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Quartonic family: Difference between revisions

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{{Technical data page}}<br><br>
{{Technical data page}}
The '''quartonic family''' of [[regular temperament|temperaments]] [[tempering out|tempers out]] the [[quartonic comma]], {{monzo| 3 -18 11 }} = 390625000/387420489.
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 11: 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 &amp; 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 &amp; 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 &amp; 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 &amp; 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 &amp; 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 &amp; 265) with 1/5-octave period.
* [[Mirkwai comma|16875/16807]] gives quintiquart (80 & 265) with 1/5-octave period.


== Septimal quartonic ==
== Septimal quartonic ==
Line 34: 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 49: 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 62: 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 75: 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 88: 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 101: 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 114: 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 127: 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 140: 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 153: 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 166: 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 179: 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 192: 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 205: 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 218: 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 233: 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 246: 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 259: 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 272: 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 285: 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 298: 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 311: 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 326: 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 339: 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 352: 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 365: 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 380: 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 395: 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 410: 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 423: 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 436: 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 449: 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 462: 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 477: 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]]
Retrieved from "https://en.xen.wiki/w/Quartonic_family"