Schismatic family: Difference between revisions
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{{Main| Garibaldi }} | {{Main| Garibaldi }} | ||
Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave ( | Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 90: | Line 90: | ||
=== Cassandra === | === Cassandra === | ||
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup. | Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, though it comes with a higher complexity. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 546: | Line 546: | ||
{{Main| Pontiac }} | {{Main| Pontiac }} | ||
Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple augmented third (C- | Pontiac tempers out the [[ragisma]], rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^<sup>3</sup>A). | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
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=== Helenoid === | === Helenoid === | ||
Helenoid may be described as {{nowrap| 53 & 118 }}, and is closely related to the helenus temperament, differing only by the mapping of 7. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 681: | Line 681: | ||
=== Ponta === | === Ponta === | ||
Ponta tempers out [[540/539]] and may be described as {{nowrap| 171 & 224 }}. [[224edo]] itself makes for an excellent tuning. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 740: | Line 740: | ||
=== Pontic === | === Pontic === | ||
Pontic temperament tempers out [[441/440]] and may be described as {{nowrap| 118 & 171 }}. [[289edo]] may be recommended as a tuning. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 829: | Line 829: | ||
=== Bipont === | === Bipont === | ||
Bipont tempers out the [[3025/3024|lehmerisma (3025/3024)]] and the [[9801/9800|kalisma (9801/9800)]]. It may be described as {{nowrap| 118 & 224 }}. It has a period of half octave and a ploidacot signature of diploid monocot. [[342edo]] may be recommended as a tuning. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 938: | Line 938: | ||
== Grackle == | == Grackle == | ||
Grackle tempers out {{monzo| -44 26 0 1 }} | Grackle tempers out {{monzo| -44 26 0 1 }} so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,084: | Line 1,084: | ||
See [[Archytas clan #Schism]]. | See [[Archytas clan #Schism]]. | ||
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh ( | Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used. | ||
== Bischismic == | == Bischismic == | ||
Bischismic tempers out 3136/3125, the [[hemimean comma]], as well as 321489/320000, the [[varunisma]], and may be described as the {{nowrap| 118 & 130 }} temperament. The octave is split in halves, so the [[ploidacot]] of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds [[7/4]] by a stack of two [[10/9]]'s plus a semi-octave period, and in the [[11-limit]], it simply finds [[11/8]] by a stack of three [[10/9]]'s. [[248edo]] and [[378edo]] make for excellent tunings in both cases. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,184: | Line 1,186: | ||
== Kleischismic == | == Kleischismic == | ||
Kleischismic tempers out 1500625/1492992, the [[uniwiz comma]], and may be described as the {{nowrap| 94 & 118 }} temperament. The generator is a infrafifth, two of which plus a semi-octave period make the [[3/1|3rd]] [[harmonic]]; its [[ploidacot]] is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of [[9/5]]. Kleischismic then finds [[7/4]] by that minus a [[36/35]] quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for [[16/11]] and the quartertone stands in for [[33/32]] in the [[11-limit]]. [[212edo]] and [[330edo]] in the 330e val may be recommended as tunings. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,277: | Line 1,281: | ||
== Salsa == | == Salsa == | ||
Salsa tempers out 245/243, the [[sensamagic comma]], and may be described as the {{nowrap| 41 & 65 }} temperament. It has a neutral third as a generator; its [[ploidacot]] is dicot. In fact it is related to [[hemififths]], from which this less accurate temperament only differs by the mapping of [[5/1|5]]. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,325: | Line 1,331: | ||
== Hemischis == | == Hemischis == | ||
Hemischis tempers out 6144/6125, the [[porwell comma]], as well as 19683/19600, the [[cataharry comma]], and may be described as the {{nowrap| 53 & 130 }} temperament. Its [[ploidacot]] is alpha-dicot. | |||
The [[S-expression]]-based comma list for 13-limit hemischis is {[[540/539|S12/S14]], [[676/675|S13/S15 = S26]], [[729/728|S27]], [[4096/4095|S64]], ([[4225/4224|S65]])}. Tempering out [[169/168]] ({{S|13}}), [[225/224]] ({{S|15}}) or [[625/624]] ({{S|25}}) leads to [[53edo]] while tempering out [[24192/24167]] ([[S-expression|S12/S13]]), [[10985/10976]] ([[S-expression|S13/S14]]), [[43904/43875]] ([[S-expression|S14/S15]]) or [[2401/2400]] ([[S-expression|S49]]) leads to [[130edo]] and implies S12, S13, S14, and S15 are tempered together. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,358: | Line 1,368: | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
| Line 1,423: | Line 1,431: | ||
== Term == | == Term == | ||
Term tempers out the [[landscape comma]], mapping | Term tempers out the [[landscape comma]], mapping [[63/50]] to the 1/3-octave period. It can be described as {{nowrap| 12 & 171 }}, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma]], with three kleismas making a comma, so this temperament may be useful for modeling that. [[171edo]] makes for an excellent tuning. | ||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,447: | Line 1,455: | ||
=== Terminal === | === Terminal === | ||
Terminal tempers out 441/440 and 4375/4356, and may be described as {{nowrap| 159 & 171 }}. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,494: | Line 1,502: | ||
=== Terminator === | === Terminator === | ||
Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,539: | Line 1,549: | ||
=== Semiterm === | === Semiterm === | ||
The semiterm temperament ({{nowrap| 12 & 342 }} | The semiterm temperament tempers out [[9801/9800]] (kalisma) as well as [[151263/151250]] (odiheim comma), and may be described as {{nowrap| 12 & 342 }}. It has a period of 1/6 octave and its ploidacot is hexaploid monocot. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,574: | Line 1,584: | ||
=== Hemiterm === | === Hemiterm === | ||
The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid beta-dicot. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,620: | Line 1,632: | ||
== Altinex == | == Altinex == | ||
Altinex is an alternative to [[#Hemiterm|hemiterm]] and may be described as {{nowrap| 24 & 159 }}. [[159edo]] itself makes for a recommendable tuning. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,668: | Line 1,682: | ||
== Squirrel == | == Squirrel == | ||
Squirrel tempers out 686/675, the [[sengic comma]], and may be described as {{nowrap| 29 & 36 }}. It has a [[~]][[11/10]] generator, three of which give the fourth ([[4/3]]), and thirteen of which give [[7/4]] with octave reduction. Its [[ploidacot]] is omega-tricot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,717: | Line 1,731: | ||
== Tertiaschis == | == Tertiaschis == | ||
Tertiaschis may be described as {{nowrap| 94 & 159 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 1071875/1062882 for prime 7. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,781: | Line 1,795: | ||
== Countertertiaschis == | == Countertertiaschis == | ||
Countertertiaschis may be described as {{nowrap| 159 & 224 }}. It has a [[~]][[11/10]] generator, sharing the same 2.3.5.11 subgroup with [[#Squirrel|squirrel]], but tempers out 244140625/243045684 for prime 7. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,830: | Line 1,844: | ||
== Quadrant == | == Quadrant == | ||
Quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be decribed as the {{nowrap| 12 & 212 }} temperament; its ploidacot is tetraploid monocot. Just as [[#Term|term]] equates the syntonic~Pythagorean comma with three [[marvel comma]]s, quadrant equates the syntonic~Pythagorean comma with four. A [[septimal comma]] is then found as a stack of five marvel commas. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,880: | Line 1,894: | ||
== Sesquiquartififths == | == Sesquiquartififths == | ||
Sesquiquartififths tempers out 2401/2400, the [[breedsma]], and may be described as the {{nowrap| 41 & 171 }} temperament. It splits the fifth into four; its [[ploidacot]] is thus tetracot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 1,902: | Line 1,918: | ||
=== Sesquart === | === Sesquart === | ||
Sesquart is the main [[11-limit|11-]] and [[13-limit]] extension of sesquiquartififths of practical interest, as it identifies the neutral third with [[11/9]], which is realized in [[41edo]], [[89edo]], [[130edo]], and [[171edo]] also makes for a possible tuning. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 1,931: | Line 1,949: | ||
Badness (Sintel): 0.925 | Badness (Sintel): 0.925 | ||
===== | ===== Heartia ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 243/242, 364/363, 441/440 | Comma list: 243/242, 256/255, 273/272, 364/363, 441/440 | ||
Mapping: {{mapping| 1 1 7 5 2 -2 | Mapping: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175. | * CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}} | ||
{{Optimal ET sequence|legend=0| 41, | {{Optimal ET sequence|legend=0| 41, 89, 130g }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.45 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 243/242, | Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440 | ||
Mapping: {{mapping| 1 1 7 5 2 -2 | Mapping: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175. | * CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}} | ||
{{Optimal ET sequence|legend=0| 41, | {{Optimal ET sequence|legend=0| 41, 89, 130g }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.40 | ||
===== | ===== Sesquartia ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: 243/242 | Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575 | ||
Mapping: {{mapping| 1 1 7 5 2 -2 | Mapping: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 41, 130, 171 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.18 | ||
===== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: 243/242, | Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594 | ||
Mapping: {{mapping| 1 1 7 5 2 -2 | Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~ | * CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}} | ||
{{Optimal ET sequence|legend=0| 41, | {{Optimal ET sequence|legend=0| 41, 130, 171 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.24 | ||
====== | ====== 23-limit ====== | ||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19.23 | ||
Comma list: | Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594 | ||
Mapping: {{mapping| 1 1 7 5 2 -2 | Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 -6 | 0 4 -32 -15 10 39 69 -12 72 }} | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1199. | * WE: ~2 = 1199.9606{{c}}, ~21/19 = 175.4067{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175. | * CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}} | ||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 41i, 130, 171 }} | ||
Badness (Sintel): 1. | Badness (Sintel): 1.36 | ||
===== Hearty ===== | ===== Hearty ===== | ||
Revision as of 09:32, 31 December 2025
- This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.
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 syntonic comma (81/80), or alternatively put, the difference between a just major third and a Pythagorean diminished fourth.
Schismic, schismatic, a.k.a. helmholtz
The 5-limit version of the temperament is a microtemperament, called schismic, schismatic, or helmholtz. The generator is a fifth, flattened by a fraction of a schisma, and 5/4 is represented by a diminished fourth. This defies the tradition of tertian harmony, as the just major triad on C is C–F♭–G, for example. One may want to adopt an additional module of accidentals such as arrows to represent the comma step, allowing them to write the chord above as C–vE–G.
As a 5-limit system, schismic is far more accurate than meantone but still with manageable complexity. 53edo is a possible tuning for schismic, 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 schismic, with pure major thirds and fifths flattened by 1/8 schisma. Since 1/8 of a schisma is 0.244 ¢, this falls into the range of microtempering. You could also try 1/9 schisma, with pure minor thirds and a minutely better fifth, 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. Simply leaving the fifths just would also make for a viable tuning, thus collapsing schismic to a simple relabeling of the 3-limit.
Subgroup: 2.3.5
Comma list: 32805/32768
Mapping: [⟨1 0 15], ⟨0 1 -8]]
- mapping generators: ~2, ~3
- WE: ~2 = 1200.0749 ¢, ~3/2 = 701.7797 ¢
- error map: ⟨+0.075 -0.100 -0.027]
- CWE: ~2 = 1200.0000{{c]}, ~3/2 = 701.7308 ¢
- error map: ⟨0.000 -0.224 -0.160]
- 5-odd-limit diamond monotone: ~3/2 = [685.714, 705.882] (4\7 to 10\17)
- 5-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955] (1/8-comma to untempered)
Optimal ET sequence: 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc
Badness (Sintel): 0.0999
Overview to 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.
- Hemischis adds [-34 25 0 -2⟩ and has a hemififth generator.
- 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 discussed elsewhere include:
- Guiron (+1029/1024) → Gamelismic clan
- Pogo (+118098/117649) → Stearnsmic clan
The schismatic family boasts a variety of remarkable extensions to subgroups in high prime limits. These are listed at the bottom of this page, in #Subgroup extensions.
Garibaldi
Garibaldi tempers out the garischisma, equating the septimal comma with both the syntonic comma and the Pythagorean comma. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its S-expression-based comma list is {S8/S9, S15}.
Subgroup: 2.3.5.7
Comma list: 225/224, 3125/3087
Mapping: [⟨1 0 15 25], ⟨0 1 -8 -14]]
- WE: ~2 = 1200.1233 ¢, ~3/2 = 702.1573 ¢
- error map: ⟨+0.123 +0.326 -2.709 +2.328]
- CWE: ~2 = 1200.0000{{c]}, ~3/2 = 702.0774 ¢
- error map: ⟨0.000 +0.122 -2.933 +2.090]
- 7-odd-limit: ~3/2 = [2/3 1/15 0 -1/15⟩
- [[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⟩]
- unchanged-interval (eigenmonzo) basis: 2.7/3
- 9-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩
- [[1 0 0 0⟩, [25/16 1/8 0 -1/16⟩, [5/2 -1 0 1/2⟩, [25/8 -7/4 0 7/8⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/7
- 7- and 9-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
Optimal ET sequence: 12, 29, 41, 53, 94
Badness (Sintel): 0.548
Cassandra
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, though it comes with a higher complexity.
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]]
Optimal tunings:
- WE: ~2 = 1200.3089 ¢, ~3/2 = 702.3377 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1562 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [9/16 1/8 0 -1/16⟩
- unchanged-interval (eigenmonzo) basis: 2.9/7
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
- 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915]
Optimal ET sequence: 12e, 41, 53, 94, 229c
Badness (Sintel): 0.906
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]]
Optimal tunings:
- WE: ~2 = 1200.1703 ¢, ~3/2 = 702.2122 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1135 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [19/34 0 0 -1/34 0 1/34⟩
- unchanged-interval (eigenmonzo) basis: 2.13/7
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [701.887, 702.439] (31\53 to 24\41)
- 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 703.597]
- 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597]
Optimal ET sequence: 41, 53, 94, 429ccdeef, 523ccdeef
Badness (Sintel): 0.854
Cassie
Subgroup: 2.3.5.7.11.13.17
Comma list: 120/119, 154/153, 225/224, 273/272, 325/324
Mapping: [⟨1 0 15 25 -33 -28 -7], ⟨0 1 -8 -14 23 20 7]]
Optimal tunings:
- WE: ~2 = 1199.8140 ¢, ~3/2 = 701.9833 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0909 ¢
Optimal ET sequence: 12e, 41, 53, 94g
Badness (Sintel): 1.19
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 120/119, 154/153, 171/170, 190/189, 225/224, 273/272
Mapping: [⟨1 0 15 25 -33 -28 -7 9], ⟨0 1 -8 -14 23 20 7 -3]]
Optimal tunings:
- WE: ~2 = 1199.9556 ¢, ~3/2 = 702.0530 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0787 ¢
Optimal ET sequence: 12e, 41, 53
Badness (Sintel): 1.11
Cassandric
Subgroup: 2.3.5.7.11.13.17
Comma list: 225/224, 275/273, 325/324, 375/374, 385/384
Mapping: [⟨1 0 15 25 -33 -28 77], ⟨0 1 -8 -14 23 20 -46]]
Optimal tunings:
- WE: ~2 = 1200.0046 ¢, ~3/2 = 702.2167 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0962 ¢
Optimal ET sequence: 41g, 53, 94
Badness (Sintel): 1.18
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 190/189, 209/208, 225/224, 275/273, 325/324, 375/374
Mapping: [⟨1 0 15 25 -33 -28 77 9], ⟨0 1 -8 -14 23 20 -46 -3]]
Optimal tunings:
- WE: ~2 = 1200.2910 ¢, ~3/2 = 702.2681 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0967 ¢
Optimal ET sequence: 41g, 53, 94
Badness (Sintel): 1.07
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 190/189, 209/208, 225/224, 253/252, 275/273, 325/324, 375/374
Mapping: [⟨1 0 15 25 -33 -28 77 9 60], ⟨0 1 -8 -14 23 20 -46 -3 -35]]
Optimal tunings:
- WE: ~2 = 1200.2970 ¢, ~3/2 = 702.2697 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.0943 ¢
Optimal ET sequence: 41g, 53, 94
Badness (Sintel): 1.08
Cassander
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 225/224, 275/273, 325/324, 385/384
Mapping: [⟨1 0 15 25 -33 -28 -72], ⟨0 1 -8 -14 23 20 48]]
Optimal tunings:
- WE: ~2 = 1200.1986 ¢, ~3/2 = 702.2598 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1455 ¢
Optimal ET sequence: 41, 53g, 94
Badness (Sintel): 1.14
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 170/169, 190/189, 209/208, 225/224, 275/273, 325/324
Mapping: [⟨1 0 15 25 -33 -28 -72 9], ⟨0 1 -8 -14 23 20 48 -3]]
Optimal tunings:
- WE: ~2 = 1200.3057 ¢, ~3/2 = 702.3138 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.1373 ¢
Optimal ET sequence: 41, 53g, 94
Badness (Sintel): 1.07
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]]
Optimal tunings:
- WE: ~2 = 1200.1917 ¢, ~3/2 = 702.4836 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3599 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [3/5 1/10 0 0 -1/20⟩
- unchanged-interval (eigenmonzo) basis: 2.11/9
Tuning ranges:
- 11-odd-limit diamond monotone: ~3/2 = [700.000, 703.448] (7\12 to 17\29)
- 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
Optimal ET sequence: 12, 29, 41
Badness (Sintel): 0.779
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]]
Optimal tunings:
- WE: ~2 = 1200.3031 ¢, ~3/2 = 702.7368 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.5420 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [14/23 2/23 0 0 0 -1/23⟩
- unchanged-interval (eigenmonzo) basis: 2.13/9
Tuning ranges:
- 13- and 15-odd-limit diamond monotone: ~3/2 = [702.439, 703.448] (24\41 to 17\29)
- 13-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377]
- 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377]
Optimal ET sequence: 12f, 29, 41
Badness (Sintel): 0.857
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 120/119, 189/187, 196/195
Mapping: [⟨1 0 15 25 32 37 -7], ⟨0 1 -8 -14 -18 -21 7]]
Optimal tunings:
- WE: ~2 = 1199.1984 ¢, ~3/2 = 701.8424 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3384 ¢
Optimal ET sequence: 12f, 29, 41
Badness (Sintel): 1.19
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 120/119, 133/132, 189/187, 196/195
Mapping: [⟨1 0 15 25 32 37 -7 9], ⟨0 1 -8 -14 -18 -21 7 -3]]
Optimal tunings:
- WE: ~2 = 1199.5242 ¢, ~3/2 = 702.0783 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.3711 ¢
Optimal ET sequence: 12f, 29, 41
Badness (Sintel): 1.17
Schisicosiennic
Subgroup: 2.3.5.7.11.13.17
Comma list: 100/99, 105/104, 154/153, 170/169, 196/195
Mapping: [⟨1 0 15 25 32 37 58], ⟨0 1 -8 -14 -18 -21 -34]]
Optimal tunings:
- WE: ~2 = 1200.6122 ¢, ~3/2 = 703.0830 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6968 ¢
Optimal ET sequence: 12fg, 29g, 41, 70cd
Badness (Sintel): 1.11
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 100/99, 105/104, 133/132, 154/153, 170/169, 190/189
Mapping: [⟨1 0 15 25 32 37 58 9], ⟨0 1 -8 -14 -18 -21 -34 -3]]
Optimal tunings:
- WE: ~2 = 1200.7981 ¢, ~3/2 = 703.2199 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.7221 ¢
Optimal ET sequence: 12fg, 29g, 41, 70cd
Badness (Sintel): 1.09
Schisicosiennoid
Subgroup: 2.3.5.7.11.13.17
Comma list: 85/84, 100/99, 105/104, 119/117, 221/220
Mapping: [⟨1 0 15 25 32 37 12], ⟨0 1 -8 -14 -18 -21 -5]]
Optimal tunings:
- WE: ~2 = 1201.3146 ¢, ~3/2 = 703.4864 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6491 ¢
Optimal ET sequence: 12f, 29g, 41g
Badness (Sintel): 1.06
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 85/84, 100/99, 105/104, 119/117, 133/132, 153/152
Mapping: [⟨1 0 15 25 32 37 12 9], ⟨0 1 -8 -14 -18 -21 -5 -3]]
Optimal tunings:
- WE: ~2 = 1201.3140 ¢, ~3/2 = 703.4860 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 702.6578 ¢
Optimal ET sequence: 12f, 29g, 41g
Badness (Sintel): 1.02
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]]
Optimal tunings:
- WE: ~2 = 1199.7097 ¢, ~3/2 = 701.5554 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7370 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
- unchanged-interval (eigenmonzo) basis: 2.11/9
Optimal ET sequence: 12, 41e, 53, 118d
Badness (Sintel): 1.18
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]]
Optimal tunings:
- WE: ~2 = 1199.7370 ¢, ~3/2 = 701.5937 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7570 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [19/32 1/16 0 0 -1/32⟩
- unchanged-interval (eigenmonzo) basis: 2.11/9
Optimal ET sequence: 12f, …, 41ef, 53, 118d
Badness (Sintel): 1.09
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 99/98, 120/119, 176/175, 275/273, 442/441
Mapping: [⟨1 0 15 25 51 56 -7], ⟨0 1 -8 -14 -30 -33 7]]
Optimal tunings:
- WE: ~2 = 1199.2895 ¢, ~3/2 = 701.2643 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.6967 ¢
Optimal ET sequence: 12f, 53, 65d, 118dg
Badness (Sintel): 1.21
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 99/98, 120/119, 176/175, 190/189, 209/208, 247/245
Mapping: [⟨1 0 15 25 51 56 -7 9], ⟨0 1 -8 -14 -30 -33 7 -3]]
Optimal tunings:
- WE: ~2 = 1199.5280 ¢, ~3/2 = 701.4290 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7149 ¢
Optimal ET sequence: 12f, 53, 65d
Badness (Sintel): 1.18
Karadeniz
Subgroup: 2.3.5.7.11
Comma list: 225/224, 243/242, 3125/3087
Mapping: [⟨1 1 7 11 2], ⟨0 2 -16 -28 5]]
- mapping generators: ~2, ~11/9
Optimal tunings:
- WE: ~2 = 1199.7351 ¢, ~11/9 = 350.9167 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.9995 ¢
Optimal ET sequence: 24d, 41, 65d, 106, 147
Badness (Sintel): 1.37
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 225/224, 243/242, 325/324, 640/637
Mapping: [⟨1 1 7 11 2 -8], ⟨0 2 -16 -28 5 40]]
Optimal tunings:
- WE: ~2 = 1199.3042 ¢, ~11/9 = 350.7533 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 350.9686 ¢
Optimal ET sequence: 24d, 41, 65d, 106f
Badness (Sintel): 1.34
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
Optimal tunings:
- WE: ~2 = 1200.7303 ¢, ~110/63 = 951.6605 ¢
- CWE: ~2 = 1200.0000 ¢, ~110/63 = 951.0604 ¢
Optimal ET sequence: 24d, 29, 53, 82e, 135ee
Badness (Sintel): 1.68
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]]
Optimal tunings:
- WE: ~2 = 1200.8146 ¢, ~26/15 = 951.7273 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 951.0574 ¢
Optimal ET sequence: 24d, 29, 53, 82e, 135eef
Badness (Sintel): 1.13
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
Optimal tunings:
- WE: ~2 = 1200.1997 ¢, ~11/10 = 166.0018 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9786 ¢
Optimal ET sequence: 29, 65d, 94
Badness (Sintel): 1.92
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]]
Optimal tunings:
- WE: ~2 = 1200.1224 ¢, ~11/10 = 165.9800 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 165.9659 ¢
Optimal ET sequence: 29, 65d, 94
Badness (Sintel): 1.40
Pontiac
Pontiac tempers out the ragisma, rendering a very accurate 7-limit microtemperament. The 7/4 is found at +39 fifths, represented by the quintuple-augmented third (C-E𝄪𝄪♯), or triple-up major sixth (C-^3A).
Subgroup: 2.3.5.7
Comma list: 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59], ⟨0 1 -8 39]]
- WE: ~2 = 1200.0989 ¢, ~3/2 = 701.8145 ¢
- error map: ⟨+0.099 -0.042 -0.138 -0.038]
- CWE: ~2 = 1200.0000{{c]}, ~3/2 = 701.7579 ¢
- error map: ⟨0.000 -0.197 -0.377 -0.268]
- 7-odd-limit: ~3/2 = [27/47 0 -1/47 1/47⟩
- [[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⟩]
- unchanged-interval (eigenmonzo) basis: 2.7/5
- 9-odd-limit: ~3/2 = [1/2 1/5 -1/10⟩
- [[1 0 0 0⟩, [3/2 1/5 -1/10 0⟩, [3 -8/5 4/5 0⟩, [-1/2 39/5 -39/10 0⟩]
- unchanged-interval (eigenmonzo) basis: 2.9/5
- 7- and 9-odd-limit diamond monotone: ~3/2 = [701.538, 701.886] (38\65 to 31\53)
- 7- and 9-odd-limit diamond tradeoff: ~3/2 = [701.711, 701.955]
Optimal ET sequence: 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd
Badness (Sintel): 0.358
Helenoid
Helenoid may be described as 53 & 118, and is closely related to the helenus temperament, differing only by the mapping of 7.
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]]
Optimal tunings:
- WE: ~2 = 1200.3277 ¢, ~3/2 = 701.9135 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7223 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [41/69 0 0 1/69 -1/69⟩
- unchanged-interval (eigenmonzo) basis: 2.11/7
Optimal ET sequence: 53, 118, 289e, 407de
Badness (Sintel): 1.28
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]]
Optimal tunings:
- WE: ~2 = 1200.1780 ¢, ~3/2 = 701.8491 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7446 ¢
Minimax tuning:
- 13- and 15-odd-limit: ~3/2 = [43/72 0 0 1/72 -1/72⟩
- unchanged-interval (eigenmonzo) basis: 2.13/7
Optimal ET sequence: 53, 118, 171e
Badness (Sintel): 1.39
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 352/351, 385/384, 561/560, 625/624, 729/728
Mapping: [⟨1 0 15 -59 51 56 -91], ⟨0 1 -8 39 -30 -33 60]]
Optimal tunings:
- WE: ~2 = 1200.1645 ¢, ~3/2 = 701.8385 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7425 ¢
Minimax tuning:
- 17-odd-limit: ~3/2 = [18/31 0 0 0 0 -1/93 1/93⟩
- unchanged-interval (eigenmonzo) basis: 2.17/13
Optimal ET sequence: 53, 118, 171e
Badness (Sintel): 1.47
Helena
Subgroup: 2.3.5.7.11.13
Comma list: 169/168, 325/324, 385/384, 3146/3125
Mapping: [⟨1 0 15 -59 51 -28], ⟨0 1 -8 39 -30 20]]
Optimal tunings:
- WE: ~2 = 1200.5227 ¢, ~3/2 = 702.0456 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7418 ¢
Optimal ET sequence: 53, 118f, 171ef
Badness (Sintel): 1.50
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 169/168, 273/272, 325/324, 385/384, 3146/3125
Mapping: [⟨1 0 15 -59 51 -28 -91], ⟨0 1 -8 39 -30 20 60]]
Optimal tunings:
- WE: ~2 = 1200.4988 ¢, ~3/2 = 702.0218 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7332 ¢
Optimal ET sequence: 53, 118f, 171ef
Badness (Sintel): 1.56
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 169/168, 273/272, 286/285, 325/324, 385/384, 627/625
Mapping: [⟨1 0 15 -59 51 -28 -91 9], ⟨0 1 -8 39 -30 20 60 -3]]
Optimal tunings:
- WE: ~2 = 1200.5185 ¢, ~3/2 = 702.0323 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7318 ¢
Optimal ET sequence: 53, 118f, 171ef
Badness (Sintel): 1.33
Ponta
Ponta tempers out 540/539 and may be described as 171 & 224. 224edo itself makes for an excellent tuning.
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]]
Optimal tunings:
- WE: ~2 = 1199.9814 ¢, ~3/2 = 701.7725 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7834 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
- unchanged-interval (eigenmonzo) basis: 2.11/7
Optimal ET sequence: 53, 171, 224
Badness (Sintel): 1.61
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 625/624, 729/728, 2200/2197
Mapping: [⟨1 0 15 -59 135 56], ⟨0 1 -8 39 -83 -33]]
Optimal tunings:
- WE: ~2 = 1199.9601 ¢, ~3/2 = 701.7610 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7845 ¢
Minimax tuning:
- 13 and 15-odd-limit: ~3/2 = [36/61 0 0 1/122 -1/122⟩
- unchanged-interval (eigenmonzo) basis: 2.11/7
Optimal ET sequence: 53, 171, 224
Badness (Sintel): 0.976
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 375/374, 540/539, 625/624, 729/728, 2200/2197
Mapping: [⟨1 0 15 -59 135 56 -91], ⟨0 1 -8 39 -83 -33 60]]
Optimal tunings:
- WE: ~2 = 1199.8850 ¢, ~3/2 = 701.7101 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7775 ¢
Minimax tuning:
- 17-odd-limit: ~3/2 = [83/143 0 0 0 -1/143 0 1/143⟩
- unchanged-interval (eigenmonzo) basis: 2.17/11
Optimal ET sequence: 53, 171, 224, 395e, 619eg
Badness (Sintel): 1.16
Pontic
Pontic temperament tempers out 441/440 and may be described as 118 & 171. 289edo may be recommended as a tuning.
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]]
Optimal tunings:
- WE: ~2 = 1200.1259 ¢, ~3/2 = 701.7980 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7256 ¢
Minimax tuning:
- 11-odd-limit: ~3/2 = [6/11 0 0 0 1/88⟩
- unchanged-interval (eigenmonzo) basis: 2.11
Optimal ET sequence: 53e, 118, 289, 407d
Badness (Sintel): 1.64
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]]
Optimal tunings:
- WE: ~2 = 1199.9254 ¢, ~3/2 = 701.6945 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7378 ¢
Minimax tuning:
- 13 and 15-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
- unchanged-interval (eigenmonzo) basis: 2.13/11
Optimal ET sequence: 53e, 118, 171, 289f
Badness (Sintel): 1.87
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 441/440, 595/594, 625/624, 729/728, 2880/2873
Mapping: [⟨1 0 15 -59 -136 56 -91], ⟨0 1 -8 39 88 -33 60]]
Optimal tunings:
- WE: ~2 = 1199.9454 ¢, ~3/2 = 701.7085 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7401 ¢
Minimax tuning:
- 17-odd-limit: ~3/2 = [71/121 0 0 0 1/121 -1/121⟩
- unchanged-interval (eigenmonzo) basis: 2.13/11
Optimal ET sequence: 53e, 118, 171, 289f
Badness (Sintel): 1.51
Pontoid
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 4375/4374, 32805/32768
Mapping: [⟨1 0 15 -59 -136 -215], ⟨0 1 -8 39 88 138]]
Optimal tunings:
- WE: ~2 = 1200.0897 ¢, ~3/2 = 701.7874 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7356 ¢
Optimal ET sequence: 53ef, 118f, 171, 289
Badness (Sintel): 2.07
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 1156/1155, 32805/32768
Mapping: [⟨1 0 15 -59 -136 -215 -91], ⟨0 1 -8 39 88 138 60]]
Optimal tunings:
- WE: ~2 = 1200.1045 ¢, ~3/2 = 701.7962 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7359 ¢
Optimal ET sequence: 53ef, 118f, 171, 289, 460e, 749defg
Badness (Sintel): 1.50
Bipont
Bipont tempers out the lehmerisma (3025/3024) and the kalisma (9801/9800). It may be described as 118 & 224. It has a period of half octave and a ploidacot signature of diploid monocot. 342edo may be recommended as a tuning.
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4375/4374, 32805/32768
Mapping: [⟨2 0 30 -118 -85], ⟨0 1 -8 39 29]]
- mapping generators: ~99/70, ~3
Optimal tunings:
- WE: ~99/70 = 600.0500 ¢, ~3/2 = 701.8153 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7584 ¢
Optimal ET sequence: 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde
Badness (Sintel): 0.484
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 729/728, 1575/1573, 4096/4095
Mapping: [⟨2 0 30 -118 -85 112], ⟨0 1 -8 39 29 -33]]
Optimal tunings:
- WE: ~99/70 = 599.9939 ¢, ~3/2 = 701.7657 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7728 ¢
Optimal ET sequence: 106, 118, 224, 566f, 790f
Badness (Sintel): 1.25
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 625/624, 729/728, 1089/1088, 1225/1224, 2880/2873
Mapping: [⟨2 0 30 -118 -85 112 -182], ⟨0 1 -8 39 29 -33 60]]
Optimal tunings:
- WE: ~99/70 = 599.9839 ¢, ~3/2 = 701.7463 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7649 ¢
Optimal ET sequence: 106g, 118, 224, 342, 566f
Badness (Sintel): 1.38
Counterbipont
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 3025/3024, 32805/32768
Mapping: [⟨2 0 30 -118 -85 -243], ⟨0 1 -8 39 29 79]]
Optimal tunings:
- WE: ~99/70 = 600.0405 ¢, ~3/2 = 701.8160 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7697 ¢
Optimal ET sequence: 106f, 118f, 224, 342f, 566, 1356cf
Badness (Sintel): 1.06
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 32805/32768
Mapping: [⟨2 0 30 -118 -85 -243 -182], ⟨0 1 -8 39 29 79 60]]
Optimal tunings:
- WE: ~99/70 = 600.0336 ¢, ~3/2 = 701.8031 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7647 ¢
Optimal ET sequence: 106fg, 118f, 224, 342f, 566
Badness (Sintel): 1.29
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 936/935, 1089/1088, 1225/1224, 1540/1539, 4875/4864
Mapping: [⟨2 0 30 -118 -85 -243 -182 -169], ⟨0 1 -8 39 29 79 60 56]]
Optimal tunings:
- WE: ~99/70 = 600.0243 ¢, ~3/2 = 701.7891 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.7613 ¢
Optimal ET sequence: 106fgh, 118f, 224, 342f, 566h, 908fgh
Badness (Sintel): 1.35
Quadrapont
Subgroup: 2.3.5.7.11.13
Comma list: 3025/3024, 4225/4224, 4375/4374, 32805/32768
Mapping: [⟨4 0 60 -236 -170 -131], ⟨0 1 -8 39 29 23]]
- mapping generators: ~208/175, ~3
Optimal tunings:
- WE: ~208/175 = 300.0229 ¢, ~3/2 = 701.8097 ¢
- CWE: ~208/175 = 300.0000 ¢, ~3/2 = 701.7578 ¢
Optimal ET sequence: 224, 460, 684, 2276cde, 2960cde
Badness (Sintel): 0.869
Grackle
Grackle tempers out [-44 26 0 1⟩ so 7/4 is found at -26 fifths, represented by the triple-diminished ninth (C–D𝄫𝄫) or double-down minor seventh (C–vvB♭). Two comma steps are required to bend the Pythagorean minor seventh to the septimal one.
Subgroup: 2.3.5.7
Comma list: 126/125, 32805/32768
Mapping: [⟨1 0 15 44], ⟨0 1 -8 -26]]
- mapping generators: ~2, ~3
- WE: ~2 = 1199.7974 ¢, ~3/2 = 701.1210 ¢
- error map: ⟨-0.203 -1.037 +3.300 -1.618]
- CWE: ~2 = 1200.0000{{c]}, ~3/2 = 701.2465 ¢
- error map: ⟨0.000 -0.709 +3.715 -1.234]
- 7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
- 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7
Optimal ET sequence: 12, …, 65, 77, 166c
Badness (Sintel): 1.78
11-limit
Subgroup: 2.3.5.7.11
Comma list: 126/125, 176/175, 32805/32768
Mapping: [⟨1 0 15 44 70], ⟨0 1 -8 -26 -42]]
Optimal tunings:
- WE: ~2 = 1199.7077 ¢, ~3/2 = 701.0017 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.1804 ¢
Optimal ET sequence: 12, 65e, 77, 89, 166c
Badness (Sintel): 1.62
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 196/195, 5445/5408
Mapping: [⟨1 0 15 44 70 75], ⟨0 1 -8 -26 -42 -45]]
Optimal tunings:
- WE: ~2 = 1199.7782 ¢, ~3/2 = 701.0966 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2319 ¢
Optimal ET sequence: 12f, 65ef, 77, 166cf
Badness (Sintel): 1.56
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 126/125, 176/175, 196/195, 256/255, 2904/2873
Mapping: [⟨1 0 15 44 70 75 -7], ⟨0 1 -8 -26 -42 -45 7]]
Optimal tunings:
- WE: ~2 = 1199.5839 ¢, ~3/2 = 700.9632 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2137 ¢
Optimal ET sequence: 12f, 77, 89f, 166cf
Badness (Sintel): 1.52
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 126/125, 171/170, 176/175, 196/195, 209/208, 324/323
Mapping: [⟨1 0 15 44 70 75 -7 9], ⟨0 1 -8 -26 -42 -45 7 -3]]
Optimal tunings:
- WE: ~2 = 1199.7146 ¢, ~3/2 = 701.0500 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2212 ¢
Optimal ET sequence: 12f, 77, 166cf
Badness (Sintel): 1.40
Grackloid
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 176/175, 729/728, 1287/1280
Mapping: [⟨1 0 15 44 70 -47], ⟨0 1 -8 -26 -42 32]]
Optimal tunings:
- WE: ~2 = 1200.0060 ¢, ~3/2 = 701.2202 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.2167 ¢
Optimal ET sequence: 12, 77, 166c
Badness (Sintel): 2.00
Grack
Subgroup: 2.3.5.7.11
Comma list: 126/125, 245/242, 896/891
Mapping: [⟨1 0 15 44 51], ⟨0 1 -8 -26 -30]]
Optimal tunings:
- WE: ~2 = 1199.8388 ¢, ~3/2 = 701.3071 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4068 ¢
Optimal ET sequence: 12, 53d, 65, 77e
Badness (Sintel): 1.85
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 126/125, 196/195, 245/242, 832/825
Mapping: [⟨1 0 15 44 51 75], ⟨0 1 -8 -26 -30 -45]]
Optimal tunings:
- WE: ~2 = 1199.7329 ¢, ~3/2 = 701.1918 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.3555 ¢
Optimal ET sequence: 12f, 53dff, 65f, 77e
Badness (Sintel): 1.84
Catahelenic
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 126/125, 245/242, 352/351
Mapping: [⟨1 0 15 44 51 56], ⟨0 1 -8 -26 -30 -33]]
Optimal tunings:
- WE: ~2 = 1199.8928 ¢, ~3/2 = 701.4664 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.5327 ¢
Optimal ET sequence: 12f, …, 53d, 65
Badness (Sintel): 2.01
Schism
Schism is a relatively low-accuracy extension as it tempers out the septimal comma. The 7/4 is found at -2 fifths, represented by the minor seventh (C–B♭). 12edo is recommendable tuning, though 29edo (29d val), 41edo (41d val), and 53edo (53d val) can be used.
Bischismic
Bischismic tempers out 3136/3125, the hemimean comma, as well as 321489/320000, the varunisma, and may be described as the 118 & 130 temperament. The octave is split in halves, so the ploidacot of this temperament is diploid monocot. In schismic, -10 fifths make the interval class of 10/9. Bischismic then finds 7/4 by a stack of two 10/9's plus a semi-octave period, and in the 11-limit, it simply finds 11/8 by a stack of three 10/9's. 248edo and 378edo make for excellent tunings in both cases.
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
- WE: ~567/400 = 600.0072 ¢, ~3/2 = 701.6005 ¢
- error map: ⟨+0.014 -0.340 +0.982 -0.629]
- CWE: ~567/400 = 600.0000 ¢, ~3/2 = 701.5915 ¢
- error map: ⟨0.000 -0.364 +0.954 -0.656]
- 7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
- 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7
Optimal ET sequence: 12, …, 106d, 118, 130, 248, 378
Badness (Sintel): 1.39
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]]
Optimal tunings:
- WE: ~99/70 = 600.0165 ¢, ~3/2 = 701.6316 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.6110 ¢
Optimal ET sequence: 12, …, 106de, 118, 130, 248
Badness (Sintel): 0.931
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 441/440, 729/728, 1001/1000, 3136/3125
Mapping: [⟨2 0 30 69 102 -75], ⟨0 1 -8 -20 -30 26]]
Optimal tunings:
- WE: ~99/70 = 599.9610 ¢, ~3/2 = 701.5445 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.5908 ¢
Optimal ET sequence: 12, 118, 130, 248, 378
Badness (Sintel): 1.19
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 441/440, 561/560, 729/728, 3136/3125
Mapping: [⟨2 0 30 69 102 -75 5], ⟨0 1 -8 -20 -30 26 1]]
Optimal tunings:
- WE: ~99/70 = 600.0331 ¢, ~3/2 = 701.6387 ¢
- CWE: ~99/70 = 600.0000 ¢, ~3/2 = 701.5994 ¢
Optimal ET sequence: 12, 118, 130, 248g
Badness (Sintel): 1.49
Bischis
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 364/363, 441/440, 3136/3125
Mapping: [⟨2 0 30 69 102 131], ⟨0 1 -8 -20 -30 -39]]
Optimal tunings:
- WE: ~55/39 = 599.9766 ¢, ~3/2 = 701.5380 ¢
- CWE: ~55/39 = 600.0000 ¢, ~3/2 = 701.5670 ¢
Optimal ET sequence: 12f, 106deff, 118f, 130
Badness (Sintel): 1.21
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125
Mapping: [⟨2 0 30 69 102 131 5], ⟨0 1 -8 -20 -30 -39 1]]
Optimal tunings:
- WE: ~55/39 = 600.0997 ¢, ~3/2 = 701.7114 ¢
- CWE: ~55/39 = 600.0000 ¢, ~3/2 = 701.5899 ¢
Optimal ET sequence: 12f, 106deff, 118f, 130, 248fg
Badness (Sintel): 1.37
Kleischismic
Kleischismic tempers out 1500625/1492992, the uniwiz comma, and may be described as the 94 & 118 temperament. The generator is a infrafifth, two of which plus a semi-octave period make the 3rd harmonic; its ploidacot is thus diploid alpha-dicot. In schismic, 10 fifths make the interval class of 9/5. Kleischismic then finds 7/4 by that minus a 36/35 quartertone, which is the aforementioned generator minus a semi-octave period. The generator stands in for 16/11 and the quartertone stands in for 33/32 in the 11-limit. 212edo and 330edo in the 330e val may be recommended as tunings.
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
- WE: ~1225/864 = 600.1246 ¢, ~35/24 = 651.0550 ¢ (~36/35 = 50.9304 ¢)
- error map: ⟨+0.249 +0.280 -0.453 -0.650]
- CWE: ~1225/864 = 600.0000 ¢, ~35/24 = 650.9204 ¢ (~36/35 = 50.9204 ¢)
- error map: ⟨0.000 -0.114 -1.041 -1.338]
Optimal ET sequence: 24, 94, 118, 212, 330, 542d, 872cdd, 1414ccddd
Badness (Sintel): 2.80
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]]
Optimal tunings:
- WE: ~99/70 = 600.1645 ¢, ~35/24 = 651.0963 ¢ (~36/35 = 50.9319 ¢)
- CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9184 ¢ (~36/35 = 50.9184 ¢)
Optimal ET sequence: 24, 94, 118, 212, 330e, 542dee, 872cddeee
Badness (Sintel): 1.21
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]]
Optimal tunings:
- WE: ~99/70 = 600.0696 ¢, ~35/24 = 651.0136 ¢ (~36/35 = 50.9440 ¢)
- CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9378 ¢ (~36/35 = 50.9378 ¢)
Optimal ET sequence: 24, 94, 118, 212f
Badness (Sintel): 1.56
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 170/169, 289/288, 352/351, 385/384, 561/560
Mapping: [⟨2 1 22 -15 8 15 6], ⟨0 2 -16 19 -1 -7 2]]
Optimal tunings:
- WE: ~99/70 = 600.1134 ¢, ~35/24 = 651.0646 ¢ (~36/35 = 50.9512 ¢)
- CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9414 ¢ (~36/35 = 50.9414 ¢)
Optimal ET sequence: 24, 94, 118
Badness (Sintel): 1.30
Kleischis
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 1573/1568, 14641/14580
Mapping: [⟨2 1 22 -15 8 -36], ⟨0 2 -16 19 -1 40]]
Optimal tunings:
- WE: ~99/70 = 600.1909 ¢, ~35/24 = 651.1578 ¢ (~36/35 = 50.9670 ¢)
- CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9541 ¢ (~36/35 = 50.9541 ¢)
Optimal ET sequence: 24f, 94, 118f, 212
Badness (Sintel): 1.55
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 289/288, 325/324, 385/384, 442/441, 14641/14580
Mapping: [⟨2 1 22 -15 8 -36 6], ⟨0 2 -16 19 -1 40 2]]
Optimal tunings:
- WE: ~99/70 = 600.2190 ¢, ~35/24 = 651.1578 ¢ (~36/35 = 50.9670 ¢)
- CWE: ~99/70 = 600.0000 ¢, ~35/24 = 650.9518 ¢ (~36/35 = 50.9518 ¢)
Optimal ET sequence: 24f, 94, 118f, 212g
Badness (Sintel): 1.26
Salsa
Salsa tempers out 245/243, the sensamagic comma, and may be described as the 41 & 65 temperament. It has a neutral third as a generator; its ploidacot is dicot. In fact it is related to hemififths, from which this less accurate temperament only differs by the mapping of 5.
Subgroup: 2.3.5.7
Comma list: 245/243, 32805/32768
Mapping: [⟨1 1 7 -1], ⟨0 2 -16 13]]
- mapping generators: ~2, ~128/105
- WE: ~2 = 1200.7707 ¢, ~128/105 = 351.2748 ¢
- error map: ⟨+0.771 +1.365 -1.315 -3.024]
- CWE: ~2 = 1200.0000 ¢, ~128/105 = 351.0471 ¢
- error map: ⟨0.000 +0.139 -3.068 -5.213]
Optimal ET sequence: 17, 24, 41, 106d, 147d, 188cd
Badness (Sintel): 2.03
11-limit
Subgroup: 2.3.5.7.11
Comma list: 243/242, 245/242, 385/384
Mapping: [⟨1 1 7 -1 2], ⟨0 2 -16 13 5]]
Optimal tunings:
- WE: ~2 = 1200.3891 ¢, ~11/9 = 351.1275 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.0141 ¢
Optimal ET sequence: 17, 24, 41, 106d
Badness (Sintel): 1.30
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 105/104, 144/143, 243/242, 245/242
Mapping: [⟨1 1 7 -1 2 4], ⟨0 2 -16 13 5 -1]]
Optimal tunings:
- WE: ~2 = 1199.9362{c}}, ~11/9 = 351.0061 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/9 = 351.0247 ¢
Optimal ET sequence: 17, 24, 41
Badness (Sintel): 1.27
Hemischis
Hemischis tempers out 6144/6125, the porwell comma, as well as 19683/19600, the cataharry comma, and may be described as the 53 & 130 temperament. Its ploidacot is alpha-dicot.
The S-expression-based comma list for 13-limit hemischis is {S12/S14, S13/S15 = S26, S27, S64, (S65)}. Tempering out 169/168 (S13), 225/224 (S15) or 625/624 (S25) leads to 53edo while tempering out 24192/24167 (S12/S13), 10985/10976 (S13/S14), 43904/43875 (S14/S15) or 2401/2400 (S49) leads to 130edo and implies S12, S13, S14, and S15 are tempered together.
Subgroup: 2.3.5.7
Comma list: 6144/6125, 19683/19600
Mapping: [⟨1 0 15 -17], ⟨0 2 -16 25]]
- mapping generators: ~2, ~140/81
- WE: ~2 = 1199.8579 ¢, ~140/81 = 951.6847 ¢
- error map: ⟨-0.142 -0.586 +0.600 +0.708]
- CWE: ~2 = 1200.0000 ¢, ~140/81 = 951.7966 ¢
- error map: ⟨0.000 -0.362 +0.941 +1.088]
Optimal ET sequence: 24, 53, 130, 183, 313
Badness (Sintel): 1.16
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 5632/5625, 8019/8000
Mapping: [⟨1 0 15 -17 51], ⟨0 2 -16 25 -60]]
Optimal tunings:
- WE: ~2 = 1199.8482 ¢, ~140/81 = 950.6809 ¢
- CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.8020 ¢
Optimal ET sequence: 53, 130, 183, 313, 809cd
Badness (Sintel): 1.20
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 351/350, 540/539, 676/675, 4096/4095
Mapping: [⟨1 0 15 -17 51 14], ⟨0 2 -16 25 -60 -13]]
Optimal tunings:
- WE: ~2 = 1199.9140 ¢, ~140/81 = 950.7324 ¢
- CWE: ~2 = 1200.0000 ¢, ~140/81 = 950.8010 ¢
Optimal ET sequence: 53, 130, 183, 313
Badness (Sintel): 0.860
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 351/350, 442/441, 561/560, 676/675, 4096/4095
Mapping: [⟨1 0 15 -17 51 14 -49], ⟨0 2 -16 25 -60 -13 67]]
Optimal tunings:
- WE: ~2 = 1199.9740 ¢, ~26/15 = 950.7894 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8100 ¢
Optimal ET sequence: 53, 130, 183, 496d
Badness (Sintel): 1.07
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 4096/4095
Mapping: [⟨1 0 15 -17 51 14 -49 9], ⟨0 2 -16 25 -60 -13 67 -6]]
Optimal tunings:
- WE: ~2 = 1200.0464 ¢, ~26/15 = 950.8459 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8091 ¢
Optimal ET sequence: 53, 130, 183, 313h
Badness (Sintel): 1.11
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 351/350, 442/441, 456/455, 561/560, 676/675, 736/735, 4096/4095
Mapping: [⟨1 0 15 -17 51 14 -49 9 -24], ⟨0 2 -16 25 -60 -13 67 -6 36]]
Optimal tunings:
- WE: ~2 = 1200.0215 ¢, ~26/15 = 950.8239 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8069 ¢
Optimal ET sequence: 53, 130, 183, 313h
Badness (Sintel): 1.06
- Music
Term
Term tempers out the landscape comma, mapping 63/50 to the 1/3-octave period. It can be described as 12 & 171, and is the unique temperament that equates a syntonic~Pythagorean comma with a stack of three marvel commas. A septimal comma is then found as a stack of four marvel commas. In some 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a kleisma, with three kleismas making a comma, so this temperament may be useful for modeling that. 171edo makes for an excellent tuning.
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
- WE: ~63/50 = 400.0257 ¢, ~3/2 = 701.7873 ¢
- error map: ⟨+0.077 -0.091 -0.072 +0.031]
- CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.7383 ¢
- error map: ⟨0.000 -0.217 -0.220 -0.115]
- 7-odd-limit unchanged-interval (eigenmonzo) basis): 2.5/3
- 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7
Optimal ET sequence: 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722
Badness (Sintel): 0.505
Terminal
Terminal tempers out 441/440 and 4375/4356, and may be described as 159 & 171. In this temperament, 44/35 and 63/50 are represented as one period of 1/3 octave.
Subgroup: 2.3.5.7.11
Comma list: 441/440, 4375/4356, 32805/32768
Mapping: [⟨3 0 45 94 134], ⟨0 1 -8 -18 -26]]
Optimal tunings:
- WE: ~44/35 = 400.0464 ¢, ~3/2 = 701.9053 ¢
- CWE: ~44/35 = 400.0000 ¢, ~3/2 = 701.8178 ¢
Optimal ET sequence: 12, …, 159, 330
Badness (Sintel): 1.97
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 625/624, 13720/13689
Mapping: [⟨3 0 45 94 134 168], ⟨0 1 -8 -18 -26 -33]]
Optimal tunings:
- WE: ~44/35 = 400.0449 ¢, ~3/2 = 701.8995 ¢
- CWE: ~44/35 = 400.0000 ¢, ~3/2 = 701.8156 ¢
Optimal ET sequence: 12f, …, 159, 330
Badness (Sintel): 1.53
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619
Mapping: [⟨3 0 45 94 134 168 -2], ⟨0 1 -8 -18 -26 -33 3]]
Optimal tunings:
- WE: ~34/27 = 400.0195 ¢, ~3/2 = 701.8439 ¢
- CWE: ~34/27 = 400.0000 ¢, ~3/2 = 701.8081 ¢
Optimal ET sequence: 12f, 159, 171, 330
Badness (Sintel): 1.38
Terminator
Terminator tempers out 540/539, and may be described as 171 & 183.
Subgroup: 2.3.5.7.11
Comma list: 540/539, 32805/32768, 137781/137500
Mapping: [⟨3 0 45 94 -137], ⟨0 1 -8 -18 31]]
Optimal tunings:
- WE: ~63/50 = 399.9677 ¢, ~3/2 = 701.6278 ¢
- CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6846 ¢
Optimal ET sequence: 12e, 171, 183, 354, 537, 891de
Badness (Sintel): 2.21
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4096/4095, 31250/31213
Mapping: [⟨3 0 45 94 -137 -103], ⟨0 1 -8 -18 31 24]]
Optimal tunings:
- WE: ~63/50 = 399.9731 ¢, ~3/2 = 701.6414 ¢
- CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6881 ¢
Optimal ET sequence: 12e, 171, 183, 354, 891de
Badness (Sintel): 1.47
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 540/539, 729/728, 936/935, 1156/1155, 4096/4095
Mapping: [⟨3 0 45 94 -137 -103 -2], ⟨0 1 -8 -18 31 24 3]]
Optimal tunings:
- WE: ~63/50 = 399.9757 ¢, ~3/2 = 701.6458 ¢
- CWE: ~63/50 = 400.0000 ¢, ~3/2 = 701.6881 ¢
Optimal ET sequence: 12e, 171, 183, 354, 891de
Badness (Sintel): 1.04
Semiterm
The semiterm temperament tempers out 9801/9800 (kalisma) as well as 151263/151250 (odiheim comma), and may be described as 12 & 342. It has a period of 1/6 octave and its ploidacot is hexaploid monocot.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 32805/32768, 151263/151250
Mapping: [⟨6 0 90 188 287], ⟨0 1 -8 -18 -28]]
- mapping generators: ~55/49, ~3
Optimal tunings:
- WE: ~55/49 = 200.0134 ¢, ~3/2 = 701.7931 ¢
- CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.7426 ¢
Optimal ET sequence: 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde
Badness (Sintel): 0.973
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375
Mapping: [⟨6 0 90 188 287 355], ⟨0 1 -8 -18 -28 -35]]
Optimal tunings:
- WE: ~55/49 = 200.0083 ¢, ~3/2 = 701.7549 ¢
- CWE: ~55/49 = 200.0000 ¢, ~3/2 = 701.7238 ¢
Optimal ET sequence: 12f, 330eff, 342f, 696f *
* optimal patent val: 354
Badness (Sintel): 1.85
Hemiterm
The hemiterm temperament tempers out 3025/3024 (lehmerisma), and may be described as 159 & 183. Its ploidacot is triploid beta-dicot.
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 32805/32768, 102487/102400
Mapping: [⟨3 0 45 94 8], ⟨0 2 -16 -36 1]]
- mapping generators: ~63/50, ~693/400
Optimal tunings:
- WE: ~63/50 = 400.0309 ¢, ~693/400 = 950.9458 ¢ (~12/11 = 150.8841 ¢)
- CWE: ~63/50 = 400.0000 ¢, ~693/400 = 950.8707 ¢ (~12/11 = 150.8707 ¢)
Optimal ET sequence: 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce
Badness (Sintel): 0.684
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712
Mapping: [⟨3 0 45 94 8 42], ⟨0 2 -16 -36 1 -13]]
Optimal tunings:
- WE: ~63/50 = 400.0541 ¢, ~26/15 = 951.0013 ¢ (~12/11 = 150.8932 ¢)
- CWE: ~63/50 = 400.0000 ¢, ~26/15 = 950.8696 ¢ (~12/11 = 150.8696 ¢)
Optimal ET sequence: 24d, 159, 183, 342f
Badness (Sintel): 1.30
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264
Mapping: [⟨3 0 45 94 8 42 -2], ⟨0 2 -16 -36 1 -13 6]]
Optimal tunings:
- WE: ~34/27 = 400.0373 ¢, ~26/15 = 950.9556 ¢ (~12/11 = 150.8809 ¢)
- CWE: ~34/27 = 400.0000 ¢, ~26/15 = 950.8652 ¢ (~12/11 = 150.8652 ¢)
Optimal ET sequence: 24d, 159, 183, 342f, 525f
Badness (Sintel): 1.14
Altinex
Altinex is an alternative to hemiterm and may be described as 24 & 159. 159edo itself makes for a recommendable tuning.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 367653125/362797056
Mapping: [⟨3 0 45 -32], ⟨0 2 -16 17]]
- mapping generators: ~1536/1225, ~34300/19683
- WE: ~1536/1225 = 400.1360 ¢, ~34300/19683 = 951.2867 ¢
- error map: ⟨+0.408 +0.618 -0.781 -1.304]
- CWE: ~1536/1225 = 400.0000 ¢, ~34300/19683 = 950.9638 ¢
- error map: ⟨0.000 -0.027 -1.735 -2.441]
Optimal ET sequence: 24, 135, 159, 612ccdd
Badness (Sintel): 10.7
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 14700/14641, 19712/19683
Mapping: [⟨3 0 45 -32 8], ⟨0 2 -16 17 1]]
Optimal tunings:
- WE: ~44/35 = 400.1156 ¢, ~121/70 = 951.2377 ¢
- CWE: ~44/35 = 400.0000 ¢, ~121/70 = 950.9634 ¢
Optimal ET sequence: 24, 135, 159
Badness (Sintel): 3.35
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 385/384, 676/675, 19712/19683
Mapping: [⟨3 0 45 -32 8 42], ⟨0 2 -16 17 1 -13]]
Optimal tunings:
- WE: ~44/35 = 400.1396 ¢, ~26/15 = 951.2799 ¢
- CWE: ~44/35 = 400.0000 ¢, ~26/15 = 950.9462 ¢
Optimal ET sequence: 24, 135f, 159
Badness (Sintel): 2.27
Squirrel
Squirrel tempers out 686/675, the sengic comma, and may be described as 29 & 36. It has a ~11/10 generator, three of which give the fourth (4/3), and thirteen of which give 7/4 with octave reduction. Its ploidacot is omega-tricot.
Subgroup: 2.3.5.7
Comma list: 686/675, 32805/32768
Mapping: [⟨1 2 -1 1], ⟨0 -3 24 13]]
- WE: ~2 = 1200.7408 ¢, ~160/147 = 166.2424 ¢
- error map: ⟨+0.741 +0.799 +2.763 -6.934]
- CWE: ~2 = 1200.0000 ¢, ~160/147 = 166.1597 ¢
- error map: ⟨0.000 -0.434 +1.518 -8.750]
Optimal ET sequence: 29, 36, 65
Badness (Sintel): 4.42
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]]
Optimal tunings:
- WE: ~2 = 1200.6379 ¢, ~11/10 = 166.1853 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.1157 ¢
Optimal ET sequence: 29, 36, 65
Badness (Sintel): 2.26
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]]
Optimal tunings:
- WE: ~2 = 1201.1361 ¢, ~11/10 = 166.2110 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0833 ¢
Optimal ET sequence: 29, 65f, 94df
Badness (Sintel): 1.81
Tertiaschis
Tertiaschis may be described as 94 & 159. It has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with squirrel, but tempers out 1071875/1062882 for prime 7.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 1071875/1062882
Mapping: [⟨1 2 -1 10], ⟨0 -3 24 -52]]
- WE: ~2 = 1200.3627 ¢, ~192/175 = 166.0691 ¢
- error map: ⟨+0.363 +0.563 -1.019 -0.790]
- CWE: ~2 = 1200.0000 ¢, ~192/175 = 166.0172 ¢
- error map: ⟨0.000 -0.007 -1.901 -1.720]
Optimal ET sequence: 65, 94, 159, 253, 412cd
Badness (Sintel): 5.36
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]]
Optimal tunings:
- WE: ~2 = 1200.3379 ¢, ~11/10 = 166.0638 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0167 ¢
Optimal ET sequence: 65, 94, 159, 253, 412cd, 665ccde
Badness (Sintel): 2.07
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]]
Optimal tunings:
- WE: ~2 = 1200.3467 ¢, ~11/10 = 166.0635 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0142 ¢
Optimal ET sequence: 65f, 94, 159, 253, 412cdf, 665ccdef
Badness (Sintel): 1.52
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]]
Optimal tunings:
- WE: ~2 = 1200.3019 ¢, ~11/10 = 166.0535 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0114 ¢
Optimal ET sequence: 65f, 94, 159, 253
Badness (Sintel): 1.35
Countertertiaschis
Countertertiaschis may be described as 159 & 224. It has a ~11/10 generator, sharing the same 2.3.5.11 subgroup with squirrel, but tempers out 244140625/243045684 for prime 7.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 244140625/243045684
Mapping: [⟨1 2 -1 -12], ⟨0 -3 24 107]]
- WE: ~2 = 1200.1265 ¢, ~625/567 = 166.0797 ¢
- error map: ⟨+0.127 +0.059 -0.529 +0.178]
- CWE: ~2 = 1200.0000 ¢, ~625/567 = 166.0632 ¢
- error map: ⟨0.000 -0.145 -0.797 -0.065]
Optimal ET sequence: 65d, 159, 224, 383, 607
Badness (Sintel): 4.76
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 4000/3993, 32805/32768
Mapping: [⟨1 2 -1 -12 0], ⟨0 -3 24 107 25]]
Optimal tunings:
- WE: ~2 = 1200.0804 ¢, ~11/10 = 166.0739 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0634 ¢
Optimal ET sequence: 65d, 159, 224, 383, 607
Badness (Sintel): 1.62
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976
Mapping: [⟨1 2 -1 -12 0 -10], ⟨0 -3 24 107 25 99]]
Optimal tunings:
- WE: ~2 = 1200.0805 ¢, ~11/10 = 166.0740 ¢
- CWE: ~2 = 1200.0000 ¢, ~11/10 = 166.0635 ¢
Optimal ET sequence: 65d, 159, 224, 383, 607
Badness (Sintel): 1.01
Quadrant
Quadrant tempers out 390625/388962, the dimcomp comma, and maps 25/21 to the 1/4-octave period. It may be decribed as the 12 & 212 temperament; its ploidacot is tetraploid monocot. Just as term equates the syntonic~Pythagorean comma with three marvel commas, quadrant equates the syntonic~Pythagorean comma with four. A septimal comma is then found as a stack of five marvel commas.
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
- WE: ~2 = 300.0255 ¢, ~3/2 = 701.8831 ¢
- error map: ⟨+0.102 +0.030 -0.664 +0.462]
- CWE: ~2 = 300.0000 ¢, ~3/2 = 701.8180 ¢
- error map: ⟨0.000 -0.137 -0.858 +0.268]
Optimal ET sequence: 12, …, 200, 212, 224, 436, 660
Badness (Sintel): 2.79
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]]
Optimal tunings:
- WE: ~25/21 = 300.0244 ¢, ~3/2 = 701.8759 ¢
- CWE: ~25/21 = 300.0000 ¢, ~3/2 = 701.8145 ¢
Optimal ET sequence: 12, …, 212, 224, 436, 660
Badness (Sintel): 1.51
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]]
Optimal tunings:
- WE: ~25/21 = 300.0234 ¢, ~3/2 = 701.8707 ¢
- CWE: ~25/21 = 300.0000 ¢, ~3/2 = 701.8123 ¢
Optimal ET sequence: 12f, …, 212, 224, 436, 660
Badness (Sintel): 1.13
Sesquiquartififths
Sesquiquartififths tempers out 2401/2400, the breedsma, and may be described as the 41 & 171 temperament. It splits the fifth into four; its ploidacot is thus tetracot.
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
- WE: ~2 = 1200.0846 ¢, ~448/405 = 175.4460 ¢
- error map: ⟨+0.085 -0.086 +0.007 -0.093]
- CWE: ~2 = 1200.0000 ¢, ~448/405 = 175.4320 ¢
- error map: ⟨0.000 -0.227 -0.137 -0.306]
- 7-odd-limit unchanged-interval (eigenmonzo) basis: 2.7/3
- 9-odd-limit unchanged-interval (eigenmonzo) basis: 2.9/7
Optimal ET sequence: 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd
Badness (Sintel): 0.285
Sesquart
Sesquart is the main 11- and 13-limit extension of sesquiquartififths of practical interest, as it identifies the neutral third with 11/9, which is realized in 41edo, 89edo, 130edo, and 171edo also makes for a possible tuning.
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]]
Optimal tunings:
- WE: ~2 = 1199.8171 ¢, ~256/231 = 175.3793 ¢
- CWE: ~2 = 1200.0000 ¢, ~256/231 = 175.4081 ¢
Optimal ET sequence: 41, 89, 130, 301e, 431e
Badness (Sintel): 0.969
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]]
Optimal tunings:
- WE: ~2 = 1199.8352 ¢, ~72/65 = 175.3852 ¢
- CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.4095 ¢
Optimal ET sequence: 41, 89, 130, 301e, 431e
Badness (Sintel): 0.925
Heartia
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 256/255, 273/272, 364/363, 441/440
Mapping: [⟨1 1 7 5 2 -2 0], ⟨0 4 -32 -15 10 39 28]]
Optimal tunings:
- WE: ~2 = 1199.6422 ¢, ~72/65 = 175.3338 ¢
- CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3857 ¢
Optimal ET sequence: 41, 89, 130g
Badness (Sintel): 1.45
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 171/170, 243/242, 256/255, 273/272, 324/323, 441/440
Mapping: [⟨1 1 7 5 2 -2 0 6], ⟨0 4 -32 -15 10 39 28 -12]]
Optimal tunings:
- WE: ~2 = 1199.7499 ¢, ~21/19 = 175.3432 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.3797 ¢
Optimal ET sequence: 41, 89, 130g
Badness (Sintel): 1.40
Sesquartia
Subgroup: 2.3.5.7.11.13.17
Comma list: 243/242, 364/363, 441/440, 595/594, 3584/3575
Mapping: [⟨1 1 7 5 2 -2 -6], ⟨0 4 -32 -15 10 39 69]]
Optimal tunings:
- WE: ~2 = 1199.8902 ¢, ~72/65 = 175.4077 ¢
- CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.4234 ¢
Optimal ET sequence: 41, 130, 171
Badness (Sintel): 1.18
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594
Mapping: [⟨1 1 7 5 2 -2 -6 6], ⟨0 4 -32 -15 10 39 69 -12]]
Optimal tunings:
- WE: ~2 = 1199.9864 ¢, ~21/19 = 175.4169 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.4189 ¢
Optimal ET sequence: 41, 130, 171
Badness (Sintel): 1.24
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 243/242, 323/322, 361/360, 364/363, 441/440, 456/455, 595/594
Mapping: [⟨1 1 7 5 2 -2 -6 6 -6], ⟨0 4 -32 -15 10 39 69 -12 72]]
Optimal tunings:
- WE: ~2 = 1199.9606 ¢, ~21/19 = 175.4067 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/19 = 175.4123 ¢
Optimal ET sequence: 41i, 130, 171
Badness (Sintel): 1.36
Hearty
Subgroup: 2.3.5.7.11.13.17
Comma list: 221/220, 243/242, 364/363, 441/440, 1632/1625
Mapping: [⟨1 1 7 5 2 -2 13], ⟨0 4 -32 -15 10 39 -61]]
Optimal tunings:
- WE: ~2 = 1199.9458 ¢, ~72/65 = 175.3689 ¢
- CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3770 ¢
Optimal ET sequence: 41g, 89, 130
Badness (Sintel): 1.56
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 221/220, 243/242, 361/360, 364/363, 441/440, 456/455
Mapping: [⟨1 1 7 5 2 -2 13 6], ⟨0 4 -32 -15 10 39 -61 -12]]
Optimal tunings:
- WE: ~2 = 1200.0114 ¢, ~72/65 = 175.3783 ¢
- CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3765 ¢
Optimal ET sequence: 41g, 89, 130
Badness (Sintel): 1.39
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 221/220, 243/242, 276/275, 323/322, 361/360, 364/363, 441/440
Mapping: [⟨1 1 7 5 2 -2 13 6 13], ⟨0 4 -32 -15 10 39 -61 -12 -58]]
Optimal tunings:
- WE: ~2 = 1200.0122 ¢, ~72/65 = 175.3782 ¢
- CWE: ~2 = 1200.0000 ¢, ~72/65 = 175.3763 ¢
Optimal ET sequence: 41g, 89, 130
Badness (Sintel): 1.37
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]]
- mapping generators: ~99/70, ~448/405
Optimal tunings:
- WE: ~99/70 = 600.0429 ¢, ~448/405 = 175.4474 ¢
- CWE: ~99/70 = 600.0000 ¢, ~448/405 = 175.4334 ¢
Optimal ET sequence: 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee
Badness (Sintel): 0.561
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
- WE: ~2 = 1200.1039 ¢, ~243/224 = 140.3620 ¢
- error map: ⟨+0.104 -0.041 -0.067 -0.137]
- CWE: ~2 = 1200.0000 ¢, ~243/224 = 140.3496 ¢
- error map: ⟨0.000 -0.207 -0.296 -0.436]
Optimal ET sequence: 17, 77, 94, 171
Badness (Sintel): 0.777
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 1331/1323, 19712/19683
Mapping: [⟨1 1 7 0 1], ⟨0 5 -40 24 21]]
Optimal tunings:
- WE: ~2 = 1200.3103 ¢, ~88/81 = 140.4011 ¢
- CWE: ~2 = 1200.0000 ¢, ~88/81 = 140.3649 ¢
Optimal ET sequence: 17, 77, 94, 171e, 265e
Badness (Sintel): 2.10
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 352/351, 385/384, 729/728, 1331/1323
Mapping: [⟨1 1 7 0 1 3], ⟨0 5 -40 24 21 6]]
Optimal tunings:
- WE: ~2 = 1200.1840 ¢, ~13/12 = 140.3840 ¢
- CWE: ~2 = 1200.0000 ¢, ~13/12 = 140.3627 ¢
Optimal ET sequence: 17, 77, 94, 171e
Badness (Sintel): 1.57
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
- WE: ~2 = 1200.0032 ¢, ~56/45 = 380.3557 ¢
- error map: ⟨+0.003 -0.177 -0.493 +0.898]
- CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3546 ¢
- error map: ⟨0.000 -0.182 -0.498 +0.890]
Optimal ET sequence: 41, 142, 183, 224
Badness (Sintel): 1.82
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]]
Optimal tunings:
- WE: ~2 = 1199.9709 ¢, ~56/45 = 380.3423 ¢
- CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3517 ¢
Optimal ET sequence: 41, 142, 183, 224, 631d, 855d
Badness (Sintel): 1.04
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]]
Optimal tunings:
- WE: ~2 = 1199.9663 ¢, ~56/45 = 380.3403 ¢
- CWE: ~2 = 1200.0000 ¢, ~56/45 = 380.3509 ¢
Optimal ET sequence: 41, 142, 183, 224, 631d, 855d
Badness (Sintel): 0.884
Quintilipyth
The quintilipyth temperament (12 & 253, formerly quintilischis) slices the pythagorean fourth (4/3) into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 9765625/9680832
Mapping: [⟨1 2 -1 -4], ⟨0 -5 40 82]]
- mapping generators: ~2, ~625/588
- WE: ~2 = 1200.1138 ¢, ~625/588 = 99.6347 ¢
- error map: ⟨+0.114 +0.099 -1.041 +0.761]
- CWE: ~2 = 1200.0000 ¢, ~625/588 = 99.6265 ¢
- error map: ⟨0.000 -0.087 -1.255 +0.544]
Optimal ET sequence: 12, …, 253, 265
Badness (Sintel): 6.43
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]]
Optimal tunings:
- WE: ~2 = 1200.1503 ¢, ~35/33 = 99.6287 ¢
- CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6176 ¢
Optimal ET sequence: 12, …, 253, 265, 518c
Badness (Sintel): 3.74
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]]
Optimal tunings:
- WE: ~2 = 1200.1774 ¢, ~35/33 = 99.6267 ¢
- CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6134 ¢
Optimal ET sequence: 12f, …, 241cdef, 253
Badness (Sintel): 2.86
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]]
Optimal tunings:
- WE: ~2 = 1200.1745 ¢, ~18/17 = 99.6265 ¢
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6131 ¢
Optimal ET sequence: 12f, 241cdef, 253
Badness (Sintel): 2.34
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]]
Optimal tunings:
- WE: ~2 = 1200.0713 ¢, ~18/17 = 99.6208 ¢
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6152 ¢
Optimal ET sequence: 12f, 253, 265
Badness (Sintel): 2.32
Quintaschis
The quintaschis temperament (12 & 289) slices the fourth (4/3) into five semitones and tempers out 49009212/48828125 in the 7-limit.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 49009212/48828125
Mapping: [⟨1 2 -1 -5], ⟨0 -5 40 94]]
- WE: ~2 = 1200.0536 ¢, ~200/189 = 99.6684 ¢
- error map: ⟨+0.054 -0.190 +0.370 -0.262]
- CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6645 ¢
- error map: ⟨0.000 -0.277 +0.266 -0.363]
Optimal ET sequence: 12, …, 289, 301, 590, 891, 1192
Badness (Sintel): 3.36
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 32805/32768, 1953125/1951488
Mapping: [⟨1 2 -1 -5 -8], ⟨0 -5 40 94 138]]
Optimal tunings:
- WE: ~2 = 1200.0988 ¢, ~35/33 = 99.6613 ¢
- CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6540 ¢
Optimal ET sequence: 12, …, 277d, 289
Badness (Sintel): 3.69
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 364/363, 441/440, 32805/32768, 109512/109375
Mapping: [⟨1 2 -1 -5 -8 -11], ⟨0 -5 40 94 138 177]]
Optimal tunings:
- WE: ~2 = 1200.0625 ¢, ~35/33 = 99.6630 ¢
- CWE: ~2 = 1200.0000 ¢, ~35/33 = 99.6583 ¢
Optimal ET sequence: 12f, …, 277dff, 289
Badness (Sintel): 3.07
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 364/363, 441/440, 595/594, 3757/3750, 32805/32768
Mapping: [⟨1 2 -1 -5 -8 -11 5], ⟨0 -5 40 94 138 177 -11]]
Optimal tunings:
- WE: ~2 = 1200.1286 ¢, ~18/17 = 99.6668 ¢
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6568 ¢
Optimal ET sequence: 12f, 277dff, 289
Badness (Sintel): 2.58
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 364/363, 441/440, 476/475, 595/594, 3757/3750, 6885/6859
Mapping: [⟨1 2 -1 -5 -8 -11 5 4], ⟨0 -5 40 94 138 177 -11 3]]
Optimal tunings:
- WE: ~2 = 1200.0289 ¢, ~18/17 = 99.6609 ¢
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6586 ¢
Badness (Sintel): 2.56
Quintahelenic
Subgroup: 2.3.5.7.11
Comma list: 5632/5625, 8019/8000, 151263/151250
Mapping: [⟨1 2 -1 -5 -9], ⟨0 -5 40 94 150]]
Optimal tunings:
- WE: ~2 = 1200.0195 ¢, ~200/189 = 99.6723 ¢
- CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6709 ¢
Optimal ET sequence: 12, …, 289e, 301, 915
Badness (Sintel): 2.72
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000
Mapping: [⟨1 2 -1 -5 -9 -11], ⟨0 -5 40 94 150 177]]
Optimal tunings:
- WE: ~2 = 1200.0442 ¢, ~200/189 = 99.6709 ¢
- CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6675 ¢
Optimal ET sequence: 12f, …, 289e, 301
Badness (Sintel): 2.30
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 561/560, 833/832, 847/845, 1701/1700, 3757/3750
Mapping: [⟨1 2 -1 -5 -9 -11 5], ⟨0 -5 40 94 150 177 -11]]
Optimal tunings:
- WE: ~2 = 1200.1227 ¢, ~200/189 = 99.6753 ¢
- CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6658 ¢
Optimal ET sequence: 12f, 289e, 301
Badness (Sintel): 2.06
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700
Mapping: [⟨1 2 -1 -5 -9 -11 5 4], ⟨0 -5 40 94 150 177 -11 3]]
Optimal tunings:
- WE: ~2 = 1200.0230 ¢, ~200/189 = 99.6694 ¢
- CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6676 ¢
Badness (Sintel): 2.24
Quintahelenoid
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436
Mapping: [⟨1 2 -1 -5 -9 14], ⟨0 -5 40 94 150 -124]]
Optimal tunings:
- WE: ~2 = 1199.9919 ¢, ~200/189 = 99.6712 ¢
- CWE: ~2 = 1200.0000 ¢, ~200/189 = 99.6718 ¢
Optimal ET sequence: 12, 301, 614, 915
Badness (Sintel): 2.73
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 561/560, 729/728, 1001/1000, 4096/4095, 14161/14157
Mapping: [⟨1 2 -1 -5 -9 14 5], ⟨0 -5 40 94 150 -124 -11]]
Optimal tunings:
- WE: ~2 = 1200.0469 ¢, ~18/17 = 99.6749 ¢
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6710 ¢
Badness (Sintel): 2.44
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 476/475, 561/560, 729/728, 1001/1000, 4096/4095, 6144/6137
Mapping: [⟨1 2 -1 -5 -9 14 5 4], ⟨0 -5 40 94 150 -124 -11 3]]
Optimal tunings:
- WE: ~2 = 1199.9925 ¢, ~18/17 = 99.6710 ¢
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6716 ¢
Badness (Sintel): 2.41
Sextilifourths
The sextilifourths (130 & 159, also known as sextilischis, formerly sextilififths) temperament 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: 32805/32768, 235298/234375
Mapping: [⟨1 2 -1 -1], ⟨0 -6 48 55]]
- mapping generators: ~2, ~21/20
- WE: ~2 = 1200.0987 ¢, ~21/20 = 83.0599 ¢
- error map: ⟨+0.099 -0.117 +0.462 -0.630]
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0543 ¢
- error map: ⟨0.000 -0.281 +0.295 -0.837]
Optimal ET sequence: 29, 72cd, 101, 130, 289, 419
Badness (Sintel): 2.75
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]]
Optimal tunings:
- WE: ~2 = 1200.0424 ¢, ~21/20 = 83.0520 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0497 ¢
Optimal ET sequence: 29, 72cde, 101e, 130, 289
Badness (Sintel): 1.50
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]]
Optimal tunings:
- WE: ~2 = 1200.1056 ¢, ~21/20 = 83.0566 ¢
- CWE: ~2 = 1200.0000 ¢, ~21/20 = 83.0508 ¢
Optimal ET sequence: 29, 72cdef, 101e, 130, 289
Badness (Sintel): 1.04
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
- WE: ~42875/39366 = 150.0048 ¢, ~3/2 = 701.7356 ¢
- error map: ⟨+0.039 -0.181 +0.071 +0.127]
- CWE: ~42875/39366 = 150.0000 ¢, ~3/2 = 701.7134 ¢
- error map: ⟨0.000 -0.242 -0.021 +0.022]
Optimal ET sequence: 24, …, 224, 472, 696, 1168
Badness (Sintel): 3.98
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]]
Optimal tunings:
- WE: ~12/11 = 150.0010 ¢, ~3/2 = 701.7177 ¢
- CWE: ~12/11 = 150.0000 ¢, ~3/2 = 701.7131 ¢
Optimal ET sequence: 24, …, 224, 472, 696, 1168
Badness (Sintel): 1.48
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]]
Optimal tunings:
- WE: ~12/11 = 149.9957 ¢, ~3/2 = 701.7046 ¢
- CWE: ~12/11 = 150.0000 ¢, ~3/2 = 701.7247 ¢
Optimal ET sequence: 24, 224, 472, 696
Badness (Sintel): 1.26
Nonant
The nonant temperament (36 & 135) has a period of 1/9 octave and tempers out the septimal ennealimma, [-11 -9 0 9⟩.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 40353607/40310784
Mapping: [⟨9 0 135 11], ⟨0 1 -8 1]]
- mapping generators: ~2592/2401, ~3
- WE: ~2592/2401 = 133.3442 ¢, ~3/2 = 701.8000 ¢
- error map: ⟨+0.098 -0.057 -0.027 -0.141]
- CWE: ~2592/2401 = 133.3333 ¢, ~3/2 = 701.7384 ¢
- error map: ⟨0.000 -0.217 -0.221 -0.421]
Optimal ET sequence: 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd
Badness (Sintel): 1.77
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 32805/32768, 42875/42592
Mapping: [⟨9 0 135 11 131], ⟨0 1 -8 1 -7]]
Optimal tunings:
- WE: ~242/225 = 133.3308 ¢, ~3/2 = 701.8205 ¢
- CWE: ~242/225 = 133.3333 ¢, ~3/2 = 701.8351 ¢
Optimal ET sequence: 36, 135, 171
Badness (Sintel): 4.20
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 4096/4095, 16807/16731
Mapping: [⟨9 0 135 11 131 -38], ⟨0 1 -8 1 -7 5]]
Optimal tunings:
- WE: ~242/225 = 133.3180 ¢, ~3/2 = 701.6956 ¢
- CWE: ~242/225 = 133.3333 ¢, ~3/2 = 701.7800 ¢
Optimal ET sequence: 36, 99cf, 135, 171
Badness (Sintel): 3.15
Septant
The septant temperament (224 & 301) 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 0 105 -56], ⟨0 1 -8 7]]
- mapping generators: ~8575/7776, ~3
- WE: ~8575/7776 = 171.4303 ¢, ~3/2 = 701.7091 ¢
- error map: ⟨+0.012 -0.234 +0.096 +0.265]
- CWE: ~8575/7776 = 171.4286 ¢, ~3/2 = 701.7022 ¢
- error map: ⟨0.000 -0.253 +0.069 +0.232]
Optimal ET sequence: 77, 147, 224, 301, 525, 826, 1351
Badness (Sintel): 2.81
11-limit
Subgroup: 2.3.5.7.11
Comma list: 3025/3024, 24057/24010, 32805/32768
Mapping: [⟨7 0 105 -56 -120], ⟨0 1 -8 7 13]]
Optimal tunings:
- WE: ~495/448 = 171.4334 ¢, ~3/2 = 701.7387 ¢
- CWE: ~495/448 = 171.4286 ¢, ~3/2 = 701.7198 ¢
Optimal ET sequence: 77, 147, 224, 301, 525
Badness (Sintel): 1.46
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 729/728, 1716/1715, 2200/2197, 3025/3024
Mapping: [⟨7 0 105 -56 -120 37], ⟨0 1 -8 7 13 -1]]
Optimal tunings:
- WE: ~495/448 = 171.4282 ¢, ~3/2 = 701.7229 ¢
- CWE: ~495/448 = 171.4286 ¢, ~3/2 = 701.7242 ¢
Optimal ET sequence: 77, 147, 224, 525, 1274f
Badness (Sintel): 1.02
Septiquarschis
The septiquarschis temperament (89 & 94) splits septimal minor seventh (7/4) into four generators and tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma).
Subgroup: 2.3.5.7
Comma list: 32805/32768, 829440/823543
Mapping: [⟨1 -4 47 6], ⟨0 7 56 -4]]
- mapping generators: ~2, ~256/147
- WE: ~2 = 1199.8855 ¢, ~256/147 = 957.2944 ¢
- error map: ⟨-0.114 -0.436 -0.182 +1.310]
- CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3867 ¢
- error map: ⟨0.000 -0.248 +0.032 +1.627]
Optimal ET sequence: 89, 94, 183, 460d, 643d
Badness (Sintel): 4.73
11-limit
Subgroup: 2.3.5.7.11
Comma list: 540/539, 15488/15435, 32805/32768
Mapping: [⟨1 -4 47 6 25], ⟨0 7 56 -4 -27]]
Optimal tunings:
- WE: ~2 = 1199.9430 ¢, ~256/147 = 957.3390 ¢
- CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3849 ¢
Optimal ET sequence: 89, 94, 183, 460d
Badness (Sintel): 1.72
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 540/539, 729/728, 1573/1568, 4096/4095
Mapping: [⟨1 -4 47 6 25 -33], ⟨0 7 56 -4 -27 46]]
Optimal tunings:
- WE: ~2 = 1200.0058 ¢, ~256/147 = 957.3946 ¢
- CWE: ~2 = 1200.0000 ¢, ~256/147 = 957.3900 ¢
Optimal ET sequence: 89, 94, 183, 277, 460d
Badness (Sintel): 1.46
Tridecafifths
Tridecafifths divides the perfect 3/2 into 13 quartertones.
Subgroup: 2.3.5.7
Comma list: 32805/32768, [-14 -1 -9 13⟩
Mapping: [⟨1 1 7 6], ⟨0 13 -104 -71]]
- mapping generators: ~2, ~1323/1280
- WE: ~2 = 1200.1431 ¢, ~1323/1280 = 53.9838 ¢
- error map: ⟨+0.143 -0.023 +0.375 -0.816]
- CWE: ~2 = 1200.0000 ¢, ~1323/1280 = 53.9764 ¢
- error map: ⟨0.000 -0.261 -0.221 -0.421]
Optimal ET sequence: 89, 200, 289
Badness (Sintel): 10.9
11-limit
Subgroup: 2.3.5.7.11
Comma list: 441/440, 32805/32768, 55296000/55240493
Mapping: [⟨1 1 7 6 4], ⟨0 13 -104 -71 -12]]
Optimal tunings:
- WE: ~2 = 1200.0311 ¢, ~33/32 = 53.9766 ¢
- CWE: ~2 = 1200.0000 ¢, ~33/32 = 53.9750 ¢
Optimal ET sequence: 89, 200, 289
Badness (Sintel): 4.23
Subgroup extensions
Photia (2.3.5.17)
Subgroup: 2.3.5.17
Comma list: 256/255, 1458/1445
Subgroup-val mapping: [⟨1 0 15 -7], ⟨0 1 -8 7]]
Gencom mapping: [⟨1 0 15 0 0 0 -7], ⟨0 1 -8 0 0 0 7]]
- mapping generators: ~2, ~3
- WE: ~2 = 1199.5471 ¢, ~3/2 = 701.2262 ¢
- error map: ⟨-0.453 -1.182 +0.706 +3.628]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4976 ¢
- error map: ⟨0.000 -0.457 +1.705 +5.528]
Optimal ET sequence: 12, 41, 53, 65, 207g, 272gg
Badness (Sintel): 0.479
2.3.5.17.19 subgroup
Subgroup: 2.3.5.17.19
Comma list: 171/170, 256/255, 324/323
Subgroup-val mapping: [⟨1 0 15 -7 9], ⟨0 1 -8 7 -3]]
Gencom mapping: [⟨1 0 15 0 0 0 -7 9], ⟨0 1 -8 0 0 0 7 -3]]
Optimal tunings:
- WE: ~2 = 1199.7225 ¢, ~3/2 = 701.3077 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.4754 ¢
Optimal ET sequence: 12, 41, 53, 65, 142g
Badness (Sintel): 0.332
Nestoria (2.3.5.19)
- See also: No-elevens subgroup temperaments #Garibaldia and #Pontia
The S-expression-based comma list of this temperament is {S16/S18, S19 , (S15/S20)}. Strangely, despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is actually sharper than the fifth in optimal schismic. This is likely due to its optimization considering intervals like 19/10 and 19/15.
Subgroup: 2.3.5.19
Comma list: 361/360, 513/512
Subgroup-val mapping: [⟨1 0 15 9], ⟨0 1 -8 -3]]
Gencom mapping: [⟨1 0 15 0 0 0 0 9], ⟨0 1 -8 0 0 0 0 -3]]
- mapping generators: ~2, ~3
- WE: ~2 = 1200.2250 ¢, ~3/2 = 701.8776 ¢
- error map: ⟨+0.225 +0.148 +0.240 -1.796]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.7307 ¢
- error map: ⟨0.000 -0.224 -0.159 -2.705]
Optimal ET sequence: 12, 29, 41, 53, 118, 171, 460hh, 631hh
Badness (Sintel): 0.126
Taylor (2.3.5.13)
This is a 2.3.5.13 subgroup restriction of 13-limit hemischis.
Subgroup: 2.3.5.13
Comma list: 676/675, 32805/32768
Subgroup-val mapping: [⟨1 0 15 14], ⟨0 2 -16 -13]]
Gencom mapping: [⟨1 0 15 0 0 14], ⟨0 2 -16 0 0 -13]]
- mapping generators: ~2, ~26/15
- WE: ~2 = 1200.1497 ¢, ~26/15 = 950.9740 ¢
- error map: ⟨+0.150 -0.007 +0.348 -1.094]
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8493 ¢
- error map: ⟨0.000 -0.256 +0.098 -1.568]
Optimal ET sequence: 24, 53, 130, 183, 236, 525f, 761ff
Badness (Sintel): 0.334
Dakota (2.3.5.13.19)
Subgroup: 2.3.5.13.19
Comma list: 361/360, 513/512, 676/675
Subgroup-val mapping: [⟨1 0 15 14 9], ⟨0 2 -16 -13 -6]]
Optimal tunings:
- WE: ~2 = 1200.2611 ¢, ~26/15 = 951.0703 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8532 ¢
Optimal ET sequence: 24, 29, 53, 130, 183, 236h, 289h
Badness (Sintel): 0.262
2.3.5.13.19.37 subgroup
Subgroup: 2.3.5.13.19.37
Comma list: 361/360, 481/480, 513/512, 676/675
Subgroup-val mapping: [⟨1 0 15 14 9 6], ⟨0 2 -16 -13 -6 -1]]
Optimal tunings:
- WE: ~2 = 1200.2987 ¢, ~26/15 = 951.1060 ¢
- CWE: ~2 = 1200.0000 ¢, ~26/15 = 950.8595 ¢
Optimal ET sequence: 24, 29, 53, 183, 236h, 289hl, 631fhhll
Badness (Sintel): 0.223
Quintilischis (2.3.5.17)
- For full 17- and 19-limit extensions, see #Quintilipyth or #Quintaschis.
Subgroup: 2.3.5.17
Comma list: 32805/32768, 1419857/1417176
Subgroup-val mapping: [⟨1 2 -1 5], ⟨0 -5 40 -11]]
Gencom mapping: [⟨1 2 -1 0 0 0 5], ⟨0 -5 40 0 0 0 -11]]
- mapping generators: ~2, ~18/17
- WE: ~2 = 1200.1370 ¢, ~18/17 = 99.6602 ¢
- error map: ⟨+0.137 +0.018 -0.042 -0.533]
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6499 ¢
- error map: ⟨0.000 -0.205 -0.317 -1.104]
Optimal ET sequence: 12, …, 253, 265, 277, 289, 566g, 855g
Badness (Sintel): 1.34
2.3.5.17.19 subgroup
Subgroup: 2.3.5.17.19
Comma list: 4624/4617, 6144/6137, 6885/6859
Subgroup-val mapping: [⟨1 2 -1 5 4], ⟨0 -5 40 -11 3]]
Gencom mapping: [⟨1 2 -1 0 0 0 5 4], ⟨0 -5 40 0 0 0 -11 3]]
Optimal tunings:
- WE: ~2 = 1200.0350 ¢, ~18/17 = 99.6550 ¢
- CWE: ~2 = 1200.0000 ¢, ~18/17 = 99.6520 ¢
Optimal ET sequence: 12, …, 253, 265, 277, 289
Badness (Sintel): 1.17