma | ||
| Line 195: | Line 191: | ||
|- | |- | ||
| style="text-align:center;" | 703125/702464 | | style="text-align:center;" | 703125/702464 | ||
| | | {{Monzo| -11 2 7 -3 }} | ||
| style="text-align:right;" | 1.63 | | style="text-align:right;" | 1.63 | ||
| style="text-align:center;" | Meter | | style="text-align:center;" | Meter | ||
| Line 202: | Line 198: | ||
|- | |- | ||
| style="text-align:center;" | 250047/250000 | | style="text-align:center;" | 250047/250000 | ||
| | | {{Monzo| -4 6 -6 3 }} | ||
| style="text-align:right;" | 0.33 | | style="text-align:right;" | 0.33 | ||
| style="text-align:center;" | Landscape Comma | | style="text-align:center;" | Landscape Comma | ||
| Line 209: | Line 205: | ||
|- | |- | ||
| style="text-align:center;" | 99/98 | | style="text-align:center;" | 99/98 | ||
| | | {{Monzo| -1 2 0 -2 1 }} | ||
| style="text-align:right;" | 17.58 | | style="text-align:right;" | 17.58 | ||
| style="text-align:center;" | Mothwellsma | | style="text-align:center;" | Mothwellsma | ||
| Line 216: | Line 212: | ||
|- | |- | ||
| style="text-align:center;" | 100/99 | | style="text-align:center;" | 100/99 | ||
| | | {{Monzo| 2 -2 2 0 -1 }} | ||
| style="text-align:right;" | 17.40 | | style="text-align:right;" | 17.40 | ||
| style="text-align:center;" | Ptolemisma | | style="text-align:center;" | Ptolemisma | ||
| Line 223: | Line 219: | ||
|- | |- | ||
| style="text-align:center;" | 176/175 | | style="text-align:center;" | 176/175 | ||
| | | {{Monzo| 4 0 -2 -1 1 }} | ||
| style="text-align:right;" | 9.86 | | style="text-align:right;" | 9.86 | ||
| style="text-align:center;" | Valinorsma | | style="text-align:center;" | Valinorsma | ||
| Line 230: | Line 226: | ||
|- | |- | ||
| style="text-align:center;" | 896/891 | | style="text-align:center;" | 896/891 | ||
| | | {{Monzo| 7 -4 0 1 -1 }} | ||
| style="text-align:right;" | 9.69 | | style="text-align:right;" | 9.69 | ||
| style="text-align:center;" | Pentacircle | | style="text-align:center;" | Pentacircle | ||
| Line 237: | Line 233: | ||
|- | |- | ||
| style="text-align:center;" | 441/440 | | style="text-align:center;" | 441/440 | ||
| | | {{Monzo| -3 2 -1 2 -1 }} | ||
| style="text-align:right;" | 3.93 | | style="text-align:right;" | 3.93 | ||
| style="text-align:center;" | Werckisma | | style="text-align:center;" | Werckisma | ||
| Line 244: | Line 240: | ||
|- | |- | ||
| style="text-align:center;" | 9801/9800 | | style="text-align:center;" | 9801/9800 | ||
| | | {{Monzo| -3 4 -2 -2 2 }} | ||
| style="text-align:right;" | 0.18 | | style="text-align:right;" | 0.18 | ||
| style="text-align:center;" | Kalisma | | style="text-align:center;" | Kalisma | ||
| Line 251: | Line 247: | ||
|- | |- | ||
| style="text-align:center;" | 91/90 | | style="text-align:center;" | 91/90 | ||
| | | {{Monzo| -1 -2 -1 1 0 1 }} | ||
| style="text-align:right;" | 19.13 | | style="text-align:right;" | 19.13 | ||
| style="text-align:center;" | Superleap | | style="text-align:center;" | Superleap | ||
| Line 259: | Line 255: | ||
=Scales= | =Scales= | ||
* [[MOS in 12edo]] | |||
=Intervals= | =Intervals= | ||
| Line 271: | Line 267: | ||
[[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]] | [[File:12ed2-19-001e.svg|alt=alt : Your browser has no SVG support.]] | ||
[[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]] | [[:File:12ed2-19-001e.svg|12ed2-19-001e.svg]] | ||
[[Category:12edo]] | |||
[[Category:edo]] | [[Category:edo]] | ||
[[Category:meantone]] | [[Category:meantone]] | ||
| Line 280: | Line 278: | ||
The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error). | The following table shows how [[Just-24|some prominent just intervals]] are represented in 12edo (ordered by absolute error). | ||
{| class="wikitable" | {| class="wikitable" style="text-align:center;" | ||
|- | |- | ||
! Interval, complement | |||
! Error (abs., in [[cent|cents]]) | |||
|- | |- | ||
| [[4/3]], [[3/2]] | |||
| 1.955 | |||
|- | |- | ||
| [[9/8]], [[16/9]] | |||
| 3.910 | |||
|- | |- | ||
| [[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 | |||
|- | |- | ||
| [[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 | |||
|- | |- | ||
| [[11/8]], [[16/11]] | |||
| 48.682 | |||
|- | |- | ||
| [[12/11]], [[11/6]] | |||
| 50.637 | |||
|- | |- | ||
| [[15/13]], [[26/15]] | |||
| 52.259 | |||
|- | |- | ||
| [[11/9]], [[18/11]] | |||
| 52.592 | |||
|- | |- | ||
| [[13/10]], [[20/13]] | |||
| 54.214 | |||
|- | |- | ||
| [[14/13]], [[13/7]] | |||
| 71.702 | |||
|- | |- | ||
| [[13/11]], [[22/13]] | |||
| 89.210 | |||
|} | |} | ||
[[Category:edo]] | |||
<!-- interwiki --> | |||
[[de:12edo]] | |||
[[es:12 EDO]] | |||