Quartonic family: Difference between revisions

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The '''quartonic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quartonic comma]], {{monzo| 3 -18 11 }} = 390625000/387420489.
{{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 ==
== Quartonic ==
The name "quartonic" means quarter-tone, which is the [[generator]] of this temperament.
The name ''quartonic'' means quarter-tone, which is the [[generator]] of this temperament.


[[Subgroup]]: 2.3.5
[[Subgroup]]: 2.3.5
Line 10: Line 11:
{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}
{{Mapping|legend=1| 1 2 3 | 0 -11 -18 }}


[[Optimal tuning]] ([[CTE]]): 2 = 1\1, ~250/243 = 45.2368
[[Optimal tuning]] ([[CTE]]): 2 = 1200.0000{{c}}, ~250/243 = 45.2368{{c}}


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


[[Badness]]: 0.117250
[[Badness]] (Smith): 0.117250


=== Overview to extensions ===
=== Overview to 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. It also tempers out [[4375/4374]].  
* [[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.
* [[Hemimage comma|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.
* [[Landscape comma|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.
* [[Dimcomp comma|390625/388962]] or [[canousma|4802000/4782969]] gives quartiquart (80 & 212) with 1/4-octave period.
* [[Mirkwai comma|16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.
* [[Mirkwai comma|16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.


== Septimal quartonic ==
== Septimal quartonic ==
Line 33: Line 34:
{{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]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~36/35 = 45.2652{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~36/35 = 45.2652


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


[[Badness]]: 0.042632
[[Badness]] (Smith): 0.042632


=== 11-limit ===
=== 11-limit ===
Line 48: Line 47:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1674{{c}}


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


Badness: 0.034031
Badness (Smith): 0.034031


==== 13-limit ====
==== 13-limit ====
Line 61: Line 60:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.1632{{c}}


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


Badness: 0.023875
Badness (Smith): 0.023875


=== Quarto ===
=== Quarto ===
Line 74: Line 73:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.4022{{c}}


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


Badness: 0.041786
Badness (Smith): 0.041786


==== 13-limit ====
==== 13-limit ====
Line 87: Line 86:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~36/35 = 45.3857{{c}}


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


Badness: 0.027692
Badness (Smith): 0.027692


=== Quartz ===
=== Quartz ===
Line 100: Line 99:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3313{{c}}


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


Badness: 0.053285
Badness (Smith): 0.053285


==== 13-limit ====
==== 13-limit ====
Line 113: Line 112:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~33/32 = 45.3168{{c}}


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


Badness: 0.028818
Badness (Smith): 0.028818


=== Biquartonic ===
=== Biquartonic ===
Line 126: Line 125:
Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}
Mapping: {{mapping| 2 4 6 6 7 | 0 -11 -18 -5 -1 }}


Optimal tuning (CTE): ~99/70 = 1\2, ~36/35 = 45.2678
Optimal tuning (CTE): ~99/70 = 600.0000{{c}}, ~36/35 = 45.2678{{c}}


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


Badness: 0.060737
Badness (Smith): 0.060737


==== 13-limit ====
==== 13-limit ====
Line 139: Line 138:
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 tuning (CTE): ~55/39 = 600.0000{{c}}, ~40/39 = 45.2544{{c}}


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


Badness: 0.039891
Badness (Smith): 0.039891


==== 17-limit ====
==== 17-limit ====
Line 152: Line 151:
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 tuning (CTE): ~17/12 = 600.0000{{c}}, ~34/33 = 45.2397{{c}}


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


Badness: 0.028112
Badness (Smith): 0.028112


==== 19-limit ====
==== 19-limit ====
Line 165: Line 164:
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 tuning (CTE): ~17/12 = 600.000{{c}}, ~39/38 = 45.222{{c}}


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


Badness: 0.0213
Badness (Smith): 0.0213


=== Yarm ===
=== Yarm ===
Line 178: Line 177:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0880{{c}}


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


Badness: 0.099950
Badness (Smith): 0.099950


==== 13-limit ====
==== 13-limit ====
Line 191: Line 190:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0842{{c}}


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


Badness: 0.061645
Badness (Smith): 0.061645


==== 17-limit ====
==== 17-limit ====
Line 204: Line 203:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0840{{c}}


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


Badness: 0.046718
Badness (Smith): 0.046718


== Yarman I ==
== Yarman I ==
Line 217: Line 216:
{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}
{{Mapping|legend=1| 1 2 3 4 | 0 -33 -54 -95 }}


{{Multival|legend=1| 33 54 95 9 58 69 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~126/125 = 15.0714{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~126/125 = 15.0714


{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 398, 637 }}
{{Optimal ET sequence|legend=1| 79d, 80, 159, 239, 398, 637 }}


[[Badness]]: 0.193315
[[Badness]] (Smith): 0.193315


=== 11-limit ===
=== 11-limit ===
Line 232: Line 229:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0724{{c}}


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


Badness: 0.049170
Badness (Smith): 0.049170


=== 13-limit ===
=== 13-limit ===
Line 245: Line 242:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0707{{c}}


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


Badness: 0.040929
Badness (Smith): 0.040929


=== 17-limit ===
=== 17-limit ===
Line 258: Line 255:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0706{{c}}


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


Badness: 0.031015
Badness (Smith): 0.031015


=== 19-limit ===
=== 19-limit ===
Line 271: Line 268:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0683{{c}}


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


Badness: 0.023193
Badness (Smith): 0.023193


=== 23-limit ===
=== 23-limit ===
Line 284: Line 281:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0676{{c}}


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


Badness: 0.017682
Badness (Smith): 0.017682


=== 29-limit ===
=== 29-limit ===
Line 297: Line 294:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~91/90 = 15.0667{{c}}


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


Badness: 0.014289
Badness (Smith): 0.014289


== Yarman II ==
== Yarman II ==
Line 310: Line 307:
{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}
{{Mapping|legend=1| 1 2 3 2 | 0 -33 -54 64 }}


{{Multival|legend=1| 33 54 -64 9 -194 -300 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.0000{{c}}, ~875/864 = 15.0995{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~875/864 = 15.0995


{{Optimal ET sequence|legend=1| 79, 159 }}
{{Optimal ET sequence|legend=1| 79, 159 }}


[[Badness]]: 0.655487
[[Badness]] (Smith): 0.655487


=== 11-limit ===
=== 11-limit ===
Line 325: Line 320:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0982{{c}}


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


Badness: 0.143477
Badness (Smith): 0.143477


=== 13-limit ===
=== 13-limit ===
Line 338: Line 333:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0952{{c}}


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


Badness: 0.068150
Badness (Smith): 0.068150


=== 17-limit ===
=== 17-limit ===
Line 351: Line 346:
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 tuning (CTE): ~2 = 1200.0000{{c}}, ~100/99 = 15.0950{{c}}


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


Badness: 0.051019
Badness (Smith): 0.051019


== Tertiseptisix ==
== Tertiseptisix ==
Line 364: Line 359:
{{Mapping|legend=1| 1 13 21 15 | 0 -44 -72 -47 }}
{{Mapping|legend=1| 1 13 21 15 | 0 -44 -72 -47 }}


{{Multival|legend=1| 44 72 47 12 -49 -93 }}
[[Optimal tuning]] ([[CTE]]): ~2 = 1200.000{{c}}, ~875/729 = 311.308{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1\1, ~875/729 = 311.308


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


[[Badness]]: 0.155952
[[Badness]] (Smith): 0.155952


== Triquart ==
== Triquart ==
Line 379: Line 372:
{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}
{{Mapping|legend=1| 3 6 9 10 | 0 -11 -18 -14 }}


{{Multival|legend=1| 33 54 42 9 -26 -54 }}
[[Optimal tuning]] ([[CTE]]): ~63/50 = 400.0000{{c}}, ~250/243 = 45.2083{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~63/50 = 1\3, ~250/243 = 45.2083


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


[[Badness]]: 0.170062
[[Badness]] (Smith): 0.170062


== Quartiquart ==
== Quartiquart ==
Line 394: Line 385:
{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}
{{Mapping|legend=1| 4 8 12 15 | 0 -11 -18 -25 }}


{{Multival|legend=1| 44 72 100 12 35 30 }}
[[Optimal tuning]] ([[CTE]]): ~25/21 = 300.0000{{c}}, ~250/243 = 45.2411{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~25/21 = 1\4, ~250/243 = 45.2411


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


[[Badness]]: 0.199116
[[Badness]] (Smith): 0.199116


=== 11-limit ===
=== 11-limit ===
Line 409: Line 398:
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 tuning (CTE): ~25/21 = 300.0000{{c}}, ~77/75 = 45.2303{{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 (Smith): 0.062450


=== 13-limit ===
=== 13-limit ===
Line 422: Line 411:
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, ~40/39 = 45.2243
Optimal tuning (CTE): ~25/21 = 300.0000{{c}}, ~40/39 = 45.2243{{c}}


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


Badness: 0.045028
Badness (Smith): 0.045028


=== 17-limit ===
=== 17-limit ===
Line 435: Line 424:
Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}
Mapping: {{mapping| 4 8 12 15 17 16 18 | 0 -11 -18 -25 -21 -8 -11 }}


Optimal tuning (CTE): ~25/21 = 1\4, ~40/39 = 45.218
Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~40/39 = 45.218{{c}}


Optimal ET sequence: {{Optimal ET sequence| 52cdeg, 80, 132deg, 212g }}
{{Optimal ET sequence|legend=0| 52cdeg, 80, 132deg, 212g }}


Badness: 0.0312
Badness (Smith): 0.0312


=== 19-limit ===
=== 19-limit ===
Line 448: Line 437:
Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}
Mapping: {{mapping| 4 8 12 15 17 16 18 20 | 0 -11 -18 -25 -21 -8 -11 20 }}


Optimal tuning (CTE): ~25/21 = 1\4, ~39/38 = 45.210
Optimal tuning (CTE): ~25/21 = 300.000{{c}}, ~39/38 = 45.210{{c}}


Optimal ET sequence: {{Optimal ET sequence| 52cdegh, 80, 132degh, 212gh }}
{{Optimal ET sequence|legend=0| 52cdegh, 80, 132degh, 212gh }}


Badness: 0.0224
Badness (Smith): 0.0224


== Quintiquart ==
== Quintiquart ==
Line 461: Line 450:
{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}
{{Mapping|legend=1| 5 10 15 18 | 0 -11 -18 -21 }}


{{Multival|legend=1| 55 90 105 15 12 -9 }}
[[Optimal tuning]] ([[CTE]]): ~35721/31250 = 240.0000{{c}}, ~250/243 = 45.2563{{c}}
 
[[Optimal tuning]] ([[CTE]]): ~35721/31250 = 1\5, ~250/243 = 45.2563


{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d, 875cd }}
{{Optimal ET sequence|legend=1| 80, 185c, 265, 610d, 875cd }}


[[Badness]]: 0.357387
[[Badness]] (Smith): 0.357387


=== 11-limit ===
=== 11-limit ===
Line 476: Line 463:
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 tuning (CTE): ~8019/7000 = 240.0000{{c}}, ~250/243 = 45.2624{{c}}


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


Badness: 0.103496
Badness (Smith): 0.103496


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

Latest revision as of 15:46, 2 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 means quarter-tone, which is the generator of this temperament.

Subgroup: 2.3.5

Comma list: 390625000/387420489

Mapping: [⟨1 2 3], ⟨0 -11 -18]]

Optimal tuning (CTE): 2 = 1200.0000 ¢, ~250/243 = 45.2368 ¢

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

Badness (Smith): 0.117250

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

Subgroup: 2.3.5.7

Comma list: 1728/1715, 4000/3969

Mapping: [⟨1 2 3 3], ⟨0 -11 -18 -5]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~36/35 = 45.2652 ¢

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

Badness (Smith): 0.042632

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 tuning (CTE): ~2 = 1200.0000 ¢, ~36/35 = 45.1674 ¢

Optimal ET sequence: 26e, 27e, 53, 80

Badness (Smith): 0.034031

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 tuning (CTE): ~2 = 1200.0000 ¢, ~36/35 = 45.1632 ¢

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

Badness (Smith): 0.023875

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 tuning (CTE): ~2 = 1200.0000 ¢, ~36/35 = 45.4022 ¢

Optimal ET sequence: 26, 53e, 132ee

Badness (Smith): 0.041786

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 tuning (CTE): ~2 = 1200.0000 ¢, ~36/35 = 45.3857 ¢

Optimal ET sequence: 26, 53e, 132ee

Badness (Smith): 0.027692

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 tuning (CTE): ~2 = 1200.0000 ¢, ~33/32 = 45.3313 ¢

Optimal ET sequence: 26, 53

Badness (Smith): 0.053285

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 tuning (CTE): ~2 = 1200.0000 ¢, ~33/32 = 45.3168 ¢

Optimal ET sequence: 26, 53

Badness (Smith): 0.028818

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]]

Optimal tuning (CTE): ~99/70 = 600.0000 ¢, ~36/35 = 45.2678 ¢

Optimal ET sequence: 26, 54c, 80, 106

Badness (Smith): 0.060737

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 tuning (CTE): ~55/39 = 600.0000 ¢, ~40/39 = 45.2544 ¢

Optimal ET sequence: 26, 54c, 80, 106

Badness (Smith): 0.039891

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 tuning (CTE): ~17/12 = 600.0000 ¢, ~34/33 = 45.2397 ¢

Optimal ET sequence: 26, 54c, 80, 106

Badness (Smith): 0.028112

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 tuning (CTE): ~17/12 = 600.000 ¢, ~39/38 = 45.222 ¢

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

Badness (Smith): 0.0213

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0880 ¢

Optimal ET sequence: 79, 80, 159d

Badness (Smith): 0.099950

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0842 ¢

Optimal ET sequence: 79, 80, 159d

Badness (Smith): 0.061645

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0840 ¢

Optimal ET sequence: 79, 80, 159d

Badness (Smith): 0.046718

Yarman I

Subgroup: 2.3.5.7

Comma list: 10976/10935, 244140625/243045684

Mapping: [⟨1 2 3 4], ⟨0 -33 -54 -95]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~126/125 = 15.0714 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398, 637

Badness (Smith): 0.193315

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0724 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398, 637, 1035bd

Badness (Smith): 0.049170

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0707 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f, 637ff

Badness (Smith): 0.040929

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0706 ¢

Optimal ET sequence: 79d, 80, 159, 239, 398f, 637ff

Badness (Smith): 0.031015

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0683 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 637ffh

Badness (Smith): 0.023193

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0676 ¢

Optimal ET sequence: 79dh, 80, 159, 239, 637ffhi

Badness (Smith): 0.017682

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 tuning (CTE): ~2 = 1200.0000 ¢, ~91/90 = 15.0667 ¢

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

Badness (Smith): 0.014289

Yarman II

Subgroup: 2.3.5.7

Comma list: 5359375/5308416, 390625000/387420489

Mapping: [⟨1 2 3 2], ⟨0 -33 -54 64]]

Optimal tuning (CTE): ~2 = 1200.0000 ¢, ~875/864 = 15.0995 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.655487

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0982 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.143477

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0952 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.068150

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 tuning (CTE): ~2 = 1200.0000 ¢, ~100/99 = 15.0950 ¢

Optimal ET sequence: 79, 159

Badness (Smith): 0.051019

Tertiseptisix

Subgroup: 2.3.5.7

Comma list: 2401/2400, 390625000/387420489

Mapping: [⟨1 13 21 15], ⟨0 -44 -72 -47]]

Optimal tuning (CTE): ~2 = 1200.000 ¢, ~875/729 = 311.308 ¢

Optimal ET sequence: 27, 131bccd, 158cd, 185c, 212, 239, 451

Badness (Smith): 0.155952

Triquart

Subgroup: 2.3.5.7

Comma list: 117649/116640, 250047/250000

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

Optimal tuning (CTE): ~63/50 = 400.0000 ¢, ~250/243 = 45.2083 ¢

Optimal ET sequence: 27, 105cd, 132d, 159, 186, 345d

Badness (Smith): 0.170062

Quartiquart

Subgroup: 2.3.5.7

Comma list: 390625/388962, 4802000/4782969

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

Optimal tuning (CTE): ~25/21 = 300.0000 ¢, ~250/243 = 45.2411 ¢

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

Badness (Smith): 0.199116

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 tuning (CTE): ~25/21 = 300.0000 ¢, ~77/75 = 45.2303 ¢

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

Badness (Smith): 0.062450

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 tuning (CTE): ~25/21 = 300.0000 ¢, ~40/39 = 45.2243 ¢

Optimal ET sequence: 80, 132de, 212

Badness (Smith): 0.045028

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 tuning (CTE): ~25/21 = 300.000 ¢, ~40/39 = 45.218 ¢

Optimal ET sequence: 52cdeg, 80, 132deg, 212g

Badness (Smith): 0.0312

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 tuning (CTE): ~25/21 = 300.000 ¢, ~39/38 = 45.210 ¢

Optimal ET sequence: 52cdegh, 80, 132degh, 212gh

Badness (Smith): 0.0224

Quintiquart

Subgroup: 2.3.5.7

Comma list: 16875/16807, 390625000/387420489

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

Optimal tuning (CTE): ~35721/31250 = 240.0000 ¢, ~250/243 = 45.2563 ¢

Optimal ET sequence: 80, 185c, 265, 610d, 875cd

Badness (Smith): 0.357387

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 tuning (CTE): ~8019/7000 = 240.0000 ¢, ~250/243 = 45.2624 ¢

Optimal ET sequence: 80, 185c, 265, 610de, 875cde

Badness (Smith): 0.103496

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