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Quartonic family: Difference between revisions

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Cleanup. Review the optimal ET sequences for 5- and 7-limit quartonic
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Temperaments of the '''quartonic family''' temper out {{monzo| 3 -18 11 }} = 390625000/387420489 (saleyo comma).
The '''quartonic family''' of [[regular temperament|temperaments]] [[tempering out|temper 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
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[[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.
* [[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.
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{{Multival|legend=1| 11 18 5 3 -23 -39 }}
{{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 }}
{{Optimal ET sequence|legend=1| 26, 27, 53, 80, 133d, 186d, 319dd }}


[[Badness]]: 0.042632
[[Badness]]: 0.042632
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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, 80 }}
Optimal ET sequence: {{Optimal ET sequence| 26e, 27e, 53, 80 }}
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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, 80, 133d, 186d }}
Optimal ET sequence: {{Optimal ET sequence| 26e, 27e, 53, 80, 133d, 186d }}
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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 }}
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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 }}
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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 }}
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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 }}
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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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{{Multival|legend=1| 33 54 95 9 58 69 }}
{{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 }}
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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 }}
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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 }}
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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 }}
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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 }}
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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 }}
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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 }}
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{{Multival|legend=1| 33 54 -64 9 -194 -300 }}
{{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 }}
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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 }}
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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 }}
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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 }}
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{{Multival|legend=1| 44 72 47 12 -49 -93 }}
{{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 }}
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{{Multival|legend=1| 33 54 42 9 -26 -54 }}
{{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 }}
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{{Multival|legend=1| 55 90 105 15 12 -9 }}
{{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 }}
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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 }}
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