Diaschismic family: Difference between revisions
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The 5-limit parent comma for the '''diaschismic family''' is 2048/2025, the [[ | 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 <<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. | ||
[[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [600.000 to 720.000] (2 to 5) | [[Tuning_Ranges_of_Regular_Temperaments|valid range]]: [600.000 to 720.000] (2 to 5) | ||
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EDOs: 34, 46, 80, 206c, 286bc | EDOs: 34, 46, 80, 206c, 286bc | ||
==Seven limit children== | == Seven limit children == | ||
The second comma of the [[Normal_lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from 64/63 and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar 245/243, the sensamagic comma. The pajara, diaschismic 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 [[ | The second comma of the [[Normal_lists|normal comma list]] defines which 7-limit family member we are looking at. Pajara derives from [[64/63]] and is a popular and well-known choice. Diaschismic adds 2097152/2066715 to obtain 7-limit harmony by more complex methods, but with greater accuracy. Keen adds 2240/2187, echidna 1728/1715 and shrutar [[245/243]], the sensamagic comma. The pajara, diaschismic 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. Adding 4375/4374 does no significant tuning damage, so for that we keep the 5-limit label srutal. | ||
=Srutal= | = Srutal = | ||
Commas: 2048/2025, 4375/4374 | Commas: 2048/2025, 4375/4374 | ||
| Line 37: | Line 37: | ||
Badness: 0.0915 | Badness: 0.0915 | ||
==11-limit== | == 11-limit == | ||
Commas: 176/175, 896/891, 1331/1323 | Commas: 176/175, 896/891, 1331/1323 | ||
| Line 54: | Line 54: | ||
Badness: 0.0353 | Badness: 0.0353 | ||
==13-limit== | == 13-limit == | ||
Commas: 169/168, 176/175, 325/324, 364/363 | Commas: 169/168, 176/175, 325/324, 364/363 | ||
| Line 71: | Line 71: | ||
Badness: 0.0253 | Badness: 0.0253 | ||
=Pajara= | = Pajara = | ||
{{main| Pajara }} | |||
Pajara, with wedgie <<2 -4 -4 -11 -12 2|| is closely associated with 22et (not to mention [[ | Pajara, with wedgie <<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 <34 54 79 96| and 56 with the 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 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. | ||
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Badness: 0.0200 | Badness: 0.0200 | ||
==11-limit== | == 11-limit == | ||
Commas: 50/49, 64/63, 99/98 | Commas: 50/49, 64/63, 99/98 | ||
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Badness: 0.0203 | Badness: 0.0203 | ||
==13-limit== | === 13-limit === | ||
Commas: 50/49, 64/63, 65/63, 99/98 | Commas: 50/49, 64/63, 65/63, 99/98 | ||
| Line 128: | Line 128: | ||
Badness: 0.0276 | Badness: 0.0276 | ||
==Pajarous== | == Pajarous == | ||
Commas: 50/49, 55/54, 64/63 | Commas: 50/49, 55/54, 64/63 | ||
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Badness: 0.0283 | Badness: 0.0283 | ||
===13-limit=== | === 13-limit === | ||
Commas: 50/49, 55/54, 64/63, 65/63 | Commas: 50/49, 55/54, 64/63, 65/63 | ||
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Badness: 0.0252 | Badness: 0.0252 | ||
==Pajaric== | === Pajaro === | ||
Commas: 40/39, 50/49, 55/54, 64/63 | |||
POTE generator ~3/2 = 710.818 | |||
Map: [<2 0 11 12 -9 17|, <0 1 -2 -2 5 -3|] | |||
EDOs: 10, 22f, 32f, 54f | |||
Badness: 0.0274 | |||
== Pajaric == | |||
Commas: 45/44, 50/49, 56/55 | Commas: 45/44, 50/49, 56/55 | ||
| Line 173: | Line 184: | ||
Badness: 0.0238 | Badness: 0.0238 | ||
===13-limit=== | === 13-limit === | ||
Commas: 40/39, 45/44, 50/49, 56/55 | Commas: 40/39, 45/44, 50/49, 56/55 | ||
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Badness: 0.0205 | Badness: 0.0205 | ||
== Hemipaj == | |||
==Hemipaj== | |||
Commas: 50/49, 64/63, 121/120 | Commas: 50/49, 64/63, 121/120 | ||
| Line 206: | Line 206: | ||
Badness: 0.0389 | Badness: 0.0389 | ||
=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 <<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]] 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 wedgie <<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]] 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. | ||
| Line 221: | Line 221: | ||
Badness: 0.0379 | Badness: 0.0379 | ||
==11-limit== | == 11-limit == | ||
Commas: 126/125, 176/175, 896/891 | Commas: 126/125, 176/175, 896/891 | ||
| Line 230: | Line 230: | ||
EDOs: 46, 58, 104c, 162ce | EDOs: 46, 58, 104c, 162ce | ||
==13-limit== | == 13-limit == | ||
Commas: 126/125, 196/195, 364/363, 2048/2025 | Commas: 126/125, 196/195, 364/363, 2048/2025 | ||
| Line 239: | Line 239: | ||
EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]] | EDOs: [[46edo|46]], [[58edo|58]], [[104edo|104c]], [[162edo|162cef]] | ||
==17-limit== | == 17-limit == | ||
Commas: 126/125, 136/135, 176/175, 196/195, 256/255 | Commas: 126/125, 136/135, 176/175, 196/195, 256/255 | ||
| Line 248: | Line 248: | ||
EDOs: 46, 58, 104c | EDOs: 46, 58, 104c | ||
=Keen= | = Keen = | ||
Keen adds 875/864 as well as 2240/2187 to the set of commas, and has wedgie <<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, <<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, and has wedgie <<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, <<2 -4 18 -12 ...||, is really more interesting, adding 100/99 and 385/384 to the commas. | ||
Commas: 2048/2025 | Commas: 875/864, 2048/2025 | ||
[[POTE_tuning|POTE generator]]: 707.571 | [[POTE_tuning|POTE generator]]: 707.571 | ||
| Line 259: | Line 259: | ||
EDOs: 22, 56, 78, 134b, 212b, 290b | EDOs: 22, 56, 78, 134b, 212b, 290b | ||
==11-limit== | == 11-limit == | ||
Commas: 100/99, 385/384, 1232/1215 | Commas: 100/99, 385/384, 1232/1215 | ||
| Line 268: | Line 268: | ||
EDOs: 22, 56, 78, 212bf, 290bf | EDOs: 22, 56, 78, 212bf, 290bf | ||
=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. | ||
| Line 283: | Line 283: | ||
Badness: 0.0565 | Badness: 0.0565 | ||
==11-limit== | == 11-limit == | ||
Commas: 176/175, 896/891, 1375/1372 | Commas: 176/175, 896/891, 1375/1372 | ||
| Line 294: | Line 294: | ||
Badness: 0.0402 | Badness: 0.0402 | ||
==13-limit== | == 13-limit == | ||
Commas: 176/175, 325/324, 640/637, 896/891 | Commas: 176/175, 325/324, 640/637, 896/891 | ||
| Line 305: | Line 305: | ||
Badness: 0.0411 | Badness: 0.0411 | ||
=Echidna= | = Echidna = | ||
Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie <<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 <138 219 321 388|. | Echidna adds 1728/1715 to the commas and takes 9/7 as a generator. It has a wedgie <<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 <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. | ||
Commas: 2048/2025 | Commas: 1728/1715, 2048/2025 | ||
[[POTE_tuning|POTE generator]]: 434.856 | [[POTE_tuning|POTE generator]]: 434.856 | ||
| Line 320: | Line 320: | ||
Badness: 0.0580 | Badness: 0.0580 | ||
==11-limit== | == 11-limit == | ||
Commas: 176/175, 896/891 | Commas: 176/175, 540/539, 896/891 | ||
11-limit minimax | 11-limit minimax | ||
| Line 340: | Line 340: | ||
Badness: 0.0260 | Badness: 0.0260 | ||
==13-limit== | == 13-limit == | ||
Commas: 176/175, 351/350, 364/363, 540/539 | Commas: 176/175, 351/350, 364/363, 540/539 | ||
| Line 351: | Line 351: | ||
Badness: 0.0237 | Badness: 0.0237 | ||
==17-limit== | == 17-limit == | ||
Commas: 136/135, 176/175, 221/220, 256/255, 540/539 | Commas: 136/135, 176/175, 221/220, 256/255, 540/539 | ||
| Line 362: | Line 362: | ||
Badness: 0.0203 | Badness: 0.0203 | ||
=Echidnic= | = Echidnic = | ||
Commas: 686/675, 1029/1024 | Commas: 686/675, 1029/1024 | ||
| Line 373: | Line 373: | ||
Badness: 0.0722 | Badness: 0.0722 | ||
==11-limit== | == 11-limit == | ||
Commas: 385/384, 441/440, 686/675 | Commas: 385/384, 441/440, 686/675 | ||
| Line 384: | Line 384: | ||
Badness: 0.0451 | Badness: 0.0451 | ||
==13-limit== | == 13-limit == | ||
Commas: 91/90, 169/168, 385/384, 441/440 | Commas: 91/90, 169/168, 385/384, 441/440 | ||
| Line 401: | Line 401: | ||
(the description says "lemba" which has a similar scale structure but different mapping for 5) | (the description says "lemba" which has a similar scale structure but different mapping for 5) | ||
=Shrutar= | = Shrutar = | ||
Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie <<4 -8 14 -22 11 55||, 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. [[ | Shrutar adds 245/243 to the commas, and also tempers out 6144/6125. With wedgie <<4 -8 14 -22 11 55||, 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^(1/7), 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^(1/7) 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^(1/7) generator can again be used as tunings. | ||
Commas: 2048/2025 | Commas: 245/243, 2048/2025 | ||
[[POTE_tuning|POTE generator]]: 52.811 | [[POTE_tuning|POTE generator]]: 52.811 | ||
| Line 414: | Line 414: | ||
EDOs: 22, 46, 68, 182b, 250bc | EDOs: 22, 46, 68, 182b, 250bc | ||
==11-limit== | == 11-limit == | ||
Commas: | Commas: 121/120, 176/175, 245/243 | ||
[[POTE_tuning|POTE generator]]: 52.680 | [[POTE_tuning|POTE generator]]: 52.680 | ||
| Line 423: | Line 423: | ||
EDOs: 22, 46, 68, 114, 296bce, 410bce | EDOs: 22, 46, 68, 114, 296bce, 410bce | ||
==13-limit== | == 13-limit == | ||
Commas: 121/120, 176/175, 196/195, 245/243 | Commas: 121/120, 176/175, 196/195, 245/243 | ||
| Line 434: | Line 434: | ||
Badness: 0.0281 | Badness: 0.0281 | ||
==17-limit== | == 17-limit == | ||
Commas: 121/120, 136/135, 154/153, 176/175, 196/195 | Commas: 121/120, 136/135, 154/153, 176/175, 196/195 | ||
| Line 445: | Line 445: | ||
Badness: 0.0187 | Badness: 0.0187 | ||
==19-limit== | == 19-limit == | ||
Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342 | Commas: 121/120, 136/135, 154/153, 176/175, 196/195, 343/342 | ||
| Line 456: | Line 456: | ||
Badness: 0.0175 | Badness: 0.0175 | ||
=Sruti= | = Sruti = | ||
Commas: 2048/2025, 19683/19600 | Commas: 2048/2025, 19683/19600 | ||
| Line 469: | Line 469: | ||
Badness: 0.1174 | Badness: 0.1174 | ||
==11-limit== | == 11-limit == | ||
Commas: 176/175, 243/242, 896/891 | Commas: 176/175, 243/242, 896/891 | ||
| Line 480: | Line 480: | ||
Badness: 0.0415 | Badness: 0.0415 | ||
==13-limit== | == 13-limit == | ||
Commas: 144/143, 176/175, 351/350, 676/675 | Commas: 144/143, 176/175, 351/350, 676/675 | ||
| Line 491: | Line 491: | ||
Badness: 0.0238 | Badness: 0.0238 | ||
=Anguirus= | = Anguirus = | ||
Commas: 49/48, 2048/2025 | Commas: 49/48, 2048/2025 | ||
| Line 504: | Line 504: | ||
Badness: 0.0780 | Badness: 0.0780 | ||
==11-limit== | == 11-limit == | ||
Commas: 49/48, 56/55, 243/242 | Commas: 49/48, 56/55, 243/242 | ||
| Line 515: | Line 515: | ||
Badness: 0.0493 | Badness: 0.0493 | ||
==13-limit== | == 13-limit == | ||
Commas: 49/48 56/55 91/90 352/351 | Commas: 49/48, 56/55, 91/90, 352/351 | ||
POTE generator: ~8/7 = 247.691 | POTE generator: ~8/7 = 247.691 | ||
| Line 526: | Line 526: | ||
Badness: 0.0308 | Badness: 0.0308 | ||
=Shru= | = Shru = | ||
Commas: 392/375, 1323/1280 | Commas: 392/375, 1323/1280 | ||
| Line 539: | Line 539: | ||
Badness: 0.1576 | Badness: 0.1576 | ||
==11-limit== | == 11-limit == | ||
Commas: 56/55, 77/75, 1323/1280 | Commas: 56/55, 77/75, 1323/1280 | ||
| Line 550: | Line 550: | ||
Badness: 0.0635 | Badness: 0.0635 | ||
==13-limit== | == 13-limit == | ||
Commas: 56/55, 77/75, 105/104, 507/500 | Commas: 56/55, 77/75, 105/104, 507/500 | ||
| Line 564: | Line 564: | ||
[[Category:Temperament family]] | [[Category:Temperament family]] | ||
[[Category:Diaschismic]] | [[Category:Diaschismic]] | ||
[[Category:Rank 2]] | |||
[[Category:Todo: | [[Category:Todo:Add definition]] | ||
[[Category:Todo: | [[Category:Todo:Review]] | ||