Schismatic family: Difference between revisions
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The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields {{multival| 1 -8 -15 }} for the [[wedgie]]. This tells us the generator is a fifth and [[5/4]] is represented by a diminished fourth. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768. | The 5-limit parent comma for the '''schismatic''' (or '''schismic''') '''family''' is the [[schisma]] of 32805/32768, which is the amount by which the [[Pythagorean comma]] exceeds the [[Didymus comma]] (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its [[monzo]] is {{monzo| -15 8 1 }}, and flipping that yields {{multival| 1 -8 -15 }} for the [[wedgie]]. This tells us the generator is a fifth and [[5/4]] is represented by a diminished fourth. In fact, 10 = (4/3)<sup>8</sup> × 32805/32768. | ||
= Schismatic aka Helmholtz = | == Schismatic aka Helmholtz == | ||
The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''Helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut. | The 5-limit version of the temperament is a [[microtemperament]], sometimes called '''Helmholtz''', '''schismic''' or '''schismatic''', which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. [[53edo]] is a possible tuning for schismatic, but you need [[118edo]] if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut. | ||
| Line 29: | Line 29: | ||
[[Badness]]: 0.004259 | [[Badness]]: 0.004259 | ||
== 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. | ||
* Garibaldi adds [[garischisma|{{monzo|25 -14 0 -1}}]], | * Garibaldi adds [[garischisma|{{monzo|25 -14 0 -1}}]], | ||
| Line 44: | Line 44: | ||
Temperaments not discussed here include [[Sensamagic clan #Salsa|salsa]], [[Gamelismic clan #Guiron|guiron]], [[Porwell temperaments #Hemischis|hemischis]] and [[Turkish maqam music temperaments #Karadeniz temperament|karadeniz]]. | Temperaments not discussed here include [[Sensamagic clan #Salsa|salsa]], [[Gamelismic clan #Guiron|guiron]], [[Porwell temperaments #Hemischis|hemischis]] and [[Turkish maqam music temperaments #Karadeniz temperament|karadeniz]]. | ||
= Garibaldi = | == Garibaldi == | ||
{{main| Garibaldi temperament }} | {{main| Garibaldi temperament }} | ||
| Line 76: | Line 76: | ||
[[Badness]]: 0.021644 | [[Badness]]: 0.021644 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 96: | Line 96: | ||
Badness: 0.027396 | Badness: 0.027396 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 111: | Line 111: | ||
Badness: 0.020676 | Badness: 0.020676 | ||
== Andromeda == | === Andromeda === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 126: | Line 126: | ||
Badness: 0.023556 | Badness: 0.023556 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 141: | Line 141: | ||
Badness: 0.020749 | Badness: 0.020749 | ||
== Helenus == | === Helenus === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 156: | Line 156: | ||
Badness: 0.035637 | Badness: 0.035637 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 171: | Line 171: | ||
Badness: 0.026284 | Badness: 0.026284 | ||
== Hemigari == | === Hemigari === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 186: | Line 186: | ||
Badness: 0.050681 | Badness: 0.050681 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 201: | Line 201: | ||
Badness: 0.027464 | Badness: 0.027464 | ||
== Sanjaab == | === Sanjaab === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 216: | Line 216: | ||
Badness: 0.058040 | Badness: 0.058040 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 231: | Line 231: | ||
Badness: 0.033849 | Badness: 0.033849 | ||
= Schism = | == Schism == | ||
{{see also| Archytas clan #Schism }} | {{see also| Archytas clan #Schism }} | ||
| Line 250: | Line 250: | ||
[[Badness]]: 0.056648 | [[Badness]]: 0.056648 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 265: | Line 265: | ||
Badness: 0.037482 | Badness: 0.037482 | ||
= Pontiac = | == Pontiac == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 295: | Line 295: | ||
[[Badness]]: 0.014133 | [[Badness]]: 0.014133 | ||
== Helenoid == | === Helenoid === | ||
The ''helenoid'' temperament (53&118, named by [[User:Xenllium|Xenllium]]) is closely related to the helenus temperament, but with the [[4375/4374|ragisma]] rather than the [[225/224|marvel comma]] tempered out. | The ''helenoid'' temperament (53&118, named by [[User:Xenllium|Xenllium]]) is closely related to the helenus temperament, but with the [[4375/4374|ragisma]] rather than the [[225/224|marvel comma]] tempered out. | ||
| Line 310: | Line 310: | ||
Badness: 0.038863 | Badness: 0.038863 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 323: | Line 323: | ||
Badness: 0.033677 | Badness: 0.033677 | ||
== Ponta == | === Ponta === | ||
The ''ponta'' temperament (53&224, named by [[User:Xenllium|Xenllium]]) tempers out the [[540/539|swetisma]] and the ragisma. | The ''ponta'' temperament (53&224, named by [[User:Xenllium|Xenllium]]) tempers out the [[540/539|swetisma]] and the ragisma. | ||
| Line 338: | Line 338: | ||
Badness: 0.048692 | Badness: 0.048692 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 351: | Line 351: | ||
Badness: 0.023616 | Badness: 0.023616 | ||
== Pontic == | === Pontic === | ||
The ''pontic'' temperament (118&171, named by [[User:Xenllium|Xenllium]]) tempers out the [[441/440|werckisma]] and the ragisma. | The ''pontic'' temperament (118&171, named by [[User:Xenllium|Xenllium]]) tempers out the [[441/440|werckisma]] and the ragisma. | ||
| Line 366: | Line 366: | ||
Badness: 0.049573 | Badness: 0.049573 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 379: | Line 379: | ||
Badness: 0.045308 | Badness: 0.045308 | ||
== Bipont == | === Bipont === | ||
The ''bipont'' temperament (118&224, named by [[User:Xenllium|Xenllium]]) has a period of half octave and tempers out the [[3025/3024|lehmerisma]], 3025/3024 and the [[9801/9800|kalisma]], 9801/9800. | The ''bipont'' temperament (118&224, named by [[User:Xenllium|Xenllium]]) has a period of half octave and tempers out the [[3025/3024|lehmerisma]], 3025/3024 and the [[9801/9800|kalisma]], 9801/9800. | ||
| Line 394: | Line 394: | ||
Badness: 0.014629 | Badness: 0.014629 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 407: | Line 407: | ||
Badness: 0.030172 | Badness: 0.030172 | ||
=== Counterbipont === | ==== Counterbipont ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 420: | Line 420: | ||
Badness: 0.025547 | Badness: 0.025547 | ||
=== Quadrapont === | ==== Quadrapont ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 433: | Line 433: | ||
Badness: 0.021025 | Badness: 0.021025 | ||
= Grackle = | == Grackle == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 454: | Line 454: | ||
[[Badness]]: 0.070407 | [[Badness]]: 0.070407 | ||
= Bischismic = | == Bischismic == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 475: | Line 475: | ||
[[Badness]]: 0.054744 | [[Badness]]: 0.054744 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 490: | Line 490: | ||
Badness: 0.028160 | Badness: 0.028160 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 505: | Line 505: | ||
Badness: 0.028722 | Badness: 0.028722 | ||
= Kleischismic = | == Kleischismic == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 522: | Line 522: | ||
[[Badness]]: 0.110583 | [[Badness]]: 0.110583 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 537: | Line 537: | ||
Badness: 0.036749 | Badness: 0.036749 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 552: | Line 552: | ||
Badness: 0.037640 | Badness: 0.037640 | ||
= Pogo = | == Pogo == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 569: | Line 569: | ||
[[Badness]]: 0.079635 | [[Badness]]: 0.079635 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 584: | Line 584: | ||
Badness: 0.031857 | Badness: 0.031857 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 599: | Line 599: | ||
Badness: 0.017514 | Badness: 0.017514 | ||
= Squirrel = | == Squirrel == | ||
The squirrel temperament (29&36) has an 11/10 generator, three of which give the fourth (4/3), and thirteen of which give 7/4 with octave reduction. | The squirrel temperament (29&36) has an 11/10 generator, three of which give the fourth (4/3), and thirteen of which give 7/4 with octave reduction. | ||
| Line 616: | Line 616: | ||
[[Badness]]: 0.174705 | [[Badness]]: 0.174705 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 629: | Line 629: | ||
Badness: 0.068310 | Badness: 0.068310 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 642: | Line 642: | ||
Badness: 0.043750 | Badness: 0.043750 | ||
= Tertiaschis = | == Tertiaschis == | ||
The ''tertiaschis'' temperament (94&159, named by [[User:Xenllium|Xenllium]]) has an 11/10 generator and tempers out 1071785/1062882. | The ''tertiaschis'' temperament (94&159, named by [[User:Xenllium|Xenllium]]) has an 11/10 generator and tempers out 1071785/1062882. | ||
| Line 659: | Line 659: | ||
[[Badness]]: 0.211859 | [[Badness]]: 0.211859 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 672: | Line 672: | ||
Badness: 0.061336 | Badness: 0.061336 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 685: | Line 685: | ||
Badness: 0.036700 | Badness: 0.036700 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 698: | Line 698: | ||
Badness: 0.026504 | Badness: 0.026504 | ||
= Term = | == Term == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 719: | Line 719: | ||
[[Badness]]: 0.019950 | [[Badness]]: 0.019950 | ||
== Terminal == | === Terminal === | ||
The ''terminal'' temperament (12&159, named by [[User:Xenllium|Xenllium]]) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave. | The ''terminal'' temperament (12&159, named by [[User:Xenllium|Xenllium]]) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave. | ||
| Line 734: | Line 734: | ||
Badness: 0.059502 | Badness: 0.059502 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 747: | Line 747: | ||
Badness: 0.037082 | Badness: 0.037082 | ||
=== 17-limit === | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 760: | Line 760: | ||
Badness: 0.027073 | Badness: 0.027073 | ||
== Hemiterm == | === Hemiterm === | ||
The ''hemiterm'' temperament (12&342, named by [[User:Xenllium|Xenllium]]) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma). | The ''hemiterm'' temperament (12&342, named by [[User:Xenllium|Xenllium]]) has a period of 1/6 octave and tempers out [[9801/9800]] (kalisma) and 151263/151250 (odiheim comma). | ||
| Line 775: | Line 775: | ||
Badness: 0.029438 | Badness: 0.029438 | ||
= Sesquiquartififths = | == Sesquiquartififths == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 796: | Line 796: | ||
[[Badness]]: 0.011244 | [[Badness]]: 0.011244 | ||
== Sesquart == | === Sesquart === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 811: | Line 811: | ||
Badness: 0.029306 | Badness: 0.029306 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 824: | Line 824: | ||
Badness: 0.022396 | Badness: 0.022396 | ||
== Bisesqui == | === Bisesqui === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 837: | Line 837: | ||
Badness: 0.016968 | Badness: 0.016968 | ||
= Quintilischis = | == Quintilischis == | ||
The ''quintilischis'' temperament (12&253, named by [[User:Xenllium|Xenllium]]) slices the fourth (4/3) into five semitones. | The ''quintilischis'' temperament (12&253, named by [[User:Xenllium|Xenllium]]) slices the fourth (4/3) into five semitones. | ||
| Line 854: | Line 854: | ||
[[Badness]]: 0.253966 | [[Badness]]: 0.253966 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 867: | Line 867: | ||
Badness: 0.113044 | Badness: 0.113044 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 880: | Line 880: | ||
Badness: 0.069127 | Badness: 0.069127 | ||
== 17-limit == | === 17-limit === | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 893: | Line 893: | ||
Badness: 0.045992 | Badness: 0.045992 | ||
== 19-limit == | === 19-limit === | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 906: | Line 906: | ||
Badness: 0.038155 | Badness: 0.038155 | ||
= Sextilififths = | == Sextilififths == | ||
The sextilififths or ''sextilischis'' (the latter is named by [[User:Xenllium|Xenllium]]) slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21. | The sextilififths or ''sextilischis'' (the latter is named by [[User:Xenllium|Xenllium]]) slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21. | ||
| Line 925: | Line 925: | ||
[[Badness]]: 0.108794 | [[Badness]]: 0.108794 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 940: | Line 940: | ||
Badness: 0.045457 | Badness: 0.045457 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 955: | Line 955: | ||
Badness: 0.025276 | Badness: 0.025276 | ||
= Tsaharuk = | == Tsaharuk == | ||
{{See also|Tsaharuk}} | {{See also|Tsaharuk}} | ||
| Line 974: | Line 974: | ||
[[Badness]]: 0.030697 | [[Badness]]: 0.030697 | ||
= Quanharuk = | == Quanharuk == | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 991: | Line 991: | ||
[[Badness]]: 0.071950 | [[Badness]]: 0.071950 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,006: | Line 1,006: | ||
Badness: 0.031549 | Badness: 0.031549 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,021: | Line 1,021: | ||
Badness: 0.021392 | Badness: 0.021392 | ||
= Quadrant = | == Quadrant == | ||
The ''quadrant'' temperament (12&224, named by [[User:Xenllium|Xenllium]]) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave. | The ''quadrant'' temperament (12&224, named by [[User:Xenllium|Xenllium]]) has a period of quarter octave and tempers out the [[dimcomp comma]], 390625/388962. In this temperament, 25/21 is mapped into quarter octave. | ||
| Line 1,040: | Line 1,040: | ||
[[Badness]]: 0.110242 | [[Badness]]: 0.110242 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,053: | Line 1,053: | ||
Badness: 0.045738 | Badness: 0.045738 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,066: | Line 1,066: | ||
Badness: 0.027243 | Badness: 0.027243 | ||
= Septant = | == Septant == | ||
The ''septant'' temperament (224&301, named by [[User:Xenllium|Xenllium]]) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo|47 -7 -7 -7}}. | The ''septant'' temperament (224&301, named by [[User:Xenllium|Xenllium]]) has a period of 1/7 octave and tempers out the [[akjaysma]], {{monzo|47 -7 -7 -7}}. | ||
| Line 1,083: | Line 1,083: | ||
[[Badness]]: 0.111142 | [[Badness]]: 0.111142 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,096: | Line 1,096: | ||
Badness: 0.044122 | Badness: 0.044122 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,109: | Line 1,109: | ||
Badness: 0.024706 | Badness: 0.024706 | ||
= Octant = | == Octant == | ||
The octant temperament (224&472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively. | The octant temperament (224&472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively. | ||
| Line 1,128: | Line 1,128: | ||
[[Badness]]: 0.157186 | [[Badness]]: 0.157186 | ||
== 11-limit == | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,143: | Line 1,143: | ||
Badness: 0.044778 | Badness: 0.044778 | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Revision as of 21:19, 1 June 2021
The 5-limit parent comma for the schismatic (or schismic) family is the schisma of 32805/32768, which is the amount by which the Pythagorean comma exceeds the Didymus comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth. Its monzo is [-15 8 1⟩, and flipping that yields ⟨⟨ 1 -8 -15 ]] for the wedgie. This tells us the generator is a fifth and 5/4 is represented by a diminished fourth. In fact, 10 = (4/3)8 × 32805/32768.
Schismatic aka Helmholtz
The 5-limit version of the temperament is a microtemperament, sometimes called Helmholtz, schismic or schismatic, which flattens the fifth by a fraction of a schisma, but some other members of the family are less accurate. As a 5-limit system, it is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismatic, but you need 118edo if you want to get the full effect. In exact analogy with 1/4 comma meantone there is also 1/8 schismatic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 cents, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better 5th, or 2/17 schisma, with both thirds flat by 1/17 of a schisma, although the differences would be very hard to distinguish unless using a large gamut.
Subgroup: 2.3.5
Comma list: 32805/32768
Mapping: [⟨1 0 15], ⟨0 1 -8]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.736
- diamond monotone range: ~3/2 = [700.000, 705.882] (7\12 to 10\17)
- diamond tradeoff range: ~3/2 = [701.711, 701.955]
- diamond monotone and tradeoff range: ~3/2 = [701.711, 701.955]
Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc
Badness: 0.004259
Seven-limit extensions
The second comma of the normal comma list defines which 7-limit family member we are looking at.
- Garibaldi adds [25 -14 0 -1⟩,
- Grackle adds [-44 26 0 1⟩,
- Schism adds [6 -2 0 -1⟩,
- Pontiac adds [-59 39 0 -1⟩.
Those all have a fifth as generator.
- Bischismic adds [-69 40 0 2⟩ and has a fifth generator with a half-octave period.
- Guiron adds [-10 1 0 3⟩, with an 8/7 generator, three of which give the fifth.
- Term adds [-94 54 0 3⟩ with a 1/3 octave period.
- Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.
Temperaments not discussed here include salsa, guiron, hemischis and karadeniz.
Garibaldi
Subgroup: 2.3.5.7
Comma list: 225/224, 3125/3087
Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 -14 -15 -25 -10 ]]
POTE generator: ~3/2 = 702.085
- [[1 0 0 0⟩, [5/3 1/15 0 -1/15⟩, [5/3 -8/15 0 8/15⟩, [5/3 -14/15 0 14/15⟩]
- Eigenmonzos: 2, 7/6
- [[1 0 0 0⟩, [25/16 1/8 0 -1/16⟩, [5/2 -1 0 1/2⟩, [25/8 -7/4 0 7/8⟩]
- Eigenmonzos: 2, 9/7
- diamond monotone range: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- diamond tradeoff range: ~3/2 = [701.711, 702.915]
- diamond monotone and tradeoff range: ~3/2 = [701.711, 702.915]
Badness: 0.021644
11-limit
Subgroup: 2.3.5.7.11
Comma list: 225/224, 385/384, 2200/2187
Mapping: [⟨1 0 15 25 -33], ⟨0 1 -8 -14 23]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.157
Minimax tuning:
- 11-odd-limit
- [[1 0 0 0 0⟩, [25/16 1/8 0 -1/16 0⟩, [5/2 -1 0 1/2 0⟩, [25/8 -7/4 0 7/8 0⟩, [47/16 23/8 0 -23/16 0⟩]
- Eigenmonzos: 2, 9/7
Vals: Template:Val list
Badness: 0.027396
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 325/324, 385/384
Mapping: [⟨1 0 15 25 -33 -28], ⟨0 1 -8 -14 23 20]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.113
Vals: Template:Val list
Badness: 0.020676
Andromeda
Subgroup: 2.3.5.7.11
Comma list: 100/99, 225/224, 245/242
Mapping: [⟨1 0 15 25 32], ⟨0 1 -8 -14 -18]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.321
Vals: Template:Val list
Badness: 0.023556
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 100/99, 105/104, 196/195, 245/242
Mapping: [⟨1 0 15 25 32 37], ⟨0 1 -8 -14 -18 -21]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 702.559
Vals: Template:Val list
Badness: 0.020749
Helenus
Subgroup: 2.3.5.7.11
Comma list: 99/98, 176/175, 3125/3087
Mapping: [⟨1 0 15 25 51], ⟨0 1 -8 -14 -30]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.725
Vals: Template:Val list
Badness: 0.035637
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 99/98, 176/175, 275/273, 847/845
Mapping: [⟨1 0 15 25 51 56], ⟨0 1 -8 -14 -30 -33]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.747
Vals: Template:Val list
Badness: 0.026284
Hemigari
Subgroup: 2.3.5.7.11
Comma list: 121/120, 225/224, 3125/3087
Mapping: [⟨1 0 15 25 9], ⟨0 2 -16 -28 -7]]
Mapping generators: ~2, ~110/63
POTE generator: ~63/55 = 248.918
Vals: Template:Val list
Badness: 0.050681
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 121/120, 169/168, 225/224, 275/273
Mapping: [⟨1 0 15 25 9 14], ⟨0 2 -16 -28 -7 -13]]
Mapping generators: ~2, ~26/15
POTE generator: ~15/13 = 248.918
Vals: Template:Val list
Badness: 0.027464
Sanjaab
Subgroup: 2.3.5.7.11
Comma list: 225/224, 1331/1323, 3125/3087
Mapping: [⟨1 2 -1 -3 0], ⟨0 -3 24 42 25]]
Mapping generators: ~2, ~11/10
POTE generator: ~11/10 = 165.974
Vals: Template:Val list
Badness: 0.058040
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 275/273, 847/845, 1331/1323
Mapping: [⟨1 2 -1 -3 0 -1], ⟨0 -3 24 42 25 34]]
Mapping generators: ~2, ~11/10
POTE generator: ~11/10 = 165.963
Vals: Template:Val list
Badness: 0.033849
Schism
Subgroup: 2.3.5.7
Comma list: 64/63, 360/343
Mapping: [⟨1 0 15 6], ⟨0 1 -8 -2]]
Mapping generators: ~2, ~3
POTE generator: ~3/2 = 701.556
Wedgie: ⟨⟨ 1 -8 -2 -15 -6 18 ]]
Badness: 0.056648
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 64/63, 99/98
Mapping: [⟨1 0 15 6 13], ⟨0 1 -8 -2 -6]]
Mapping generators: ~2, ~3
POTE generator ~3/2 = 702.136
Vals: Template:Val list
Badness: 0.037482
Pontiac
Subgroup: 2.3.5.7
Comma list: 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 39 -15 59 113 ]]
POTE generator: ~3/2 = 701.757
- [[1 0 0 0⟩, [74/47 0 -1/47 1/47⟩, [113/47 0 8/47 -8/47⟩, [113/47 0 -39/47 39/47⟩]
- Eigenmonzos: 2, 7/5
- [[1 0 0 0⟩, [3/2 1/5 -1/10 0⟩, [3 -8/5 4/5 0⟩, [-1/2 39/5 -39/10 0⟩]
- Eigenmonzos: 2, 10/9
- diamond monotone range: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
- diamond tradeoff range: ~3/2 = [701.711, 701.955]
- diamond monotone and tradeoff range: ~3/2 = [701.711, 701.886]
Badness: 0.014133
Helenoid
The helenoid temperament (53&118, named by Xenllium) is closely related to the helenus temperament, but with the ragisma rather than the marvel comma tempered out.
Subgroup: 2.3.5.7.11
Comma list: 385/384, 3388/3375, 4375/4374
Mapping: [⟨1 0 15 -59 51], ⟨0 1 -8 39 -30]]
POTE generator: ~3/2 = 701.722
Vals: Template:Val list
Badness: 0.038863
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 625/624, 729/728
Mapping: [⟨1 0 15 -59 51 56], ⟨0 1 -8 39 -30 -33]]
POTE generator: ~3/2 = 701.745
Vals: Template:Val list
Badness: 0.033677
Ponta
The ponta temperament (53&224, named by Xenllium) tempers out the swetisma and the ragisma.
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59 135], ⟨0 1 -8 39 -83]]
POTE generator: ~3/2 = 701.783
Vals: Template:Val list
Badness: 0.048692
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 4096/4095
Mapping: [⟨1 0 15 -59 135 56], ⟨0 1 -8 39 -83 -33]]
POTE generator: ~3/2 = 701.784
Vals: Template:Val list
Badness: 0.023616
Pontic
The pontic temperament (118&171, named by Xenllium) tempers out the werckisma and the ragisma.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59 -136], ⟨0 1 -8 39 88]]
POTE generator: ~3/2 = 701.724
Vals: Template:Val list
Badness: 0.049573
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 625/624, 729/728, 3584/3575
Mapping: [⟨1 0 15 -59 -136 56], ⟨0 1 -8 39 88 -33]]
POTE generator: ~3/2 = 701.738
Vals: Template:Val list
Badness: 0.045308
Bipont
The bipont temperament (118&224, named by Xenllium) has a period of half octave and tempers out the lehmerisma, 3025/3024 and the kalisma, 9801/9800.
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 32805/32768
Mapping: [⟨2 3 6 -1 2], ⟨0 1 -8 39 29]]
POTE generator: ~3/2 = 701.757
Vals: Template:Val list
Badness: 0.014629
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 729/728, 1575/1573, 4096/4095
Mapping: [⟨2 3 6 -1 2 13], ⟨0 1 -8 39 29 -33]]
POTE generator: ~3/2 = 701.773
Vals: Template:Val list
Badness: 0.030172
Counterbipont
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768
Mapping: [⟨2 3 6 -1 2 -6], ⟨0 1 -8 39 29 79]]
POTE generator: ~3/2 = 701.769
Vals: Template:Val list
Badness: 0.025547
Quadrapont
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768
Mapping: [⟨4 6 12 -2 4 7], ⟨0 1 -8 39 29 23]]
POTE generator: ~3/2 = 701.756
Vals: Template:Val list
Badness: 0.021025
Grackle
Subgroup: 2.3.5.7
Comma list: 126/125, 32805/32768
Mapping: [⟨1 0 15 -44], ⟨0 1 -8 -26]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 -8 -26 -15 -44 -38 ]]
POTE generator: ~3/2 = 701.239
- 7-odd-limit eigenmonzos: 2, 7/6
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.070407
Bischismic
Subgroup: 2.3.5.7
Comma list: 3136/3125, 32805/32768
Mapping: [⟨2 0 30 69], ⟨0 1 -8 -20]]
Mapping generators: ~567/400, ~3
Wedgie: ⟨⟨ 2 -16 -40 -30 -69 -48 ]]
POTE generator: ~3/2 = 701.592
- 7-odd-limit eigenmonzos: 2, 7/6
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.054744
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 3136/3125, 8019/8000
Mapping: [⟨2 0 30 69 102], ⟨0 1 -8 -20 -30]]
Mapping generators: ~99/70, ~3
POTE generator: ~3/2 = 701.612
Vals: Template:Val list
Badness: 0.028160
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 1001/1000, 3136/3125
Mapping generators: ~99/70, ~3
Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]
POTE generator: ~3/2 = 701.590
Vals: Template:Val list
Badness: 0.028722
Kleischismic
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1500625/1492992
Mapping: [⟨2 1 22 -15], ⟨0 2 -16 19]]
Mapping generators: ~1225/864, ~35/24
Wedgie: ⟨⟨ 4 -32 38 -60 49 178 ]]
POTE generator: ~36/35 = 50.920
Badness: 0.110583
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 9801/9800, 14641/14580
Mapping: [⟨2 1 22 -15 8], ⟨0 2 -16 19 -1]]
Mapping generators: ~99/70, ~16/11
POTE generator: ~36/35 = 50.918
Vals: Template:Val list
Badness: 0.036749
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1575/1573
Mapping: [⟨2 1 22 -15 8 15], ⟨0 2 -16 19 -1 -7]]
Mapping generators: ~99/70, ~16/11
POTE generator: ~36/35 = 50.938
Vals: Template:Val list
Badness: 0.037640
Pogo
Subgroup: 2.3.5.7
Comma list: 32805/32768, 118098/117649
Mapping: [⟨2 1 22 2], ⟨0 3 -24 5]]
Mapping generators: ~343/243, ~9/7
Wedgie: ⟨⟨ 6 -48 10 -90 -1 158 ]]
POTE generator: ~9/7 = 433.901
Badness: 0.079635
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 4000/3993, 32805/32768
Mapping: [⟨2 1 22 2 25], ⟨0 3 -24 5 -25]]
Mapping generators: ~99/70, ~9/7
POTE generator: ~9/7 = 433.911
Vals: Template:Val list
Badness: 0.031857
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4000/3993, 4225/4224
Mapping: [⟨2 1 22 2 25 -2], ⟨0 3 -24 5 -25 13]]
Mapping generators: ~99/70, ~9/7
POTE generator: ~9/7 = 433.911
Vals: Template:Val list
Badness: 0.017514
Squirrel
The squirrel temperament (29&36) has an 11/10 generator, three of which give the fourth (4/3), and thirteen of which give 7/4 with octave reduction.
Subgroup: 2.3.5.7
Comma list: 686/675, 32805/32768
Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]
Wedgie: ⟨⟨ 3 -24 -13 -45 -29 37 ]]
POTE generator: ~160/147 = 166.140
Badness: 0.174705
11-limit
Subgroup: 2.3.5.7.11
Comma list: 245/242, 686/675, 896/891
Mapping: [⟨1 2 -1 1 0], ⟨0 -3 24 13 25]]
POTE generator: ~11/10 = 166.097
Vals: Template:Val list
Badness: 0.068310
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 91/90, 169/168, 245/242, 896/891
Mapping: [⟨1 2 -1 1 0 3], ⟨0 -3 24 13 25 5]]
POTE generator: ~11/10 = 166.054
Vals: Template:Val list
Badness: 0.043750
Tertiaschis
The tertiaschis temperament (94&159, named by Xenllium) has an 11/10 generator and tempers out 1071785/1062882.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1071875/1062882
Mapping: [⟨1 2 -1 10], ⟨0 -3 24 -52]]
Wedgie: ⟨⟨ 3 -24 52 -45 74 188 ]]
POTE generator: ~192/175 = 166.019
Badness: 0.211859
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 4000/3993, 19712/19683
Mapping: [⟨1 2 -1 10 0], ⟨0 -3 24 -52 25]]
POTE generator: ~11/10 = 166.017
Vals: Template:Val list
Badness: 0.061336
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1575/1573, 10985/10976
Mapping: [⟨1 2 -1 10 0 12], ⟨0 -3 24 -52 25 -60]]
POTE generator: ~11/10 = 166.016
Vals: Template:Val list
Badness: 0.036700
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 325/324, 375/374, 385/384, 595/594, 10985/10976
Mapping: [⟨1 2 -1 10 0 12 -2], ⟨0 -3 24 -52 25 -60 44]]
POTE generator: ~11/10 = 166.012
Vals: Template:Val list
Badness: 0.026504
Term
Subgroup: 2.3.5.7
Comma list: 32805/32768, 250047/250000
Mapping: [⟨3 0 45 94], ⟨0 1 -8 -18]]
Mapping generators: ~63/50, ~3
Wedgie: ⟨⟨ 3 -24 -54 -45 -94 -58 ]]
POTE generator: ~3/2 = 701.742
- 7-odd-limit eigenmonzos: 2, 6/5
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.019950
Terminal
The terminal temperament (12&159, named by Xenllium) tempers out 441/440 and 4375/4356. In this temperament, 44/35 and 63/50 is represented as one period of 1/3 octave.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4356, 32805/32768
Mapping: [⟨3 5 5 4 4], ⟨0 -1 8 18 26]]
POTE generator: ~35/33 = 98.176
Vals: Template:Val list
Badness: 0.059502
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 625/624, 13720/13689
Mapping: [⟨3 5 5 4 4 3], ⟨0 -1 8 18 26 33]]
POTE generator: ~35/33 = 98.179
Vals: Template:Val list
Badness: 0.037082
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
Mapping: [⟨3 5 5 4 4 3 13], ⟨0 -1 8 18 26 33 -3]]
POTE generator: ~18/17 = 98.190
Vals: Template:Val list
Badness: 0.027073
Hemiterm
The hemiterm temperament (12&342, named by Xenllium) has a period of 1/6 octave and tempers out 9801/9800 (kalisma) and 151263/151250 (odiheim comma).
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 32805/32768, 151263/151250
Mapping: [⟨6 10 10 8 7], ⟨0 -1 8 18 28]]
POTE generator: ~35/33 = 98.254
Vals: Template:Val list
Badness: 0.029438
Sesquiquartififths
Subgroup: 2.3.5.7
Comma list: 2401/2400, 32805/32768
Mapping: [⟨1 1 7 5], ⟨0 4 -32 -15]]
Mapping generators: ~2, ~448/405
Wedgie: ⟨⟨ 4 -32 -15 -60 -35 55 ]]
POTE generator: ~448/405 = 175.434
- 7-odd-limit eigenmonzos: 2, 7/6
- 9-odd-limit eigenmonzos: 2, 9/7
Badness: 0.011244
Sesquart
Subgroup: 2.3.5.7.11
Comma list: 243/242, 441/440, 16384/16335
Mapping: [⟨1 1 7 5 2], ⟨0 4 -32 -15 10]]
Mapping generators: ~2, ~256/231
POTE generator: ~256/231 = 175.406
Vals: Template:Val list
Badness: 0.029306
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 243/242, 364/363, 441/440, 3584/3575
Mapping: [⟨1 1 7 5 2 -2], ⟨0 4 -32 -15 10 39]]
POTE generator: ~72/65 = 175.409
Vals: Template:Val list
Badness: 0.022396
Bisesqui
Subgroup: 2.3.5.7.11
Comma list: 2401/2400, 9801/9800, 32805/32768
Mapping: [⟨2 2 14 10 23], ⟨0 4 -32 -15 -55]]
POTE generator: ~448/405 = 175.435
Vals: Template:Val list
Badness: 0.016968
Quintilischis
The quintilischis temperament (12&253, named by Xenllium) slices the fourth (4/3) into five semitones.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 9765625/9680832
Mapping: [⟨1 2 -1 -4], ⟨0 -5 40 82]]
Wedgie: ⟨⟨ 5 -40 -82 -75 -144 -78 ]]
POTE generator: ~625/588 = 99.625
Badness: 0.253966
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 4375/4356, 32805/32768
Mapping: [⟨1 2 -1 -4 -7], ⟨0 -5 40 82 126]]
POTE generator: ~35/33 = 99.616
Vals: Template:Val list
Badness: 0.113044
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647
Mapping: [⟨1 2 -1 -4 -7 -9], ⟨0 -5 40 82 126 153]]
POTE generator: ~35/33 = 99.612
Vals: Template:Val list
Badness: 0.069127
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 595/594, 833/832, 1375/1372, 8624/8619
Mapping: [⟨1 2 -1 -4 -7 -9 5], ⟨0 -5 40 82 126 153 -11]]
POTE generator: ~18/17 = 99.612
Vals: Template:Val list
Badness: 0.045992
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971
Mapping: [⟨1 2 -1 -4 -7 -9 5 4], ⟨0 -5 40 82 126 153 -11 3]]
POTE generator: ~18/17 = 99.615
Vals: Template:Val list
Badness: 0.038155
Sextilififths
The sextilififths or sextilischis (the latter is named by Xenllium) slices the fourth (4/3) into six small semitones, which serves as both 21/20 and 22/21.
Subgroup: 2.3.5.7
Comma list: 32768/32805, 235298/234375
Mapping: [⟨1 2 -1 -1], ⟨0 -6 48 55]]
Mapping generators: ~2, ~21/20
Wedgie: ⟨⟨ 6 -48 -55 -90 -104 7 ]]
POTE generator: ~21/20 = 83.053
Badness: 0.108794
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4000/3993, 235298/234375
Mapping: [⟨1 2 -1 -1 0], ⟨0 -6 48 55 50]]
Mapping generators: ~2, ~21/20
POTE generator: ~21/20 = 83.049
Vals: Template:Val list
Badness: 0.045457
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 676/675, 10985/10976
Mapping: [⟨1 2 -1 -1 0 1], ⟨0 -6 48 55 50 39]]
Mapping generators: ~2, ~21/20
POTE generator: ~21/20 = 83.049
Vals: Template:Val list
Badness: 0.025276
Tsaharuk
Subgroup: 2.3.5.7
Comma list: 32805/32768, 420175/419904
Mapping: [⟨1 1 7 0], ⟨0 5 -40 24]]
Mapping generators: ~2, ~243/224
Wedgie: ⟨⟨ 5 -40 24 -75 24 168 ]]
POTE generator: ~243/224 = 140.350
Badness: 0.030697
Quanharuk
Subgroup: 2.3.5.7
Comma list: 16875/16807, 32805/32768
Mapping: [⟨1 0 15 12], ⟨0 5 -40 -29]]
Mapping generators: ~2, ~56/45
Wedgie: ⟨⟨ 5 -40 -29 -75 -60 45 ]]
POTE generator: ~56/45 = 380.355
Badness: 0.071950
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 1375/1372, 32805/32768
Mapping: [⟨1 0 15 12 -7], ⟨0 5 -40 -29 33]]
Mapping generators: ~2, ~56/45
POTE generator: ~56/45 = 380.352
Vals: Template:Val list
Badness: 0.031549
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1375/1372, 4096/4095
Mapping: [⟨1 0 15 12 -7 -15], ⟨0 5 -40 -29 33 59]]
Mapping generators: ~2, ~56/45
POTE generator: ~56/45 = 380.351
Vals: Template:Val list
Badness: 0.021392
Quadrant
The quadrant temperament (12&224, named by Xenllium) has a period of quarter octave and tempers out the dimcomp comma, 390625/388962. In this temperament, 25/21 is mapped into quarter octave.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 390625/388962
Mapping: [⟨4 0 60 119], ⟨0 1 -8 -17]]
Mapping generators: ~25/21, ~3
Wedgie: ⟨⟨ 4 -32 -68 -60 -119 -68 ]]
POTE generator: ~28/25 = 198.177
Badness: 0.110242
11-limit
Subgroup: 2.3.5.7.11
Comma list: 1375/1372, 6250/6237, 32805/32768
Mapping: [⟨4 0 60 119 185], ⟨0 1 -8 -17 -27]]
POTE generator: ~28/25 = 198.181
Vals: Template:Val list
Badness: 0.045738
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1375/1372, 2080/2079, 10648/10647
Mapping: [⟨4 0 60 119 185 224], ⟨0 1 -8 -17 -27 -33]]
POTE generator: ~28/25 = 198.184
Vals: Template:Val list
Badness: 0.027243
Septant
The septant temperament (224&301, named by Xenllium) has a period of 1/7 octave and tempers out the akjaysma, [47 -7 -7 -7⟩.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 516560652/514714375
Mapping: [⟨7 11 17 19], ⟨0 1 -8 7]]
Wedgie: ⟨⟨ 7 -56 49 -105 58 271 ]]
POTE generator: ~3/2 = 701.702
Badness: 0.111142
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 24057/24010, 32805/32768
Mapping: [⟨7 11 17 19 23], ⟨0 1 -8 7 13]]
POTE generator: ~3/2 = 701.719
Vals: Template:Val list
Badness: 0.044122
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
Mapping: [⟨7 11 17 19 23 26], ⟨0 1 -8 7 13 -1]]
POTE generator: ~3/2 = 701.724
Vals: Template:Val list
Badness: 0.024706
Octant
The octant temperament (224&472) has a period of 1/8 octave. In this temperament, 12/11, 35/27, and 99/70 are mapped into 1\8, 3\8, and 4\8 respectively.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 2259436291848/2251875390625
Mapping: [⟨8 0 120 -117], ⟨0 1 -8 11]]
Mapping generators: ~42875/39366, ~3
Wedgie: ⟨⟨ 8 -64 88 -120 117 384 ]]
POTE generator: ~3/2 = 701.713
Badness: 0.157186
11-limit
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 32805/32768, 46656/46585
Mapping: [⟨8 0 120 -117 15], ⟨0 1 -8 11 1]]
Mapping generators: ~12/11, ~3
POTE generator: ~3/2 = 701.713
Vals: Template:Val list
Badness: 0.044778
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1575/1573, 2200/2197, 6656/6655
Mapping: [⟨8 0 120 -117 15 93], ⟨0 1 -8 11 1 -5]]
Mapping generators: ~12/11, ~3
POTE generator: ~3/2 = 701.725
Vals: Template:Val list
Badness: 0.030425