Schismatic family: Difference between revisions
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{{interwiki | {{interwiki | ||
| en = Schismatic family | |||
| de = Schismatische Temperaturen | | de = Schismatische Temperaturen | ||
| es = | | es = | ||
| ja = | | ja = | ||
}} | }} | ||
{{Technical data page}} | {{Technical data page}} | ||
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 [[ | 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 [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]]. | ||
== Schismic, schismatic, a.k.a. helmholtz == | |||
{{Main| Schismic }} | |||
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 {{w|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. | |||
The 5-limit version of the temperament is a [[microtemperament]], | |||
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{{cent}}, 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 | [[Subgroup]]: 2.3.5 | ||
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{{Mapping|legend=1| 1 0 15 | 0 1 -8 }} | {{Mapping|legend=1| 1 0 15 | 0 1 -8 }} | ||
: mapping generators: ~2, ~3 | |||
: | |||
[[Optimal tuning]]s: | [[Optimal tuning]]s: | ||
* [[ | * [[WE]]: ~2 = 1200.0749{{c}}, ~3/2 = 701.7797{{c}} | ||
* [[ | : [[error map]]: {{val| +0.075 -0.100 -0.027 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7308{{c}} | |||
: error map: {{val| 0.000 -0.224 -0.160 }} | |||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
* 5-odd-limit [[diamond monotone]]: ~3/2 = [685.714, 705.882] (4\7 to 10\17) | * [[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) | * 5-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] (1/8-comma to untempered) | ||
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }} | {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 289, 460, 749, 3456bc, 4205bc, 4954bc, 5703bbc, 6452bbcc }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.0999 | ||
=== Overview to extensions === | === Overview to extensions === | ||
The second comma of the [[ | The second comma of the [[normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at. [[#Garibaldi|Garibaldi]] adds [[garischisma|{{monzo| 25 -14 0 -1 }}]], [[#Grackle|grackle]] adds {{monzo| -44 26 0 1 }}, [[#Pontiac|pontiac]] adds {{monzo| -59 39 0 -1 }}, and [[#Schism|schism]] adds [[64/63|{{monzo| 6 -2 0 -1 }}]]. Those all have a fifth as generator. | ||
[[#Bischismic|Bischismic]] adds {{monzo| -69 40 0 2 }} and has a fifth generator with a half-octave period. [[#Salsa|Salsa]] adds [[parahemif comma|{{monzo| 15 -13 0 2 }}]] and has a hemififth generator. [[#Hemischis|Hemischis]] adds {{monzo| -34 25 0 -2 }} and has a hemitwelfth generator. [[Gamelismic clan #Guiron|Guiron]] adds [[1029/1024|{{monzo| -10 1 0 3 }}]], with an ~8/7 generator, three of which give the fifth. [[#Term|Term]] adds {{monzo| -94 54 0 3 }} with a 1/3-octave period. [[#Squirrel|Squirrel]], [[#Tertiaschis|tertiaschis]], and [[#Countertertiaschis|countertertiaschis]] each has a generator that is 1/3 of the fourth. [[#Quadrant|Quadrant]] adds {{monzo| -119 68 0 4 }} with a 1/4-octave period. [[#Kleischismic|Kleischismic]] adds {{monzo| 49 -38 0 4 }} with a half-octave period and also a bisect generator. [[#Sesquiquartififths|Sesquiquartififths]] adds {{monzo| -35 15 0 4 }} and slices the fifth in four. | |||
Temperaments involving larger splits include [[#Tsaharuk|tsaharuk]], [[#Quanharuk|quanharuk]], [[#Quintilipyth|quintilipyth]], [[#Quintaschis|quintaschis]], [[#Altinex|altinex]], [[Stearnsmic clan #Pogo|pogo]], [[#Sextilifourths|sextilifourths]], [[#Septant|septant]], [[#Octant|octant]], [[#Nonant|nonant]], [[#Septiquarschis|septiquarschis]], and [[#Tridecafifths|tridecafifths]]. Those split the schismic structure into five to thirteen parts. | |||
Those | |||
Temperaments discussed elsewhere include: | |||
* [[ | * ''[[Guiron]]'' (+1029/1024) → [[Gamelismic clan #Guiron|Gamelismic clan]] | ||
* ''[[Pogo]]'' (+118098/117649) → [[Stearnsmic clan #Pogo|Stearnsmic clan]] | |||
* [[ | |||
Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths. | |||
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 [[ | 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 == | ||
{{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 65: | Line 62: | ||
{{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }} | {{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }} | ||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.1233{{c}}, ~3/2 = 702.1573{{c}} | |||
{{ | : [[error map]]: {{val| +0.123 +0.326 -2.709 +2.328 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 702.0774{{c}} | |||
[[ | : error map: {{val| 0.000 +0.122 -2.933 +2.090 }} | ||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }} | * [[7-odd-limit]]: ~3/2 = {{monzo| 2/3 1/15 0 -1/15 }} | ||
: {{monzo list| 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 }} | : {{monzo list| 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 }} | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3 | ||
* [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }} | * [[9-odd-limit]]: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }} | ||
: {{monzo list| 1 0 0 0 | 25/16 1/8 0 -1/16 | 5/2 -1 0 1/2 | 25/8 -7/4 0 7/8 }} | : {{monzo list| 1 0 0 0 | 25/16 1/8 0 -1/16 | 5/2 -1 0 1/2 | 25/8 -7/4 0 7/8 }} | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/7 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
| Line 83: | Line 80: | ||
* 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915] | * 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 702.915] | ||
{{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94 | {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 94 }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.548 | ||
=== Cassandra === | === Cassandra === | ||
Cassandra is one of the best | 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, even though it comes with a much higher complexity. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 96: | Line 93: | ||
Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }} | Mapping: {{mapping| 1 0 15 25 -33 | 0 1 -8 -14 23 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3089{{c}}, ~3/2 = 702.3377{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1562{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }} | * 11-odd-limit: ~3/2 = {{monzo| 9/16 1/8 0 -1/16 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.9/7 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 106: | Line 105: | ||
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915] | * 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 702.915] | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12e, 41, 53, 94, 229c }} | ||
Badness: 0. | Badness (Sintel): 0.906 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 117: | Line 116: | ||
Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }} | Mapping: {{mapping| 1 0 15 25 -33 -28 | 0 1 -8 -14 23 20 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1703{{c}}, ~3/2 = 702.2122{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1135{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 19/34 0 0 -1/34 0 1/34 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/7 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 128: | Line 129: | ||
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597] | * 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 703.597] | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 53, 94, 429ccdeef, 523ccdeef }} | ||
Badness: 0. | Badness (Sintel): 0.854 | ||
===== Cassie ===== | ===== Cassie ===== | ||
| Line 139: | Line 140: | ||
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }} | Mapping: {{mapping| 1 0 15 25 -33 -28 -7 | 0 1 -8 -14 23 20 7 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8140{{c}}, ~3/2 = 701.9833{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0909{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }} | ||
Badness: | Badness (Sintel): 1.19 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 152: | Line 155: | ||
Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }} | Mapping: {{mapping| 1 0 15 25 -33 -28 -7 9 | 0 1 -8 -14 23 20 7 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9556{{c}}, ~3/2 = 702.0530{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0787{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12e, 41, 53 }} | ||
Badness: | Badness (Sintel): 1.11 | ||
===== Cassandric ===== | ===== Cassandric ===== | ||
| Line 165: | Line 170: | ||
Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }} | Mapping: {{mapping| 1 0 15 25 -33 -28 77 | 0 1 -8 -14 23 20 -46 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0046{{c}}, ~3/2 = 702.2167{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0962{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41g, 53, 94 }} | ||
Badness: | Badness (Sintel): 1.18 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 179: | Line 186: | ||
Optimal tunings: | Optimal tunings: | ||
* ~2 = 1200.2910, ~3/2 = | * WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}} | ||
* ~2 = | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0967{{c}} | ||
{{Optimal ET sequence|legend=1| 41g, 53, 94 | {{Optimal ET sequence|legend=1| 41g, 53, 94 }} | ||
Badness: | Badness (Sintel): 1.07 | ||
====== 23-limit ====== | ====== 23-limit ====== | ||
| Line 195: | Line 201: | ||
Optimal tunings: | Optimal tunings: | ||
* ~2 = 1200.2970, ~3/2 = | * WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}} | ||
* ~2 = | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}} | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41g, 53, 94 }} | ||
Badness: | Badness (Sintel): 1.08 | ||
===== Cassander ===== | ===== Cassander ===== | ||
| Line 210: | Line 215: | ||
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }} | Mapping: {{mapping| 1 0 15 25 -33 -28 -72 | 0 1 -8 -14 23 20 48 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1986{{c}}, ~3/2 = 702.2598{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1455{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 53g, 94 }} | ||
Badness: | Badness (Sintel): 1.14 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 223: | Line 230: | ||
Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }} | Mapping: {{mapping| 1 0 15 25 -33 -28 -72 9 | 0 1 -8 -14 23 20 48 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3057{{c}}, ~3/2 = 702.3138{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.1373{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 53g, 94 }} | ||
Badness: | Badness (Sintel): 1.07 | ||
=== Andromeda === | === Andromeda === | ||
| Line 236: | Line 245: | ||
Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }} | Mapping: {{mapping| 1 0 15 25 32 | 0 1 -8 -14 -18 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1917{{c}}, ~3/2 = 702.4836{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3599{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }} | * 11-odd-limit: ~3/2 = {{monzo| 3/5 1/10 0 0 -1/20 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 246: | Line 257: | ||
* 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377] | * 11-odd-limit diamond tradeoff: ~3/2 = [701.711, 704.377] | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, 29, 41 }} | ||
Badness: 0. | Badness (Sintel): 0.779 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 257: | Line 268: | ||
Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }} | Mapping: {{mapping| 1 0 15 25 32 37 | 0 1 -8 -14 -18 -21 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3031{{c}}, ~3/2 = 702.7368{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.5420{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 14/23 2/23 0 0 0 -1/23 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/9 | ||
Tuning ranges: | Tuning ranges: | ||
| Line 268: | Line 281: | ||
* 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377] | * 15-odd-limit diamond tradeoff: ~3/2 = [701.676, 704.377] | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 29, 41 }} | ||
Badness: 0. | Badness (Sintel): 0.857 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 279: | Line 292: | ||
Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }} | Mapping: {{mapping| 1 0 15 25 32 37 -7 | 0 1 -8 -14 -18 -21 7 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.1984{{c}}, ~3/2 = 701.8424{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3384{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 29, 41 }} | ||
Badness: | Badness (Sintel): 1.19 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 292: | Line 307: | ||
Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }} | Mapping: {{mapping| 1 0 15 25 32 37 -7 9 | 0 1 -8 -14 -18 -21 7 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.5242{{c}}, ~3/2 = 702.0783{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.3711{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 29, 41 }} | ||
Badness: | Badness (Sintel): 1.17 | ||
===== Schisicosiennic ===== | ===== Schisicosiennic ===== | ||
| Line 305: | Line 322: | ||
Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }} | Mapping: {{mapping| 1 0 15 25 32 37 58 | 0 1 -8 -14 -18 -21 -34 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.6122{{c}}, ~3/2 = 703.0830{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6968{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }} | ||
Badness: | Badness (Sintel): 1.11 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 318: | Line 337: | ||
Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }} | Mapping: {{mapping| 1 0 15 25 32 37 58 9 | 0 1 -8 -14 -18 -21 -34 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.7981{{c}}, ~3/2 = 703.2199{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.7221{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }} | ||
Badness: | Badness (Sintel): 1.09 | ||
===== Schisicosiennoid ===== | ===== Schisicosiennoid ===== | ||
| Line 331: | Line 352: | ||
Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }} | Mapping: {{mapping| 1 0 15 25 32 37 12 | 0 1 -8 -14 -18 -21 -5 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.3146{{c}}, ~3/2 = 703.4864{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6491{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 29g, 41g }} | ||
Badness: | Badness (Sintel): 1.06 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 344: | Line 367: | ||
Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }} | Mapping: {{mapping| 1 0 15 25 32 37 12 9 | 0 1 -8 -14 -18 -21 -5 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1201.3140{{c}}, ~3/2 = 703.4860{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.6578{{c}} | |||
{{Optimal ET sequence|legend=1| 12f, 29g, 41g | {{Optimal ET sequence|legend=1| 12f, 29g, 41g }} | ||
Badness: | Badness (Sintel): 1.02 | ||
=== Helenus === | === Helenus === | ||
| Line 357: | Line 382: | ||
Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }} | Mapping: {{mapping| 1 0 15 25 51 | 0 1 -8 -14 -30 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7097{{c}}, ~3/2 = 701.5554{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7370{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }} | * 11-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, 41e, 53, 118d }} | ||
Badness: | Badness (Sintel): 1.18 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 374: | Line 401: | ||
Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }} | Mapping: {{mapping| 1 0 15 25 51 56 | 0 1 -8 -14 -30 -33 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7370{{c}}, ~3/2 = 701.5937{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7570{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 19/32 1/16 0 0 -1/32 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, …, 41ef, 53, 118d }} | ||
Badness: | Badness (Sintel): 1.09 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 391: | Line 420: | ||
Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }} | Mapping: {{mapping| 1 0 15 25 51 56 -7 | 0 1 -8 -14 -30 -33 7 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.2895{{c}}, ~3/2 = 701.2643{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.6967{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }} | ||
Badness: | Badness (Sintel): 1.21 | ||
==== 19-limit ==== | ==== 19-limit ==== | ||
| Line 404: | Line 435: | ||
Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }} | Mapping: {{mapping| 1 0 15 25 51 56 -7 9 | 0 1 -8 -14 -30 -33 7 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.5280{{c}}, ~3/2 = 701.4290{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7149{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 53, 65d }} | ||
Badness: | Badness (Sintel): 1.18 | ||
=== Karadeniz === | |||
{{See also| Turkish maqam music temperaments #Karadeniz temperament }} | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 225/224, 243/242, 3125/3087 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 7 11 2 | 0 2 -16 -28 5 }} | ||
: mapping generators: ~2, ~11/9 | |||
: | Optimal tunings: | ||
* WE: ~2 = 1199.7351{{c}}, ~11/9 = 350.9167{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9995{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 24d, 41, 65d, 106, 147 }} | ||
Badness: | Badness (Sintel): 1.37 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 225/224, 243/242, 325/324, 640/637 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 7 11 2 -8 | 0 2 -16 -28 5 40 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.3042{{c}}, ~11/9 = 350.7533{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 350.9686{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 24d, 41, 65d, 106f }} | ||
Badness: | Badness (Sintel): 1.34 | ||
=== Hemigari === | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 225/224 | Comma list: 121/120, 225/224, 3125/3087 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 15 25 9 | 0 2 -16 -28 -7 }} | ||
: mapping generators: ~2, ~110/63 | |||
: | Optimal tunings: | ||
* WE: ~2 = 1200.7303{{c}}, ~110/63 = 951.6605{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~110/63 = 951.0604{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135ee }} | ||
Badness: | Badness (Sintel): 1.68 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 121/120, 169/168, 225/224, 275/273 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 15 25 9 14 | 0 2 -16 -28 -7 -13 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.8146{{c}}, ~26/15 = 951.7273{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 951.0574{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }} | ||
Badness: | Badness (Sintel): 1.13 | ||
=== Sanjaab === | === Sanjaab === | ||
| Line 474: | Line 513: | ||
Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }} | Mapping: {{mapping| 1 2 -1 -3 0 | 0 -3 24 42 25 }} | ||
: mapping generators: ~2, ~11/10 | : mapping generators: ~2, ~11/10 | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1997{{c}}, ~11/10 = 166.0018{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9786{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 29, 65d, 94 }} | ||
Badness: | Badness (Sintel): 1.92 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 490: | Line 530: | ||
Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }} | Mapping: {{mapping| 1 2 -1 -3 0 -1 | 0 -3 24 42 25 34 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1224{{c}}, ~11/10 = 165.9800{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 165.9659{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 29, 65d, 94 }} | ||
Badness | Badness (Sintel): 1.40 | ||
== Pontiac == | == Pontiac == | ||
{{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 | ||
| Line 543: | Line 549: | ||
{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }} | {{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }} | ||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.0989{{c}}, ~3/2 = 701.8145{{c}} | |||
[[ | : [[error map]]: {{val| +0.099 -0.042 -0.138 -0.038 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7579{{c}} | |||
: error map: {{val| 0.000 -0.197 -0.377 -0.268 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }} | * [[7-odd-limit]]: ~3/2 = {{monzo| 27/47 0 -1/47 1/47 }} | ||
: [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}] | : [{{monzo| 1 0 0 0 }}, {{monzo| 74/47 0 -1/47 1/47 }}, {{monzo| 113/47 0 8/47 -8/47 }}, {{monzo| 113/47 0 -39/47 39/47 }}] | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5 | ||
* [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }} | * [[9-odd-limit]]: ~3/2 = {{monzo| 1/2 1/5 -1/10 }} | ||
: [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}] | : [{{monzo| 1 0 0 0 }}, {{monzo| 3/2 1/5 -1/10 0 }}, {{monzo| 3 -8/5 4/5 0 }}, {{monzo| -1/2 39/5 -39/10 0 }}] | ||
: [[ | : [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5 | ||
[[Tuning ranges]]: | [[Tuning ranges]]: | ||
* 7- and 9-odd-limit [[diamond monotone]]: ~3/2 = [701.538, 701.886] (38\65 to 31\53) | * 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] | * 7- and 9-odd-limit [[diamond tradeoff]]: ~3/2 = [701.711, 701.955] | ||
{{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, | {{Optimal ET sequence|legend=1| 53, 118, 171, 1592c, 1763c, …, 2960cd, 3131bcd }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.358 | ||
=== 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 573: | Line 580: | ||
Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }} | Mapping: {{mapping| 1 0 15 -59 51 | 0 1 -8 39 -30 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3277{{c}}, ~3/2 = 701.9135{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7223{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }} | * 11-odd-limit: ~3/2 = {{monzo| 41/69 0 0 1/69 -1/69 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/7 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 118, 289e, 407de }} | ||
Badness: | Badness (Sintel): 1.28 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 590: | Line 599: | ||
Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }} | Mapping: {{mapping| 1 0 15 -59 51 56 | 0 1 -8 39 -30 -33 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1780{{c}}, ~3/2 = 701.8491{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7446{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }} | * 13- and 15-odd-limit: ~3/2 = {{monzo| 43/72 0 0 1/72 -1/72 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/7 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 118, 171e }} | ||
Badness: | Badness (Sintel): 1.39 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 607: | Line 618: | ||
Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }} | Mapping: {{mapping| 1 0 15 -59 51 56 -91 | 0 1 -8 39 -30 -33 60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1645{{c}}, ~3/2 = 701.8385{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7425{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }} | * 17-odd-limit: ~3/2 = {{monzo| 18/31 0 0 0 0 -1/93 1/93 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.17/13 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 118, 171e }} | ||
Badness: | Badness (Sintel): 1.47 | ||
==== Helena ==== | ==== Helena ==== | ||
| Line 624: | Line 637: | ||
Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }} | Mapping: {{mapping| 1 0 15 -59 51 -28 | 0 1 -8 39 -30 20 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.5227{{c}}, ~3/2 = 702.0456{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7418{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 118f, 171ef }} | ||
Badness: | Badness (Sintel): 1.50 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 637: | Line 652: | ||
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }} | Mapping: {{mapping| 1 0 15 -59 51 -28 -91 | 0 1 -8 39 -30 20 60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.4988{{c}}, ~3/2 = 702.0218{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7332{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 118f, 171ef }} | ||
Badness: | Badness (Sintel): 1.56 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 650: | Line 667: | ||
Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }} | Mapping: {{mapping| 1 0 15 -59 51 -28 -91 9 | 0 1 -8 39 -30 20 60 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.5185{{c}}, ~3/2 = 702.0323{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7318{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 118f, 171ef }} | ||
Badness: | Badness (Sintel): 1.33 | ||
=== 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 665: | Line 684: | ||
Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }} | Mapping: {{mapping| 1 0 15 -59 135 | 0 1 -8 39 -83 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9814{{c}}, ~3/2 = 701.7725{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7834{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }} | * 11-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/7 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 171, 224 }} | ||
Badness: | Badness (Sintel): 1.61 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 682: | Line 703: | ||
Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }} | Mapping: {{mapping| 1 0 15 -59 135 56 | 0 1 -8 39 -83 -33 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9601{{c}}, ~3/2 = 701.7610{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7845{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }} | * 13 and 15-odd-limit: ~3/2 = {{monzo| 36/61 0 0 1/122 -1/122 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11/7 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 171, 224 }} | ||
Badness: 0. | Badness (Sintel): 0.976 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 699: | Line 722: | ||
Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }} | Mapping: {{mapping| 1 0 15 -59 135 56 -91 | 0 1 -8 39 -83 -33 60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8850{{c}}, ~3/2 = 701.7101{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7775{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }} | * 17-odd-limit: ~3/2 = {{monzo| 83/143 0 0 0 -1/143 0 1/143 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.17/11 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 171, 224, 395e, 619eg }} | ||
Badness: | Badness (Sintel): 1.16 | ||
=== 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 718: | Line 743: | ||
Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }} | Mapping: {{mapping| 1 0 15 -59 -136 | 0 1 -8 39 88 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1259{{c}}, ~3/2 = 701.7980{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7256{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }} | * 11-odd-limit: ~3/2 = {{monzo| 6/11 0 0 0 1/88 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.11 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53e, 118, 289, 407d }} | ||
Badness: | Badness (Sintel): 1.64 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 735: | Line 762: | ||
Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }} | Mapping: {{mapping| 1 0 15 -59 -136 56 | 0 1 -8 39 88 -33 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9254{{c}}, ~3/2 = 701.6945{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7378{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }} | * 13 and 15-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/11 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }} | ||
Badness: | Badness (Sintel): 1.87 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 752: | Line 781: | ||
Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }} | Mapping: {{mapping| 1 0 15 -59 -136 56 -91 | 0 1 -8 39 88 -33 60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9454{{c}}, ~3/2 = 701.7085{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7401{{c}} | |||
Minimax tuning: | Minimax tuning: | ||
* 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }} | * 17-odd-limit: ~3/2 = {{monzo| 71/121 0 0 0 1/121 -1/121 }} | ||
: | : unchanged-interval (eigenmonzo) basis: 2.13/11 | ||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }} | ||
Badness: | Badness (Sintel): 1.51 | ||
==== Pontoid ==== | ==== Pontoid ==== | ||
| Line 769: | Line 800: | ||
Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }} | Mapping: {{mapping| 1 0 15 -59 -136 -215 | 0 1 -8 39 88 138 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0897{{c}}, ~3/2 = 701.7874{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7356{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }} | ||
Badness: | Badness (Sintel): 2.07 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 782: | Line 815: | ||
Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }} | Mapping: {{mapping| 1 0 15 -59 -136 -215 -91 | 0 1 -8 39 88 138 60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1045{{c}}, ~3/2 = 701.7962{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.7359{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289, 460e, 749defg }} | ||
Badness: | Badness (Sintel): 1.50 | ||
=== 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 796: | Line 831: | ||
Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }} | Mapping: {{mapping| 2 0 30 -118 -85 | 0 1 -8 39 29 }} | ||
: mapping generators: ~99/70, ~3 | |||
: | Optimal tunings: | ||
* WE: ~99/70 = 600.0500{{c}}, ~3/2 = 701.8153{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7584{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }} | ||
Badness (Sintel): 0.484 | |||
Badness: 0. | |||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 812: | Line 848: | ||
Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }} | Mapping: {{mapping| 2 0 30 -118 -85 112 | 0 1 -8 39 29 -33 }} | ||
Optimal tunings: | |||
* WE: ~99/70 = 599.9939{{c}}, ~3/2 = 701.7657{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7728{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }} | ||
Badness: | Badness (Sintel): 1.25 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 827: | Line 863: | ||
Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }} | Mapping: {{mapping| 2 0 30 -118 -85 112 -182 | 0 1 -8 39 29 -33 60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 599.9839{{c}}, ~3/2 = 701.7463{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7649{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }} | ||
Badness: | Badness (Sintel): 1.38 | ||
==== Counterbipont ==== | ==== Counterbipont ==== | ||
| Line 840: | Line 878: | ||
Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }} | Mapping: {{mapping| 2 0 30 -118 -85 -243 | 0 1 -8 39 29 79 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.0405{{c}}, ~3/2 = 701.8160{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7697{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }} | ||
Badness: | Badness (Sintel): 1.06 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 853: | Line 893: | ||
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }} | Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 | 0 1 -8 39 29 79 60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.0336{{c}}, ~3/2 = 701.8031{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7647{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }} | ||
Badness: | Badness (Sintel): 1.29 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 866: | Line 908: | ||
Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }} | Mapping: {{mapping| 2 0 30 -118 -85 -243 -182 -169 | 0 1 -8 39 29 79 60 56 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~99/70 = 600.0243{{c}}, ~3/2 = 701.7891{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.7613{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }} | ||
Badness: | Badness (Sintel): 1.35 | ||
==== Quadrapont ==== | ==== Quadrapont ==== | ||
| Line 878: | Line 922: | ||
Mapping: {{mapping| 4 0 60 -236 -170 -131 | 0 1 -8 39 29 23 }} | Mapping: {{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{{c}}, ~3/2 = 701.8097{{c}} | |||
* CWE: ~208/175 = 300.0000{{c}}, ~3/2 = 701.7578{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }} | ||
Badness: 0. | Badness (Sintel): 0.869 | ||
== 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 895: | Line 940: | ||
{{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }} | {{Mapping|legend=1| 1 0 15 44 | 0 1 -8 -26 }} | ||
: mapping generators: ~2, ~3 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.7974{{c}}, ~3/2 = 701.1210{{c}} | |||
{{ | : [[error map]]: {{val| -0.203 -1.037 +3.300 -1.618 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.2465{{c}} | |||
: error map: {{val| 0.000 -0.709 +3.715 -1.234 }} | |||
[[Minimax tuning]]: | [[Minimax tuning]]: | ||
* [[7-odd-limit]] | * [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3 | ||
* [[9-odd-limit]] | * [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7 | ||
{{Optimal ET sequence|legend=1| 12, | {{Optimal ET sequence|legend=1| 12, …, 65, 77, 166c }} | ||
[[Badness]]: | [[Badness]] (Sintel): 1.78 | ||
=== 11-limit === | === 11-limit === | ||
| Line 917: | Line 963: | ||
Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }} | Mapping: {{mapping| 1 0 15 44 70 | 0 1 -8 -26 -42 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7077{{c}}, ~3/2 = 701.0017{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.1804{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }} | ||
Badness: | Badness (Sintel): 1.62 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 930: | Line 978: | ||
Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }} | Mapping: {{mapping| 1 0 15 44 70 75 | 0 1 -8 -26 -42 -45 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7782{{c}}, ~3/2 = 701.0966{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2319{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }} | ||
Badness: | Badness (Sintel): 1.56 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 943: | Line 993: | ||
Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }} | Mapping: {{mapping| 1 0 15 44 70 75 -7 | 0 1 -8 -26 -42 -45 7 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.5839{{c}}, ~3/2 = 700.9632{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2137{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }} | ||
Badness: | Badness (Sintel): 1.52 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 956: | Line 1,008: | ||
Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }} | Mapping: {{mapping| 1 0 15 44 70 75 -7 9 | 0 1 -8 -26 -42 -45 7 -3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7146{{c}}, ~3/2 = 701.0500{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2212{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 77, 166cf }} | ||
Badness: | Badness (Sintel): 1.40 | ||
==== Grackloid ==== | ==== Grackloid ==== | ||
| Line 969: | Line 1,023: | ||
Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }} | Mapping: {{mapping| 1 0 15 44 70 -47 | 0 1 -8 -26 -42 32 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0060{{c}}, ~3/2 = 701.2202{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.2167{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, 77, 166c }} | ||
Badness: | Badness (Sintel): 2.00 | ||
=== Grack === | === Grack === | ||
| Line 982: | Line 1,038: | ||
Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }} | Mapping: {{mapping| 1 0 15 44 51 | 0 1 -8 -26 -30 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8388{{c}}, ~3/2 = 701.3071{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4068{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }} | ||
Badness: | Badness (Sintel): 1.85 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 995: | Line 1,053: | ||
Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }} | Mapping: {{mapping| 1 0 15 44 51 75 | 0 1 -8 -26 -30 -45 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.7329{{c}}, ~3/2 = 701.1918{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.3555{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }} | ||
Badness: | Badness (Sintel): 1.84 | ||
==== Catahelenic ==== | ==== Catahelenic ==== | ||
| Line 1,008: | Line 1,068: | ||
Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }} | Mapping: {{mapping| 1 0 15 44 51 56 | 0 1 -8 -26 -30 -33 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8928{{c}}, ~3/2 = 701.4664{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.5327{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, …, 53d, 65 }} | ||
Badness: 0. | Badness (Sintel): 2.01 | ||
== Quasipyth == | |||
Named by [[Xenllium]] in 2026, quasipyth tempers out {{monzo| 109 -67 0 -1 }}, the [[nanisma]], as well as the [[catasyc comma]], 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 32805/32768, 390625/387072 | ||
{{Mapping|legend=1| | {{Mapping|legend=1| 1 0 15 109 | 0 1 -8 -67 }} | ||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.2569{{c}}, ~3/2 = 702.1149{{c}} | |||
: [[error map]]: {{val| +0.2569 +0.4168 -1.4342 +0.2685 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.9615{{c}} | |||
[[Optimal tuning]] | : error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }} | ||
[[ | |||
* [[ | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }} | ||
[[Badness]]: | [[Badness]] (Sintel): 5.04 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 19712/19683, 78125/77616 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 0 15 109 -117 | 0 1 -8 -67 76 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3283{{c}}, ~3/2 = 702.1636{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9713{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }} | ||
Badness: | Badness (Sintel): 3.83 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 325/324, 385/384, 2200/2197, 19712/19683 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 0 15 109 -117 -28 | 0 1 -8 -67 76 20 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3229{{c}}, ~3/2 = 702.1603{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.9714{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 200, 253, 559ce }} | ||
Badness: | Badness (Sintel): 2.13 | ||
== | == 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 (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 {{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 | |||
[[Comma list]]: 3136/3125, 32805/32768 | |||
{{Mapping|legend=1| 2 0 30 69 | 0 1 -8 -20 }} | |||
: mapping generators: ~567/400, ~3 | |||
==== | [[Optimal tuning]]s: | ||
* [[WE]]: ~567/400 = 600.0072{{c}}, ~3/2 = 701.6005{{c}} | |||
: [[error map]]: {{val| +0.014 -0.340 +0.982 -0.629 }} | |||
* [[CWE]]: ~567/400 = 600.0000{{c}}, ~3/2 = 701.5915{{c}} | |||
: error map: {{val| 0.000 -0.364 +0.954 -0.656 }} | |||
[[Minimax tuning]]: | |||
* [[7-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.7/3 | |||
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7 | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 0 30 69 102 | 0 1 -8 -20 -30 }} | |||
Optimal tunings: | |||
* WE: ~99/70 = 600.0165{{c}}, ~3/2 = 701.6316{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.6110{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 30 69 102 -75 | 0 1 -8 -20 -30 26 }} | |||
Optimal tunings: | |||
* WE: ~99/70 = 599.9610{{c}}, ~3/2 = 701.5445{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5908{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 30 69 102 -75 5 | 0 1 -8 -20 -30 26 1 }} | ||
Optimal tunings: | |||
* WE: ~99/70 = 600.0331{{c}}, ~3/2 = 701.6387{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~3/2 = 701.5994{{c}} | |||
= | {{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }} | ||
Optimal tunings: | |||
* WE: ~55/39 = 599.9766{{c}}, ~3/2 = 701.5380{{c}} | |||
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5670{{c}} | |||
= | {{Optimal ET sequence|legend=0| 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: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }} | ||
Optimal tunings: | |||
* WE: ~55/39 = 600.0997{{c}}, ~3/2 = 701.7114{{c}} | |||
* CWE: ~55/39 = 600.0000{{c}}, ~3/2 = 701.5899{{c}} | |||
= | {{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }} | ||
Badness (Sintel): 1.37 | |||
== 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 | |||
[[Comma list]]: 32805/32768, 1500625/1492992 | |||
{{Mapping|legend=1| 2 1 22 -15 | 0 2 -16 19 }} | |||
: mapping generators: ~1225/864, ~35/24 | |||
==== | [[Optimal tuning]]s: | ||
* [[WE]]: ~1225/864 = 600.1246{{c}}, ~35/24 = 651.0550{{c}} (~36/35 = 50.9304{{c}}) | |||
: [[error map]]: {{val| +0.249 +0.280 -0.453 -0.650 }} | |||
* [[CWE]]: ~1225/864 = 600.0000{{c}}, ~35/24 = 650.9204{{c}} (~36/35 = 50.9204{{c}}) | |||
: error map: {{val| 0.000 -0.114 -1.041 -1.338 }} | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 2 1 22 -15 8 | 0 2 -16 19 -1 }} | |||
=== | Optimal tunings: | ||
* WE: ~99/70 = 600.1645{{c}}, ~35/24 = 651.0963{{c}} (~36/35 = 50.9319{{c}}) | |||
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9184{{c}} (~36/35 = 50.9184{{c}}) | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 22 -15 8 15 | 0 2 -16 19 -1 -7 }} | |||
== | Optimal tunings: | ||
* WE: ~99/70 = 600.0696{{c}}, ~35/24 = 651.0136{{c}} (~36/35 = 50.9440{{c}}) | |||
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9378{{c}} (~36/35 = 50.9378{{c}}) | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 22 -15 8 15 6 | 0 2 -16 19 -1 -7 2 }} | |||
{{ | Optimal tunings: | ||
* WE: ~99/70 = 600.1134{{c}}, ~35/24 = 651.0646{{c}} (~36/35 = 50.9512{{c}}) | |||
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9414{{c}} (~36/35 = 50.9414{{c}}) | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 22 -15 8 -36 | 0 2 -16 19 -1 40 }} | |||
{{ | Optimal tunings: | ||
* WE: ~99/70 = 600.1909{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}}) | |||
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9541{{c}} (~36/35 = 50.9541{{c}}) | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 2 1 22 -15 8 -36 6 | 0 2 -16 19 -1 40 2 }} | |||
{{ | Optimal tunings: | ||
* WE: ~99/70 = 600.2190{{c}}, ~35/24 = 651.1578{{c}} (~36/35 = 50.9670{{c}}) | |||
* CWE: ~99/70 = 600.0000{{c}}, ~35/24 = 650.9518{{c}} (~36/35 = 50.9518{{c}}) | |||
{{Optimal ET sequence|legend=0| 24f, 94, 118f, 212g }} | |||
Badness (Sintel): 1.26 | |||
== 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 | |||
[[Comma list]]: 245/243, 32805/32768 | |||
{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }} | |||
: mapping generators: ~2, ~128/105 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.7707{{c}}, ~128/105 = 351.2748{{c}} | |||
: [[error map]]: {{val| +0.771 +1.365 -1.315 -3.024 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~128/105 = 351.0471{{c}} | |||
: error map: {{val| 0.000 +0.139 -3.068 -5.213 }} | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.3891{{c}}, ~11/9 = 351.1275{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0141{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1199.9362{{c}}, ~11/9 = 351.0061{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/9 = 351.0247{{c}} | |||
{{Optimal ET sequence|legend=0| 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 {{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 | |||
[[Comma list]]: 6144/6125, 19683/19600 | |||
{{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }} | |||
: mapping generators: ~2, ~140/81 | |||
[[Optimal tuning]]s: | |||
* | * [[WE]]: ~2 = 1199.8579{{c}}, ~140/81 = 951.6847{{c}} | ||
: [[error map]]: {{val| -0.142 -0.586 +0.600 +0.708 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~140/81 = 951.7966{{c}} | |||
: error map: {{val| 0.000 -0.362 +0.941 +1.088 }} | |||
{{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }} | |||
{{Optimal ET sequence|legend=1| | |||
[[Badness]]: | [[Badness]] (Sintel): 1.16 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 540/539, 5632/5625, 8019/8000 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 15 -17 51 | 0 2 -16 25 -60 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8482{{c}}, ~140/81 = 950.6809{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8020{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 130, 183, 313, 809cd }} | ||
Badness: | Badness (Sintel): 1.20 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 351/350, 540/539, 676/675, 4096/4095 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 15 -17 51 14 | 0 2 -16 25 -60 -13 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9140{{c}}, ~140/81 = 950.7324{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~140/81 = 950.8010{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 53, 130, 183, 313 }} | ||
Badness: 0. | 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: {{mapping| 1 0 15 -17 51 14 -49 | 0 2 -16 25 -60 -13 67 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1199.9740{{c}}, ~26/15 = 950.7894{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8100{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 15 -17 51 14 -49 9 | 0 2 -16 25 -60 -13 67 -6 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0464{{c}}, ~26/15 = 950.8459{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8091{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~26/15 = 950.8239{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8069{{c}} | |||
{{Optimal ET sequence|legend=0| 53, 130, 183, 313h }} | |||
Badness (Sintel): 1.06 | |||
; Music | |||
* ''HemischisMatic EP'' (2023) by [[User:Francium|Francium]] – [https://open.spotify.com/album/1Fx2shLclpNgFQJRw3ZHya Spotify] | [https://francium223.bandcamp.com/album/hemischismatic-ep Bandcamp] | [https://www.youtube.com/playlist?list=PLLZE7hMjEXRaiipPYK1InZBXTru_UtRsq YouTube] – 4-piece extended play | |||
== Term == | |||
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 | |||
Comma list: | [[Comma list]]: 32805/32768, 250047/250000 | ||
{{Mapping|legend=1| 3 0 45 94 | 0 1 -8 -18 }} | |||
: mapping generators: ~63/50, ~3 | |||
Optimal tuning | [[Optimal tuning]]s: | ||
* [[WE]]: ~63/50 = 400.0257{{c}}, ~3/2 = 701.7873{{c}} | |||
: [[error map]]: {{val| +0.077 -0.091 -0.072 +0.031 }} | |||
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~3/2 = 701.7383{{c}} | |||
: error map: {{val| 0.000 -0.217 -0.220 -0.115 }} | |||
{{Optimal ET sequence|legend=1| | [[Minimax tuning]]: | ||
* [[7-odd-limit]] [[eigenmonzo basis|unchanged-interval (eigenmonzo) basis)]]: 2.5/3 | |||
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7 | |||
{{Optimal ET sequence|legend=1| 12, …, 159, 171, 867, 1038, 1209, 1380, 1551, 1722 }} | |||
Badness: 0. | [[Badness]] (Sintel): 0.505 | ||
== | === 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 | |||
Comma list: 441/440, 4375/4356, 32805/32768 | |||
{{ | Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }} | ||
Optimal tunings: | |||
* WE: ~44/35 = 400.0464{{c}}, ~3/2 = 701.9053{{c}} | |||
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8178{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, …, 159, 330 }} | ||
Badness (Sintel): 1.97 | |||
=== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 364/363, 441/440, 625/624, 13720/13689 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 3 0 45 94 134 168 | 0 1 -8 -18 -26 -33 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~44/35 = 400.0449{{c}}, ~3/2 = 701.8995{{c}} | |||
* CWE: ~44/35 = 400.0000{{c}}, ~3/2 = 701.8156{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, …, 159, 330 }} | ||
Badness: | Badness (Sintel): 1.53 | ||
=== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 364/363, 375/374, 441/440, 595/594, 8624/8619 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~34/27 = 400.0195{{c}}, ~3/2 = 701.8439{{c}} | |||
* CWE: ~34/27 = 400.0000{{c}}, ~3/2 = 701.8081{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 159, 171, 330 }} | ||
Badness: | Badness (Sintel): 1.38 | ||
== | === Terminator === | ||
{{ | Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}. | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 540/539, 32805/32768, 137781/137500 | |||
Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }} | |||
{{ | Optimal tunings: | ||
* WE: ~63/50 = 399.9677{{c}}, ~3/2 = 701.6278{{c}} | |||
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6846{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }} | |||
== | Optimal tunings: | ||
* WE: ~63/50 = 399.9731{{c}}, ~3/2 = 701.6414{{c}} | |||
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 0 45 94 -137 -103 -2 | 0 1 -8 -18 31 24 3 }} | |||
== | Optimal tunings: | ||
* WE: ~63/50 = 399.9757{{c}}, ~3/2 = 701.6458{{c}} | |||
* CWE: ~63/50 = 400.0000{{c}}, ~3/2 = 701.6881{{c}} | |||
{{Optimal ET sequence|legend=0| 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 {{nowrap| 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: {{mapping| 6 0 90 188 287 | 0 1 -8 -18 -28 }} | |||
: mapping generators: ~55/49, ~3 | |||
Optimal tunings: | |||
* WE: ~55/49 = 200.0134{{c}}, ~3/2 = 701.7931{{c}} | |||
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7426{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }} | ||
Optimal tunings: | |||
* WE: ~55/49 = 200.0083{{c}}, ~3/2 = 701.7549{{c}} | |||
* CWE: ~55/49 = 200.0000{{c}}, ~3/2 = 701.7238{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} * | |||
* | |||
<nowiki>*</nowiki> optimal patent val: [[354edo|354]] | |||
Badness (Sintel): 1.85 | |||
=== | === Hemiterm === | ||
The | The hemiterm temperament tempers out [[3025/3024]] (lehmerisma), and may be described as {{nowrap| 159 & 183 }}. Its ploidacot is triploid alpha-dicot. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 3025/3024, 32805/32768, 102487/102400 | ||
Mapping: {{mapping| 3 0 45 94 | Mapping: {{mapping| 3 0 45 94 8 | 0 2 -16 -36 1 }} | ||
: mapping generators: ~63/50, ~693/400 | |||
Optimal | Optimal tunings: | ||
* WE: ~63/50 = 400.0309{{c}}, ~693/400 = 950.9458{{c}} (~12/11 = 150.8841{{c}}) | |||
* CWE: ~63/50 = 400.0000{{c}}, ~693/400 = 950.8707{{c}} (~12/11 = 150.8707{{c}}) | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 24d, 159, 183, 342, 1209, 1551, 1893e, 2235ce }} | ||
Badness: 0. | Badness (Sintel): 0.684 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 676/675, 1001/1000, 3025/3024, 19773/19712 | ||
Mapping: {{mapping| 3 0 45 94 | Mapping: {{mapping| 3 0 45 94 8 42 | 0 2 -16 -36 1 -13 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~63/50 = 400.0541{{c}}, ~26/15 = 951.0013{{c}} (~12/11 = 150.8932{{c}}) | |||
* CWE: ~63/50 = 400.0000{{c}}, ~26/15 = 950.8696{{c}} (~12/11 = 150.8696{{c}}) | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 24d, 159, 183, 342f }} | ||
Badness: | Badness (Sintel): 1.30 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 676/675, 715/714, 936/935, 1001/1000, 11271/11264 | ||
Mapping: {{mapping| 3 0 45 94 | Mapping: {{mapping| 3 0 45 94 8 42 -2 | 0 2 -16 -36 1 -13 6 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~34/27 = 400.0373{{c}}, ~26/15 = 950.9556{{c}} (~12/11 = 150.8809{{c}}) | |||
* CWE: ~34/27 = 400.0000{{c}}, ~26/15 = 950.8652{{c}} (~12/11 = 150.8652{{c}}) | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 24d, 159, 183, 342f, 525f }} | ||
Badness: | Badness (Sintel): 1.14 | ||
== | == Altinex == | ||
Named by [[Aura]] in 2021, 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 | |||
[[Comma list]]: 32805/32768, 367653125/362797056 | |||
{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }} | |||
: mapping generators: ~1536/1225, ~34300/19683 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~1536/1225 = 400.1360{{c}}, ~34300/19683 = 951.2867{{c}} | |||
: [[error map]]: {{val| +0.408 +0.618 -0.781 -1.304 }} | |||
* [[CWE]]: ~1536/1225 = 400.0000{{c}}, ~34300/19683 = 950.9638{{c}} | |||
: error map: {{val| 0.000 -0.027 -1.735 -2.441 }} | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }} | |||
{{ | Optimal tunings: | ||
* WE: ~44/35 = 400.1156{{c}}, ~121/70 = 951.2377{{c}} | |||
* CWE: ~44/35 = 400.0000{{c}}, ~121/70 = 950.9634{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 3 0 45 -32 8 42 | 0 2 -16 17 1 -13 }} | |||
{{ | Optimal tunings: | ||
* WE: ~44/35 = 400.1396{{c}}, ~26/15 = 951.2799{{c}} | |||
* CWE: ~44/35 = 400.0000{{c}}, ~26/15 = 950.9462{{c}} | |||
{{Optimal ET sequence|legend=0| 24, 135f, 159 }} | |||
Badness (Sintel): 2.27 | |||
== 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 | |||
[[Comma list]]: 686/675, 32805/32768 | |||
{{Mapping|legend=1| 1 2 -1 1 | 0 -3 24 13 }} | |||
Optimal tuning | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.7408{{c}}, ~160/147 = 166.2424{{c}} | |||
: [[error map]]: {{val| +0.741 +0.799 +2.763 -6.934 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~160/147 = 166.1597{{c}} | |||
: error map: {{val| 0.000 -0.434 +1.518 -8.750 }} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=1| 29, 36, 65 }} | ||
Badness: | [[Badness]] (Sintel): 4.42 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 245/242, 686/675, 896/891 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 2 -1 1 0 | 0 -3 24 13 25 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.6379{{c}}, ~11/10 = 166.1853{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.1157{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 1 0 3 | 0 -3 24 13 25 5 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1201.1361{{c}}, ~11/10 = 166.2110{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0833{{c}} | |||
{{Optimal ET sequence|legend=0| 29, 65f, 94df }} | |||
Badness (Sintel): 1.81 | |||
{{ | == Tertiaschis == | ||
Named by [[Xenllium]] in 2021, 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 | |||
[[Comma list]]: 32805/32768, 1071875/1062882 | |||
{{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.3627{{c}}, ~192/175 = 166.0691{{c}} | |||
: [[error map]]: {{val| +0.363 +0.563 -1.019 -0.790 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~192/175 = 166.0172{{c}} | |||
: error map: {{val| 0.000 -0.007 -1.901 -1.720 }} | |||
Optimal | {{Optimal ET sequence|legend=1| 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: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.3467{{c}}, ~11/10 = 166.0635{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0142{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 10 0 12 -2 | 0 -3 24 -52 25 -60 44 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.3019{{c}}, ~11/10 = 166.0535{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0114{{c}} | |||
{{Optimal ET sequence|legend=1| 65f, 94, 159, 253 }} | |||
Badness (Sintel): 1.35 | |||
== Countertertiaschis == | |||
Named by [[Flora Canou]] in 2021, 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 | |||
[[Comma list]]: 32805/32768, 244140625/243045684 | |||
{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }} | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.1265{{c}}, ~625/567 = 166.0797{{c}} | |||
: [[error map]]: {{val| +0.127 +0.059 -0.529 +0.178 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/567 = 166.0632{{c}} | |||
: error map: {{val| 0.000 -0.145 -0.797 -0.065 }} | |||
Optimal | {{Optimal ET sequence|legend=1| 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: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.0804{{c}}, ~11/10 = 166.0739{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0634{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.0805{{c}}, ~11/10 = 166.0740{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0635{{c}} | |||
= | {{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }} | ||
Badness (Sintel): 1.01 | |||
== Quadrant == | |||
Named by [[Xenllium]] in 2021, quadrant tempers out 390625/388962, the [[dimcomp comma]], and maps [[25/21]] to the 1/4-octave period. It may be described 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 | |||
[[Comma list]]: 32805/32768, 390625/388962 | |||
{{Mapping|legend=1| 4 0 60 119 | 0 1 -8 -17 }} | |||
: mapping generators: ~25/21, ~3 | |||
== | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 300.0255{{c}}, ~3/2 = 701.8831{{c}} | |||
: [[error map]]: {{val| +0.102 +0.030 -0.664 +0.462 }} | |||
* [[CWE]]: ~2 = 300.0000{{c}}, ~3/2 = 701.8180{{c}} | |||
: error map: {{val| 0.000 -0.137 -0.858 +0.268 }} | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 4 0 60 119 185 | 0 1 -8 -17 -27 }} | |||
== | Optimal tunings: | ||
* WE: ~25/21 = 300.0244{{c}}, ~3/2 = 701.8759{{c}} | |||
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8145{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 4 0 60 119 185 224 | 0 1 -8 -17 -27 -33 }} | |||
== | Optimal tunings: | ||
* WE: ~25/21 = 300.0234{{c}}, ~3/2 = 701.8707{{c}} | |||
* CWE: ~25/21 = 300.0000{{c}}, ~3/2 = 701.8123{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, …, 212, 224, 436, 660 }} | |||
Badness (Sintel): 1.13 | |||
== 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 | |||
[[Comma list]]: 2401/2400, 32805/32768 | |||
= | {{Mapping|legend=1| 1 1 7 5 | 0 4 -32 -15 }} | ||
: mapping generators: ~2, ~448/405 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.0846{{c}}, ~448/405 = 175.4460{{c}} | |||
: [[error map]]: {{val| +0.085 -0.086 +0.007 -0.093 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~448/405 = 175.4320{{c}} | |||
: error map: {{val| 0.000 -0.227 -0.137 -0.306 }} | |||
[[Minimax tuning]]: | |||
* [[7-odd-limit]] [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/3 | |||
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7 | |||
Optimal | {{Optimal ET sequence|legend=1| 41, 89, 130, 171, 814, 985, 1156, 1327, 1498, 2825bd }} | ||
[[Badness]] (Sintel): 0.285 | |||
=== 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 | |||
Comma list: 243/242, | Comma list: 243/242, 441/440, 16384/16335 | ||
Mapping: {{mapping| 1 1 7 5 2 | Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8171{{c}}, ~256/231 = 175.3793{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~256/231 = 175.4081{{c}} | |||
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }} | |||
Badness: 0. | Badness (Sintel): 0.969 | ||
==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 243/242, 364/363, 441/440, 3584/3575 | ||
Mapping: {{mapping| 1 1 7 5 2 -2 | Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.8352{{c}}, ~72/65 = 175.3852{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4095{{c}} | |||
{{Optimal ET sequence|legend=0| 41, 89, 130, 301e, 431e }} | |||
Badness: 0. | Badness (Sintel): 0.925 | ||
===== | ===== Heartia ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | 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 | Optimal tunings: | ||
* WE: ~2 = 1199.6422{{c}}, ~72/65 = 175.3338{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3857{{c}} | |||
{{Optimal ET sequence|legend=0| 41, 89, 130g }} | |||
Badness: | 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: | 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 | Optimal tunings: | ||
* WE: ~2 = 1199.7499{{c}}, ~21/19 = 175.3432{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.3797{{c}} | |||
{{Optimal ET sequence|legend=0| 41, 89, 130g }} | |||
Badness: | Badness (Sintel): 1.40 | ||
===== | ===== Sesquartia ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | 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 | Optimal tunings: | ||
* WE: ~2 = 1199.8902{{c}}, ~72/65 = 175.4077{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.4234{{c}} | |||
{{Optimal ET sequence|legend=0| 41, 130, 171 }} | |||
Badness: | Badness (Sintel): 1.18 | ||
=== | ====== 19-limit ====== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 243/242, 361/360, 364/363, 441/440, 456/455, 595/594 | ||
Mapping: {{mapping| 2 2 | Mapping: {{mapping| 1 1 7 5 2 -2 -6 6 | 0 4 -32 -15 10 39 69 -12 }} | ||
: | Optimal tunings: | ||
* WE: ~2 = 1199.9864{{c}}, ~21/19 = 175.4169{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4189{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~21/19 = 175.4067{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~21/19 = 175.4123{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 7 5 2 -2 13 | 0 4 -32 -15 10 39 -61 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1199.9458{{c}}, ~72/65 = 175.3689{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3770{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 1 7 5 2 -2 13 6 | 0 4 -32 -15 10 39 -61 -12 }} | |||
== | Optimal tunings: | ||
* WE: ~2 = 1200.0114{{c}}, ~72/65 = 175.3783{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3765{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{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{{c}}, ~72/65 = 175.3782{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~72/65 = 175.3763{{c}} | |||
{{Optimal ET sequence|legend=0| 41g, 89, 130 }} | |||
Badness (Sintel): 1.37 | |||
=== Bisesqui === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 2401/2400, 9801/9800, 32805/32768 | |||
Mapping: {{mapping| 2 2 14 10 23 | 0 4 -32 -15 -55 }} | |||
: mapping generators: ~99/70, ~448/405 | |||
== | Optimal tunings: | ||
* WE: ~99/70 = 600.0429{{c}}, ~448/405 = 175.4474{{c}} | |||
* CWE: ~99/70 = 600.0000{{c}}, ~448/405 = 175.4334{{c}} | |||
{{Optimal ET sequence|legend=1| 82e, 130, 212, 342, 1156, 1498, 1840d, 5862bbccdddee }} | |||
Badness (Sintel): 0.561 | |||
== Tsaharuk == | |||
{{Main| Tsaharuk }} | |||
{{ | |||
Tsaharuk tempers out 420175/419904, the [[wizma]], and may be described as the {{nowrap| 77 & 94 }} temperament. It is generated by a slightly flat neutral second of [[~]][[13/12]], five of which make the [[3/2|perfect fifth]], so its [[ploidacot]] is pentacot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 32805/32768, | [[Comma list]]: 32805/32768, 420175/419904 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 1 7 0 | 0 5 -40 24 }} | ||
: mapping generators: ~2, ~243/224 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.1039{{c}}, ~243/224 = 140.3620{{c}} | |||
[[ | : [[error map]]: {{val| +0.104 -0.041 -0.067 -0.137 }} | ||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~243/224 = 140.3496{{c}} | |||
: error map: {{val| 0.000 -0.207 -0.296 -0.436 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 17, 77, 94, 171 }} | ||
[[Badness]]: 0. | [[Badness]] (Sintel): 0.777 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 1331/1323, 19712/19683 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3103{{c}}, ~88/81 = 140.4011{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~88/81 = 140.3649{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17, 77, 94, 171e, 265e }} | ||
Badness: | Badness (Sintel): 2.10 | ||
=== 13-limit === | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 352/351, 385/384, 729/728, 1331/1323 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1840{{c}}, ~13/12 = 140.3840{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~13/12 = 140.3627{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 17, 77, 94, 171e }} | ||
Badness: | Badness (Sintel): 1.57 | ||
== | == Quanharuk == | ||
Quanharuk tempers out 16875/16807, the [[mirkwai]] comma, and may be described as the {{nowrap| 41 & 183 }} temperament. The generator is a slightly flat major third of [[~]][[56/45]], five of which make the [[3/1|3rd]] [[harmonic]], so the [[ploidacot]] of this temperament is alpha-pentacot. [[224edo]] makes for a recommendable tuning. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 16875/16807, 32805/32768 | |||
Optimal tuning | {{Mapping|legend=1| 1 0 15 12 | 0 5 -40 -29 }} | ||
: mapping generators: ~2, ~56/45 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.0032{{c}}, ~56/45 = 380.3557{{c}} | |||
: [[error map]]: {{val| +0.003 -0.177 -0.493 +0.898 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~56/45 = 380.3546{{c}} | |||
: error map: {{val| 0.000 -0.182 -0.498 +0.890 }} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 41, 142, 183, 224 }} | ||
Badness: | [[Badness]] (Sintel): 1.82 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 540/539, 1375/1372, 32805/32768 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }} | ||
Badness: | Badness (Sintel): 1.04 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 540/539, 729/728, 1375/1372, 4096/4095 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }} | ||
Badness: 0. | Badness (Sintel): 0.884 | ||
== | == Quintilipyth == | ||
Named by [[Xenllium]] in 2021, quintilipyth (formerly ''quintilischis'') slices the [[4/3|perfect fourth]] into five semitones and tempers out the [[compass comma]] (9765625/9680832) in the [[7-limit]]. It may be described as the {{nowrap| 12 & 253 }} temperament, and its [[ploidacot]] is omega-pentacot. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 32805/32768, 9765625/9680832 | |||
{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }} | |||
: mapping generators: ~2, ~625/588 | |||
{{ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.1138{{c}}, ~625/588 = 99.6347{{c}} | |||
: [[error map]]: {{val| +0.114 +0.099 -1.041 +0.761 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~625/588 = 99.6265{{c}} | |||
: error map: {{val| 0.000 -0.087 -1.255 +0.544 }} | |||
{{Optimal ET sequence|legend=1| 12, …, 253, 265 }} | |||
[[Badness]] (Sintel): 6.43 | |||
=== 11-limit === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 1375/1372, 4375/4356, 32805/32768 | |||
Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.1503{{c}}, ~35/33 = 99.6287{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6176{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.1774{{c}}, ~35/33 = 99.6267{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6134{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 -4 -7 -9 5 | 0 -5 40 82 126 153 -11 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.1745{{c}}, ~18/17 = 99.6265{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6131{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 241cdef, 253 }} | |||
Badness (Sintel): 2.34 | |||
=== 19-limit === | |||
Subgroup: 2.3.5.7.11.13.17.19 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971 | ||
Mapping: {{mapping| 1 2 -1 - | Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 253, 265 }} | ||
Badness: | Badness (Sintel): 2.32 | ||
== | == Quintaschis == | ||
Named by [[Xenllium]] in 2021, quintaschis slices the [[4/3|perfect fourth]] into five semitones and tempers out 49009212/48828125 in the [[7-limit]]. It may be described as the {{nowrap| 12 & 289 }} temperament, and its [[ploidacot]] is omega-pentacot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 32805/32768, | [[Comma list]]: 32805/32768, 49009212/48828125 | ||
{{Mapping|legend=1| 1 2 -1 - | {{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }} | ||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.0536{{c}}, ~200/189 = 99.6684{{c}} | |||
: [[error map]]: {{val| +0.054 -0.190 +0.370 -0.262 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~200/189 = 99.6645{{c}} | |||
: error map: {{val| 0.000 -0.277 +0.266 -0.363 }} | |||
{{ | {{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }} | ||
[[ | [[Badness]] (Sintel): 3.36 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 441/440, | Comma list: 441/440, 32805/32768, 1953125/1951488 | ||
Mapping: {{mapping| 1 2 -1 - | Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0988{{c}}, ~35/33 = 99.6613{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6540{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, …, 277d, 289 }} | ||
Badness: | Badness (Sintel): 3.69 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 364/363, 441/440, | Comma list: 364/363, 441/440, 32805/32768, 109512/109375 | ||
Mapping: {{mapping| 1 2 -1 - | Mapping: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, …, 277dff, 289 }} | ||
Badness: | 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: {{mapping| 1 2 -1 -5 -8 -11 5 | 0 -5 40 94 138 177 -11 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.1286{{c}}, ~18/17 = 99.6668{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6568{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 -5 -8 -11 5 4 | 0 -5 40 94 138 177 -11 3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0289{{c}}, ~18/17 = 99.6609{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6586{{c}} | |||
{{Optimal ET sequence|legend=0| 12f, 289 }} | |||
Badness (Sintel): 2.56 | |||
=== Quintahelenic === | |||
Subgroup: 2.3.5.7.11 | |||
Comma list: 5632/5625, 8019/8000, 151263/151250 | |||
Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0195{{c}}, ~200/189 = 99.6723{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6709{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }} | |||
== | Optimal tunings: | ||
{{ | * WE: ~2 = 1200.0442{{c}}, ~200/189 = 99.6709{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6675{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 2 -1 -5 -9 -11 5 | 0 -5 40 94 150 177 -11 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.1227{{c}}, ~200/189 = 99.6753{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6658{{c}} | |||
{{Optimal ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 12f, 289e, 301 }} | ||
Badness (Sintel): 2.06 | |||
=== | ===== 19-limit ===== | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11.13.17.19 | ||
Comma list: | Comma list: 476/475, 495/494, 561/560, 833/832, 847/845, 1701/1700 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 2 -1 -5 -9 -11 5 4 | 0 -5 40 94 150 177 -11 3 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0230{{c}}, ~200/189 = 99.6694{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6676{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12f, 301 }} | ||
Badness: | Badness (Sintel): 2.24 | ||
=== | ==== Quintahelenoid ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 729/728, 1001/1000, 4096/4095, 86515/86436 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 2 -1 -5 -9 14 | 0 -5 40 94 150 -124 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9919{{c}}, ~200/189 = 99.6712{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~200/189 = 99.6718{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, 301, 614, 915 }} | ||
Badness: | 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: {{mapping| 1 2 -1 -5 -9 14 5 | 0 -5 40 94 150 -124 -11 }} | ||
: | Optimal tunings: | ||
* WE: ~2 = 1200.0469{{c}}, ~18/17 = 99.6749{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6710{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 12, 301 }} | ||
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: {{mapping| 1 2 -1 -5 -9 14 5 4 | 0 -5 40 94 150 -124 -11 3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9925{{c}}, ~18/17 = 99.6710{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6716{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 301 }} | |||
Badness (Sintel): 2.41 | |||
== Sextilifourths == | |||
Named by [[Xenllium]] in 2021, sextilifourths (also known as ''sextilischis'', formerly ''sextilififths'') slices the [[4/3|perfect fourth]] into six small semitones, which serves as both [[21/20]] and [[22/21]]. It may be described as {{nowrap| 130 & 159 }}, and its [[ploidacot]] is omega-hexacot. [[289edo]] gives a highly recommendable tuning. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[Comma list]]: 32805/32768, 235298/234375 | |||
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }} | |||
: mapping generators: ~2, ~21/20 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.0987{{c}}, ~21/20 = 83.0599{{c}} | |||
: [[error map]]: {{val| +0.099 -0.117 +0.462 -0.630 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~21/20 = 83.0543{{c}} | |||
: error map: {{val| 0.000 -0.281 +0.295 -0.837 }} | |||
{{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }} | |||
{{Optimal ET sequence|legend=1| | |||
[[Badness]]: | [[Badness]] (Sintel): 2.75 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 441/440, 4000/3993, 235298/234375 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.0424{{c}}, ~21/20 = 83.0520{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0497{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 29, 72cde, 101e, 130, 289 }} | ||
Badness: | Badness (Sintel): 1.50 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 364/363, 441/440, 676/675, 10985/10976 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.1056{{c}}, ~21/20 = 83.0566{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~21/20 = 83.0508{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 29, 72cdef, 101e, 130, 289 }} | ||
Badness: | Badness (Sintel): 1.04 | ||
== Septant == | == Septant == | ||
Named by [[Xenllium]] in 2021, septant notably tempers out the [[akjaysma]] ({{monzo| 47 -7 -7 -7 }}) and may be described as the {{nowrap| 224 & 301 }} temperament. It has a period of 1/7 octave, and its [[ploidacot]] is heptaploid monocot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 2,310: | Line 2,553: | ||
{{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }} | {{Mapping|legend=1| 7 0 105 -56 | 0 1 -8 7 }} | ||
: mapping generators: ~8575/7776, ~3 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~8575/7776 = 171.4303{{c}}, ~3/2 = 701.7091{{c}} | |||
{{ | : [[error map]]: {{val| +0.012 -0.234 +0.096 +0.265 }} | ||
* [[CWE]]: ~8575/7776 = 171.4286{{c}}, ~3/2 = 701.7022{{c}} | |||
[[ | : error map: {{val| 0.000 -0.253 +0.069 +0.232 }} | ||
{{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }} | {{Optimal ET sequence|legend=1| 77, 147, 224, 301, 525, 826, 1351 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 2.81 | ||
=== 11-limit === | === 11-limit === | ||
| Line 2,328: | Line 2,572: | ||
Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }} | Mapping: {{mapping| 7 0 105 -56 -120 | 0 1 -8 7 13 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~495/448 = 171.4334{{c}}, ~3/2 = 701.7387{{c}} | |||
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7198{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }} | ||
Badness: | Badness (Sintel): 1.46 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,341: | Line 2,587: | ||
Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }} | Mapping: {{mapping| 7 0 105 -56 -120 37 | 0 1 -8 7 13 -1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~495/448 = 171.4282{{c}}, ~3/2 = 701.7229{{c}} | |||
* CWE: ~495/448 = 171.4286{{c}}, ~3/2 = 701.7242{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 77, 147, 224, 525, 1274f }} | ||
Badness: | Badness (Sintel): 1.02 | ||
== Octant == | == Octant == | ||
Octant may be described as the {{nowrap| 224 & 248 }} temperament. It has a period of 1/8 octave, and its [[ploidacot]] is octaploid monocot. In this temperament, [[12/11]], [[35/27]], and [[99/70]] are mapped to 1\8, 3\8, and 4\8 respectively. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 2,355: | Line 2,603: | ||
{{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }} | {{Mapping|legend=1| 8 0 120 -117 | 0 1 -8 11 }} | ||
: mapping generators: ~42875/39366, ~3 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~42875/39366 = 150.0048{{c}}, ~3/2 = 701.7356{{c}} | |||
{{ | : [[error map]]: {{val| +0.039 -0.181 +0.071 +0.127 }} | ||
* [[CWE]]: ~42875/39366 = 150.0000{{c}}, ~3/2 = 701.7134{{c}} | |||
[[ | : error map: {{val| 0.000 -0.242 -0.021 +0.022 }} | ||
{{Optimal ET sequence|legend=1| 24, 224, 472, 696, 1168 }} | {{Optimal ET sequence|legend=1| 24, …, 224, 472, 696, 1168 }} | ||
[[Badness]]: | [[Badness]] (Sintel): 3.98 | ||
=== 11-limit === | === 11-limit === | ||
| Line 2,373: | Line 2,622: | ||
Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }} | Mapping: {{mapping| 8 0 120 -117 15 | 0 1 -8 11 1 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~12/11 = 150.0010{{c}}, ~3/2 = 701.7177{{c}} | |||
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7131{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }} | ||
Badness: | Badness (Sintel): 1.48 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,386: | Line 2,637: | ||
Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }} | Mapping: {{mapping| 8 0 120 -117 15 93 | 0 1 -8 11 1 -5 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~12/11 = 149.9957{{c}}, ~3/2 = 701.7046{{c}} | |||
* CWE: ~12/11 = 150.0000{{c}}, ~3/2 = 701.7247{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 24, 224, 472, 696 }} | ||
Badness: | Badness (Sintel): 1.26 | ||
== Nonant == | == Nonant == | ||
Named by [[Xenllium]] in 2023, nonant tempers out the [[septimal ennealimma]] ({{monzo| -11 -9 0 9 }}) and may be described as the {{nowrap| 36 & 171 }} temperament. It has a period of 1/9 octave, and its [[ploidacot]] is enneaploid monocot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 2,400: | Line 2,653: | ||
{{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }} | {{Mapping|legend=1| 9 0 135 11 | 0 1 -8 1 }} | ||
: mapping generators: ~2592/2401, ~3 | |||
: | [[Optimal tuning]]s: | ||
* [[WE]]: ~2592/2401 = 133.3442{{c}}, ~3/2 = 701.8000{{c}} | |||
: [[error map]]: {{val| +0.098 -0.057 -0.027 -0.141 }} | |||
* [[CWE]]: ~2592/2401 = 133.3333{{c}}, ~3/2 = 701.7384{{c}} | |||
: error map: {{val| 0.000 -0.217 -0.221 -0.421 }} | |||
{{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }} | |||
[[Badness]] (Sintel): 1.77 | |||
[[Badness]]: | |||
=== 11-limit === | === 11-limit === | ||
| Line 2,416: | Line 2,672: | ||
Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }} | Mapping: {{mapping| 9 0 135 11 131 | 0 1 -8 1 -7 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~242/225 = 133.3308{{c}}, ~3/2 = 701.8205{{c}} | |||
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.8351{{c}} | |||
{{Optimal ET sequence|legend=0| 36, 135, 171 }} | |||
Badness: | Badness (Sintel): 4.20 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,429: | Line 2,687: | ||
Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }} | Mapping: {{mapping| 9 0 135 11 131 -38 | 0 1 -8 1 -7 5 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~242/225 = 133.3180{{c}}, ~3/2 = 701.6956{{c}} | |||
* CWE: ~242/225 = 133.3333{{c}}, ~3/2 = 701.7800{{c}} | |||
{{Optimal ET sequence|legend=0| 36, 99cf, 135, 171 }} | |||
Badness: | Badness (Sintel): 3.15 | ||
== | == Septiquarschis == | ||
Named by [[Xenllium]] in 2021, septiquarschis tempers out [[829440/823543]] (mynaslender comma) and [[67108864/66706983]] (septiness comma), and may be described as the {{nowrap| 89 & 94 }} temperament. It splits septimal minor seventh ([[7/4]]) into four generators. Note that in the data below, the generator is the [[octave complement]] so that seven of them minus five octaves make a [[3/2|perfect fifth]]; its [[ploidacot]] is thus epsilon-heptacot. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 32805/32768, | [[Comma list]]: 32805/32768, 829440/823543 | ||
: | {{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }} | ||
: mapping generators: ~2, ~256/147 | |||
[[Optimal tuning]] | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1199.8855{{c}}, ~256/147 = 957.2944{{c}} | |||
: [[error map]]: {{val| -0.114 -0.436 -0.182 +1.310 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~256/147 = 957.3867{{c}} | |||
: error map: {{val| 0.000 -0.248 +0.032 +1.627 }} | |||
{{Optimal ET sequence|legend=1| 89, | {{Optimal ET sequence|legend=1| 89, 94, 183, 460d, 643d }} | ||
[[Badness]]: | [[Badness]] (Sintel): 4.73 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 540/539, 15488/15435, 32805/32768 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 -4 47 6 25 | 0 7 56 -4 -27 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1199.9430{{c}}, ~256/147 = 957.3390{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3849{{c}} | |||
{{Optimal ET sequence|legend=0| 89, 94, 183, 460d }} | |||
Badness: | Badness (Sintel): 1.72 | ||
== | === 13-limit === | ||
== | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 540/539, 729/728, 1573/1568, 4096/4095 | |||
Mapping: {{mapping| 1 -4 47 6 25 -33 | 0 7 56 -4 -27 46 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.0058{{c}}, ~256/147 = 957.3946{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~256/147 = 957.3900{{c}} | |||
{{ | {{Optimal ET sequence|legend=0| 89, 94, 183, 277, 460d }} | ||
: | Badness (Sintel): 1.46 | ||
[[ | == Tridecafifths == | ||
Named by [[Eliora]] in 2023, tridecafifths may be described as the {{nowrap| 89 & 200 }} temperament. It divides the [[3/2|perfect fifth]] into thirteen quartertones, so its [[ploidacot]] is 13-cot. [[289edo]] gives a highly recommendable tuning. | |||
[[Subgroup]]: 2.3.5.7 | |||
[[ | [[Comma list]]: 32805/32768, {{monzo| -14 -1 -9 13 }} | ||
= | {{Mapping|legend=1| 1 1 7 6 | 0 13 -104 -71 }} | ||
: mapping generators: ~2, ~1323/1280 | |||
[[ | [[Optimal tuning]]s: | ||
* [[WE]]: ~2 = 1200.1431{{c}}, ~1323/1280 = 53.9838{{c}} | |||
: [[error map]]: {{val| +0.143 -0.023 +0.375 -0.816 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1323/1280 = 53.9764{{c}} | |||
: error map: {{val| 0.000 -0.261 -0.221 -0.421 }} | |||
{{ | {{Optimal ET sequence|legend=1| 89, 200, 289 }} | ||
[[Badness]] (Sintel): 10.9 | |||
: | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 441/440, 32805/32768, 55296000/55240493 | |||
{{ | Mapping: {{mapping| 1 1 7 6 4 | 0 13 -104 -71 -12 }} | ||
Optimal tunings: | |||
* WE: ~2 = 1200.0311{{c}}, ~33/32 = 53.9766{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 53.9750{{c}} | |||
= | {{Optimal ET sequence|legend=0| 89, 200, 289 }} | ||
Badness (Sintel): 4.23 | |||
[[ | == Subgroup extensions == | ||
=== Maqamschismic (2.3.5.11) === | |||
Proposed by [[Eufalesio]] in 2026, maqamschismic is equivalent to the no-7 [[cassandra]]. The 2.3.5.11.13 subgroup adds [[352/351]] to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning [[maqam]]at (especially the Turkish version), as outlined by [[Ozan Yarman]]. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 [[helenus]]. | |||
Subgroup: 2.3.5.11 | |||
Comma list: 2200/2187, 4125/4096 | |||
: mapping | Subgroup-val mapping: {{mapping| 1 0 15 -33 | 0 1 -8 23 }} | ||
{{ | Optimal tunings: | ||
* WE: ~2 = 1200.5458{{c}} ~3/2 = 702.4021{{c}} | |||
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0906{{c}} | |||
{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e, 241ce, 335ce }} | |||
Badness (Sintel): 1.34 | |||
==== 2.3.5.11.13 subgroup ==== | |||
Subgroup: 2.3.5.11.13 | |||
Comma list: 325/324, 352/351, 4125/4096 | |||
Subgroup-val mapping: {{mapping| 1 0 15 -33 -28 | 0 1 -8 23 20 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.4565{{c}} ~3/2 = 702.3057{{c}} | |||
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 702.0485{{c}} | |||
{{Optimal ET sequence|legend=0| 12e, …, 41, 53, 94, 147e }} | |||
Badness (Sintel): 0.862 | |||
=== Tridecaschismic (2.3.5.13) === | |||
Proposed by [[Eufalesio]] in 2026, tridecaschismic adds the [[325/324|marveltwin comma]] to the comma list, or equivalently, the [[tridecapyth comma]]. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 [[restriction]] of 13-limit [[cassandra]]. | |||
Subgroup: 2.3.5.13 | |||
: | Comma list: 325/324, 32805/32768 | ||
Subgroup-val mapping: {{mapping| 1 0 15 -28 | 0 1 -8 20 }} | |||
{{Optimal ET sequence|legend= | Optimal tunings: | ||
* WE: ~2 = 1200.3326{{c}} ~3/2 = 702.1092{{c}} | |||
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9189{{c}} | |||
{{Optimal ET sequence|legend=0| 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff }} | |||
Badness (Sintel): 0.582 | |||
==== 2.3.5.13.19 subgroup ==== | |||
Subgroup: 2.3.5.13.19 | |||
Comma list: 325/324, 361/360, 513/512 | |||
Subgroup-val mapping: {{mapping| 1 0 15 -28 9 | 0 1 -8 20 -3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.4236{{c}}, ~3/2 = 702.1510{{c}} | |||
* CWE: 2 = 1200.0000{{c}}, ~3/2 = 701.9064{{c}} | |||
{{Optimal ET sequence|legend=0| 12, …, 41, 53 }} | |||
=== | Badness (Sintel): 0.354 | ||
=== Photia (2.3.5.17) === | |||
{{See also| No-elevens subgroup temperaments #Garibaldia }} | |||
[[Subgroup]]: 2.3.5.17 | [[Subgroup]]: 2.3.5.17 | ||
[[Comma list]]: | [[Comma list]]: 256/255, 1458/1445 | ||
{{Mapping|legend=2| 1 0 15 -7 | 0 1 -8 7 }} | |||
{{Mapping|legend=3| 1 0 15 0 0 0 -7 | 0 1 -8 0 0 0 7 }} | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1199.5471{{c}}, ~3/2 = 701.2262{{c}} | |||
: [[error map]]: {{val| -0.453 -1.182 +0.706 +3.628 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.4976{{c}} | |||
: error map: {{val| 0.000 -0.457 +1.705 +5.528 }} | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 15 -7 9 | 0 1 -8 7 -3 }} | ||
Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }} | |||
{{ | Optimal tunings: | ||
* WE: ~2 = 1199.7225{{c}}, ~3/2 = 701.3077{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 701.4754{{c}} | |||
{{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }} | |||
Badness (Sintel): 0.332 | |||
[[ | === Nestoria (2.3.5.19) === | ||
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]'' | |||
Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime [[19/1|19]] being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like [[19/10]] and [[19/15]]. However, the dyadic tuning sensitivity of [[19/16]] suggests using tunings like [[65edo]] and [[77edo]] to optimize in favour of prime 19, as [[53edo]] is already arguably undertempered for it despite being the flattest tuning appearing in the optimal ET sequence. | |||
[[Subgroup]]: 2.3.5.19 | |||
[[Comma list]]: 361/360, 513/512 | |||
[[Optimal tuning]] ([[ | {{Mapping|legend=2| 1 0 15 9 | 0 1 -8 -3 }} | ||
{{Mapping|legend=3| 1 0 15 0 0 0 0 9 | 0 1 -8 0 0 0 0 -3 }} | |||
: mapping generators: ~2, ~3 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.2250{{c}}, ~3/2 = 701.8776{{c}} | |||
: [[error map]]: {{val| +0.225 +0.148 +0.240 -1.796 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~3/2 = 701.7307{{c}} | |||
: error map: {{val| 0.000 -0.224 -0.159 -2.705 }} | |||
{{Optimal ET sequence|legend=1| 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 | |||
{{Mapping|legend=2| 1 0 15 14 | 0 2 -16 -13 }} | |||
{{Mapping|legend=3| 1 0 15 0 0 14 | 0 2 -16 0 0 -13 }} | |||
: mapping generators: ~2, ~26/15 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.1497{{c}}, ~26/15 = 950.9740{{c}} | |||
: [[error map]]: {{val| +0.150 -0.007 +0.348 -1.094 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~26/15 = 950.8493{{c}} | |||
: error map: {{val| 0.000 -0.256 +0.098 -1.568 }} | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.2611{{c}}, ~26/15 = 951.0703{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8532{{c}} | |||
{{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.2987{{c}}, ~26/15 = 951.1060{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~26/15 = 950.8595{{c}} | |||
{{Optimal ET sequence|legend=0| 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 | |||
{{Mapping|legend=2| 1 2 -1 5 | 0 -5 40 -11 }} | |||
{{Mapping|legend=3| 1 2 -1 0 0 0 5 | 0 -5 40 0 0 0 -11 }} | |||
: mapping generators: ~2, ~18/17 | |||
[[Optimal tuning]]s: | |||
* [[WE]]: ~2 = 1200.1370{{c}}, ~18/17 = 99.6602{{c}} | |||
: [[error map]]: {{val| +0.137 +0.018 -0.042 -0.533 }} | |||
* [[CWE]]: ~2 = 1200.0000{{c}}, ~18/17 = 99.6499{{c}} | |||
: error map: {{val| 0.000 -0.205 -0.317 -1.104 }} | |||
{{Optimal ET sequence|legend=1| 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: {{mapping| 1 2 -1 5 4 | 0 -5 40 -11 3 }} | |||
Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0350{{c}}, ~18/17 = 99.6550{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6520{{c}} | |||
{{Optimal ET sequence|legend= | {{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }} | ||
Badness (Sintel): 1.17 | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category:Schismatic family| ]] <!-- main article --> | [[Category:Schismatic family| ]] <!-- main article --> | ||
[[Category:Rank 2]] | [[Category:Rank 2]] | ||
Latest revision as of 14:58, 5 June 2026
- 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 ¢, ~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⟩, pontiac adds [-59 39 0 -1⟩, and schism adds [6 -2 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. Salsa adds [15 -13 0 2⟩ and has a hemififth generator. Hemischis adds [-34 25 0 -2⟩ and has a hemitwelfth 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. Squirrel, tertiaschis, and countertertiaschis each has a generator that is 1/3 of the fourth. Quadrant adds [-119 68 0 4⟩ with a 1/4-octave period. Kleischismic adds [49 -38 0 4⟩ with a half-octave period and also a bisect generator. Sesquiquartififths adds [-35 15 0 4⟩ and slices the fifth in four.
Temperaments involving larger splits include tsaharuk, quanharuk, quintilipyth, quintaschis, altinex, pogo, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths. Those split the schismic structure into five to thirteen parts.
Temperaments discussed elsewhere include:
- Guiron (+1029/1024) → Gamelismic clan
- Pogo (+118098/117649) → Stearnsmic clan
Considered below are garibaldi, pontiac, grackle, schism, bischismic, kleischismic, salsa, hemischis, term, altinex, squirrel, tertiaschis, countertertiaschis, quadrant, sesquiquartififths, tsaharuk, quanharuk, quintilipyth, quintaschis, sextilifourths, septant, octant, nonant, septiquarschis, and tridecafifths.
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 ¢, ~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, even though it comes with a much 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 ¢, ~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 ¢, ~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
Quasipyth
Named by Xenllium in 2026, quasipyth tempers out [109 -67 0 -1⟩, the nanisma, as well as the catasyc comma, 390625/387072. The 7/4 is found at −67 fifths, represented by the nonuple-diminished thirteenth.
Subgroup: 2.3.5.7
Comma list: 32805/32768, 390625/387072
Mapping: [⟨1 0 15 109], ⟨0 1 -8 -67]]
- WE: ~2 = 1200.2569 ¢, ~3/2 = 702.1149 ¢
- error map: ⟨+0.2569 +0.4168 -1.4342 +0.2685]
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9615 ¢
- error map: ⟨0.0000 +0.0065 -2.0054 -0.2437]
Optimal ET sequence: 53, 147d, 200, 253, 306c, 559c
Badness (Sintel): 5.04
11-limit
Subgroup: 2.3.5.7.11
Comma list: 385/384, 19712/19683, 78125/77616
Mapping: [⟨1 0 15 109 -117], ⟨0 1 -8 -67 76]]
Optimal tunings:
- WE: ~2 = 1200.3283 ¢, ~3/2 = 702.1636 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9713 ¢
Optimal ET sequence: 53, 200, 253, 559ce
Badness (Sintel): 3.83
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 325/324, 385/384, 2200/2197, 19712/19683
Mapping: [⟨1 0 15 109 -117 -28], ⟨0 1 -8 -67 76 20]]
Optimal tunings:
- WE: ~2 = 1200.3229 ¢, ~3/2 = 702.1603 ¢
- CWE: ~2 = 1200.0000 ¢, ~3/2 = 701.9714 ¢
Optimal ET sequence: 53, 200, 253, 559ce
Badness (Sintel): 2.13
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 ¢, ~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 alpha-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
Named by Aura in 2021, 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
Named by Xenllium in 2021, 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
Named by Flora Canou in 2021, 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
Named by Xenllium in 2021, quadrant tempers out 390625/388962, the dimcomp comma, and maps 25/21 to the 1/4-octave period. It may be described 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
Tsaharuk tempers out 420175/419904, the wizma, and may be described as the 77 & 94 temperament. It is generated by a slightly flat neutral second of ~13/12, five of which make the perfect fifth, so its ploidacot is pentacot.
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
Quanharuk tempers out 16875/16807, the mirkwai comma, and may be described as the 41 & 183 temperament. The generator is a slightly flat major third of ~56/45, five of which make the 3rd harmonic, so the ploidacot of this temperament is alpha-pentacot. 224edo makes for a recommendable tuning.
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
Named by Xenllium in 2021, quintilipyth (formerly quintilischis) slices the perfect fourth into five semitones and tempers out the compass comma (9765625/9680832) in the 7-limit. It may be described as the 12 & 253 temperament, and its ploidacot is omega-pentacot.
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
Named by Xenllium in 2021, quintaschis slices the perfect fourth into five semitones and tempers out 49009212/48828125 in the 7-limit. It may be described as the 12 & 289 temperament, and its ploidacot is omega-pentacot.
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
Named by Xenllium in 2021, sextilifourths (also known as sextilischis, formerly sextilififths) slices the perfect fourth into six small semitones, which serves as both 21/20 and 22/21. It may be described as 130 & 159, and its ploidacot is omega-hexacot. 289edo gives a highly recommendable tuning.
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
Septant
Named by Xenllium in 2021, septant notably tempers out the akjaysma ([47 -7 -7 -7⟩) and may be described as the 224 & 301 temperament. It has a period of 1/7 octave, and its ploidacot is heptaploid monocot.
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
Octant
Octant may be described as the 224 & 248 temperament. It has a period of 1/8 octave, and its ploidacot is octaploid monocot. In this temperament, 12/11, 35/27, and 99/70 are mapped to 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
Named by Xenllium in 2023, nonant tempers out the septimal ennealimma ([-11 -9 0 9⟩) and may be described as the 36 & 171 temperament. It has a period of 1/9 octave, and its ploidacot is enneaploid monocot.
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
Septiquarschis
Named by Xenllium in 2021, septiquarschis tempers out 829440/823543 (mynaslender comma) and 67108864/66706983 (septiness comma), and may be described as the 89 & 94 temperament. It splits septimal minor seventh (7/4) into four generators. Note that in the data below, the generator is the octave complement so that seven of them minus five octaves make a perfect fifth; its ploidacot is thus epsilon-heptacot.
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
Named by Eliora in 2023, tridecafifths may be described as the 89 & 200 temperament. It divides the perfect fifth into thirteen quartertones, so its ploidacot is 13-cot. 289edo gives a highly recommendable tuning.
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
Maqamschismic (2.3.5.11)
Proposed by Eufalesio in 2026, maqamschismic is equivalent to the no-7 cassandra. The 2.3.5.11.13 subgroup adds 352/351 to the comma list and tempers 11/9~39/32 together (and 16/13~27/22), providing a very simple framework for tuning maqamat (especially the Turkish version), as outlined by Ozan Yarman. 41edo and 53edo are simplest, but 94edo is more optimized. It is only slightly worse than the no-7 helenus.
Subgroup: 2.3.5.11
Comma list: 2200/2187, 4125/4096
Subgroup-val mapping: [⟨1 0 15 -33], ⟨0 1 -8 23]]
Optimal tunings:
- WE: ~2 = 1200.5458 ¢ ~3/2 = 702.4021 ¢
- CWE: 2 = 1200.0000 ¢, ~3/2 = 702.0906 ¢
Optimal ET sequence: 12e, …, 41, 53, 94, 147e, 241ce, 335ce
Badness (Sintel): 1.34
2.3.5.11.13 subgroup
Subgroup: 2.3.5.11.13
Comma list: 325/324, 352/351, 4125/4096
Subgroup-val mapping: [⟨1 0 15 -33 -28], ⟨0 1 -8 23 20]]
Optimal tunings:
- WE: ~2 = 1200.4565 ¢ ~3/2 = 702.3057 ¢
- CWE: 2 = 1200.0000 ¢, ~3/2 = 702.0485 ¢
Optimal ET sequence: 12e, …, 41, 53, 94, 147e
Badness (Sintel): 0.862
Tridecaschismic (2.3.5.13)
Proposed by Eufalesio in 2026, tridecaschismic adds the marveltwin comma to the comma list, or equivalently, the tridecapyth comma. It benefits from a fifth that is just, or practically indistinguishable from just, like in 53edo. It is one of the lowest badness schismic extensions. It is also equivalent to the 2.3.5.13 restriction of 13-limit cassandra.
Subgroup: 2.3.5.13
Comma list: 325/324, 32805/32768
Subgroup-val mapping: [⟨1 0 15 -28], ⟨0 1 -8 20]]
Optimal tunings:
- WE: ~2 = 1200.3326 ¢ ~3/2 = 702.1092 ¢
- CWE: 2 = 1200.0000 ¢, ~3/2 = 701.9189 ¢
Optimal ET sequence: 12, …, 41, 53, 412cf, 465cf, …, 783ccff, 836ccfff
Badness (Sintel): 0.582
2.3.5.13.19 subgroup
Subgroup: 2.3.5.13.19
Comma list: 325/324, 361/360, 513/512
Subgroup-val mapping: [⟨1 0 15 -28 9], ⟨0 1 -8 20 -3]]
Optimal tunings:
- WE: ~2 = 1200.4236 ¢, ~3/2 = 702.1510 ¢
- CWE: 2 = 1200.0000 ¢, ~3/2 = 701.9064 ¢
Optimal ET sequence: 12, …, 41, 53
Badness (Sintel): 0.354
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
Nestoria is notable for having one of the lowest-badness subgroup extensions of schismic. Note that despite prime 19 being optimized by a flatter fifth, the fifth in optimal tunings of nestoria is generally not flatter than the fifth in optimal schismic due to its optimization considering intervals like 19/10 and 19/15. However, the dyadic tuning sensitivity of 19/16 suggests using tunings like 65edo and 77edo to optimize in favour of prime 19, as 53edo is already arguably undertempered for it despite being the flattest tuning appearing in the optimal ET sequence.
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