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Diaschismic family: Difference between revisions

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= Srutal (12&34, aka diaschismic) =
= Srutal (12&34, aka diaschismic) =
Subgroup: 2.3.5


[[Comma list]]: 2048/2025
[[Comma list]]: 2048/2025
Line 31: Line 33:


= Srutal =
= Srutal =
Subgroup: 2.3.5.7
[[Comma list]]: 2048/2025, 4375/4374
[[Comma list]]: 2048/2025, 4375/4374


Line 44: Line 48:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 176/175, 896/891, 1331/1323
Comma list: 176/175, 896/891, 1331/1323


Line 51: Line 57:
* strict range: ~3/2 = [704.348, 705.882]
* strict range: ~3/2 = [704.348, 705.882]


Mapping: [{{val|2 0 11 -42 -28}}, {{val|0 1 -2 15 11}}]
Mapping: [{{val| 2 0 11 -42 -28 }}, {{val| 0 1 -2 15 11 }}]


POTE generator: ~3/2 = 704.856
POTE generator: ~3/2 = 704.856
Line 60: Line 66:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 176/175, 325/324, 364/363
Comma list: 169/168, 176/175, 325/324, 364/363


Line 67: Line 75:
* strict range: ~3/2 = [704.348, 705.882]
* strict range: ~3/2 = [704.348, 705.882]


Mapping: [{{val|2 0 11 -42 -28 -18}}, {{val|0 1 -2 15 11 8}}]
Mapping: [{{val| 2 0 11 -42 -28 -18 }}, {{val| 0 1 -2 15 11 8 }}]


POTE generator: ~3/2 = 704.881
POTE generator: ~3/2 = 704.881
Line 83: Line 91:
{{main| Pajara }}
{{main| Pajara }}


Pajara, with wedgie {{multival|2 -4 -4 -11 -12 2}} is closely associated with 22et (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22et, 34 with the val {{val|34 54 79 96}} and 56 with the val {{val|56 89 130 158}} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12et and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.
Pajara is closely associated with 22edo (not to mention [[Paul Erlich]]) but other tunings are possible. The 1/2 octave period serves as both a [[10/7]] and a [[7/5]]. Aside from 22edo, 34 with the val {{val| 34 54 79 96 }} and 56 with the val {{val| 56 89 130 158 }} are are interesting alternatives, with more accpetable fifths, and a tetrad which is more clearly a dominant seventh. As such, they are closer to the tuning of 12edo and of common practice Western music in general, while retaining the distictiveness of a sharp fifth.


Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Pajara extends nicely to an 11-limit version, for which the 56 tuning can be used, but a good alternative is to make the major thirds pure by setting the fifth to be 706.843 cents. Now 99/98, 100/99, 176/175 and 896/891 are being tempered out.
Subgroup: 2.3.5.7


[[Comma list]]: 50/49, 64/63
[[Comma list]]: 50/49, 64/63


[[Mapping]]: [{{val|2 0 11 12}}, {{val|0 1 -2 -2}}]
[[Mapping]]: [{{val| 2 0 11 12 }}, {{val| 0 1 -2 -2 }}]
 
{{Multival|legend=1| 2 -4 -4 -11 -12 2 }}


[[POTE generator]]: ~3/2 = 707.048
[[POTE generator]]: ~3/2 = 707.048
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== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 50/49, 64/63, 99/98
Comma list: 50/49, 64/63, 99/98


Mapping: [{{val|2 0 11 12 26}}, {{val|0 1 -2 -2 -6}}]
Mapping: [{{val| 2 0 11 12 26 }}, {{val| 0 1 -2 -2 -6 }}]


POTE generator: ~3/2 = 706.885
POTE generator: ~3/2 = 706.885
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===  13-limit ===
===  13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 64/63, 65/63, 99/98
Comma list: 50/49, 64/63, 65/63, 99/98


Mapping: [{{val|2 0 11 12 26 1}}, {{val|0 1 -2 -2 -6 2}}]
Mapping: [{{val| 2 0 11 12 26 1 }}, {{val| 0 1 -2 -2 -6 2 }}]


POTE generator: ~3/2 = 708.919
POTE generator: ~3/2 = 708.919
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== Pajarous ==
== Pajarous ==
Subgroup: 2.3.5.7.11
Comma list: 50/49, 55/54, 64/63
Comma list: 50/49, 55/54, 64/63


Mapping: [{{val|2 0 11 12 -9}}, {{val|0 1 -2 -2 5}}]
Mapping: [{{val| 2 0 11 12 -9 }}, {{val| 0 1 -2 -2 5 }}]


POTE generator: ~3/2 = 709.578
POTE generator: ~3/2 = 709.578
Line 151: Line 169:


=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 50/49, 55/54, 64/63, 65/63
Comma list: 50/49, 55/54, 64/63, 65/63


Mapping: [{{val|2 0 11 12 -9 1}}, {{val|0 1 -2 -2 5 2}}]
Mapping: [{{val| 2 0 11 12 -9 1 }}, {{val| 0 1 -2 -2 5 2 }}]


POTE generator: ~3/2 = 710.240
POTE generator: ~3/2 = 710.240
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=== Pajaro ===
=== Pajaro ===
Subgroup: 2.3.5.7.11.13
Comma list: 40/39, 50/49, 55/54, 64/63
Comma list: 40/39, 50/49, 55/54, 64/63


Mapping: [{{val|2 0 11 12 -9 17}}, {{val|0 1 -2 -2 5 -3}}]
Mapping: [{{val| 2 0 11 12 -9 17 }}, {{val| 0 1 -2 -2 5 -3 }}]


POTE generator ~3/2 = 710.818
POTE generator ~3/2 = 710.818
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== Pajaric ==
== Pajaric ==
Subgroup: 2.3.5.7.11
Comma list: 45/44, 50/49, 56/55
Comma list: 45/44, 50/49, 56/55


Mapping: [{{val|2 0 11 12 7}}, {{val|0 1 -2 -2 0}}]
Mapping: [{{val| 2 0 11 12 7 }}, {{val| 0 1 -2 -2 0 }}]


POTE generator: ~3/2 = 705.524
POTE generator: ~3/2 = 705.524
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=== 13-limit ===
=== 13-limit ===
Subgroup: 2.3.5.7.11.13
Comma list: 40/39, 45/44, 50/49, 56/55
Comma list: 40/39, 45/44, 50/49, 56/55


Mapping: [{{val|2 0 11 12 7 17}}, {{val|0 1 -2 -2 0 -3}}]
Mapping: [{{val| 2 0 11 12 7 17 }}, {{val| 0 1 -2 -2 0 -3 }}]


POTE generator: ~3/2 = 707.442
POTE generator: ~3/2 = 707.442
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== Hemipaj ==
== Hemipaj ==
Commas: 50/49, 64/63, 121/120
Subgroup: 2.3.5.7.11


Mapping: [{{val|2 1 9 10 8}}, {{val|0 2 -4 -4 -1}}]
Comma list: 50/49, 64/63, 121/120
 
Mapping: [{{val| 2 1 9 10 8 }}, {{val| 0 2 -4 -4 -1 }}]


POTE generator: ~11/8 = 546.383
POTE generator: ~11/8 = 546.383
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= Diaschismic =
= Diaschismic =
A simpler characterization than the one given by the normal comma list is that diaschismic adds 126/125 or 5120/5103 to the set of commas, and it can also be called 46&58. However described, diaschismic has wedgie {{multival|2 -4 -16 -11 -31 -26}}, with a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo|58EDO]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58et.
A simpler characterization than the one given by the normal comma list is that diaschismic adds [[126/125]] or [[5120/5103]] to the set of commas, and it can also be called 46&58. However described, diaschismic has a 1/2 period and a sharp fifth generator like pajara, but not so sharp, giving a more accurate but more complex temperament. [[58edo]] provides an excellent tuning, but an alternative is to make [[7/4]] just by making the fifth 703.897 cents, as opposed to 703.448 cents for 58edo.


Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Diaschismic extends naturally to the 17-limit, for which the same tunings may be used, making it one of the most important of the higher limit rank two temperaments. Adding the 11-limit adds the commas 176/175, 896/891 and 441/440. The 13-limit yields 196/195, 351/350, and 364/363; the 17-limit adds 136/135, 221/220, and 442/441. If you want to explore higher limit harmonies, diaschismic is certainly one excellent way to do it; MOS of 34 notes and even more the 46 note MOS will encompass very great deal of it. Of course 46 or 58 equal provide alternatives which in many ways are similar, particularly in the case of 58.
Subgroup: 2.3.5.7


[[Comma list]]: 126/125, 2048/2025
[[Comma list]]: 126/125, 2048/2025


[[Mapping]]: [{{val|2 0 11 31}}, {{val|0 1 -2 -8}}]
[[Mapping]]: [{{val| 2 0 11 31 }}, {{val| 0 1 -2 -8 }}]
 
{{Multival|legend=1| 2 -4 -16 -11 -31 -26 }}


[[POTE generator]]: ~3/2 = 703.681
[[POTE generator]]: ~3/2 = 703.681
Line 221: Line 253:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 896/891
Comma list: 126/125, 176/175, 896/891


Mapping: [{{val|2 0 11 31 45}}, {{val|0 1 -2 -8 -12}}]
Mapping: [{{val| 2 0 11 31 45 }}, {{val| 0 1 -2 -8 -12 }}]


POTE generator: ~3/2 = 703.714
POTE generator: ~3/2 = 703.714
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== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 364/363, 2048/2025
Comma list: 126/125, 196/195, 364/363, 2048/2025


Mapping: [{{val|2 0 11 31 45 55}}, {{val|0 1 -2 -8 -12 -15}}]
Mapping: [{{val| 2 0 11 31 45 55 }}, {{val| 0 1 -2 -8 -12 -15 }}]


POTE generator: ~3/2 = 703.704
POTE generator: ~3/2 = 703.704
Line 243: Line 279:


== 17-limit (Na"Naa') ==
== 17-limit (Na"Naa') ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255
Comma list: 126/125, 136/135, 176/175, 196/195, 256/255


Mapping: [{{val|2 0 11 31 45 55 5}}, {{val|0 1 -2 -8 -12 -15 1}}]
Mapping: [{{val| 2 0 11 31 45 55 5 }}, {{val| 0 1 -2 -8 -12 -15 1 }}]


POTE generator: ~3/2 = 703.812
POTE generator: ~3/2 = 703.812
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= Keen =
= Keen =
Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie {{multival|2 -4 18 -11 23 53}}. It may also be described as the 22&56 temperament. [[78edo|78et]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival|2 -4 18 -12 ...}}, is really more interesting, adding 100/99 and 385/384 to the commas.
Keen adds 875/864 as well as 2240/2187 to the set of commas. It may also be described as the 22&56 temperament. [[78edo]] is a good tuning choice, and remains a good one in the 11-limit, where keen, {{multival|2 -4 18 -12 … }}, is really more interesting, adding 100/99 and 385/384 to the commas.
 
Subgroup: 2.3.5.7


[[Comma list]]: 875/864, 2048/2025
[[Comma list]]: 875/864, 2048/2025


[[Mapping]]: [{{val|2 0 11 -23}}, {{val|0 1 -2 9}}]
[[Mapping]]: [{{val|2 0 11 -23}}, {{val|0 1 -2 9}}]
{{Multival|legend=1| 2 -4 18 -11 23 53 }}


[[POTE generator]]: ~3/2 = 707.571
[[POTE generator]]: ~3/2 = 707.571
Line 267: Line 309:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 100/99, 385/384, 1232/1215
Comma list: 100/99, 385/384, 1232/1215


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= Bidia =
= Bidia =
Bidia adds 3136/3125 to the commas, splitting the period into 1/4 octave. It may be called the 12&56 temperament.
Bidia adds [[3136/3125]] to the commas, splitting the period into 1/4 octave. It may be called the 12&56 temperament.
 
Subgroup: 2.3.5.7


[[Comma list]]: 2048/2025, 3136/3125
[[Comma list]]: 2048/2025, 3136/3125
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== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 176/175, 896/891, 1375/1372
Comma list: 176/175, 896/891, 1375/1372


Line 304: Line 352:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 325/324, 640/637, 896/891
Comma list: 176/175, 325/324, 640/637, 896/891


Line 315: Line 365:


== 17-limit ==
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637
Comma list: 136/135, 176/175, 256/255, 325/324, 640/637


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= Echidna =
= Echidna =
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie {{multival|6 -12 10 -33 -1 57}} and may be called the 22&58 temperament. [[58edo|58et]] or [[80edo|80et]] make for good tunings, or their vals can be add to {{val|138 219 321 388}}.
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It may be called the 22&58 temperament. [[58edo]] or [[80edo]] make for good tunings, or their vals can be add to {{val| 138 219 321 388 }}.


Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.
Echidna becomes more interesting when extended to be an 11-limit temperament by adding 176/175, 896/891 or 540/539 to the commas, where the same tunings can be used as before. It then is able to represent the entire 11-limit diamond to within about six cents of error, within a compass of 24 notes. The 28 note 2MOS gives scope for this, and the 36 note MOS much more.
Subgroup: 2.3.5.7


[[Comma list]]: 1728/1715, 2048/2025
[[Comma list]]: 1728/1715, 2048/2025


[[Mapping]]: [{{val|2 1 9 2}}, {{val|0 3 -6 5}}]
[[Mapping]]: [{{val|2 1 9 2}}, {{val|0 3 -6 5}}]
{{Multival|legend=1| 6 -12 10 -33 -1 57 }}


[[POTE generator]]: ~9/7 = 434.856
[[POTE generator]]: ~9/7 = 434.856
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== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 176/175, 540/539, 896/891
Comma list: 176/175, 540/539, 896/891


Line 358: Line 416:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 176/175, 351/350, 364/363, 540/539
Comma list: 176/175, 351/350, 364/363, 540/539


Line 369: Line 429:


== 17-limit ==
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539
Comma list: 136/135, 176/175, 221/220, 256/255, 540/539


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= Echidnic =
= Echidnic =
Subgroup: 2.3.5.7
[[Comma list]]: 686/675, 1029/1024
[[Comma list]]: 686/675, 1029/1024


Line 391: Line 455:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 385/384, 441/440, 686/675
Comma list: 385/384, 441/440, 686/675


Line 402: Line 468:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 385/384, 441/440
Comma list: 91/90, 169/168, 385/384, 441/440


Line 413: Line 481:


== 17-limit ==
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255
Comma list: 91/90, 136/135, 154/153, 169/168, 256/255


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= Shrutar =
= Shrutar =
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie &lt;&lt;4 -8 14 -22 11 55||, it can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7s just.
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&amp;46. Its generator can be taken as either 36/35 or 35/24; the latter is interesting since along with 15/14 and 21/20, it connects opposite sides of a hexany. [[68edo]] makes for a good tuning, but another and excellent choice is a generator of 14<sup>(1/7)</sup>, making 7s just.


By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
By adding 121/120 or 176/175 to the commas, shrutar can be extended to the 11-limit, which loses a bit of accuracy, but picks up low-complexity 11-limit harmony, making shrutar quite an interesting 11-limit system. 68, 114 or a 14<sup>(1/7)</sup> generator can again be used as tunings.
Subgroup: 2.3.5.7


[[Comma list]]: 245/243, 2048/2025
[[Comma list]]: 245/243, 2048/2025


[[Mapping]]: [{{val| 2 1 9 -2 }}, {{val| 0 2 -4 7 }}]
[[Mapping]]: [{{val| 2 1 9 -2 }}, {{val| 0 2 -4 7 }}]
{{Multival|legend=1| 4 -8 14 -22 11 55 }}


[[POTE generator]]: ~36/35 = 52.811
[[POTE generator]]: ~36/35 = 52.811
Line 440: Line 514:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 121/120, 176/175, 245/243
Comma list: 121/120, 176/175, 245/243


Line 449: Line 525:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 176/175, 196/195, 245/243
Comma list: 121/120, 176/175, 196/195, 245/243


Line 460: Line 538:


== 17-limit ==
== 17-limit ==
Subgroup: 2.3.5.7.11.13.17
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195


Line 471: Line 551:


== 19-limit ==
== 19-limit ==
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342
Comma list: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342


Line 482: Line 564:


= Sruti =
= Sruti =
Subgroup: 2.3.5.7
[[Comma list]]: 2048/2025, 19683/19600
[[Comma list]]: 2048/2025, 19683/19600


Line 495: Line 579:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 176/175, 243/242, 896/891
Comma list: 176/175, 243/242, 896/891


Line 506: Line 592:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 144/143, 176/175, 351/350, 676/675
Comma list: 144/143, 176/175, 351/350, 676/675


Line 517: Line 605:


= Anguirus =
= Anguirus =
Subgroup: 2.3.5.7
[[Comma list]]: 49/48, 2048/2025
[[Comma list]]: 49/48, 2048/2025


Line 530: Line 620:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 49/48, 56/55, 243/242
Comma list: 49/48, 56/55, 243/242


Line 541: Line 633:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 56/55, 91/90, 352/351
Comma list: 49/48, 56/55, 91/90, 352/351


Line 552: Line 646:


= Shru =
= Shru =
Subgroup: 2.3.5.7
[[Comma list]]: 392/375, 1323/1280
[[Comma list]]: 392/375, 1323/1280


Line 565: Line 661:


== 11-limit ==
== 11-limit ==
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 1323/1280
Comma list: 56/55, 77/75, 1323/1280


Line 576: Line 674:


== 13-limit ==
== 13-limit ==
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 77/75, 105/104, 507/500
Comma list: 56/55, 77/75, 105/104, 507/500


Line 586: Line 686:
Badness: 0.0457
Badness: 0.0457


[[Category:Theory]]
[[Category:Regular temperament theory]]
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Diaschismic family| ]] <!-- main article -->
[[Category:Temperament family]]
[[Category:Temperament family]]
[[Category:Rank 2]]
[[Category:Rank 2]]


[[Category:Todo:Add definition]]
[[Category:Todo:Review]]
[[Category:Todo:Review]]
Retrieved from "https://en.xen.wiki/w/Diaschismic_family"