Schismatic family: Difference between revisions
m Units & misc. cleanup |
m unfortunately thats not how optimal ET sequences work. it's a defined thing by fumica's temperament evaluator. the standard ive been abiding by is that missing entries are to be discussed on a case-by-case basis |
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{{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 [[syntonic comma]] (81/80), or alternatively put, the difference between a [[5/4|just major third]] and a [[8192/6561|Pythagorean diminished fourth]]. | 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]]. | ||
| Line 23: | Line 17: | ||
[[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]] ( | [[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. | ||
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. | |||
Temperaments discussed elsewhere include: | 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 [[#Subgroup extensions]]. | 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]]. | ||
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{{Main| Garibaldi }} | {{Main| Garibaldi }} | ||
Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double diminished octave ( | Garibaldi tempers out the [[garischisma]], equating the [[64/63|septimal comma]] with both the [[syntonic comma]] and the [[Pythagorean comma]]. The 7/4 is found at -14 fifths, represented by the double-diminished octave (C–C𝄫), or down-minor seventh (C-vB♭) with the down-arrow representing the comma step. It necessitates a sharper fifth than pure. Its [[S-expression]]-based comma list is {[[5120/5103|S8/S9]], [[225/224|S15]]}. | ||
[[147edo]] (from 94+53) is a [[patent val]] tuning with 5 and 7 very close to equally out-of-tune in opposite directions s.t only 7/5 and 10/7 are inconsistent in the 9-odd-limit, so might be considered for (EG) a 41-note subset ([[12L 29s]]). | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
| Line 65: | Line 58: | ||
{{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }} | {{Mapping|legend=1| 1 0 15 25 | 0 1 -8 -14 }} | ||
[[Optimal tuning]] | [[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]]: | ||
| Line 79: | Line 76: | ||
* 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]] ( | [[Badness]] (Sintel): 0.548 | ||
=== Cassandra === | === Cassandra === | ||
Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup. | Cassandra is one of the best extensions of garibaldi to the 11- and 13-limit as well as the 2.3.5.7.11.13.19 subgroup, even though it comes with a much higher complexity. | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
| Line 92: | Line 89: | ||
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: | ||
| Line 102: | Line 101: | ||
* 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=0| 41, 53, 94, 229c | {{Optimal ET sequence|legend=0| 12e, 41, 53, 94, 229c }} | ||
Badness ( | Badness (Sintel): 0.906 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 113: | Line 112: | ||
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: | ||
| Line 126: | Line 127: | ||
{{Optimal ET sequence|legend=0| 41, 53, 94, 429ccdeef, 523ccdeef }} | {{Optimal ET sequence|legend=0| 41, 53, 94, 429ccdeef, 523ccdeef }} | ||
Badness (Sintel): 0. | Badness (Sintel): 0.854 | ||
===== Cassie ===== | ===== Cassie ===== | ||
| Line 135: | Line 136: | ||
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=0| 41, 53, 94g }} | {{Optimal ET sequence|legend=0| 12e, 41, 53, 94g }} | ||
Badness ( | Badness (Sintel): 1.19 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 148: | Line 151: | ||
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=0| 41, 53 | {{Optimal ET sequence|legend=0| 12e, 41, 53 }} | ||
Badness ( | Badness (Sintel): 1.11 | ||
===== Cassandric ===== | ===== Cassandric ===== | ||
| Line 161: | Line 166: | ||
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=0| 41g, 53, 94 | {{Optimal ET sequence|legend=0| 41g, 53, 94 }} | ||
Badness ( | Badness (Sintel): 1.18 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 176: | Line 183: | ||
Optimal tunings: | Optimal tunings: | ||
* WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}} | * WE: ~2 = 1200.2910{{c}}, ~3/2 = 702.2681{{c}} | ||
* CWE: ~2 = 1200. | * 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 193: | Line 199: | ||
* WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}} | * WE: ~2 = 1200.2970{{c}}, ~3/2 = 702.2697{{c}} | ||
* CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}} | * CWE: ~2 = 1200.0000{{c}}, ~3/2 = 702.0943{{c}} | ||
{{Optimal ET sequence|legend=0| 41g, 53, 94 }} | {{Optimal ET sequence|legend=0| 41g, 53, 94 }} | ||
Badness ( | Badness (Sintel): 1.08 | ||
===== Cassander ===== | ===== Cassander ===== | ||
| Line 206: | Line 211: | ||
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=0| 41, 53g, 94 }} | {{Optimal ET sequence|legend=0| 41, 53g, 94 }} | ||
Badness ( | Badness (Sintel): 1.14 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 219: | Line 226: | ||
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=0| 41, 53g, 94 }} | {{Optimal ET sequence|legend=0| 41, 53g, 94 }} | ||
Badness ( | Badness (Sintel): 1.07 | ||
=== Andromeda === | === Andromeda === | ||
| Line 232: | Line 241: | ||
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: | ||
| Line 242: | Line 253: | ||
* 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=0| 12, 29, 41 | {{Optimal ET sequence|legend=0| 12, 29, 41 }} | ||
Badness ( | Badness (Sintel): 0.779 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 253: | Line 264: | ||
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: | ||
| Line 264: | Line 277: | ||
* 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=0| 12f, 29, 41 | {{Optimal ET sequence|legend=0| 12f, 29, 41 }} | ||
Badness ( | Badness (Sintel): 0.857 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 275: | Line 288: | ||
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=0| 12f, 29, 41 }} | {{Optimal ET sequence|legend=0| 12f, 29, 41 }} | ||
Badness ( | Badness (Sintel): 1.19 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 288: | Line 303: | ||
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=0| 12f, 29, 41 }} | {{Optimal ET sequence|legend=0| 12f, 29, 41 }} | ||
Badness ( | Badness (Sintel): 1.17 | ||
===== Schisicosiennic ===== | ===== Schisicosiennic ===== | ||
| Line 301: | Line 318: | ||
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=0| 12fg, 29g, 41, 70cd | {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }} | ||
Badness ( | Badness (Sintel): 1.11 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 314: | Line 333: | ||
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=0| 12fg, 29g, 41, 70cd | {{Optimal ET sequence|legend=0| 12fg, 29g, 41, 70cd }} | ||
Badness ( | Badness (Sintel): 1.09 | ||
===== Schisicosiennoid ===== | ===== Schisicosiennoid ===== | ||
| Line 327: | Line 348: | ||
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=0| 12f, 29g, 41g | {{Optimal ET sequence|legend=0| 12f, 29g, 41g }} | ||
Badness ( | Badness (Sintel): 1.06 | ||
====== 19-limit ====== | ====== 19-limit ====== | ||
| Line 340: | Line 363: | ||
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 353: | Line 378: | ||
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: | ||
| Line 359: | Line 386: | ||
: unchanged-interval (eigenmonzo) basis: 2.11/9 | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
{{Optimal ET sequence|legend=0| 12, 41e, 53, 118d | {{Optimal ET sequence|legend=0| 12, 41e, 53, 118d }} | ||
Badness ( | Badness (Sintel): 1.18 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 370: | Line 397: | ||
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: | ||
| Line 376: | Line 405: | ||
: unchanged-interval (eigenmonzo) basis: 2.11/9 | : unchanged-interval (eigenmonzo) basis: 2.11/9 | ||
{{Optimal ET sequence|legend=0| 12f, 41ef, 53, 118d | {{Optimal ET sequence|legend=0| 12f, …, 41ef, 53, 118d }} | ||
Badness ( | Badness (Sintel): 1.09 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 387: | Line 416: | ||
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=0| 12f | {{Optimal ET sequence|legend=0| 12f, 53, 65d, 118dg }} | ||
Badness ( | Badness (Sintel): 1.21 | ||
==== 19-limit ==== | ==== 19-limit ==== | ||
| Line 400: | Line 431: | ||
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=0| 12f, 53, 65d }} | |||
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, ~ | : mapping generators: ~2, ~11/9 | ||
Optimal | 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=0| | {{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=0| | {{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, ~ | : mapping generators: ~2, ~110/63 | ||
Optimal | 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=0| | {{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=0| | {{Optimal ET sequence|legend=0| 24d, 29, 53, 82e, 135eef }} | ||
Badness ( | Badness (Sintel): 1.13 | ||
=== Sanjaab === | === Sanjaab === | ||
| Line 470: | Line 511: | ||
: 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 483: | Line 526: | ||
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=0| 29, 65d, 94 }} | {{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 505: | Line 545: | ||
{{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }} | {{Mapping|legend=1| 1 0 15 -59 | 0 1 -8 39 }} | ||
[[Optimal tuning]] | [[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]]: | ||
| Line 519: | Line 563: | ||
* 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]] ( | [[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 532: | Line 576: | ||
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: | ||
| Line 540: | Line 586: | ||
{{Optimal ET sequence|legend=0| 53, 118, 289e, 407de }} | {{Optimal ET sequence|legend=0| 53, 118, 289e, 407de }} | ||
Badness ( | Badness (Sintel): 1.28 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 549: | Line 595: | ||
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: | ||
| Line 557: | Line 605: | ||
{{Optimal ET sequence|legend=0| 53, 118, 171e }} | {{Optimal ET sequence|legend=0| 53, 118, 171e }} | ||
Badness ( | Badness (Sintel): 1.39 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 566: | Line 614: | ||
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: | ||
| Line 572: | Line 622: | ||
: unchanged-interval (eigenmonzo) basis: 2.17/13 | : unchanged-interval (eigenmonzo) basis: 2.17/13 | ||
{{Optimal ET sequence|legend=0| 53, 118, 171e | {{Optimal ET sequence|legend=0| 53, 118, 171e }} | ||
Badness ( | Badness (Sintel): 1.47 | ||
==== Helena ==== | ==== Helena ==== | ||
| Line 583: | Line 633: | ||
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=0| 53, 118f, 171ef }} | {{Optimal ET sequence|legend=0| 53, 118f, 171ef }} | ||
Badness ( | Badness (Sintel): 1.50 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 596: | Line 648: | ||
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=0| 53, 118f, 171ef | {{Optimal ET sequence|legend=0| 53, 118f, 171ef }} | ||
Badness ( | Badness (Sintel): 1.56 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 609: | Line 663: | ||
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=0| 53, 118f, 171ef | {{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 624: | Line 680: | ||
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: | ||
| Line 630: | Line 688: | ||
: unchanged-interval (eigenmonzo) basis: 2.11/7 | : unchanged-interval (eigenmonzo) basis: 2.11/7 | ||
{{Optimal ET sequence|legend=0| 53, 171, 224 | {{Optimal ET sequence|legend=0| 53, 171, 224 }} | ||
Badness ( | Badness (Sintel): 1.61 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 641: | Line 699: | ||
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: | ||
| Line 649: | Line 709: | ||
{{Optimal ET sequence|legend=0| 53, 171, 224 }} | {{Optimal ET sequence|legend=0| 53, 171, 224 }} | ||
Badness ( | Badness (Sintel): 0.976 | ||
==== 17-limit ==== | ==== 17-limit ==== | ||
| Line 658: | Line 718: | ||
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: | ||
| Line 666: | Line 728: | ||
{{Optimal ET sequence|legend=0| 53, 171, 224, 395e, 619eg }} | {{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 677: | Line 739: | ||
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: | ||
| Line 683: | Line 747: | ||
: unchanged-interval (eigenmonzo) basis: 2.11 | : unchanged-interval (eigenmonzo) basis: 2.11 | ||
{{Optimal ET sequence|legend=0| 53e, 118, 289, 407d | {{Optimal ET sequence|legend=0| 53e, 118, 289, 407d }} | ||
Badness ( | Badness (Sintel): 1.64 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 694: | Line 758: | ||
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: | ||
| Line 700: | Line 766: | ||
: unchanged-interval (eigenmonzo) basis: 2.13/11 | : unchanged-interval (eigenmonzo) basis: 2.13/11 | ||
{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f | {{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }} | ||
Badness ( | Badness (Sintel): 1.87 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 711: | Line 777: | ||
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: | ||
| Line 717: | Line 785: | ||
: unchanged-interval (eigenmonzo) basis: 2.13/11 | : unchanged-interval (eigenmonzo) basis: 2.13/11 | ||
{{Optimal ET sequence|legend=0| 53e, 118, 171, 289f | {{Optimal ET sequence|legend=0| 53e, 118, 171, 289f }} | ||
Badness ( | Badness (Sintel): 1.51 | ||
==== Pontoid ==== | ==== Pontoid ==== | ||
| Line 728: | Line 796: | ||
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=0| 53ef, 118f, 171, 289 | {{Optimal ET sequence|legend=0| 53ef, 118f, 171, 289 }} | ||
Badness ( | Badness (Sintel): 2.07 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 741: | Line 811: | ||
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=0| 53ef, 118f, 171, 289, 460e, 749defg }} | {{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 757: | Line 829: | ||
: mapping generators: ~99/70, ~3 | : mapping generators: ~99/70, ~3 | ||
Optimal | 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 ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }} | {{Optimal ET sequence|legend=0| 106, 118, 224, 342, 1592c, 1934ce, 2276cde, 2618cde, 2960cde }} | ||
Badness ( | Badness (Sintel): 0.484 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 770: | Line 844: | ||
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 | 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=0| 106, 118, 224, 566f, 790f }} | {{Optimal ET sequence|legend=0| 106, 118, 224, 566f, 790f }} | ||
Badness ( | Badness (Sintel): 1.25 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 783: | Line 859: | ||
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=0| 106g, 118, 224, 342, 566f }} | {{Optimal ET sequence|legend=0| 106g, 118, 224, 342, 566f }} | ||
Badness ( | Badness (Sintel): 1.38 | ||
==== Counterbipont ==== | ==== Counterbipont ==== | ||
| Line 796: | Line 874: | ||
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=0| 106f, 118f, 224, 342f, 566, 1356cf | {{Optimal ET sequence|legend=0| 106f, 118f, 224, 342f, 566, 1356cf }} | ||
Badness ( | Badness (Sintel): 1.06 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 809: | Line 889: | ||
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=0| 106fg, 118f, 224, 342f, 566 | {{Optimal ET sequence|legend=0| 106fg, 118f, 224, 342f, 566 }} | ||
Badness ( | Badness (Sintel): 1.29 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 822: | Line 904: | ||
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=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }} | {{Optimal ET sequence|legend=0| 106fgh, 118f, 224, 342f, 566h, 908fgh }} | ||
Badness ( | Badness (Sintel): 1.35 | ||
==== Quadrapont ==== | ==== Quadrapont ==== | ||
| Line 836: | Line 920: | ||
: mapping generators: ~208/175, ~3 | : mapping generators: ~208/175, ~3 | ||
Optimal | 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=0| 224, 460, 684, 2276cde, 2960cde | {{Optimal ET sequence|legend=0| 224, 460, 684, 2276cde, 2960cde }} | ||
Badness ( | 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 852: | Line 938: | ||
: mapping generators: ~2, ~3 | : mapping generators: ~2, ~3 | ||
[[Optimal tuning]] | [[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]]: | ||
| Line 858: | Line 948: | ||
* [[9-odd-limit]] unchanged-interval (eigenmonzo) basis: 2.9/7 | * [[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 869: | Line 959: | ||
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=0| 12 | {{Optimal ET sequence|legend=0| 12, 65e, 77, 89, 166c }} | ||
Badness ( | Badness (Sintel): 1.62 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 882: | Line 974: | ||
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=0| 12f | {{Optimal ET sequence|legend=0| 12f, 65ef, 77, 166cf }} | ||
Badness ( | Badness (Sintel): 1.56 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
| Line 895: | Line 989: | ||
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=0| 12f | {{Optimal ET sequence|legend=0| 12f, 77, 89f, 166cf }} | ||
Badness ( | Badness (Sintel): 1.52 | ||
===== 19-limit ===== | ===== 19-limit ===== | ||
| Line 908: | Line 1,004: | ||
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=0| 12f | {{Optimal ET sequence|legend=0| 12f, 77, 166cf }} | ||
Badness ( | Badness (Sintel): 1.40 | ||
==== Grackloid ==== | ==== Grackloid ==== | ||
| Line 921: | Line 1,019: | ||
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=0| 12 | {{Optimal ET sequence|legend=0| 12, 77, 166c }} | ||
Badness ( | Badness (Sintel): 2.00 | ||
=== Grack === | === Grack === | ||
| Line 934: | Line 1,034: | ||
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=0| 12, 53d, 65, 77e | {{Optimal ET sequence|legend=0| 12, 53d, 65, 77e }} | ||
Badness ( | Badness (Sintel): 1.85 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
| Line 947: | Line 1,049: | ||
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=0| 12f, 53dff, 65f, 77e }} | {{Optimal ET sequence|legend=0| 12f, 53dff, 65f, 77e }} | ||
Badness ( | Badness (Sintel): 1.84 | ||
==== Catahelenic ==== | ==== Catahelenic ==== | ||
| Line 960: | Line 1,064: | ||
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=0| 12f, …, 53d, 65 }} | |||
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]] | [[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}} | |||
: error map: {{val| 0.0000 +0.0065 -2.0054 -0.2437 }} | |||
{{Optimal ET sequence|legend=1| 53, 147d, 200, 253, 306c, 559c }} | |||
[[Badness]] (Sintel): 5.04 | |||
[[Badness]] ( | |||
=== 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=0| | {{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=0| | {{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 | {{Optimal ET sequence|legend=0| 12, 118, 130, 248g }} | ||
Badness (Sintel): 1.49 | |||
==== Bischis ==== | |||
==== | |||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 351/350, 364/363, 441/440, 3136/3125 | ||
Mapping: {{mapping| 2 1 | Mapping: {{mapping| 2 0 30 69 102 131 | 0 1 -8 -20 -30 -39 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130 }} | ||
Badness ( | Badness (Sintel): 1.21 | ||
===== 17-limit ===== | ===== 17-limit ===== | ||
Subgroup: 2.3.5.7.11.13.17 | Subgroup: 2.3.5.7.11.13.17 | ||
Comma list: | Comma list: 221/220, 289/288, 351/350, 441/440, 3136/3125 | ||
Mapping: {{mapping| 2 1 | Mapping: {{mapping| 2 0 30 69 102 131 5 | 0 1 -8 -20 -30 -39 1 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 12f, 106deff, 118f, 130, 248fg }} | ||
Badness ( | 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 | {{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 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 245/243, 32805/32768 | ||
{{Mapping|legend=1| 1 1 7 -1 | 0 2 -16 13 }} | |||
: mapping generators: ~2, ~128/105 | |||
Optimal tuning | [[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= | {{Optimal ET sequence|legend=1| 17, 24, 41, 106d, 147d, 188cd }} | ||
Badness ( | [[Badness]] (Sintel): 2.03 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 243/242, 245/242, 385/384 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 7 -1 2 | 0 2 -16 13 5 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 17, 24, 41, 106d }} | ||
Badness ( | Badness (Sintel): 1.30 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 105/104, 144/143, 243/242, 245/242 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 1 1 7 -1 2 4 | 0 2 -16 13 5 -1 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 17, 24, 41 }} | ||
Badness ( | 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 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: | [[Comma list]]: 6144/6125, 19683/19600 | ||
{{Mapping|legend=1| 1 | {{Mapping|legend=1| 1 0 15 -17 | 0 2 -16 25 }} | ||
: mapping generators: ~2, ~140/81 | |||
[[Optimal tuning]] | [[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| | {{Optimal ET sequence|legend=1| 24, 53, 130, 183, 313 }} | ||
[[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=0| | {{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=0| | {{Optimal ET sequence|legend=0| 53, 130, 183, 313 }} | ||
Badness ( | 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 | {{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 | {{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 tempers together the syntonic and Pythagorean commas and equates it with a stack of three [[marvel comma]]s. A [[septimal comma]] is then found as a stack of four marvel commas. In certain 7-limit adaptive-tuning practice, the marvel comma corresponds to a melodic unit called a [[kleisma #As an interval region|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|legend=1| 3 0 45 94 | 0 1 -8 -18 }} | |||
: mapping generators: ~63/50, ~3 | |||
{{ | [[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 }} | |||
[[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]] (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 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 441/440, 4375/4356, 32805/32768 | ||
Mapping: {{mapping| 1 | Mapping: {{mapping| 3 0 45 94 134 | 0 1 -8 -18 -26 }} | ||
Optimal | 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=0| | {{Optimal ET sequence|legend=0| 12, …, 159, 330 }} | ||
Badness ( | Badness (Sintel): 1.97 | ||
=== 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, 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=0| | {{Optimal ET sequence|legend=0| 12f, …, 159, 330 }} | ||
Badness ( | 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: {{mapping| 3 0 45 94 134 168 -2 | 0 1 -8 -18 -26 -33 3 }} | |||
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=0| 12f, 159, 171, 330 }} | ||
Badness (Sintel): 1.38 | |||
{{ | === Terminator === | ||
Terminator tempers out 540/539, and may be described as {{nowrap| 171 & 183 }}. | |||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 540/539, | Comma list: 540/539, 32805/32768, 137781/137500 | ||
Mapping: {{mapping| | Mapping: {{mapping| 3 0 45 94 -137 | 0 1 -8 -18 31 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 537, 891de }} | ||
Badness ( | Badness (Sintel): 2.21 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: 540/539, 729/728, | Comma list: 540/539, 729/728, 4096/4095, 31250/31213 | ||
Mapping: {{mapping| | Mapping: {{mapping| 3 0 45 94 -137 -103 | 0 1 -8 -18 31 24 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 12e, 171, 183, 354, 891de }} | ||
Badness ( | 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 | |||
Mapping: {{mapping| | |||
Optimal | 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, | {{Optimal ET sequence|legend=0| 12, …, 330e, 342, 1380, 1722, 2064, 2406c, 5154bccdde }} | ||
Badness ( | Badness (Sintel): 0.973 | ||
==== 13-limit ==== | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 1716/1715, 2080/2079, 32805/32768, 34398/34375 | ||
Mapping: {{mapping| | Mapping: {{mapping| 6 0 90 188 287 355 | 0 1 -8 -18 -28 -35 }} | ||
Optimal | 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, | {{Optimal ET sequence|legend=0| 12f, 330eff, 342f, 696f }} * | ||
<nowiki>*</nowiki> optimal patent val: [[354edo|354]] | |||
Badness (Sintel): 1.85 | |||
=== Hemiterm === | |||
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 ( | 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 | [[Subgroup]]: 2.3.5.7 | ||
Comma list: | [[Comma list]]: 32805/32768, 367653125/362797056 | ||
{{Mapping|legend=1| 3 0 45 -32 | 0 2 -16 17 }} | |||
: mapping generators: ~ | : mapping generators: ~1536/1225, ~34300/19683 | ||
Optimal tuning | [[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= | {{Optimal ET sequence|legend=1| 24, 135, 159, 612ccdd }} | ||
Badness ( | [[Badness]] (Sintel): 10.7 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 385/384, 14700/14641, 19712/19683 | ||
Mapping: {{mapping| | Mapping: {{mapping| 3 0 45 -32 8 | 0 2 -16 17 1 }} | ||
Optimal | 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| | {{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 | {{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 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[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 ( | 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. See also [[daemotertiaschis]] which is made using every other generator of tertiaschis. | |||
[[Subgroup]]: 2.3.5.7 | [[Subgroup]]: 2.3.5.7 | ||
[[Comma list]]: 32805/32768, | [[Comma list]]: 32805/32768, 1071875/1062882 | ||
{{Mapping|legend=1| | {{Mapping|legend=1| 1 2 -1 10 | 0 -3 24 -52 }} | ||
[[Optimal tuning]] | [[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 ET sequence|legend=1| | {{Optimal ET sequence|legend=1| 65, 94, 159, 253, 412cd }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 5.36 | ||
=== 11-limit === | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: 385/384, | Comma list: 385/384, 4000/3993, 19712/19683 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 2 -1 10 0 | 0 -3 24 -52 25 }} | ||
Optimal | Optimal tunings: | ||
* WE: ~2 = 1200.3379{{c}}, ~11/10 = 166.0638{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~11/10 = 166.0167{{c}} | |||
{{Optimal ET sequence|legend=0| | {{Optimal ET sequence|legend=0| 65, 94, 159, 253, 412cd, 665ccde }} | ||
Badness ( | Badness (Sintel): 2.07 | ||
=== 13-limit === | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 325/324, 385/384, 1575/1573, 10985/10976 | ||
Mapping: {{mapping| | Mapping: {{mapping| 1 2 -1 10 0 12 | 0 -3 24 -52 25 -60 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 65f, 94, 159, 253, 412cdf, 665ccdef }} | ||
Badness ( | 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 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 32805/32768, 244140625/243045684 | ||
{{Mapping|legend=1| 1 2 -1 -12 | 0 -3 24 107 }} | |||
Optimal tuning | [[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 ET sequence|legend= | {{Optimal ET sequence|legend=1| 65d, 159, 224, 383, 607 }} | ||
Badness ( | [[Badness]] (Sintel): 4.76 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 3025/3024, 4000/3993, 32805/32768 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 2 -1 -12 0 | 0 -3 24 107 25 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }} | ||
Badness ( | Badness (Sintel): 1.62 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 625/624, 1575/1573, 2080/2079, 10985/10976 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 2 -1 -12 0 -10 | 0 -3 24 107 25 99 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 65d, 159, 224, 383, 607 }} | ||
Badness ( | 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 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 | |||
Comma list: 243/242, 441/440, 16384/16335 | |||
Mapping: {{mapping| 1 1 7 5 2 | 0 4 -32 -15 10 }} | |||
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 (Sintel): 0.969 | |||
==== 13-limit ==== | |||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 243/242, 364/363, 441/440, 3584/3575 | |||
Mapping: {{mapping| 1 1 7 5 2 -2 | 0 4 -32 -15 10 39 }} | |||
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 (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: {{mapping| 1 1 7 5 2 -2 0 | 0 4 -32 -15 10 39 28 }} | |||
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 (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: {{mapping| 1 1 7 5 2 -2 0 6 | 0 4 -32 -15 10 39 28 -12 }} | |||
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 (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: {{mapping| 1 1 7 5 2 -2 -6 | 0 4 -32 -15 10 39 69 }} | |||
== | 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 (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: {{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 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 | |||
[[Comma list]]: 32805/32768, 420175/419904 | |||
= | {{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| 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: {{mapping| 1 1 7 0 1 | 0 5 -40 24 21 }} | |||
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=0| 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: {{mapping| 1 1 7 0 1 3 | 0 5 -40 24 21 6 }} | |||
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=0| 17, 77, 94, 171e }} | |||
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 | |||
{{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 | {{Optimal ET sequence|legend=1| 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: {{mapping| 1 0 15 12 -7 | 0 5 -40 -29 33 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9709{{c}}, ~56/45 = 380.3423{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3517{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 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: {{mapping| 1 0 15 12 -7 -15 | 0 5 -40 -29 33 59 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1199.9663{{c}}, ~56/45 = 380.3403{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~56/45 = 380.3509{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 41, 142, 183, 224, 631d, 855d }} | ||
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 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 32805/32768, 9765625/9680832 | ||
{{Mapping|legend=1| 1 2 -1 -4 | 0 -5 40 82 }} | |||
: mapping generators: ~2, ~625/588 | |||
Optimal tuning | [[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= | {{Optimal ET sequence|legend=1| 12, …, 253, 265 }} | ||
Badness ( | [[Badness]] (Sintel): 6.43 | ||
=== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 1375/1372, 4375/4356, 32805/32768 | ||
Mapping: {{mapping| 1 2 -1 - | Mapping: {{mapping| 1 2 -1 -4 -7 | 0 -5 40 82 126 }} | ||
Optimal | 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, | {{Optimal ET sequence|legend=0| 12, …, 253, 265, 518c }} | ||
Badness ( | Badness (Sintel): 3.74 | ||
=== | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 1375/1372, 2080/2079, 4375/4356, 10648/10647 | ||
Mapping: {{mapping| 1 2 -1 - | Mapping: {{mapping| 1 2 -1 -4 -7 -9 | 0 -5 40 82 126 153 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 12f, …, 241cdef, 253 }} | ||
Badness ( | 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 | |||
Comma list: 375/374, 400/399, 495/494, 595/594, 1375/1372, 3978/3971 | |||
Mapping: {{mapping| 1 2 -1 -4 -7 -9 5 4 | 0 -5 40 82 126 153 -11 3 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0713{{c}}, ~18/17 = 99.6208{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~18/17 = 99.6152{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 12f, 253, 265 }} | ||
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: | [[Comma list]]: 32805/32768, 49009212/48828125 | ||
{{Mapping|legend=1| 1 2 -1 -5 | 0 -5 40 94 }} | |||
Optimal tuning | [[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= | {{Optimal ET sequence|legend=1| 12, …, 289, 301, 590, 891, 1192 }} | ||
Badness ( | [[Badness]] (Sintel): 3.36 | ||
== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 441/440, 32805/32768, 1953125/1951488 | |||
Mapping: {{mapping| 1 2 -1 -5 -8 | 0 -5 40 94 138 }} | |||
{{ | 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=0| 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: {{mapping| 1 2 -1 -5 -8 -11 | 0 -5 40 94 138 177 }} | |||
Optimal tunings: | |||
* WE: ~2 = 1200.0625{{c}}, ~35/33 = 99.6630{{c}} | |||
* CWE: ~2 = 1200.0000{{c}}, ~35/33 = 99.6583{{c}} | |||
Optimal | {{Optimal ET sequence|legend=0| 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: {{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 | {{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 | Subgroup: 2.3.5.7.11 | ||
Comma list: | Comma list: 5632/5625, 8019/8000, 151263/151250 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 2 -1 -5 -9 | 0 -5 40 94 150 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 12, …, 289e, 301, 915 }} | ||
Badness ( | Badness (Sintel): 2.72 | ||
=== 13-limit === | ==== 13-limit ==== | ||
Subgroup: 2.3.5.7.11.13 | Subgroup: 2.3.5.7.11.13 | ||
Comma list: | Comma list: 847/845, 1716/1715, 5632/5625, 8019/8000 | ||
Mapping: {{mapping| 1 1 | Mapping: {{mapping| 1 2 -1 -5 -9 -11 | 0 -5 40 94 150 177 }} | ||
Optimal | 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| | {{Optimal ET sequence|legend=0| 12f, …, 289e, 301 }} | ||
Badness ( | 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 | 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=0| | {{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 | 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=0| | {{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 | {{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 | |||
Subgroup: 2.3.5.7 | |||
Comma list: | [[Comma list]]: 32805/32768, 235298/234375 | ||
{{Mapping|legend=1| 1 2 -1 -1 | 0 -6 48 55 }} | |||
: mapping generators: ~2, ~21/20 | |||
Optimal tuning | [[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= | {{Optimal ET sequence|legend=1| 29, 72cd, 101, 130, 289, 419 }} | ||
Badness ( | [[Badness]] (Sintel): 2.75 | ||
== | === 11-limit === | ||
Subgroup: 2.3.5.7.11 | |||
Comma list: 441/440, 4000/3993, 235298/234375 | |||
Mapping: {{mapping| 1 2 -1 -1 0 | 0 -6 48 55 50 }} | |||
{{ | 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=0| 29, 72cde, 101e, 130, 289 }} | |||
Badness (Sintel): 1.50 | |||
[[Badness]] ( | === 13-limit === | ||
Subgroup: 2.3.5.7.11.13 | |||
Comma list: 364/363, 441/440, 676/675, 10985/10976 | |||
Mapping: {{mapping| 1 2 -1 -1 0 1 | 0 -6 48 55 50 39 }} | |||
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=0| 29, 72cdef, 101e, 130, 289 }} | |||
Badness (Sintel): 1.04 | |||
== 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 | |||
[[Comma list]]: 32805/32768, 516560652/514714375 | |||
{{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 }} | |||
[[Badness]] (Sintel): 2.81 | |||
=== 11-limit === | === 11-limit === | ||
| Line 2,233: | Line 2,568: | ||
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=0| 77, 147, 224, 301, 525 }} | {{Optimal ET sequence|legend=0| 77, 147, 224, 301, 525 }} | ||
Badness ( | Badness (Sintel): 1.46 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,246: | Line 2,583: | ||
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=0| 77, 147, 224, 525 }} | {{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 2,262: | Line 2,601: | ||
: mapping generators: ~42875/39366, ~3 | : mapping generators: ~42875/39366, ~3 | ||
[[Optimal tuning]] | [[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 2,275: | Line 2,618: | ||
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=0| 24, 224, 472, 696, 1168 }} | {{Optimal ET sequence|legend=0| 24, …, 224, 472, 696, 1168 }} | ||
Badness ( | Badness (Sintel): 1.48 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,288: | Line 2,633: | ||
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=0| 24, 224, 472, 696 }} | {{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 2,304: | Line 2,651: | ||
: mapping generators: ~2592/2401, ~3 | : mapping generators: ~2592/2401, ~3 | ||
[[Optimal tuning]] | [[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 }} | {{Optimal ET sequence|legend=1| 36, 99c, 135, 171, 2772bd, 2943bdd, …, 5166bccddd, 5337bccddd }} | ||
[[Badness]] ( | [[Badness]] (Sintel): 1.77 | ||
=== 11-limit === | === 11-limit === | ||
| Line 2,317: | Line 2,668: | ||
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 | {{Optimal ET sequence|legend=0| 36, 135, 171 }} | ||
Badness ( | Badness (Sintel): 4.20 | ||
=== 13-limit === | === 13-limit === | ||
| Line 2,330: | Line 2,683: | ||
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 }} | {{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 | {{Mapping|legend=1| 1 -4 47 6 | 0 7 56 -4 }} | ||
: mapping generators: ~2, ~ | : 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, | {{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 | |||
== Subgroup extensions == | == Subgroup extensions == | ||
=== 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 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) === | === Photia (2.3.5.17) === | ||
{{See also| No-elevens subgroup temperaments #Garibaldia }} | {{See also| No-elevens subgroup temperaments #Garibaldia }} | ||
| Line 2,378: | Line 2,787: | ||
: mapping generators: ~2, ~3 | : mapping generators: ~2, ~3 | ||
[[Optimal tuning]] | [[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 }} | {{Optimal ET sequence|legend=1| 12, 41, 53, 65, 207g, 272gg }} | ||
[[ | [[Badness]] (Sintel): 0.479 | ||
==== 2.3.5.17.19 subgroup ==== | ==== 2.3.5.17.19 subgroup ==== | ||
| Line 2,393: | Line 2,806: | ||
Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }} | Gencom mapping: {{mapping| 1 0 15 0 0 0 -7 9 | 0 1 -8 0 0 0 7 -3 }} | ||
Optimal | 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 }} | {{Optimal ET sequence|legend=0| 12, 41, 53, 65, 142g }} | ||
Badness (Sintel): 0.332 | |||
=== Nestoria (2.3.5.19) === | === Nestoria (2.3.5.19) === | ||
: ''See also: [[No-elevens subgroup temperaments #Garibaldia]] and [[No-elevens subgroup temperaments #Pontia|#Pontia]]'' | : ''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 (especially the minor triad ~16:19:24 which is equated with the Pythagorean minor triad), as [[171edo]] is already arguably undertempered for it despite being the optimal patent val. | |||
[[Subgroup]]: 2.3.5.19 | [[Subgroup]]: 2.3.5.19 | ||
| Line 2,413: | Line 2,828: | ||
: mapping generators: ~2, ~3 | : mapping generators: ~2, ~3 | ||
[[Optimal tuning]] | [[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 }} | {{Optimal ET sequence|legend=1| 12, 29, 41, 53, 118, 171, 460hh, 631hh }} | ||
[[ | [[Badness]] (Sintel): 0.126 | ||
=== Taylor (2.3.5.13) === | === Taylor (2.3.5.13) === | ||
| Line 2,431: | Line 2,850: | ||
: mapping generators: ~2, ~26/15 | : mapping generators: ~2, ~26/15 | ||
[[Optimal tuning]] | [[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 }} | {{Optimal ET sequence|legend=1| 24, 53, 130, 183, 236, 525f, 761ff }} | ||
[[ | [[Badness]] (Sintel): 0.334 | ||
==== Dakota (2.3.5.13.19) ==== | ==== Dakota (2.3.5.13.19) ==== | ||
| Line 2,444: | Line 2,867: | ||
Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }} | Subgroup-val mapping: {{mapping| 1 0 15 14 9 | 0 2 -16 -13 -6 }} | ||
Optimal | 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 }} | {{Optimal ET sequence|legend=0| 24, 29, 53, 130, 183, 236h, 289h }} | ||
Badness ( | Badness (Sintel): 0.262 | ||
===== 2.3.5.13.19.37 subgroup ===== | ===== 2.3.5.13.19.37 subgroup ===== | ||
| Line 2,457: | Line 2,882: | ||
Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }} | Subgroup-val mapping: {{mapping| 1 0 15 14 9 6 | 0 2 -16 -13 -6 -1 }} | ||
Optimal | 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 }} | {{Optimal ET sequence|legend=0| 24, 29, 53, 183, 236h, 289hl, 631fhhll }} | ||
Badness ( | Badness (Sintel): 0.223 | ||
=== Quintilischis (2.3.5.17) === | === Quintilischis (2.3.5.17) === | ||
| Line 2,475: | Line 2,902: | ||
: mapping generators: ~2, ~18/17 | : mapping generators: ~2, ~18/17 | ||
[[Optimal tuning]] | [[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 }} | {{Optimal ET sequence|legend=1| 12, …, 253, 265, 277, 289, 566g, 855g }} | ||
[[ | [[Badness]] (Sintel): 1.34 | ||
==== 2.3.5.17.19 subgroup ==== | ==== 2.3.5.17.19 subgroup ==== | ||
| Line 2,490: | Line 2,921: | ||
Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }} | Gencom mapping: {{mapping| 1 2 -1 0 0 0 5 4 | 0 -5 40 0 0 0 -11 3 }} | ||
Optimal | 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=0| 12, 253, 265, 277, 289 }} | {{Optimal ET sequence|legend=0| 12, …, 253, 265, 277, 289 }} | ||
Badness (Sintel): 1.17 | |||
[[Category:Schismatic family| ]] <!-- main article --> | |||
[[Category:Temperament families]] | [[Category:Temperament families]] | ||
[[Category: | [[Category:Catalogs of rank-2 temperaments]] | ||