Quartonic family: Difference between revisions

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Temperaments of the '''quartonic family''' temper out {{monzo| 3 -18 11 }} = 390625000/387420489 (saleyo comma).
{{Technical data page}}
The '''quartonic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quartonic comma]] ({{monzo|legend=1| 3 -18 11 }}, [[ratio]]: 390 625 000 / 387 420 489).


== Quartonic (27 & 53) ==
== Quartonic ==
The name "quartonic" means quarter-tone, which is the generator of this temperament.
The name ''quartonic'' refers to the [[quartertone]], since a somewhat flattened quartertone (representing the triptolemaic chromatic semitone, i.e. [[porcupine comma]]) is the [[generator]] of this temperament. The [[ploidacot]] for the temperament is omega-hendecacot, with eleven generator steps giving the [[4/3|perfect fourth]]. It is a member of the [[schismic–Mercator equivalence continuum]], with equivalence number ''n'' = 11/2.  


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 8: Line 9:
[[Comma list]]: 390625000/387420489
[[Comma list]]: 390625000/387420489


[[Mapping]]: [{{val| 1 2 3 }}, {{val| 0 -11 -18 }}]
{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
: mapping generators: ~2, ~250/243


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~250/243 = 45.2368
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.9709{{c}}, ~250/243 = 45.2318{{c}}
: [[error map]]: {{val| -0.029 +0.437 -0.574 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~250/243 = 45.2349{{c}}
: error map: {{val| 0.000 +0.461 -0.541 }}


{{Optimal ET sequence|legend=1| 26, 27, 53, 292, 345, 398, 451, 504, 955, 1459bc }}
{{Optimal ET sequence|legend=1| 26, 27, 53, 239, 292, 345, 398, 451 }}


[[Badness]]: 0.117250
[[Badness]] (Sintel): 2.75


=== Overview to extensions ===
=== Overview to extensions ===
Apart from the saleyo comma, we also consider other extensions. The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at.
The second comma of the [[Normal forms #Normal forms for commas|normal comma list]] defines which 7-limit family member we are looking at.
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35.
* [[1728/1715]] or [[4000/3969]] gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out [[4375/4374]].  
* [[Hemimage comma|10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* [[10976/10935]] gives yarman I (80 & 159) and slices the quartonic generator in three.
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.  
* 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.  
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[2401/2400]] gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
* [[Landscape comma|250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[250047/250000]] gives triquart (27 & 159) with 1/3-octave period.
* [[Dimcomp comma|390625/388962]] or [[Canousma|4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[390625/388962]] or [[4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[Mirkwai comma|16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.
* [[16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.


== Septimal quartonic ==
== Septimal quartonic ==
Septimal quartonic tempers out the [[orwellisma]], the [[octagar comma]], as well as the [[ragisma]], and may be described as the {{nowrap| 26 & 27 }} temperament.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 33: Line 41:
{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}
{{Mapping|legend=1| 1 2 3 3 | 0 -11 -18 -5 }}


{{Multival|legend=1| 11 18 5 3 -23 -39 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~2 = 1199.1424{{c}}, ~36/35 = 45.1064{{c}}
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~36/35 = 45.2652
: [[error map]]: {{val| -0.858 +0.160 -0.801 +3.069 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~36/35 = 45.1881{{c}}
: error map: {{val| 0.000 +0.976 +0.301 +5.234 }}


{{Optimal ET sequence|legend=1| 26, 27, 53 }}
{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}


[[Badness]]: 0.042632
[[Badness]] (Sintel): 1.08


=== 11-limit ===
=== 11-limit ===
Line 48: Line 58:
Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}
Mapping: {{mapping| 1 2 3 3 5 | 0 -11 -18 -5 -41 }}


Optimal tuning (CTE): ~2 = 1＼1, ~36/35 = 45.1674
Optimal tunings:
* WE: ~2 = 1198.8787{{c}}, ~36/35 = 44.9994{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.0863{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26e, 27e, 53, 133d, 186d }}
{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80 }}


Badness: 0.034031
Badness (Sintel): 1.13


==== 13-limit ====
==== 13-limit ====
Line 61: Line 73:
Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}
Mapping: {{mapping| 1 2 3 3 5 4 | 0 -11 -18 -5 -41 -8 }}


Optimal tuning (CTE): ~2 = 1＼1, ~36/35 = 45.1632
Optimal tunings:
* WE: ~2 = 1199.2991{{c}}, ~36/35 = 45.0539{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.1049{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26e, 27e, 53, 133d, 186d }}
{{Optimal ET sequence|legend=0| 26e, 27e, 53, 80, 133d }}


Badness: 0.023875
Badness (Sintel): 0.987


=== Quarto ===
=== Quarto ===
Line 74: Line 88:
Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -11 -18 -5 -14 }}


Optimal tuning (CTE): ~2 = 1＼1, ~36/35 = 45.4022
Optimal tunings:
* WE: ~2 = 1198.3094{{c}}, ~36/35 = 45.0688{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2329{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 53e, 132ee }}
{{Optimal ET sequence|legend=0| 26, 27e, 53e, 80ee }}


Badness: 0.041786
Badness (Sintel): 1.38


==== 13-limit ====
==== 13-limit ====
Line 87: Line 103:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -11 -18 -5 -14 -8 }}


Optimal tuning (CTE): ~2 = 1＼1, ~36/35 = 45.3857
Optimal tunings:
* WE: ~2 = 1198.9315{{c}}, ~36/35 = 45.1650{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~36/35 = 45.2640{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 53e, 132ee }}
{{Optimal ET sequence|legend=0| 26, 27e, 53e }}


Badness: 0.027692
Badness (Sintel): 1.14


=== Quartz ===
=== Quartz ===
Line 100: Line 118:
Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}
Mapping: {{mapping| 1 2 3 3 3 | 0 -11 -18 -5 12 }}


Optimal tuning (CTE): ~2 = 1＼1, ~33/32 = 45.3313
Optimal tunings:
* WE: ~2 = 1200.5082{{c}}, ~33/32 = 45.4047{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 53 }}
{{Optimal ET sequence|legend=0| 26, 53 }}


Badness: 0.053285
Badness (Sintel): 1.76


==== 13-limit ====
==== 13-limit ====
Line 113: Line 133:
Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}
Mapping: {{mapping| 1 2 3 3 3 4 | 0 -11 -18 -5 12 -8 }}


Optimal tuning (CTE): ~2 = 1＼1, ~33/32 = 45.3168
Optimal tunings:
* WE: ~2 = 1200.5899{{c}}, ~33/32 = 45.4096{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~33/32 = 45.3739{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 53 }}
{{Optimal ET sequence|legend=0| 26, 53 }}


Badness: 0.028818
Badness (Sintel): 1.19


=== Biquartonic ===
=== Biquartonic ===
Line 125: Line 147:


Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
: mapping generators: ~99/70, ~36/35


Optimal tuning (CTE): ~99/70 = 1\2, ~36/35 = 45.2678
Optimal tunings:
* WE: ~99/70 = 599.5389{{c}}, ~36/35 = 45.0947{{c}}
* CWE: ~99/70 = 600.0000{{c}}, ~36/35 = 45.1692{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 186de, 266dde }}


Badness: 0.060737
Badness (Sintel): 2.01


==== 13-limit ====
==== 13-limit ====
Line 139: Line 164:
Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}
Mapping: {{mapping| 2 4 6 6 7 8 | 0 -11 -18 -5 -1 -8 }}


Optimal tuning (CTE): ~55/39 = 1\2, ~40/39 = 45.2544
Optimal tunings:
* WE: ~55/39 = 599.6870{{c}}, ~40/39 = 45.1293{{c}}
* CWE: ~55/39 = 600.0000{{c}}, ~40/39 = 45.1789{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 54c, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}


Badness: 0.039891
Badness (Sintel): 1.65


==== 17-limit ====
==== 17-limit ====
Line 152: Line 179:
Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}
Mapping: {{mapping| 2 4 6 6 7 8 9 | 0 -11 -18 -5 -1 -8 -11 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~34/33 = 45.2397
Optimal tunings:
* WE: ~17/12 = 599.7706{{c}}, ~40/39 = 45.1427{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~40/39 = 45.1790{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 54c, 80, 106, 292ddefg }}
{{Optimal ET sequence|legend=0| 26, 54c, 80, 106 }}


Badness: 0.028112
Badness (Sintel): 1.43


==== 19-limit ====
==== 19-limit ====
Line 165: Line 194:
Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}
Mapping: {{mapping| 2 4 6 6 7 8 9 10 | 0 -11 -18 -5 -1 -8 -11 -20 }}


Optimal tuning (CTE): ~17/12 = 1\2, ~39/38 = 45.222
Optimal tunings:
* WE: ~17/12 = 599.7848{{c}}, ~39/38 = 45.1288{{c}}
* CWE: ~17/12 = 600.0000{{c}}, ~39/38 = 45.1634{{c}}


Optimal ET sequence: {{Optimal ET sequence| 26, 54ch, 80, 106 }}
{{Optimal ET sequence|legend=0| 26, 54ch, 80, 106 }}


Badness: 0.0384524
Badness (Sintel): 1.30


=== Yarm ===
=== Yarm ===
Line 178: Line 209:
Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}
Mapping: {{mapping| 1 2 3 3 4 | 0 -33 -54 -15 -43 }}


Optimal tuning (CTE): ~2 = 1＼1, ~100/99 = 15.0880
Optimal tunings:
* WE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0363{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0617{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80, 239dd }}


Badness: 0.099950
Badness (Sintel): 3.30


==== 13-limit ====
==== 13-limit ====
Line 191: Line 224:
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}
Mapping: {{mapping| 1 2 3 3 4 4 | 0 -33 -54 -15 -43 -24 }}


Optimal tuning (CTE): ~2 = 1＼1, ~100/99 = 15.0842
Optimal tunings:
* WE: ~2 = 1199.5798{{c}}, ~100/99 = 15.0551{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0681{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80 }}


Badness: 0.061645
Badness (Sintel): 2.55


==== 17-limit ====
==== 17-limit ====
Line 204: Line 239:
Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}
Mapping: {{mapping| 1 2 3 3 4 4 4 | 0 -33 -54 -15 -43 -24 7 }}


Optimal tuning (CTE): ~2 = 1＼1, ~100/99 = 15.0840
Optimal tunings:
* WE: ~2 = 1199.6800{{c}}, ~100/99 = 15.0624{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0705{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
{{Optimal ET sequence|legend=0| 79, 80 }}


Badness: 0.046718
Badness (Sintel): 2.38


== Yarman I ==
== Yarman I ==
Yarman I is the first interpretation of [[Ozan Yarman]]'s proposed tuning for the Turkish makam.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 216: Line 255:


{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}
: mapping generators: ~2, ~126/125


{{Multival|legend=1| 33 54 95 9 58 69 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1199.8726{{c}}, ~126/125 = 15.0651{{c}}
: [[error map]]: {{val| -0.127 +0.641 -0.213 -0.523 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~126/125 = 15.0689{{c}}
: error map: {{val| 0.000 +0.771 -0.036 -0.374 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~126/125 = 15.0714
{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 637, 876b }}


{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 398, 637 }}
[[Badness]] (Sintel): 4.89
 
[[Badness]]: 0.193315


=== 11-limit ===
=== 11-limit ===
Line 232: Line 274:
Mapping: {{mapping| 1 2 3 4 4 | 0 -33 -54 -95 -43 }}
Mapping: {{mapping| 1 2 3 4 4 | 0 -33 -54 -95 -43 }}


Optimal tuning (CTE): ~2 = 1＼1, ~100/99 = 15.0724
Optimal tunings:
* WE: ~2 = 1199.8339{{c}}, ~100/99 = 15.0637{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0685{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398, 637, 1035bd }}
{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 876be }}


Badness: 0.049170
Badness (Sintel): 1.63


=== 13-limit ===
=== 13-limit ===
Line 245: Line 289:
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -33 -54 -95 -43 -24 }}
Mapping: {{mapping| 1 2 3 4 4 4 | 0 -33 -54 -95 -43 -24 }}


Optimal tuning (CTE): ~2 = 1＼1, ~91/90 = 15.0707
Optimal tunings:
* WE: ~2 = 1200.1080{{c}}, ~100/99 = 15.0766{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0737{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398f, 637ff }}
{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f }}


Badness: 0.040929
Badness (Sintel): 1.69


=== 17-limit ===
=== 17-limit ===
Line 258: Line 304:
Mapping: {{mapping| 1 2 3 4 4 4 4 | 0 -33 -54 -95 -43 -24 7 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 | 0 -33 -54 -95 -43 -24 7 }}


Optimal tuning (CTE): ~2 = 1＼1, ~91/90 = 15.0706
Optimal tunings:
* WE: ~2 = 1200.0222{{c}}, ~100/99 = 15.0718{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0713{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398f, 637ff }}
{{Optimal ET sequence|legend=0| 79d, 80, 159, 239, 398f }}


Badness: 0.031015
Badness (Sintel): 1.58


=== 19-limit ===
=== 19-limit ===
Line 271: Line 319:
Mapping: {{mapping| 1 2 3 4 4 4 4 5 | 0 -33 -54 -95 -43 -24 7 -60 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 5 | 0 -33 -54 -95 -43 -24 7 -60 }}


Optimal tuning (CTE): ~2 = 1＼1, ~91/90 = 15.0683
Optimal tunings:
* WE: ~2 = 1200.0698{{c}}, ~100/99 = 15.0722{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0706{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79dh, 80, 159, 239, 637ffh }}
{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 398fh }}


Badness: 0.023193
Badness (Sintel): 1.41


=== 23-limit ===
=== 23-limit ===
Line 284: Line 334:
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 | 0 -33 -54 -95 -43 -24 7 -60 -38 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 | 0 -33 -54 -95 -43 -24 7 -60 -38 }}


Optimal tuning (CTE): ~2 = 1＼1, ~91/90 = 15.0676
Optimal tunings:
* WE: ~2 = 1200.0946{{c}}, ~100/99 = 15.0732{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0710{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79dh, 80, 159, 239, 637ffhi }}
{{Optimal ET sequence|legend=0| 79dh, 80, 159, 239, 398fh }}


Badness: 0.017682
Badness (Sintel): 1.27


=== 29-limit ===
=== 29-limit ===
Line 297: Line 349:
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 6 | 0 -33 -54 -95 -43 -24 7 -60 -38 -91 }}
Mapping: {{mapping| 1 2 3 4 4 4 4 5 5 6 | 0 -33 -54 -95 -43 -24 7 -60 -38 -91 }}


Optimal tuning (CTE): ~2 = 1＼1, ~91/90 = 15.0667
Optimal tunings:
* WE: ~2 = 1200.0995{{c}}, ~100/99 = 15.0726{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.0703{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79dhj, 80, 159, 239 }}
{{Optimal ET sequence|legend=0| 79dhj, 80, 159, 239, 398fh }}


Badness: 0.014289
Badness (Sintel): 1.19


== Yarman II ==
== Yarman II ==
Yarman II is the other interpretation of [[Ozan Yarman]]'s proposed tuning for the Turkish makam.
[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7


Line 309: Line 365:


{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}
{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}
: mapping generators: ~2, ~875/864


{{Multival|legend=1| 33 54 -64 9 -194 -300 }}
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.4734{{c}}, ~875/864 = 15.1122{{c}}
: [[error map]]: {{val| +0.473 +0.291 -0.950 -0.701 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~875/864 = 15.1048{{c}}
: error map: {{val| 0.000 -0.412 -1.971 -2.121 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~875/864 = 15.0995
{{Optimal ET sequence|legend=1| 79, 159, 397cd, 556cd, 715ccdd }}


{{Optimal ET sequence|legend=1| 79, 159 }}
[[Badness]] (Sintel): 16.6
 
[[Badness]]: 0.655487


=== 11-limit ===
=== 11-limit ===
Line 325: Line 384:
Mapping: {{mapping| 1 2 3 2 4 | 0 -33 -54 64 -43 }}
Mapping: {{mapping| 1 2 3 2 4 | 0 -33 -54 64 -43 }}


Optimal tuning (CTE): ~2 = 1＼1, ~100/99 = 15.0982
Optimal tunings:
* WE: ~2 = 1200.4357{{c}}, ~100/99 = 15.1126{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1054{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
{{Optimal ET sequence|legend=0| 79, 159, 397cd, 556cd }}


Badness: 0.143477
Badness (Sintel): 4.74


=== 13-limit ===
=== 13-limit ===
Line 338: Line 399:
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -33 -54 64 -43 -24 }}
Mapping: {{mapping| 1 2 3 2 4 4 | 0 -33 -54 64 -43 -24 }}


Optimal tuning (CTE): ~2 = 1＼1, ~100/99 = 15.0952
Optimal tunings:
* WE: ~2 = 1200.5028{{c}}, ~100/99 = 15.1135{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1052{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
{{Optimal ET sequence|legend=0| 79, 159, 397cdff, 556cdff }}


Badness: 0.068150
Badness (Sintel): 2.82


=== 17-limit ===
=== 17-limit ===
Line 351: Line 414:
Mapping: {{mapping| 1 2 3 2 4 4 4 | 0 -33 -54 64 -43 -24 7 }}
Mapping: {{mapping| 1 2 3 2 4 4 4 | 0 -33 -54 64 -43 -24 7 }}


Optimal tuning (CTE): ~2 = 1＼1, ~100/99 = 15.0950
Optimal tunings:
* WE: ~2 = 1200.3840{{c}}, ~100/99 = 15.1086{{c}}
* CWE: ~2 = 1200.0000{{c}}, ~100/99 = 15.1025{{c}}


Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
{{Optimal ET sequence|legend=0| 79, 159 }}


Badness: 0.051019
Badness (Sintel): 2.60


== Tertiseptisix ==
== Tertiseptisix ==
Line 362: Line 427:
[[Comma list]]: 2401/2400, 390625000/387420489
[[Comma list]]: 2401/2400, 390625000/387420489


{{Mapping|legend=1| 1 13 21 15 | 0 -44 -72 -47 }}
{{Mapping|legend=1| 1 -31 -51 -32 | 0 44 72 47 }}
 
: mapping generators: ~2, ~1458/875
{{Multival|legend=1| 44 72 47 12 -49 -93 }}


[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~875/729 = 311.308
[[Optimal tuning]]s:
* [[WE]]: ~2 = 1200.0156{{c}}, ~1458/875 = 888.7028{{c}}
: [[error map]]: {{val| +0.016 +0.485 -0.507 -0.293 }}
* [[CWE]]: ~2 = 1200.0000{{c}}, ~1458/875 = 888.6915{{c}}
: error map: {{val| 0.000 +0.472 -0.523 -0.324 }}


{{Optimal ET sequence|legend=1| 27, 131bccd, 158cd, 185c, 212, 239, 451 }}
{{Optimal ET sequence|legend=1| 27, …, 212, 239, 451 }}


[[Badness]]: 0.155952
[[Badness]] (Sintel): 3.95


== Triquart ==
== Triquart ==
Line 378: Line 446:


{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}
{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}
: mapping generators: ~63/50, ~250/243


{{Multival|legend=1| 33 54 42 9 -26 -54 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~63/50 = 400.0521{{c}}, ~250/243 = 45.2373{{c}}
[[Optimal tuning]] ([[CTE]]): ~63/50 = 1＼3, ~250/243 = 45.2083
: [[error map]]: {{val| +0.156 +0.747 -0.117 -1.628 }}
* [[CWE]]: ~63/50 = 400.0000{{c}}, ~250/243 = 45.2204{{c}}
: error map: {{val| 0.000 +0.621 -0.281 -1.911 }}


{{Optimal ET sequence|legend=1| 27, 105cd, 132d, 159, 186, 345d }}
{{Optimal ET sequence|legend=1| 27, 105cd, 132d, 159, 345d, 506d }}


[[Badness]]: 0.170062
[[Badness]] (Sintel): 4.30


== Quartiquart ==
== Quartiquart ==
Line 393: Line 464:


{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}
{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}
: mapping generators: ~25/21, ~250/243


{{Multival|legend=1| 44 72 100 12 35 30 }}
[[Optimal tuning]]s:
 
* [[WE]]: ~25/21 = 299.9955{{c}}, ~250/243 = 45.2382{{c}}
[[Optimal tuning]] ([[CTE]]): ~25/21 = 1＼4, ~250/243 = 45.2411
: [[error map]]: {{val| -0.018 +0.389 -0.655 +0.152 }}
* [[CWE]]: ~25/21 = 300.0000{{c}}, ~250/243 = 45.2400{{c}}
: error map: {{val| 0.000 +0.405 -0.634 +0.174 }}


{{Optimal ET sequence|legend=1| 80, 132d, 212, 292, 504 }}
{{Optimal ET sequence|legend=1| 80, 132d, 212, 292, 504 }}


[[Badness]]: 0.199116
[[Badness]] (Sintel): 5.04


=== 11-limit ===
=== 11-limit ===
Line 409: Line 483:
Mapping: {{mapping| 4 8 12 15 17 | 0 -11 -18 -25 -21 }}
Mapping: {{mapping| 4 8 12 15 17 | 0 -11 -18 -25 -21 }}


Optimal tuning (CTE): ~25/21 = 1＼4, ~77/75 = 45.2303
Optimal tunings:
* WE: ~25/21 = 300.0140{{c}}, ~77/75 = 45.2398{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~77/75 = 45.2342{{c}}


Optimal ET sequence: {{Optimal ET sequence| 80, 132de, 212, 292, 504e }}
{{Optimal ET sequence|legend=0| 80, 132de, 212, 292, 504e }}


Badness: 0.062450
Badness (Sintel): 2.06


=== 13-limit ===
=== 13-limit ===
Line 422: Line 498:
Mapping: {{mapping| 4 8 12 15 17 16 | 0 -11 -18 -25 -21 -8 }}
Mapping: {{mapping| 4 8 12 15 17 16 | 0 -11 -18 -25 -21 -8 }}


Optimal tuning (CTE): ~25/21 = 1＼4, ~77/75 = 45.2243
Optimal tunings:
* WE: ~25/21 = 300.0805{{c}}, ~40/39 = 45.2812{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~40/39 = 45.2517{{c}}
 
{{Optimal ET sequence|legend=0| 80, 132de, 212 }}
 
Badness (Sintel): 1.86
 
=== 17-limit ===
Subgroup: 2.3.5.7.11.13.17
 
Comma list: 289/288, 325/324, 561/560, 1001/1000, 10648/10647
 
Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}
 
Optimal tunings:
* WE: ~25/21 = 300.1108{{c}}, ~40/39 = 45.2990{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~40/39 = 45.2604{{c}}
 
{{Optimal ET sequence|legend=0| 80, 132deg, 212g }}
 
Badness (Sintel): 1.59
 
=== 19-limit ===
Subgroup: 2.3.5.7.11.13.17.19
 
Comma list: 289/288, 325/324, 361/360, 561/560, 1001/1000, 1331/1330
 
Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}


Optimal ET sequence: {{Optimal ET sequence| 80, 212, 292f, 504eff }}
Optimal tunings:  
* WE: ~25/21 = 300.1212{{c}}, ~39/38 = 45.3015{{c}}
* CWE: ~25/21 = 300.0000{{c}}, ~39/38 = 45.2590{{c}}


Badness: 0.045028
{{Optimal ET sequence|legend=0| 80, 132degh, 212gh }}
 
Badness (Sintel): 1.36


== Quintiquart ==
== Quintiquart ==
Line 434: Line 542:


{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}
{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}
: mapping generators: ~35721/31250, ~250/243


{{Multival|legend=1| 55 90 105 15 12 -9 }}
[[Optimal tuning]]s:
* [[WE]]: ~35721/31250 = 239.9950{{c}}, ~250/243 = 45.2520{{c}}
: [[error map]]: {{val| -0.025 +0.223 -0.925 +0.792 }}
* [[CWE]]: ~35721/31250 = 240.0000{{c}}, ~250/243 = 45.2546{{c}}
: error map: {{val| 0.000 +0.244 -0.897 +0.826 }}


[[Optimal tuning]] ([[CTE]]): ~35721/31250 = 1＼5, ~250/243 = 45.2563
{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d }}


{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d, 875cd }}
[[Badness]] (Sintel): 9.04
 
[[Badness]]: 0.357387


=== 11-limit ===
=== 11-limit ===
Line 450: Line 561:
Mapping: {{mapping| 5 10 15 18 19 | 0 -11 -18 -21 -9 }}
Mapping: {{mapping| 5 10 15 18 19 | 0 -11 -18 -21 -9 }}


Optimal tuning (CTE): ~8019/7000 = 1＼5, ~250/243 = 45.2624
Optimal tunings:
* WE: ~8019/7000 = 239.9613{{c}}, ~250/243 = 45.2270{{c}}
* CWE: ~8019/7000 = 240.0000{{c}}, ~250/243 = 45.2460{{c}}


Optimal ET sequence: {{Optimal ET sequence| 80, 185c, 265, 610de, 875cde }}
{{Optimal ET sequence|legend=0| 80, 185c, 265, 345 }}


Badness: 0.103496
Badness (Sintel): 3.42


[[Category:Quartonic family| ]] <!-- main article -->
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Quartonic family| ]] <!-- main article -->
[[Category:Catalogs of rank-2 temperaments]]
[[Category:Rank 2]]

Latest revision as of 12:17, 14 August 2026

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

The quartonic family of temperaments tempers out the quartonic comma (monzo: [3 -18 11⟩, ratio: 390 625 000 / 387 420 489).

Quartonic

The name quartonic refers to the quartertone, since a somewhat flattened quartertone (representing the triptolemaic chromatic semitone, i.e. porcupine comma) is the generator of this temperament. The ploidacot for the temperament is omega-hendecacot, with eleven generator steps giving the perfect fourth. It is a member of the schismic–Mercator equivalence continuum, with equivalence number n = 11/2.

Subgroup: 2.3.5

Comma list: 390625000/387420489

Mapping: [⟨1 2 3], ⟨0 -11 -18]]

mapping generators: ~2, ~250/243

Optimal tunings:

  • WE: ~2 = 1199.9709 ¢, ~250/243 = 45.2318 ¢
error map: ⟨-0.029 +0.437 -0.574]
  • CWE: ~2 = 1200.0000 ¢, ~250/243 = 45.2349 ¢
error map: ⟨0.000 +0.461 -0.541]

Optimal ET sequence: 26, 27, 53, 239, 292, 345, 398, 451

Badness (Sintel): 2.75

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at.

  • 1728/1715 or 4000/3969 gives septimal quartonic, with interpretation of the generator ~36/35. It also tempers out 4375/4374.
  • 10976/10935 gives yarman I (80 & 159) and slices the quartonic generator in three.
  • 5359375/5308416 gives yarman II (79 & 159) and slices the quartonic generator in three.
  • 2401/2400 gives tertiseptisix (27 & 212) with generator ~875/729, three of them give ~12/7, and four give ~250/243 with octave reduction.
  • 250047/250000 gives triquart (27 & 159) with 1/3-octave period.
  • 390625/388962 or 4802000/4782969 gives quartiquart (80 & 212) with 1/4-octave period.
  • 16875/16807 gives quintiquart (80 & 265) with 1/5-octave period.

Septimal quartonic

Septimal quartonic tempers out the orwellisma, the octagar comma, as well as the ragisma, and may be described as the 26 & 27 temperament.

Subgroup: 2.3.5.7

Comma list: 1728/1715, 4000/3969

Mapping: [⟨1 2 3 3], ⟨0 -11 -18 -5]]

Optimal tunings:

  • WE: ~2 = 1199.1424 ¢, ~36/35 = 45.1064 ¢
error map: ⟨-0.858 +0.160 -0.801 +3.069]
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.1881 ¢
error map: ⟨0.000 +0.976 +0.301 +5.234]

Optimal ET sequence: 26, 27, 53, 80, 133d, 186d, 319dd

Badness (Sintel): 1.08

11-limit

Subgroup: 2.3.5.7.11

Comma list: 176/175, 540/539, 2200/2187

Mapping: [⟨1 2 3 3 5], ⟨0 -11 -18 -5 -41]]

Optimal tunings:

  • WE: ~2 = 1198.8787 ¢, ~36/35 = 44.9994 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.0863 ¢

Optimal ET sequence: 26e, 27e, 53, 80

Badness (Sintel): 1.13

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324, 540/539

Mapping: [⟨1 2 3 3 5 4], ⟨0 -11 -18 -5 -41 -8]]

Optimal tunings:

  • WE: ~2 = 1199.2991 ¢, ~36/35 = 45.0539 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.1049 ¢

Optimal ET sequence: 26e, 27e, 53, 80, 133d

Badness (Sintel): 0.987

Quarto

Subgroup: 2.3.5.7.11

Comma list: 100/99, 245/242, 864/847

Mapping: [⟨1 2 3 3 4], ⟨0 -11 -18 -5 -14]]

Optimal tunings:

  • WE: ~2 = 1198.3094 ¢, ~36/35 = 45.0688 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.2329 ¢

Optimal ET sequence: 26, 27e, 53e, 80ee

Badness (Sintel): 1.38

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 78/77, 100/99, 144/143, 245/242

Mapping: [⟨1 2 3 3 4 4], ⟨0 -11 -18 -5 -14 -8]]

Optimal tunings:

  • WE: ~2 = 1198.9315 ¢, ~36/35 = 45.1650 ¢
  • CWE: ~2 = 1200.0000 ¢, ~36/35 = 45.2640 ¢

Optimal ET sequence: 26, 27e, 53e

Badness (Sintel): 1.14

Quartz

Subgroup: 2.3.5.7.11

Comma list: 99/98, 385/384, 4000/3969

Mapping: [⟨1 2 3 3 3], ⟨0 -11 -18 -5 12]]

Optimal tunings:

  • WE: ~2 = 1200.5082 ¢, ~33/32 = 45.4047 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 45.3739 ¢

Optimal ET sequence: 26, 53

Badness (Sintel): 1.76

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 169/168, 275/273, 385/384

Mapping: [⟨1 2 3 3 3 4], ⟨0 -11 -18 -5 12 -8]]

Optimal tunings:

  • WE: ~2 = 1200.5899 ¢, ~33/32 = 45.4096 ¢
  • CWE: ~2 = 1200.0000 ¢, ~33/32 = 45.3739 ¢

Optimal ET sequence: 26, 53

Badness (Sintel): 1.19

Biquartonic

Subgroup: 2.3.5.7.11

Comma list: 1728/1715, 2420/2401, 2560/2541

Mapping: [⟨2 4 6 6 7], ⟨0 -11 -18 -5 -1]]

mapping generators: ~99/70, ~36/35

Optimal tunings:

  • WE: ~99/70 = 599.5389 ¢, ~36/35 = 45.0947 ¢
  • CWE: ~99/70 = 600.0000 ¢, ~36/35 = 45.1692 ¢

Optimal ET sequence: 26, 54c, 80, 186de, 266dde

Badness (Sintel): 2.01

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 364/363, 640/637

Mapping: [⟨2 4 6 6 7 8], ⟨0 -11 -18 -5 -1 -8]]

Optimal tunings:

  • WE: ~55/39 = 599.6870 ¢, ~40/39 = 45.1293 ¢
  • CWE: ~55/39 = 600.0000 ¢, ~40/39 = 45.1789 ¢

Optimal ET sequence: 26, 54c, 80, 106

Badness (Sintel): 1.65

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539

Mapping: [⟨2 4 6 6 7 8 9], ⟨0 -11 -18 -5 -1 -8 -11]]

Optimal tunings:

  • WE: ~17/12 = 599.7706 ¢, ~40/39 = 45.1427 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~40/39 = 45.1790 ¢

Optimal ET sequence: 26, 54c, 80, 106

Badness (Sintel): 1.43

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 169/168, 221/220, 289/288, 325/324, 544/539, 400/399

Mapping: [⟨2 4 6 6 7 8 9 10], ⟨0 -11 -18 -5 -1 -8 -11 -20]]

Optimal tunings:

  • WE: ~17/12 = 599.7848 ¢, ~39/38 = 45.1288 ¢
  • CWE: ~17/12 = 600.0000 ¢, ~39/38 = 45.1634 ¢

Optimal ET sequence: 26, 54ch, 80, 106

Badness (Sintel): 1.30

Yarm

Subgroup: 2.3.5.7.11

Comma list: 1331/1323, 1728/1715, 4000/3969

Mapping: [⟨1 2 3 3 4], ⟨0 -33 -54 -15 -43]]

Optimal tunings:

  • WE: ~2 = 1200.0000 ¢, ~100/99 = 15.0363 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0617 ¢

Optimal ET sequence: 79, 80, 239dd

Badness (Sintel): 3.30

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 325/324, 640/637, 1331/1323

Mapping: [⟨1 2 3 3 4 4], ⟨0 -33 -54 -15 -43 -24]]

Optimal tunings:

  • WE: ~2 = 1199.5798 ¢, ~100/99 = 15.0551 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0681 ¢

Optimal ET sequence: 79, 80

Badness (Sintel): 2.55

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 169/168, 325/324, 561/560, 640/637, 850/847

Mapping: [⟨1 2 3 3 4 4 4], ⟨0 -33 -54 -15 -43 -24 7]]

Optimal tunings:

  • WE: ~2 = 1199.6800 ¢, ~100/99 = 15.0624 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0705 ¢

Optimal ET sequence: 79, 80

Badness (Sintel): 2.38

Yarman I

Yarman I is the first interpretation of Ozan Yarman's proposed tuning for the Turkish makam.

Subgroup: 2.3.5.7

Comma list: 10976/10935, 244140625/243045684

Mapping: [⟨1 2 3 4], ⟨0 -33 -54 -95]]

mapping generators: ~2, ~126/125

Optimal tunings:

  • WE: ~2 = 1199.8726 ¢, ~126/125 = 15.0651 ¢
error map: ⟨-0.127 +0.641 -0.213 -0.523]
  • CWE: ~2 = 1200.0000 ¢, ~126/125 = 15.0689 ¢
error map: ⟨0.000 +0.771 -0.036 -0.374]

Optimal ET sequence: 79d, 80, 159, 239, 637, 876b

Badness (Sintel): 4.89

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3025/3024, 4000/3993, 10976/10935

Mapping: [⟨1 2 3 4 4], ⟨0 -33 -54 -95 -43]]

Optimal tunings:

  • WE: ~2 = 1199.8339 ¢, ~100/99 = 15.0637 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0685 ¢

Optimal ET sequence: 79d, 80, 159, 239, 876be

Badness (Sintel): 1.63

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 364/363, 1001/1000, 10976/10935

Mapping: [⟨1 2 3 4 4 4], ⟨0 -33 -54 -95 -43 -24]]

Optimal tunings:

  • WE: ~2 = 1200.1080 ¢, ~100/99 = 15.0766 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0737 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f

Badness (Sintel): 1.69

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 325/324, 364/363, 595/594, 1001/1000, 10976/10935

Mapping: [⟨1 2 3 4 4 4 4], ⟨0 -33 -54 -95 -43 -24 7]]

Optimal tunings:

  • WE: ~2 = 1200.0222 ¢, ~100/99 = 15.0718 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0713 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f

Badness (Sintel): 1.58

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 325/324, 361/360, 364/363, 595/594, 969/968, 1001/1000

Mapping: [⟨1 2 3 4 4 4 4 5], ⟨0 -33 -54 -95 -43 -24 7 -60]]

Optimal tunings:

  • WE: ~2 = 1200.0698 ¢, ~100/99 = 15.0722 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0706 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 398fh

Badness (Sintel): 1.41

23-limit

Subgroup: 2.3.5.7.11.13.17.19.23

Comma list: 325/324, 361/360, 364/363, 460/459, 507/506, 529/528, 760/759

Mapping: [⟨1 2 3 4 4 4 4 5 5], ⟨0 -33 -54 -95 -43 -24 7 -60 -38]]

Optimal tunings:

  • WE: ~2 = 1200.0946 ¢, ~100/99 = 15.0732 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0710 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 398fh

Badness (Sintel): 1.27

29-limit

Subgroup: 2.3.5.7.11.13.17.19.23.29

Comma list: 325/324, 361/360, 364/363, 406/405, 460/459, 494/493, 507/506, 529/528

Mapping: [⟨1 2 3 4 4 4 4 5 5 6], ⟨0 -33 -54 -95 -43 -24 7 -60 -38 -91]]

Optimal tunings:

  • WE: ~2 = 1200.0995 ¢, ~100/99 = 15.0726 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.0703 ¢

Optimal ET sequence: 79dhj, 80, 159, 239, 398fh

Badness (Sintel): 1.19

Yarman II

Yarman II is the other interpretation of Ozan Yarman's proposed tuning for the Turkish makam.

Subgroup: 2.3.5.7

Comma list: 5359375/5308416, 390625000/387420489

Mapping: [⟨1 2 3 2], ⟨0 -33 -54 64]]

mapping generators: ~2, ~875/864

Optimal tunings:

  • WE: ~2 = 1200.4734 ¢, ~875/864 = 15.1122 ¢
error map: ⟨+0.473 +0.291 -0.950 -0.701]
  • CWE: ~2 = 1200.0000 ¢, ~875/864 = 15.1048 ¢
error map: ⟨0.000 -0.412 -1.971 -2.121]

Optimal ET sequence: 79, 159, 397cd, 556cd, 715ccdd

Badness (Sintel): 16.6

11-limit

Subgroup: 2.3.5.7.11

Comma list: 385/384, 4000/3993, 78121827/77948684

Mapping: [⟨1 2 3 2 4], ⟨0 -33 -54 64 -43]]

Optimal tunings:

  • WE: ~2 = 1200.4357 ¢, ~100/99 = 15.1126 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.1054 ¢

Optimal ET sequence: 79, 159, 397cd, 556cd

Badness (Sintel): 4.74

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 385/384, 1575/1573, 85683/85184

Mapping: [⟨1 2 3 2 4 4], ⟨0 -33 -54 64 -43 -24]]

Optimal tunings:

  • WE: ~2 = 1200.5028 ¢, ~100/99 = 15.1135 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.1052 ¢

Optimal ET sequence: 79, 159, 397cdff, 556cdff

Badness (Sintel): 2.82

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 273/272, 325/324, 385/384, 1575/1573, 4928/4913

Mapping: [⟨1 2 3 2 4 4 4], ⟨0 -33 -54 64 -43 -24 7]]

Optimal tunings:

  • WE: ~2 = 1200.3840 ¢, ~100/99 = 15.1086 ¢
  • CWE: ~2 = 1200.0000 ¢, ~100/99 = 15.1025 ¢

Optimal ET sequence: 79, 159

Badness (Sintel): 2.60

Tertiseptisix

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625000/387420489

Mapping: [⟨1 -31 -51 -32], ⟨0 44 72 47]]

mapping generators: ~2, ~1458/875

Optimal tunings:

  • WE: ~2 = 1200.0156 ¢, ~1458/875 = 888.7028 ¢
error map: ⟨+0.016 +0.485 -0.507 -0.293]
  • CWE: ~2 = 1200.0000 ¢, ~1458/875 = 888.6915 ¢
error map: ⟨0.000 +0.472 -0.523 -0.324]

Optimal ET sequence: 27, …, 212, 239, 451

Badness (Sintel): 3.95

Triquart

Subgroup: 2.3.5.7

Comma list: 117649/116640, 250047/250000

Mapping: [⟨3 6 9 10], ⟨0 -11 -18 -14]]

mapping generators: ~63/50, ~250/243

Optimal tunings:

  • WE: ~63/50 = 400.0521 ¢, ~250/243 = 45.2373 ¢
error map: ⟨+0.156 +0.747 -0.117 -1.628]
  • CWE: ~63/50 = 400.0000 ¢, ~250/243 = 45.2204 ¢
error map: ⟨0.000 +0.621 -0.281 -1.911]

Optimal ET sequence: 27, 105cd, 132d, 159, 345d, 506d

Badness (Sintel): 4.30

Quartiquart

Subgroup: 2.3.5.7

Comma list: 390625/388962, 4802000/4782969

Mapping: [⟨4 8 12 15], ⟨0 -11 -18 -25]]

mapping generators: ~25/21, ~250/243

Optimal tunings:

  • WE: ~25/21 = 299.9955 ¢, ~250/243 = 45.2382 ¢
error map: ⟨-0.018 +0.389 -0.655 +0.152]
  • CWE: ~25/21 = 300.0000 ¢, ~250/243 = 45.2400 ¢
error map: ⟨0.000 +0.405 -0.634 +0.174]

Optimal ET sequence: 80, 132d, 212, 292, 504

Badness (Sintel): 5.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 1375/1372, 6250/6237, 14641/14580

Mapping: [⟨4 8 12 15 17], ⟨0 -11 -18 -25 -21]]

Optimal tunings:

  • WE: ~25/21 = 300.0140 ¢, ~77/75 = 45.2398 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~77/75 = 45.2342 ¢

Optimal ET sequence: 80, 132de, 212, 292, 504e

Badness (Sintel): 2.06

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 325/324, 1001/1000, 1375/1372, 10648/10647

Mapping: [⟨4 8 12 15 17 16], ⟨0 -11 -18 -25 -21 -8]]

Optimal tunings:

  • WE: ~25/21 = 300.0805 ¢, ~40/39 = 45.2812 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~40/39 = 45.2517 ¢

Optimal ET sequence: 80, 132de, 212

Badness (Sintel): 1.86

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 289/288, 325/324, 561/560, 1001/1000, 10648/10647

Mapping: [⟨4 8 12 15 17 16 18], ⟨0 -11 -18 -25 -21 -8 -11]]

Optimal tunings:

  • WE: ~25/21 = 300.1108 ¢, ~40/39 = 45.2990 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~40/39 = 45.2604 ¢

Optimal ET sequence: 80, 132deg, 212g

Badness (Sintel): 1.59

19-limit

Subgroup: 2.3.5.7.11.13.17.19

Comma list: 289/288, 325/324, 361/360, 561/560, 1001/1000, 1331/1330

Mapping: [⟨4 8 12 15 17 16 18 20], ⟨0 -11 -18 -25 -21 -8 -11 20]]

Optimal tunings:

  • WE: ~25/21 = 300.1212 ¢, ~39/38 = 45.3015 ¢
  • CWE: ~25/21 = 300.0000 ¢, ~39/38 = 45.2590 ¢

Optimal ET sequence: 80, 132degh, 212gh

Badness (Sintel): 1.36

Quintiquart

Subgroup: 2.3.5.7

Comma list: 16875/16807, 390625000/387420489

Mapping: [⟨5 10 15 18], ⟨0 -11 -18 -21]]

mapping generators: ~35721/31250, ~250/243

Optimal tunings:

  • WE: ~35721/31250 = 239.9950 ¢, ~250/243 = 45.2520 ¢
error map: ⟨-0.025 +0.223 -0.925 +0.792]
  • CWE: ~35721/31250 = 240.0000 ¢, ~250/243 = 45.2546 ¢
error map: ⟨0.000 +0.244 -0.897 +0.826]

Optimal ET sequence: 80, 185c, 265, 610d

Badness (Sintel): 9.04

11-limit

Subgroup: 2.3.5.7.11

Comma list: 540/539, 1375/1372, 390625000/387420489

Mapping: [⟨5 10 15 18 19], ⟨0 -11 -18 -21 -9]]

Optimal tunings:

  • WE: ~8019/7000 = 239.9613 ¢, ~250/243 = 45.2270 ¢
  • CWE: ~8019/7000 = 240.0000 ¢, ~250/243 = 45.2460 ¢

Optimal ET sequence: 80, 185c, 265, 345

Badness (Sintel): 3.42

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