Meantone family: Difference between revisions
No edit summary |
No edit summary |
||
| Line 430: | Line 430: | ||
Badness: 0.026288 | Badness: 0.026288 | ||
== Migration == | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 121/120, 126/125 | |||
Mapping: [{{val| 1 1 0 -3 2 }}, {{val| 0 2 8 20 5 }}] | |||
Mapping generators: ~2, ~11/9 | |||
POTE generator: ~11/9 = 348.182 | |||
Vals: {{Val list| 7d, 24d, 31, 100de, 131bdee, 162bdee }} | |||
Badness: 0.025516 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 66/65, 81/80, 121/120, 126/125 | |||
Mapping: [{{val| 1 1 0 -3 2 4 }}, {{val| 0 2 8 20 5 -1 }}] | |||
Mapping generators: ~2, ~11/9 | |||
POTE generator: ~11/9 = 348.490 | |||
Vals: {{Val list| 7d, 24d, 31, 55d }} | |||
Badness: 0.028071 | |||
== Bimeantone == | == Bimeantone == | ||
| Line 532: | Line 562: | ||
Scales: [[flattone12]] | Scales: [[flattone12]] | ||
== 13-limit == | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 551: | Line 581: | ||
Scales: [[flattone12]] | Scales: [[flattone12]] | ||
== Ptolemy == | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 81/80, 121/120, 525/512 | |||
Mapping: [{{val| 1 1 0 8 2 }}, {{val| 0 2 8 -18 5 }}] | |||
POTE generator: ~11/9 = 346.922 | |||
Vals: {{Val list| 7, 31dd, 38d, 45e, 83bcddee }} | |||
Badness: 0.058785 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 65/64, 81/80, 105/104, 121/120 | |||
Mapping: [{{val| 1 1 0 8 2 6 }}, {{val| 0 2 8 -18 5 -8 }}] | |||
POTE generator: ~11/9 = 346.910 | |||
Vals: {{Val list| 7, 31ddf, 38df, 45ef, 83bcddeeff }} | |||
Badness: 0.034316 | |||
= Dominant = | = Dominant = | ||
| Line 608: | Line 664: | ||
Vals: {{Val list| 12f, 17c, 29cdef }} | Vals: {{Val list| 12f, 17c, 29cdef }} | ||
Badness: 0. | Badness: 0.024108 | ||
=== Dominion === | === Dominion === | ||
| Line 621: | Line 677: | ||
POTE generator: ~3/2 = 704.905 | POTE generator: ~3/2 = 704.905 | ||
Badness: 0. | Badness: 0.027295 | ||
== Domineering == | == Domineering == | ||
| Line 685: | Line 741: | ||
Vals: {{Val list| 5e, 7, 12f, 19df }} | Vals: {{Val list| 5e, 7, 12f, 19df }} | ||
Badness: 0.018289 | |||
== Domination == | == Domination == | ||
| Line 697: | Line 755: | ||
Vals: {{Val list| 5e, 12e, 17c, 46cd }} | Vals: {{Val list| 5e, 12e, 17c, 46cd }} | ||
Badness: 0. | Badness: 0.036562 | ||
=== 13-limit === | === 13-limit === | ||
| Line 710: | Line 768: | ||
Vals: {{Val list| 5e, 12e, 17c }} | Vals: {{Val list| 5e, 12e, 17c }} | ||
Badness: 0. | Badness: 0.027435 | ||
== Arnold == | == Arnold == | ||
| Line 764: | Line 822: | ||
Badness: 0.021098 | Badness: 0.021098 | ||
= Sharptone = | == Maqamic == | ||
Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo|12EDO]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val. | <span style="display: block; text-align: right;">[[:de:maqamisch|Deutsch]]</span> | ||
{{main| Maqamic }} | |||
Maqamic has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation. | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 36/35, 64/63, 121/120 | |||
Mapping: [{{val| 1 1 0 4 2 }}, {{val| 0 2 8 -4 5 }}] | |||
Mapping generators: ~2, ~11/9 | |||
POTE generator: ~11/9 = 350.934 | |||
Vals: {{Val list| 7, 17c, 24d, 41cd }} | |||
Badness: 0.040240 | |||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 36/35, 64/63, 66/65, 121/120 | |||
Mapping: [{{val| 1 1 0 4 2 4 }}, {{val| 0 2 8 -4 5 -1 }}] | |||
Mapping generators: ~2, ~11/9 | |||
POTE generator: ~11/9 = 350.816 | |||
Vals: {{Val list| 7, 17c, 24d, 41cd }} | |||
Badness: 0.027214 | |||
= Sharptone = | |||
Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. [[12edo|12EDO]] tuning does sharptone about as well as such a thing can be done, of course not in its patent val. | |||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 779: | Line 872: | ||
{{Val list|legend=1| 5, 7d, 12d }} | {{Val list|legend=1| 5, 7d, 12d }} | ||
[[Badness]]: 0. | [[Badness]]: 0.024848 | ||
== Meanertone == | == Meanertone == | ||
| Line 792: | Line 885: | ||
Vals: {{Val list| 5, 7d, 12de }} | Vals: {{Val list| 5, 7d, 12de }} | ||
Badness: 0. | Badness: 0.025167 | ||
= Plutus = | = Plutus = | ||
| Line 805: | Line 898: | ||
[[POTE generator]]: ~3/2 = 682.895 | [[POTE generator]]: ~3/2 = 682.895 | ||
{{Val list|legend=1| | {{Val list|legend=1| 2cd, 5d, 7 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.045275 | ||
== 11-limit == | == 11-limit == | ||
| Line 818: | Line 911: | ||
POTE generator: ~3/2 = 685.234 | POTE generator: ~3/2 = 685.234 | ||
Vals: {{Val list| 5de, 7 }} | Vals: {{Val list| 2cde, 5de, 7 }} | ||
Badness: 0. | Badness: 0.032521 | ||
= Supermean = | = Supermean = | ||
| Line 833: | Line 926: | ||
{{Val list|legend=1| 5d, 12d, 17c, 29c }} | {{Val list|legend=1| 5d, 12d, 17c, 29c }} | ||
[[Badness]]: 0. | [[Badness]]: 0.134204 | ||
== 11-limit == | == 11-limit == | ||
| Line 844: | Line 937: | ||
POTE generator: ~3/2 = 705.096 | POTE generator: ~3/2 = 705.096 | ||
{{Val list | Vals: {{Val list| 5de, 12de, 17c, 29c }} | ||
Badness: 0. | Badness: 0.063262 | ||
== 13-limit == | == 13-limit == | ||
| Line 857: | Line 950: | ||
POTE generator: ~3/2 = 705.094 | POTE generator: ~3/2 = 705.094 | ||
{{Val list | Vals: {{Val list| 5de, 12de, 17c, 29c }} | ||
Badness: 0.040324 | |||
= Godzilla = | = Godzilla = | ||
| Line 885: | Line 980: | ||
{{Val list|legend=1| 5, 14c, 19 }} | {{Val list|legend=1| 5, 14c, 19 }} | ||
[[Badness]]: 0. | [[Badness]]: 0.026747 | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 909: | Line 1,003: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 930: | Line 1,023: | ||
== Semafour == | == Semafour == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 946: | Line 1,038: | ||
== Varan == | == Varan == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 962: | Line 1,053: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 978: | Line 1,068: | ||
== Baragon == | == Baragon == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 991: | Line 1,080: | ||
{{Val list|legend=1| 19, 24, 43d }} | {{Val list|legend=1| 19, 24, 43d }} | ||
Badness: 0. | Badness: 0.035673 | ||
= Injera = | = Injera = | ||
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera. | Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. [[38edo]], which is two parallel [[19edo]]s, is an excellent tuning for injera. | ||
| Line 1,024: | Line 1,112: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,045: | Line 1,132: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,066: | Line 1,152: | ||
=== Enjera === | === Enjera === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,082: | Line 1,167: | ||
== Injerous == | == Injerous == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,098: | Line 1,182: | ||
== Lahoh == | == Lahoh == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,133: | Line 1,216: | ||
{{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }} | {{Val list|legend=1| 7, 17c, 24, 31, 100e, 131bee }} | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
== Mohoho == | == Mohoho == | ||
Subgroup: 2.3.5.11.13 | Subgroup: 2.3.5.11.13 | ||
| Line 1,154: | Line 1,234: | ||
POTE generator: ~11/9 = 348.9155 | POTE generator: ~11/9 = 348.9155 | ||
{{Val list | Vals: {{Val list| 7, 17c, 24, 31, 55, 86ef, 141ceff }} | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
= Mohajira = | = Mohajira = | ||
{{main|Mohajira}} | {{main|Mohajira}} | ||
Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. | Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. [[31edo]] makes for an excellent (7-limit) mohajira tuning, with generator 9/31. | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 1,194: | Line 1,269: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,271: | Line 1,345: | ||
= Mohamaq = | = Mohamaq = | ||
Subgroup: 2.3.5.7 | Subgroup: 2.3.5.7 | ||
| Line 1,290: | Line 1,362: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,308: | Line 1,379: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,325: | Line 1,395: | ||
Scales: [[mohaha7]], [[mohaha10]] | Scales: [[mohaha7]], [[mohaha10]] | ||
= | = Orphic = | ||
Subgroup: 2.3.5.7 | |||
[[Comma list]]: 81/80, 5898240/5764801 | |||
[[Mapping]]: [{{val| 2 1 -4 4 }}, {{val|0 4 16 3 }}] | |||
[[Mapping]]: [{{val| 2 1 -4 4 }}, {{val|0 4 16 3 }}] | |||
Mapping generators: ~2401/1728, ~343/288 | Mapping generators: ~2401/1728, ~343/288 | ||
| Line 1,439: | Line 1,413: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,455: | Line 1,428: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,471: | Line 1,443: | ||
= Mothra = | = Mothra = | ||
Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to [[slendric]]. | Mothra splits the fifth into three 8/7 generators. It uses [[1029/1024]], the gamelisma, to accomplish this deed and also tempers out [[1728/1715]], the orwell comma. Using [[31edo]] with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to [[slendric]]. | ||
| Line 1,500: | Line 1,471: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,516: | Line 1,486: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,532: | Line 1,501: | ||
== Cynder == | == Cynder == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,548: | Line 1,516: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,564: | Line 1,531: | ||
== Mosura == | == Mosura == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,580: | Line 1,546: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,596: | Line 1,561: | ||
=== Music === | === Music === | ||
* [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil] | * [http://micro.soonlabel.com/16-ET/mothra/20141028_mothra16br4.mp3 Prelude for solo piano in mothra16, brat 4 tuning] by [http://chrisvaisvil.com/prelude-for-solo-piano-in-mothra16-brat-4-tuning/ Chris Vaisvil] | ||
= Squares = | = Squares = | ||
Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]]. | Squares splits the interval of an eleventh, or 8/3, into four supermajor third ([[9/7]]) intervals, and uses it for a generator. [[31edo]], with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out [[2401/2400]], the breedsma, as well as [[2430/2401]]. | ||
| Line 1,630: | Line 1,593: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,648: | Line 1,610: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,666: | Line 1,627: | ||
=== Agora === | === Agora === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,684: | Line 1,644: | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 1,700: | Line 1,659: | ||
==== 19-limit ==== | ==== 19-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 1,716: | Line 1,674: | ||
== Cuboctahedra == | == Cuboctahedra == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,760: | Line 1,717: | ||
== Liesel == | == Liesel == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,793: | Line 1,749: | ||
== Elisa == | == Elisa == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,809: | Line 1,764: | ||
== Lisa == | == Lisa == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,825: | Line 1,779: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,860: | Line 1,813: | ||
== 11-limit == | == 11-limit == | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,876: | Line 1,828: | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,892: | Line 1,843: | ||
== 17-limit == | == 17-limit == | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
| Line 1,908: | Line 1,858: | ||
== 19-limit == | == 19-limit == | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
| Line 2,020: | Line 1,969: | ||
= Undevigintone = | = Undevigintone = | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 2,033: | Line 1,981: | ||
{{Val list|legend=1| 19, 38d }} | {{Val list|legend=1| 19, 38d }} | ||
Badness: 0. | Badness: 0.036387 | ||
== 13-limit == | == 13-limit == | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 2,047: | Line 1,994: | ||
POTE generator: ~11/8 = 537.061 | POTE generator: ~11/8 = 537.061 | ||
{{Val list | Vals: {{Val list| 19, 38df }} | ||
Badness: 0. | Badness: 0.022933 | ||
[[Category:Regular temperament theory]] | [[Category:Regular temperament theory]] | ||
Revision as of 12:31, 30 May 2021
The 5-limit parent comma of the meantone family is the Didymus or syntonic comma, 81/80. This is the one they all temper out. The period is an octave, the generator is a fifth, and four fifths go to make up a 5/1 interval.
Meantone (12&19, 2.3.5)
Subgroup: 2.3.5
Comma list: 81/80
Mapping: [⟨1 0 -4], ⟨0 1 4]]
Mapping generators: ~2, ~3
Wedgie: ⟨⟨ 1 4 4 ]]
POTE generator: ~3/2 = 696.239
- diamond monotone range: [685.714, 720.000] (4\7 to 3\5)
- diamond purer range: [694.786, 701.955] (1/3 comma to Pythagorean)
- diamond nice range: [694.786, 701.955]
Badness: 0.007381
Scales: meantone5, meantone7, meantone12
Seven-limit extensions
The 7-limit extensions of meantone are:
- Septimal meantone, with normal comma list [[-4 4 -1⟩, [-13 10 0 -1⟩],
- Flattone, with normal list [[-4 4 -1⟩, [-17 9 0 1⟩],
- Dominant, with normal list [[-4 4 -1⟩, [6 -2 0 -1⟩],
- Sharptone, with normal list [[-4 4 -1⟩, [2 -3 0 1⟩],
- Injera, with normal list [[-4 4 -1⟩, [-7 8 0 -2⟩],
- Mohajira, with normal list [[-4 4 -1⟩, [-23 11 0 2⟩],
- Godzilla, with normal list [[-4 4 -1⟩, [-4 -1 0 2⟩],
- Mothra, with normal list [[-4 4 -1⟩, [-10 1 0 3⟩],
- Squares, with normal list [[-4 4 -1⟩, [-3 9 0 -4⟩], and
- Liese, with normal list [[-4 4 -1⟩, [-9 11 0 -3⟩].
Septimal meantone
The 7/4 of septimal meantone is the augmented sixth, C-A#, and other septimal intervals are 7/6, C-D#, the augmented second, 7/5, C-F#, the tritone, and 21/16, C-E#, the augmented third. Septimal meantone tempers out the common 7-limit commas 126/125 and 225/224 and in fact can be defined as the 7-limit temperament that tempers out any two of 81/80, 126/125 and 225/224.
Subgroup: 2.3.5.7
Comma list: 81/80, 126/125
Mapping: [⟨1 0 -4 -13], ⟨0 1 4 10]]
Wedgie: ⟨⟨ 1 4 10 4 13 12 ]]
POTE generator: ~3/2 = 696.495
- 7- and 9-odd-limit
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [-3 0 5/2 0⟩]
- Eigenmonzos: 2, 5
- diamond monotone range: [694.737, 700.000] (11\19 to 7\12)
- diamond purer range: [694.786, 701.955] (1/3 comma to Pythagorean)
- diamond nice range: [694.786, 700.000]
Algebraic generator: Cybozem, the real root of 15x3 - 10x2 - 18, 503.4257 cents. The recurrence converges quickly.
Badness: 0.013707
Scales: meantone5, meantone7, meantone12
Unidecimal meantone aka Huygens
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 126/125
Mapping: [⟨1 0 -4 -13 -25], ⟨0 1 4 10 18]]
POTE generator: ~3/2 = 696.967
Minimax tuning:
- [[1 0 0 0 0⟩, [25/16 -1/8 0 0 1/16⟩, [9/4 -1/2 0 0 1/4⟩, [21/8 -5/4 0 0 5/8⟩, [25/8 -9/4 0 0 9/8⟩]
- Eigenmonzos: 2, 11/9
Tuning ranges:
- diamond monotone range: [696.774, 700.000] (18\31 to 7\12)
- diamond purer range: [691.202, 701.955]
- diamond nice range: [696.774, 700.000]
Algebraic generator: Traverse, the positive real root of x4 + 2x - 13, or 696.9529 cents.
Vals: Template:Val list
Badness: 0.017027
- Music
Tridecimal meantone
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 99/98, 105/104
Mapping: [⟨1 0 -4 -13 -25 -20], ⟨0 1 4 10 18 15]]
POTE generator: ~3/2 = 696.642
Vals: Template:Val list
Badness: 0.018048
Grosstone
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 -25 29], ⟨0 1 4 10 18 -16]]
POTE generator: ~3/2 = 697.264
Tuning ranges:
- diamond monotone range: [696.774, 700.000] (18\31 to 7\12)
- diamond purer range: [691.202, 701.955]
- diamond nice range: [696.774, 700.000]
Vals: Template:Val list
Badness: 0.025899
Meridetone
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 81/80, 99/98, 126/125
Mapping: [⟨1 0 -4 -13 -25 -39], ⟨0 1 4 10 18 27]]
POTE generator: ~3/2 = 697.529
Vals: Template:Val list
Badness: 0.026421
Hemimeantone
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 126/125, 169/168
Mapping: [⟨1 0 -4 -13 -25 -5], ⟨0 2 8 20 36 11]]
Mapping generators: ~2, ~26/15
POTE generator: ~15/13 = 251.535
Vals: Template:Val list
Badness: 0.031433
Meanpop
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 385/384
Mapping: [⟨1 0 -4 -13 24], ⟨0 1 4 10 -13]]
Mapping generator: ~2, ~3
POTE generator: ~3/2 = 696.434
Minimax tuning:
- 11-odd-limit: 1/4 comma
- [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [-3 0 5/2 0 0⟩, [11 0 -13/4 0 0⟩]
- Eigenmonzos: 2, 5
Tuning ranges:
- diamond monotone range: [694.737, 696.774] (11\19 to 18\31)
- diamond purer range: [691.202, 701.955]
- diamond nice range: [694.737, 696.774]
Algebraic generator: Cybozem; or else Radieubiz, the real root of 3x3 + 6x - 19. Unlike Cybozem, the recurrence for Radieubiz does not converge.
Vals: Template:Val list
Badness: 0.021543
- Music
- Scott Joplin's "The Entertainer" tuned into meanpop[dead link]
- Twinkle canon – 50 edo by Claudi Meneghin
Tridecimal Meanpop
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 126/125, 144/143
Mapping: [⟨1 0 -4 -13 24 -20], ⟨0 1 4 10 -13 15]]
Mapping generator: ~2, ~3
POTE generator: ~3/2 = 696.211
Tuning ranges:
- diamond monotone range: [694.737, 696.774] (11\19 to 18\31)
- diamond purer range: [691.202, 701.955]
- diamond nice range: [694.737, 696.774]
Vals: Template:Val list
Badness: 0.020883
Meanplop
Subgroup: 2.3.5.7.11.13
Comma list: 65/64, 78/77, 81/80, 91/90
Mapping: [⟨1 0 -4 -13 24 10], ⟨0 1 4 10 -13 -4]]
POTE generator: ~3/2 = 696.202
Vals: Template:Val list
Badness: 0.027666
Meanenneadecal
Subgroup: 2.3.5.7.11
Comma list: 45/44, 56/55, 81/80
Mapping: [⟨1 0 -4 -13 -6], ⟨0 1 4 10 6]]
POTE generator: ~3/2 = 696.250
Vals: Template:Val list
Badness: 0.021423
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 56/55, 78/77, 81/80
Mapping: [⟨1 0 -4 -13 -6 -20], ⟨0 1 4 10 6 15]]
POTE generator: ~3/2 = 696.146
Vals: Template:Val list
Badness: 0.021182
Vincenzo
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10], ⟨0 1 4 10 6 -4]]
POTE generator: ~3/2 = 695.060
Vals: Template:Val list
Badness: 0.024763
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 45/44, 52/51, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12], ⟨0 1 4 10 6 -4 -5]]
POTE generator: ~3/2 = 695.858
Vals: Template:Val list
Badness: 0.025535
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12 9], ⟨0 1 4 10 6 -4 -5 -3]]
POTE generator: ~3/2 = 696.131
Vals: Template:Val list
Badness: 0.022302
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 39/38, 45/44, 52/51, 56/55, 65/64, 69/68, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14], ⟨0 1 4 10 6 -4 -5 -3 -6]]
POTE generator: ~3/2 = 696.044
Vals: Template:Val list
Badness: 0.020139
29-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8], ⟨0 1 4 10 6 -4 -5 -3 -6 -2]]
POTE generator: ~3/2 = 695.913
Vals: Template:Val list
Badness: 0.018168
31-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 81/80, 93/92
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7]]
POTE generator: ~3/2 = 695.750
Vals: Template:Val list
Badness: 0.017069
37-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9]]
POTE generator: ~3/2 = 695.603
Vals: Template:Val list
Badness: 0.016129
41-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 93/92, 124/123
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8]]
POTE generator: ~3/2 = 695.696
Vals: Template:Val list
Badness: 0.015356
43-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 124/123
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1]]
POTE generator: ~3/2 = 695.688
Vals: Template:Val list
Badness: 0.013906
47-limit
Subgroup: 2.3.5.7.11.13.17.19.23.29.31.37.41.43.47
Comma list: 39/38, 45/44, 52/51, 56/55, 58/57, 65/64, 69/68, 75/74, 81/80, 86/85, 93/92, 95/94, 124/123
Mapping: [⟨1 0 -4 -13 -6 10 12 9 14 8 16 -9 18 7 4], ⟨0 1 4 10 6 -4 -5 -3 -6 -2 -7 9 -8 -1 1]]
POTE generator: ~3/2 = 695.676
Vals: Template:Val list
Badness: 0.013818
Meanundec
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 40/39, 45/44, 56/55
Mapping: [⟨1 0 -4 -13 -6 -1], ⟨0 1 4 10 6 3]]
POTE generator: ~3/2 = 697.254
Vals: Template:Val list
Badness: 0.024243
Meanundeci
Subgroup: 2.3.5.7.11
Comma list: 33/32, 55/54, 77/75
Mapping: [⟨1 0 -4 -13 5], ⟨0 1 4 10 -1]]
POTE generator: ~3/2 = 694.689
Vals: Template:Val list
Badness: 0.031539
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 33/32, 55/54, 65/64, 77/75
Mapping: [⟨1 0 -4 -13 5 10], ⟨0 1 4 10 -1 -4]]
POTE generator: ~3/2 = 694.764
Vals: Template:Val list
Badness: 0.026288
Migration
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120, 126/125
Mapping: [⟨1 1 0 -3 2], ⟨0 2 8 20 5]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.182
Vals: Template:Val list
Badness: 0.025516
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 121/120, 126/125
Mapping: [⟨1 1 0 -3 2 4], ⟨0 2 8 20 5 -1]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.490
Vals: Template:Val list
Badness: 0.028071
Bimeantone
11/8 is mapped to half octave minus the meantone diesis.
Subgroup: 2.3.5.7.11
Comma list: 81/80, 126/125, 245/242
Mapping: [⟨2 0 -8 -26 -31], ⟨0 1 4 10 12]]
Mapping generators: ~63/44, ~3
POTE generator: ~3/2 = 696.016
Vals: Template:Val list
Badness: 0.038122
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 126/125, 245/242
Mapping: [⟨2 0 -8 -26 -31 -40], ⟨0 1 4 10 12 15]]
Mapping generators: ~55/39, ~3
POTE generator: ~3/2 = 695.836
Vals: Template:Val list
Badness: 0.028817
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 105/104, 126/125, 189/187, 221/220
Mapping: [⟨2 0 -8 -26 -31 -40 5], ⟨0 1 4 10 12 15 1]]
Mapping generators: ~17/12, ~3
POTE generator: ~3/2 = 695.783
Vals: Template:Val list
Badness: 0.022666
Flattone
In flattone, 9 generator steps of 4/3 get to the interval class for 7, meaning that 7/4 is a diminished seventh interval (C-Bbb). Other intervals are 7/6, a diminished third (C-Ebb), and 7/5, a doubly diminshed fifth (C-Gbb). Good tunings for flattone are 26EDO, 45EDO and 64EDO.
Subgroup: 2.3.5.7
Comma list: 81/80, 525/512
Mapping: [⟨1 0 -4 17], ⟨0 1 4 -9]]
Wedgie: ⟨⟨ 1 4 -9 4 -17 -32 ]]
POTE generator: ~3/2 = 693.779
- [[1 0 0 0⟩, [21/13 0 1/13 -1/13⟩, [32/13 0 4/13 -4/13⟩, [32/13 0 -9/13 9/13⟩]
- Eigenmonzos: 2, 7/5
- [[1 0 0 0⟩, [17/11 2/11 0 -1/11⟩, [24/11 8/11 0 -4/11⟩, [34/11 -18/11 0 9/11⟩]
- Eigenmonzos: 2, 9/7
- diamond monotone range: [692.308, 694.737] (15\26 to 11\19)
- diamond purer range: [692.353, 701.955]
- diamond nice range: [692.353, 694.737]
Algebraic generator: Squarto, the positive root of 8x2 - 4x - 9, at 506.3239 cents, equal to (1 + sqrt (19))/4.
Badness: 0.038553
Scales: flattone12
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 385/384
Mapping: [⟨1 0 -4 17 -6], ⟨0 1 4 -9 6]]
POTE generator: ~3/2 = 693.126
Tuning ranges:
- diamond monotone range: [692.308, 694.737] (15\26 to 11\19)
- diamond purer range: [682.502, 701.955]
- diamond nice range: [692.308, 694.737]
Vals: Template:Val list
Badness: 0.033839
Scales: flattone12
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 65/64, 78/77, 81/80
Mapping: [⟨1 0 -4 17 -6 10], ⟨0 1 4 -9 6 -4]]
POTE generator: ~3/2 = 693.058
Tuning ranges:
- diamond monotone range: [692.308, 694.737] (15\26 to 11\19)
- diamond purer range: [682.502, 701.955]
- diamond nice range: [692.308, 694.737]
Vals: Template:Val list
Badness: 0.022260
Scales: flattone12
Ptolemy
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120, 525/512
Mapping: [⟨1 1 0 8 2], ⟨0 2 8 -18 5]]
POTE generator: ~11/9 = 346.922
Vals: Template:Val list
Badness: 0.058785
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 65/64, 81/80, 105/104, 121/120
Mapping: [⟨1 1 0 8 2 6], ⟨0 2 8 -18 5 -8]]
POTE generator: ~11/9 = 346.910
Vals: Template:Val list
Badness: 0.034316
Dominant
The interval class for 7 is obtained from two fourths in succession, so that 7/4 is a minor seventh. The 7/6 interval is, like 6/5, now a minor third, and 7/5 is a diminished fifth. An excellent tuning for dominant is 12EDO, but it also works well with the Pythagorean tuning of pure 3/2 fifths, and with 29EDO, 41EDO, or 53EDO.
Subgroup: 2.3.5.7
Comma list: 36/35, 64/63
Mapping: [⟨1 0 -4 6], ⟨0 1 4 -2]]
Wedgie: ⟨⟨ 1 4 -2 4 -6 -16 ]]
POTE generator: ~3/2 = 701.573
- diamond monotone range: [700.000, 720.000] (7\12 to 3\5)
- diamond purer range: [694.786, 715.587]
- diamond nice range: [700.000, 715.587]
Badness: 0.020690
11-limit
Subgroup: 2.3.5.7.11
Comma list: 36/35, 56/55, 64/63
Mapping: [⟨1 0 -4 6 13], ⟨0 1 4 -2 -6]]
Tuning ranges:
- diamond monotone range: [700.000, 705.882] (7\12 to 10\17)
- diamond purer range: [691.202, 715.587]
- diamond nice range: [700.000, 705.882]
POTE generator: ~3/2 = 703.254
Vals: Template:Val list
Badness: 0.024180
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 56/55, 64/63, 66/65
Mapping: [⟨1 0 -4 6 13 18], ⟨0 1 4 -2 -6 -9]]
POTE generator: ~3/2 = 703.636
Tuning ranges:
- diamond monotone range: 705.882 (10\17)
- diamond purer range: [691.202, 715.587]
- diamond nice range: 705.882
Vals: Template:Val list
Badness: 0.024108
Dominion
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 56/55, 64/63
Mapping: [⟨1 0 -4 6 13 -9], ⟨0 1 4 -2 -6 8]]
Vals: Template:Val list
POTE generator: ~3/2 = 704.905
Badness: 0.027295
Domineering
Subgroup: 2.3.5.7.11
Comma list: 36/35, 45/44, 64/63
Mapping: [⟨1 0 -4 6 -6], ⟨0 1 4 -2 6]]
POTE generator: ~3/2 = 698.776
Vals: Template:Val list
Badness: 0.021978
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 45/44, 52/49, 64/63
Mapping: [⟨1 0 -4 6 -6 10], ⟨0 1 4 -2 6 -4]]
POTE generator: ~3/2 = 695.762
Vals: Template:Val list
Badness: 0.027039
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 36/35, 45/44, 51/49, 52/49, 64/63
Mapping: [⟨1 0 -4 6 -6 10 12], ⟨0 1 4 -2 6 -4 -5]]
POTE generator: ~3/2 = 696.115
Vals: Template:Val list
Badness: 0.024539
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 36/35, 39/38, 45/44, 51/49, 52/49, 57/56
Mapping: [⟨1 0 -4 6 -6 10 12 9], ⟨0 1 4 -2 6 -4 -5 -3]]
POTE generator: ~3/2 = 696.217
Vals: Template:Val list
Badness: 0.020398
Dominatrix
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 36/35, 45/44, 64/63
Mapping: [⟨1 0 -4 6 -6 -1], ⟨0 1 4 -2 6 3]]
POTE generator: ~3/2 = 698.544
Vals: Template:Val list
Badness: 0.018289
Domination
Subgroup: 2.3.5.7.11
Comma list: 36/35, 64/63, 77/75
Mapping: [⟨1 0 -4 6 -14], ⟨0 1 4 -2 11]]
POTE generator: ~3/2 = 705.004
Vals: Template:Val list
Badness: 0.036562
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 36/35, 64/63, 66/65
Mapping: [⟨1 0 -4 6 -14 -9], ⟨0 1 4 -2 11 8]]
POTE generator: ~3/2 = 705.496
Vals: Template:Val list
Badness: 0.027435
Arnold
Subgroup: 2.3.5.7.11
Comma list: 22/21, 33/32, 36/35
Mapping: [⟨1 0 -4 6 5], ⟨0 1 4 -2 -1]]
POTE generator: ~3/2 = 698.491
Vals: Template:Val list
Badness: 0.026141
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 22/21, 27/26, 33/32, 36/35
Mapping: [⟨1 0 -4 6 5 -1], ⟨0 1 4 -2 3]]
POTE generator: ~3/2 = 696.743
Vals: Template:Val list
Badness: 0.023300
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49
Mapping: [⟨1 0 -4 6 5 -1 12], ⟨0 1 4 -2 3 -5]]
POTE generator: ~3/2 = 696.978
Vals: Template:Val list
Badness: 0.024535
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 22/21, 27/26, 33/32, 36/35, 51/49, 57/56
Mapping: [⟨1 0 -4 6 5 -1 12 9], ⟨0 1 4 -2 3 -5 -3]]
POTE generator: ~3/2 = 697.068
Vals: Template:Val list
Badness: 0.021098
Maqamic
Maqamic has a hemififth generator (~11/9) and tempers out 36/35 and 121/120. It makes the most sense if viewed as an adaptive temperament, whereby 7/4 and 9/5 simply share an equivalence class in the resulting scales, but don't need to share a particular tempered "middle-of-the-road" intonation.
Subgroup: 2.3.5.7.11
Comma list: 36/35, 64/63, 121/120
Mapping: [⟨1 1 0 4 2], ⟨0 2 8 -4 5]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.934
Vals: Template:Val list
Badness: 0.040240
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 36/35, 64/63, 66/65, 121/120
Mapping: [⟨1 1 0 4 2 4], ⟨0 2 8 -4 5 -1]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.816
Vals: Template:Val list
Badness: 0.027214
Sharptone
Sharptone is a low-accuracy temperament tempering out 21/20 and 28/27. In sharptone, a 7/4 is a major sixth, a 7/6 a whole tone, and a 7/5 a fourth. Genuinely septimal sounding harmony therefore cannot be expected, but it can be used to translate, more or less, 7-limit JI into 5-limit meantone. 12EDO tuning does sharptone about as well as such a thing can be done, of course not in its patent val.
Subgroup: 2.3.5.7
Comma list: 21/20, 28/27
Mapping: [⟨1 0 -4 -2], ⟨0 1 4 3]]
Wedgie: ⟨⟨ 1 4 3 4 2 -4 ]]
POTE generator: ~3/2 = 700.140
Badness: 0.024848
Meanertone
Subgroup: 2.3.5.7.11
Comma list: 21/20, 28/27, 33/32
Mapping: [⟨1 0 -4 -2 5], ⟨0 1 4 3 -1]]
POTE generator: ~3/2 = 696.615
Vals: Template:Val list
Badness: 0.025167
Plutus
Subgroup: 2.3.5.7
Comma list: 15/14, 81/80
Mapping: [⟨1 0 -4 -5], ⟨0 1 4 5]]
Wedgie: ⟨⟨ 1 4 5 4 5 0 ]]
POTE generator: ~3/2 = 682.895
Badness: 0.045275
11-limit
Subgroup: 2.3.5.7.11
Comma list: 15/14, 22/21, 81/80
Mapping: [⟨1 0 -4 -5 -6], ⟨0 1 4 5 6]]
POTE generator: ~3/2 = 685.234
Vals: Template:Val list
Badness: 0.032521
Supermean
Subgroup: 2.3.5.7
Comma list: 81/80, 672/625
Mapping: [⟨1 0 -4 -21], ⟨0 1 4 15]]
POTE generator: ~3/2 = 704.889
Badness: 0.134204
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 81/80, 132/125
Mapping: [⟨1 0 -4 -21 -14], ⟨0 1 4 15 11]]
POTE generator: ~3/2 = 705.096
Vals: Template:Val list
Badness: 0.063262
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 26/25, 56/55, 66/65, 81/80
Mapping: [⟨1 0 -4 -21 -14 -9], ⟨0 1 4 15 11 8]]
POTE generator: ~3/2 = 705.094
Vals: Template:Val list
Badness: 0.040324
Godzilla
Godzilla tempers out 49/48, equating 8/7 with 7/6. Two of the step-and-a-quarter intervals these represent give a fourth, and so step-and-a-quarter generators generate godzilla. 19edo is close to being the optimal generator tuning; hence it can be more or less equated with taking 4\19 as a generator. MOS are of 5, 9, or 14 notes.
Subgroup: 2.3.5.7
Comma list: 49/48, 81/80
Mapping: [⟨1 0 -4 2], ⟨0 2 8 1]]
Mapping generators: ~2, ~7/4
Wedgie: ⟨⟨ 2 8 1 8 -4 -20 ]]
POTE generator: ~8/7 = 252.635
- diamond monotone range: [240.000, 257.143] (1\5 to 3\14)
- diamond purer range: [231.174, 266.871]
- diamond nice range: [240.000, 257.143]
Badness: 0.026747
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 49/48, 81/80
Mapping: [⟨1 0 -4 2 -6], ⟨0 2 8 1 12]]
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 254.027
Tuning ranges:
- diamond monotone range: [252.632, 257.143] (4\19 to 3\14)
- diamond purer range: [231.174, 266.871]
- diamond nice range: [252.632, 257.143]
Badness: 0.0290
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 49/48, 78/77, 81/80
Mapping: [⟨1 0 -4 2 -6 -5], ⟨0 2 8 1 12 11]]
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 253.603
Tuning ranges:
- diamond monotone range: 694.737 (4\19)
- diamond purer range: [621.581, 737.652]
- diamond nice range: 694.737
Badness: 0.0225
Semafour
Subgroup: 2.3.5.7.11
Comma list: 33/32, 49/48, 55/54
Mapping: [⟨1 0 -4 2 5], ⟨0 2 8 1 -2]]
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 254.042
Badness: 0.0285
Varan
Subgroup: 2.3.5.7.11
Comma list: 49/48, 77/75, 81/80
Mapping: [⟨1 0 -4 2 -10], ⟨0 2 8 1 17]]
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 251.079
Badness: 0.0396
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 66/65, 77/75, 81/80
Mapping: [⟨1 0 -4 2 -10 -5], ⟨0 2 8 1 17 11]]
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 251.165
Badness: 0.0257
Baragon
Subgroup: 2.3.5.7.11
Comma list: 49/48, 56/55, 81/80
Mapping: [⟨1 0 -4 2 9], ⟨0 2 8 1 -7]]
Mapping generators: ~2, ~7/4
POTE generator: ~8/7 = 251.173
Badness: 0.035673
Injera
Injera has a half-octave period and a generator which can be taken as a fifth or fourth, but also as a 15/14 semitone difference between a half-octave and a perfect fifth. Injera tempers out 50/49, equating 7/5 with 10/7 and giving a tritone of half an octave. A major third up from this tritone is the 7/4. 38edo, which is two parallel 19edos, is an excellent tuning for injera.
Subgroup: 2.3.5.7
Comma list: 50/49, 81/80
Mapping: [⟨2 0 -8 -7], ⟨0 1 4 4]]
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 694.375
- diamond monotone range: [685.714, 700.000] (8\14 to 7\12)
- diamond purer range: [688.957, 701.955]
- diamond nice range: [688.957, 700.000]
Wedgie: ⟨⟨ 2 8 8 8 7 -4 ]]
Badness: 0.0311
- Music
- Two Pairs of Socks (in 26edo) by Igliashon Jones
11-limit
Subgroup: 2.3.5.7.11
Comma list: 45/44, 50/49, 81/80
Mapping: [⟨2 0 -8 -7 -12], ⟨0 1 4 4 6]]
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 692.840
Tuning ranges:
- diamond monotone range: [685.714, 700.000] (8\14 to 7\12)
- diamond purer range: [682.458, 701.955]
- diamond nice range: [685.714, 700.000]
Badness: 0.0231
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 50/49, 78/77, 81/80
Mapping: [⟨2 0 -8 -7 -12 -21], ⟨0 1 4 4 6 9]]
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 692.673
Tuning ranges:
- diamond monotone range: 692.308 (15\26)
- diamond purer range: [682.458, 701.955]
- diamond nice range: 692.308 (15\26)
Badness: 0.0216
Enjera
Subgroup: 2.3.5.7.11.13
Comma list: 27/26, 40/39, 45/44, 50/49
Mapping: [⟨2 0 -8 -7 -12 -2], ⟨0 1 4 4 6 3]]
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 694.121
Badness: 0.0265
Injerous
Subgroup: 2.3.5.7.11
Comma list: 33/32, 50/49, 55/54
Mapping: [⟨2 0 -8 -7 10], ⟨0 1 4 4 -1]]
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 690.548
Badness: 0.0386
Lahoh
Subgroup: 2.3.5.7.11
Comma list: 50/49, 56/55, 81/77
Mapping: [⟨2 0 -8 -7 7], ⟨0 1 4 4 0]]
Mapping generators: ~7/5, ~3
POTE generator: ~3/2 = 699.001
Badness: 0.0431
Mohaha
Mohaha is the 2.3.5.11 subgroup temperament with a generator of a neutral third, two of which make up a fifth, and which can be taken to represent 11/9. Mohaha can be thought of, intuitively, as "meantone with quarter tones"; as is the 3/2 generator subdivided in half, so is the 25/24 chromatic semitone divided into two equal ~33/32 quarter tones (in the 2.3.5.11 subgroup). Within this paradigm, mohaha is the temperament that splits the 3/2 into two equal 11/9's, that splits the 6/5 into two equal 11/10's, and that maps four 3/2's to 5/1. It has a 7-note MOS with three larger steps and four smaller ones, going sLsLsLs.
Subgroup: 2.3.5.11
Comma list: 81/80, 121/120
Sval mapping: [⟨1 1 0 2], ⟨0 2 8 5]]
Sval mapping generators: ~2, ~11/9
Gencom mapping: [⟨1 1 0 0 2], ⟨0 2 8 0 5]]
Gencom: [2 11/9; 81/80 121/120]
POTE generator: ~11/9 = 348.0938
Mohoho
Subgroup: 2.3.5.11.13
Comma list: 66/65, 81/80, 121/120
Sval mapping: [⟨1 1 0 2 4], ⟨0 2 8 5 -1]]
Sval mapping generators: ~2, ~11/9
Gencom mapping: [⟨1 1 0 0 2 4], ⟨0 2 8 0 5 -1]]
Gencom: [2 11/9; 66/65 81/80 121/120]
POTE generator: ~11/9 = 348.9155
Vals: Template:Val list
Mohajira
Mohajira can be viewed as derived from mohaha which maps the interval one quarter tone flat of 16/9 to 7/4, although mohajira really makes more sense as an 11-limit temperament. It tempers out 6144/6125, the porwell comma. 31edo makes for an excellent (7-limit) mohajira tuning, with generator 9/31.
Subgroup: 2.3.5.7
Comma list: 81/80, 6144/6125
Mapping: [⟨1 1 0 6], ⟨0 2 8 -11]]
Mapping generators: ~2, ~128/105
Wedgie: ⟨⟨ 2 8 -11 8 -23 -48 ]]
POTE generator: ~128/105 = 348.415
- 7- and 9-odd-limit: 1/4 comma
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [6 0 -11/8 0⟩]
- Eigenmonzos: 2, 5
Algebraic generator: Mohabis, real root of 3x3 - 3x2 - 1, 348.6067 cents. Corresponding recurrence converges quickly.
Badness: 0.0557
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 121/120, 176/175
Mapping: [⟨1 1 0 6 2], ⟨0 2 8 -11 5]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.477
Minimax tuning:
- 11-odd-limit: 1/4 comma
- [[1 0 0 0 0⟩, [1 0 1/4 0 0⟩, [0 0 1 0 0⟩, [6 0 -11/8 0 0⟩, [2 0 5/8 0 0⟩]
- Eigenmonzos: 2, 5
Badness: 0.0261
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 105/104, 121/120
Mapping: [⟨1 1 0 6 2 4], ⟨0 2 8 -11 5 -1]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.558
Badness: 0.0234
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 66/65, 81/80, 105/104, 121/120, 154/153
Mapping: [⟨1 1 0 6 2 4 7], ⟨0 2 8 -11 5 -1 -10]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.736
Badness: 0.0206
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 66/65, 77/76, 81/80, 96/95, 105/104, 153/152
Mapping: [⟨1 1 0 6 2 4 7 6], ⟨0 2 8 -11 5 -1 -10 -6]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 348.810
Badness: 0.0173
Mohamaq
Subgroup: 2.3.5.7
Comma list: 81/80, 392/375
Mapping: [⟨1 1 0 -1], ⟨0 2 8 13]]
Mapping generators: ~2, ~25/21
POTE generator: ~25/21 = 350.586
Badness: 0.0777
11-limit
Subgroup: 2.3.5.7.11
Comma list: 56/55, 77/75, 243/242
Mapping: [⟨1 1 0 -1 2], ⟨0 2 8 13 5]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.565
Badness: 0.0362
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 66/65, 77/75, 243/242
Mapping: [⟨1 1 0 -1 2 4], ⟨0 2 8 13 5 -1]]
Mapping generators: ~2, ~11/9
POTE generator: ~11/9 = 350.745
Badness: 0.0287
Orphic
Subgroup: 2.3.5.7
Comma list: 81/80, 5898240/5764801
Mapping: [⟨2 1 -4 4], ⟨0 4 16 3]]
Mapping generators: ~2401/1728, ~343/288
Wedgie: ⟨⟨ 8 32 6 32 -13 -76 ]]
POTE generator: ~7/6 = 275.794
Badness: 0.2588
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 73728/73205
Mapping: [⟨2 1 -4 4 8], ⟨0 4 16 3 -2]]
Mapping generators: ~363/256, ~77/64
POTE generator: ~7/6 = 275.762
Badness: 0.1015
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 144/143, 2200/2197
Mapping: [⟨2 1 -4 4 8 2], ⟨0 4 16 3 -2 10]]
Mapping generators: ~55/39, ~63/52
POTE generator: ~7/6 = 275.774
Badness: 0.0535
Mothra
Mothra splits the fifth into three 8/7 generators. It uses 1029/1024, the gamelisma, to accomplish this deed and also tempers out 1728/1715, the orwell comma. Using 31edo with a generator of 6/31 is an excellent tuning choice. Once again something other than a MOS should be used as a scale to get the most out of mothra. In the 2.3.7-limit, mothra is identical to slendric.
Note that mothra can also be called cynder in the 7-limit, which can be a little confusing sometimes.
Subgroup: 2.3.5.7
Comma list: 81/80, 1029/1024
Mapping: [⟨1 1 0 3], ⟨0 3 12 -1]]
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.193
Algebraic generator: Rabrindanath, largest real root of x8 - 3x2 + 1, or 232.0774 cents.
Wedgie: ⟨⟨3 12 -1 12 -10 -36]]
- 7- and 9-odd-limit: 1/4 comma
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3 0 -1/12 0⟩]
- Eigenmonzos: 2, 5
Badness: 0.0371
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 385/384
Mapping: [⟨1 1 0 3 5], ⟨0 3 12 -1 -8]]
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.031
Badness: 0.0256
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 105/104, 144/143
Mapping: [⟨1 1 0 3 5 1], ⟨0 3 12 -1 -8 14]]
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 231.811
Badness: 0.0240
Cynder
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 1029/1024
Mapping: [⟨1 1 0 3 0], ⟨0 3 12 -1 18]]
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 231.317
Badness: 0.0557
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 78/77, 81/80, 640/637
Mapping: [⟨1 1 0 3 0 1], ⟨0 3 12 -1 18 14]]
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.293
Badness: 0.0341
Mosura
Subgroup: 2.3.5.7.11
Comma list: 81/80, 176/175, 540/539
Mapping: [⟨1 1 0 3 -1], ⟨0 3 12 -1 23]]
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.419
Badness: 0.0313
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 144/143, 176/175, 196/195
Mapping: [⟨1 1 0 3 -1 7], ⟨0 3 12 -1 23 -17]]
Mapping generators: ~2, ~8/7
POTE generator: ~8/7 = 232.640
Badness: 0.0369
Music
Squares
Squares splits the interval of an eleventh, or 8/3, into four supermajor third (9/7) intervals, and uses it for a generator. 31edo, with a generator of 11/31, makes for a good squares tuning, with 8, 11, and 14 note MOS available. Squares tempers out 2401/2400, the breedsma, as well as 2430/2401.
Subgroup: 2.3.5.7
Comma list: 81/80, 2401/2400
Mapping: [⟨1 3 8 6], ⟨0 -4 -16 -9]]
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 425.942
- 7- and 9-odd-limit: 1/4 comma
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [3/2 0 9/16 0⟩]
- Eigenmonzos: 2, 5
Algebraic generator: Sceptre2, the positive root of 9x2 + x - 16, or (sqrt (577) - 1)/18, which is 425.9311 cents.
Badness: 0.0460
Scales: skwares8, skwares11, skwares14
- Music
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 121/120
Mapping: [⟨1 3 8 6 7], ⟨0 -4 -16 -9 -10]]
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 425.957
Badness: 0.0216
Scales: skwares8, skwares11, skwares14
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 66/65, 81/80, 99/98, 121/120
Mapping: [⟨1 3 8 6 7 3], ⟨0 -4 -16 -9 -10 2]]
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 425.550
Badness: 0.0255
Scales: skwares8, skwares11, skwares14
Agora
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 99/98, 105/104, 121/120
Mapping: [⟨1 3 8 6 7 14], ⟨0 -4 -16 -9 -10 -29]]
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 426.276
Badness: 0.0245
Scales: skwares8, skwares11, skwares14
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 81/80, 99/98, 105/104, 120/119, 121/119
Mapping: [⟨1 3 8 6 7 14 8], ⟨0 -4 -16 -9 -10 -29 -11]]
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 426.187
Scales: skwares8, skwares11, skwares14
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 77/76, 81/80, 99/98, 105/104, 120/119, 121/119
Mapping: [⟨1 3 8 6 7 14 8 11], ⟨0 -4 -16 -9 -10 -29 -11 -19]]
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 426.225
Scales: skwares8, skwares11, skwares14
Cuboctahedra
Subgroup: 2.3.5.7.11
Comma list: 81/80, 385/384, 1375/1372
Mapping: [⟨1 3 8 6 -4], ⟨0 -4 -16 -9 21]]
Mapping generators: ~2, ~9/7
POTE generator: ~9/7 = 425.993
Badness: 0.0568
Scales: skwares8, skwares11, skwares14
Liese
Liese splits the twelfth interval of 3/1 into three generators of 10/7, using the comma 1029/1000. It also tempers out 686/675, the senga. 74edo makes for a good liese tuning, though 19edo can be used. The tuning is well-supplied with MOS: 7, 9, 11, 13, 15, 17, 19, 36, 55.
Subgroup: 2.3.5.7
Comma list: 81/80, 686/675
Mapping: [⟨1 0 -4 -3], ⟨0 3 12 11]]
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 632.406
Minimax tuning:
- 7- and 9-odd-limit: 1/4 comma
- [[1 0 0 0⟩, [1 0 1/4 0⟩, [0 0 1 0⟩, [2/3 0 11/12 0⟩]
- Eigenmonzos: 2, 5
Algebraic generator: Radix, the real root of x5 - 2x4 + 2x3 - 2x2 + 2x - 2, also a root of x6 - x5 - 2. The recurrence converges.
Badness: 0.0467
Liesel
Subgroup: 2.3.5.7.11
Comma list: 56/55, 81/80, 540/539
Mapping: [⟨1 0 -4 -3 4], ⟨0 3 12 11 -1]]
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 633.073
Badness: 0.0407
13-limit
Liesel is a very natural 13-limit tuning, given the generator is so near 13/9.
Subgroup: 2.3.5.7.11.13
Comma list: 56/55, 78/77, 81/80, 91/90
Mapping: [⟨1 0 -4 -3 4 0], ⟨0 3 12 11 -1 7]]
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 633.042
Badness: 0.0273
Elisa
Subgroup: 2.3.5.7.11
Comma list: 77/75, 81/80, 99/98
Mapping: [⟨1 0 -4 -3 -5], ⟨0 3 12 11 -1 16]]
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 633.061
Badness: 0.0416
Lisa
Subgroup: 2.3.5.7.11
Comma list: 45/44, 81/80, 343/330
Mapping: [⟨1 0 -4 -3 -6], ⟨0 3 12 11 -1 18]]
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 631.370
Badness: 0.0548
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 45/44, 81/80, 91/88, 147/143
Mapping: [⟨1 0 -4 -3 -6 0], ⟨0 3 12 11 -1 18 7]]
Mapping generators: ~2, ~10/7
POTE generator: ~10/7 = 631.221
Badness: 0.0361
Jerome
Jerome is related to Hieronymus' tuning; the Hieronymus generator is 51/20, or 139.316 cents. While the generator represents both 13/12 and 12/11, the POTE and Hieronymus generators are close to 13/12 in size.
Subgroup: 2.3.5.7
Comma list: 81/80, 17280/16807
Mapping: [⟨1 1 0 2], ⟨0 5 20 7]]
Mapping generators: ~2, ~54/49
Wedgie: ⟨⟨ 5 30 7 20 -3 -40 ]]
POTE generator: ~54/49 = 139.343
Badness: 0.1087
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 99/98, 864/847
Mapping: [⟨1 1 0 2 3], ⟨0 5 20 7 4]]
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.428
Badness: 0.0479
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 78/77, 81/80, 99/98, 144/143
Mapping: [⟨1 1 0 2 3 3], ⟨0 5 20 7 4 6]]
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.387
Badness: 0.0293
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 78/77, 81/80, 99/98, 144/143, 189/187
Mapping: [⟨1 1 0 2 3 3 2], ⟨0 5 20 7 4 6 18]]
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.362
Badness: 0.0209
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 78/77, 81/80, 99/98, 120/119, 135/133, 144/143
Mapping: [⟨1 1 0 2 3 3 1], ⟨0 5 20 7 4 6 28]]
Mapping generators: ~2, ~12/11
POTE generator: ~12/11 = 139.313
Badness: 0.0182
Cloudtone
The cloudtone temperament (5&50, named by Xenllium) tempers out the cloudy comma, 16807/16384 and the syntonic comma, 81/80 in the 7-limit. It can be extended to the 11- and 13-limit by adding 385/384 and 105/104 to the comma list in this order.
Subgroup: 2.3.5.7
Comma list: 81/80, 16807/16384
Mapping: [⟨5 0 -20 14], ⟨0 1 4 0]]
Mapping generators: ~8/7, ~3
Wedgie: ⟨⟨ 5 20 0 20 -14 -56 ]]
POTE generator: ~3/2 = 695.720
Badness: 0.102256
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 385/384, 2401/2376
Mapping: [⟨5 0 -20 14 41], ⟨0 1 4 0 -3]]
Mapping generators: ~8/7, ~3
POTE generator: ~3/2 = 696.536
Vals: Template:Val list
Badness: 0.070378
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 144/143, 2401/2376
Mapping: [⟨5 0 -20 14 41 -21], ⟨0 1 4 0 -3 5]]
Mapping generators: ~8/7, ~3
POTE generator: ~3/2 = 696.162
Vals: Template:Val list
Badness: 0.048829
Meanmag
Subgroup: 2.3.5.7
Comma list: 81/80, 3125/3072
Mapping: [⟨19 30 44 0], ⟨0 0 0 1]]
Mapping generators: ~25/24, ~7
Wedgie: ⟨⟨0 0 19 0 30 44]]
POTE generator: ~8/7 = 238.396
Badness: 0.077023
11-limit
Subgroup: 2.3.5.7.11
Comma list: 81/80, 385/384, 625/616
Mapping: [⟨19 30 44 0 119], ⟨0 0 0 1 -1]]
Mapping generators: ~25/24, ~7
POTE generator: ~8/7 = 233.486
Badness: 0.066829
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 81/80, 105/104, 144/143, 625/616
Mapping: [⟨19 30 44 0 119 17], ⟨0 0 0 1 -1 1]]
Mapping generators: ~25/24, ~7
POTE generator: ~8/7 = 234.890
Badness: 0.045844
Undevigintone
Subgroup: 2.3.5.7.11
Comma list: 49/48, 81/80, 126/125
Mapping: [⟨19 30 44 53 0], ⟨0 0 0 0 1]]
Mapping generators: ~21/20, ~11
POTE generator: ~11/8 = 538.047
Badness: 0.036387
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 49/48, 65/64, 81/80, 126/125
Mapping: [⟨19 30 44 53 0 70], ⟨0 0 0 0 1 0]]
Mapping generators: ~21/20, ~11
POTE generator: ~11/8 = 537.061
Vals: Template:Val list
Badness: 0.022933