Diaschismic family: Difference between revisions
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The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. | The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[diaschisma]]. Its monzo is {{monzo| 11 -4 -2 }}, and flipping that yields {{multival| 2 -4 -11 }} for the wedgie for 5-limit '''diaschismic''', or '''srutal''', temperament. This tells us the period is half an octave, the [[Wikipedia: Greatest common divisor|GCD]] of 2 and -4, and that the generator is a fifth. Three periods gives 1800 cents, and decreasing this by two fifths gives the major third. [[34edo]] is a good tuning choice, with [[46edo]], [[56edo]], [[58edo]] or [[80edo]] being other possibilities. Both [[12edo]] and [[22edo]] support it, and retuning them to a MOS of diaschismic gives two scale possibilities. | ||
= Srutal (12&34, aka diaschismic) = | == Srutal (12&34, aka diaschismic) == | ||
Subgroup: 2.3.5 | Subgroup: 2.3.5 | ||
| Line 20: | Line 20: | ||
[[Badness]]: 0.019915 | [[Badness]]: 0.019915 | ||
== Seven limit extensions == | === Seven limit extensions === | ||
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. | The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. | ||
| Line 32: | Line 32: | ||
Pajara, diaschismic, srutal and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. | Pajara, diaschismic, srutal and keen keep the same 1/2 octave period and fifth generator, but shrutar has a generator of a quarter-tone (which can be taken as [[36/35]], the septimal quarter-tone) and echidna has a generator of 9/7. | ||
= Srutal = | == Srutal == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 52: | Line 52: | ||
[[Badness]]: 0.091504 | [[Badness]]: 0.091504 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 70: | Line 70: | ||
Badness: 0.035315 | Badness: 0.035315 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 88: | Line 88: | ||
Badness: 0.025286 | Badness: 0.025286 | ||
= Pajara = | == Pajara == | ||
{{main| Pajara }} | {{main| Pajara }} | ||
| Line 114: | Line 114: | ||
[[Badness]]: 0.020033 | [[Badness]]: 0.020033 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 132: | Line 132: | ||
Badness: 0.020343 | Badness: 0.020343 | ||
=== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
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Badness: 0.027642 | Badness: 0.027642 | ||
== Pajarous == | === Pajarous === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.028349 | Badness: 0.028349 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 181: | Line 181: | ||
Badness: 0.025176 | Badness: 0.025176 | ||
=== Pajaro === | ==== Pajaro ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 194: | Line 194: | ||
Badness: 0.0274 | Badness: 0.0274 | ||
== Pajaric == | === Pajaric === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 207: | Line 207: | ||
Badness: 0.023798 | Badness: 0.023798 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 220: | Line 220: | ||
Badness: 0.0205 | Badness: 0.0205 | ||
== Hemipaj == | === Hemipaj === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
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Badness: 0.038890 | Badness: 0.038890 | ||
= 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 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. | 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. | ||
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[[Badness]]: 0.037914 | [[Badness]]: 0.037914 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 265: | Line 265: | ||
Badness: 0.025034 | Badness: 0.025034 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 278: | Line 278: | ||
Badness: 0.018926 | Badness: 0.018926 | ||
== 17-limit (Na"Naa') == | === 17-limit (Na"Naa') === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
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Badness: 0.016425 | Badness: 0.016425 | ||
= Keen = | == Keen == | ||
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. | 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. | ||
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[[Badness]]: 0.083971 | [[Badness]]: 0.083971 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 321: | Line 321: | ||
Badness: 0.045270 | Badness: 0.045270 | ||
= 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. | ||
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[[Badness]]: 0.056474 | [[Badness]]: 0.056474 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 351: | Line 351: | ||
Badness: 0.040191 | Badness: 0.040191 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 364: | Line 364: | ||
Badness: 0.041137 | Badness: 0.041137 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 377: | Line 377: | ||
Badness: 0.028631 | Badness: 0.028631 | ||
= Echidna = | == Echidna == | ||
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 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 }}. | ||
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[[Badness]]: 0.058033 | [[Badness]]: 0.058033 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 415: | Line 415: | ||
Badness: 0.025987 | Badness: 0.025987 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 428: | Line 428: | ||
Badness: 0.023679 | Badness: 0.023679 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 441: | Line 441: | ||
Badness: 0.020273 | Badness: 0.020273 | ||
= Echidnic = | == Echidnic == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 454: | Line 454: | ||
[[Badness]]: 0.072246 | [[Badness]]: 0.072246 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 467: | Line 467: | ||
Badness: 0.045127 | Badness: 0.045127 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 480: | Line 480: | ||
Badness: 0.028874 | Badness: 0.028874 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 493: | Line 493: | ||
Badness: 0.019304 | Badness: 0.019304 | ||
== Compositions == | === Compositions === | ||
* [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]] | * [https://untwelve.org/competition/2011 ''A Stiff Shot of Turpentine''] [https://untwelve.org/static/audio/competition/2011/Kosmorsky-A_Stiff_Shot_of_Turpentine.mp3 play] by [[Peter Kosmorsky]] | ||
= Shrutar = | == Shrutar == | ||
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. It can also be described as 22&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&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. | ||
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{{Val list|legend=1| 22, 46, 68, 182b, 250bc }} | {{Val list|legend=1| 22, 46, 68, 182b, 250bc }} | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 524: | Line 524: | ||
Vals: {{Val list| 22, 46, 68, 114, 296bce, 410bce }} | Vals: {{Val list| 22, 46, 68, 114, 296bce, 410bce }} | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 537: | Line 537: | ||
Badness: 0.0281 | Badness: 0.0281 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 550: | Line 550: | ||
Badness: 0.0187 | Badness: 0.0187 | ||
== 19-limit == | === 19-limit === | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 563: | Line 563: | ||
Badness: 0.0175 | Badness: 0.0175 | ||
= Sruti = | == Sruti == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 578: | Line 578: | ||
[[Badness]]: 0.1174 | [[Badness]]: 0.1174 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 591: | Line 591: | ||
Badness: 0.0415 | Badness: 0.0415 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 604: | Line 604: | ||
Badness: 0.0238 | Badness: 0.0238 | ||
= Anguirus = | == Anguirus == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 619: | Line 619: | ||
[[Badness]]: 0.0780 | [[Badness]]: 0.0780 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 632: | Line 632: | ||
Badness: 0.0493 | Badness: 0.0493 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 645: | Line 645: | ||
Badness: 0.0308 | Badness: 0.0308 | ||
= Shru = | == Shru == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 660: | Line 660: | ||
[[Badness]]: 0.1576 | [[Badness]]: 0.1576 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 673: | Line 673: | ||
Badness: 0.0635 | Badness: 0.0635 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||