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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| 3 -18 11 }} = 390625000/387420489.


== Quartonic (27 & 53) ==
== 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 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 }}


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


{{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]]: 0.117250


=== 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 lists #Normal interval list|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.
* [[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.  
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{{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 = 1\1, ~36/35 = 45.2652


[[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 }}


[[Badness]]: 0.042632
[[Badness]]: 0.042632
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 = 1\1, ~36/35 = 45.1674


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


Badness: 0.034031
Badness: 0.034031
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 = 1\1, ~36/35 = 45.1632


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


Badness: 0.023875
Badness: 0.023875
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 = 1\1, ~36/35 = 45.4022


Optimal ET sequence: {{Optimal ET sequence| 26, 53e, 132ee }}
Optimal ET sequence: {{Optimal ET sequence| 26, 53e, 132ee }}
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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 = 1\1, ~36/35 = 45.3857


Optimal ET sequence: {{Optimal ET sequence| 26, 53e, 132ee }}
Optimal ET sequence: {{Optimal ET sequence| 26, 53e, 132ee }}
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 = 1\1, ~33/32 = 45.3313


Optimal ET sequence: {{Optimal ET sequence| 26, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 26, 53 }}
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 = 1\1, ~33/32 = 45.3168


Optimal ET sequence: {{Optimal ET sequence| 26, 53 }}
Optimal ET sequence: {{Optimal ET sequence| 26, 53 }}
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 = 1\2, ~36/35 = 45.2678


Optimal ET sequence: {{Optimal ET sequence| 26, 54c, 80, 106 }}
Optimal ET sequence: {{Optimal ET sequence| 26, 54c, 80, 106 }}
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, ~36/35 = 45.2544
Optimal tuning (CTE): ~55/39 = 1\2, ~40/39 = 45.2544


Optimal ET sequence: {{Optimal ET sequence| 26, 54c, 80, 106 }}
Optimal ET sequence: {{Optimal ET sequence| 26, 54c, 80, 106 }}
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 = 1\2, ~34/33 = 45.2397


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


Badness: 0.028112
Badness: 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: {{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 ET sequence: {{Optimal ET sequence| 26, 54ch, 80, 106 }}
Badness: 0.0213


=== Yarm ===
=== Yarm ===
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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 = 1\1, ~100/99 = 15.0880


Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
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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 = 1\1, ~100/99 = 15.0842


Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
Line 191: 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 = 1\1, ~100/99 = 15.0840


Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
Optimal ET sequence: {{Optimal ET sequence| 79, 80, 159d }}
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{{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 = 1\1, ~126/125 = 15.0714
 
[[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 }}
Line 219: 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 = 1\1, ~100/99 = 15.0724


Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398, 637, 1035bd }}
Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398, 637, 1035bd }}
Line 232: 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 = 1\1, ~91/90 = 15.0707


Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398f, 637ff }}
Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398f, 637ff }}
Line 245: 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 = 1\1, ~91/90 = 15.0706


Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398f, 637ff }}
Optimal ET sequence: {{Optimal ET sequence| 79d, 80, 159, 239, 398f, 637ff }}
Line 258: 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 = 1\1, ~91/90 = 15.0683


Optimal ET sequence: {{Optimal ET sequence| 79dh, 80, 159, 239, 637ffh }}
Optimal ET sequence: {{Optimal ET sequence| 79dh, 80, 159, 239, 637ffh }}
Line 271: 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 = 1\1, ~91/90 = 15.0676


Optimal ET sequence: {{Optimal ET sequence| 79dh, 80, 159, 239, 637ffhi }}
Optimal ET sequence: {{Optimal ET sequence| 79dh, 80, 159, 239, 637ffhi }}
Line 284: 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 = 1\1, ~91/90 = 15.0667


Optimal ET sequence: {{Optimal ET sequence| 79dhj, 80, 159, 239 }}
Optimal ET sequence: {{Optimal ET sequence| 79dhj, 80, 159, 239 }}
Line 297: 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 = 1\1, ~875/864 = 15.0995
 
[[Optimal tuning]] ([[CTE]]): ~2 = 1＼1, ~875/864 = 15.0995


{{Optimal ET sequence|legend=1| 79, 159 }}
{{Optimal ET sequence|legend=1| 79, 159 }}
Line 312: 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 = 1\1, ~100/99 = 15.0982


Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
Line 325: 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 = 1\1, ~100/99 = 15.0952


Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
Line 338: 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 = 1\1, ~100/99 = 15.0950


Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
Optimal ET sequence: {{Optimal ET sequence| 79, 159 }}
Line 351: 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 = 1\1, ~875/729 = 311.308
 
[[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 }}
Line 366: 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 = 1\3, ~250/243 = 45.2083
 
[[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 }}
Line 381: 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 = 1\4, ~250/243 = 45.2411
 
[[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 }}
Line 396: 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 = 1\4, ~77/75 = 45.2303


Optimal ET sequence: {{Optimal ET sequence| 80, 132de, 212, 292, 504e }}
Optimal ET sequence: {{Optimal ET sequence| 80, 132de, 212, 292, 504e }}
Line 409: 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, ~77/75 = 45.2243
Optimal tuning (CTE): ~25/21 = 1\4, ~40/39 = 45.2243


Optimal ET sequence: {{Optimal ET sequence| 80, 212, 292f, 504eff }}
Optimal ET sequence: {{Optimal ET sequence| 80, 132de, 212 }}


Badness: 0.045028
Badness: 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: {{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 ET sequence: {{Optimal ET sequence| 52cdeg, 80, 132deg, 212g }}
Badness: 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: {{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 ET sequence: {{Optimal ET sequence| 52cdegh, 80, 132degh, 212gh }}
Badness: 0.0224


== Quintiquart ==
== Quintiquart ==
Line 422: 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 = 1\5, ~250/243 = 45.2563
 
[[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 }}
Line 437: 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 = 1\5, ~250/243 = 45.2624


Optimal ET sequence: {{Optimal ET sequence| 80, 185c, 265, 610de, 875cde }}
Optimal ET sequence: {{Optimal ET sequence| 80, 185c, 265, 610de, 875cde }}
Line 444: Line 470:


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[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"