Dicot family: Difference between revisions

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The [[5-limit]] parent [[comma]] for the dicot family is [[25/24]], the classic chromatic semitone. Its [[monzo]] is {{monzo| -3 -1 2 }}, and flipping that yields {{wedgie| 2 1 -3}} for the [[wedgie]]. This tells us the generator is a third (major and minor mean the same thing), and that two thirds gives a fifth. In fact, (5/4)^2 = 3/2 * 25/24. Possible tunings for dicot are [[7edo]], [[17edo]], [[24edo]] using the val {{val|24 38 55}} (24c) and [[31edo]] using the val {{val|31 49 71}} (31c). In a sense, what dicot is all about is using neutral thirds and pretending that's 5-limit, and like any temperament which seems to involve pretending, dicot is at the edge of what can sensibly be called a temperament at all. In other words, it is an [[exotemperament]].
The [[5-limit]] parent [[comma]] for the '''dicot family''' is [[25/24]], the classical chromatic semitone. Its [[monzo]] is {{monzo| -3 -1 2 }}, and flipping that yields {{multival| 2 1 -3 }} for the [[wedgie]]. This tells us the [[generator]] is a classical third (major and minor mean the same thing), and that two such thirds give a fifth. In fact, (5/4)<sup>2</sup> = (3/2)(25/24).  
 
Possible tunings for dicot are [[7edo]], [[17edo]], [[24edo]] using the val {{val| 24 38 55 }} (24c) and [[31edo]] using the val {{val| 31 49 71 }} (31c). In a sense, what dicot is all about is using neutral thirds and pretending that is 5-limit, and like any temperament which seems to involve pretending, dicot is at the edge of what can sensibly be called a temperament at all. In other words, it is an [[exotemperament]].


== Dicot ==
== Dicot ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 25/24
[[Comma list]]: 25/24


[[Mapping]]: [{{val|1 1 2}}, {{val|0 2 1}}]
{{Mapping|legend=1| 1 1 2 | 0 2 1 }}


[[POTE tuning|POTE generator]]: ~5/4 = 348.594
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 348.594


[[Tuning ranges]]:  
[[Tuning ranges]]:  
* 5-odd-limit [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* 5-odd-limit [[diamond monotone]]: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314]
* 5-odd-limit [[diamond tradeoff]]: ~5/4 = [315.641, 386.314] (full comma to untempered)
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]
* 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]


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[[Badness]]: 0.013028
[[Badness]]: 0.013028


=== Seven limit children ===
=== Overview to extensions ===
The second comma of the [[Normal_lists|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot, with wedgie {{wedgie|2 1 3 -3 -1 4}} adds 36/35, sharp with wedgie {{wedgie|2 1 6 -3 4 11}} adds 28/27, and dichotic with wedgie {{wedgie|2 1 -4 -3 -12 -12}} ads 64/63, all retaining the same period and generator. Decimal with wedgie {{wedgie|4 2 2 -6 -8 -1}} adds 49/48, sidi with wedgie {{wedgie|4 2 9 -3 6 15}} adds 245/243, and jamesbond with wedgie {{wedgie|0 0 7 0 11 16}} adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.
The second comma of the [[Normal lists|normal comma list]] defines which [[7-limit]] family member we are looking at. Septimal dicot, with wedgie {{multival| 2 1 3 -3 -1 4 }} adds 36/35, sharp with wedgie {{multival| 2 1 6 -3 4 11 }} adds 28/27, and dichotic with wedgie {{multival| 2 1 -4 -3 -12 -12 }} adds 64/63, all retaining the same period and generator.  
 
Decimal with wedgie {{multival| 4 2 2 -6 -8 -1 }} adds 49/48, sidi with wedgie {{multival| 4 2 9 -3 6 15 }} adds 245/243, and jamesbond with wedgie {{multival| 0 0 7 0 11 16 }} adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.


== Septimal dicot ==
== Septimal dicot ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 15/14, 25/24
[[Comma list]]: 15/14, 25/24


[[Mapping]]: [{{val|1 1 2 2}}, {{val|0 2 1 3}}]
{{Mapping|legend=1| 1 1 2 2 | 0 2 1 3 }}


[[Wedgie]]: {{wedgie|2 1 3 -3 -1 4}}
{{Multival|legend=1| 2 1 3 -3 -1 4 }}


[[POTE generator]]: ~5/4 = 336.381
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 336.381


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
Line 42: Line 46:
Comma list: 15/14, 22/21, 25/24
Comma list: 15/14, 22/21, 25/24


Mapping: [{{val|1 1 2 2 2}}, {{val|0 2 1 3 5}}]
Mapping: {{mapping| 1 1 2 2 2 | 0 2 1 3 5 }}


POTE generator: ~5/4 = 342.125
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 342.125


{{Optimal ET sequence|legend=1| 3de, 4e, 7 }}
{{Optimal ET sequence|legend=1| 3de, 4e, 7 }}
Line 55: Line 59:
Comma list: 15/14, 25/24, 33/32
Comma list: 15/14, 25/24, 33/32


Mapping: [{{val|1 1 2 2 4}}, {{val|0 2 1 3 -2}}]
Mapping: {{mapping| 1 1 2 2 4 | 0 2 1 3 -2 }}


POTE generator: ~5/4 = 336.051
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.051


{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
{{Optimal ET sequence|legend=1| 3d, 4, 7, 18bc, 25bccd }}
Line 68: Line 72:
Comma list: 15/14, 25/24, 33/32, 40/39
Comma list: 15/14, 25/24, 33/32, 40/39


Mapping: [{{val|1 1 2 2 4 4}}, {{val|0 2 1 3 -2 -1}}]
Mapping: {{mapping| 1 1 2 2 4 4 | 0 2 1 3 -2 -1 }}


POTE generator: ~5/4 = 338.846
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 338.846


{{Optimal ET sequence|legend=1| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
{{Optimal ET sequence|legend=1| 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef }}
Line 77: Line 81:


== Flat ==
== Flat ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 21/20, 25/24
[[Comma list]]: 21/20, 25/24


[[Mapping]]: [{{val|1 1 2 3}}, {{val|0 2 1 -1}}]
{{Mapping|legend=1| 1 1 2 3 | 0 2 1 -1 }}


{{Multival|legend=1|2 1 -1 -3 -7 -5}}
{{Multival|legend=1|2 1 -1 -3 -7 -5}}


[[POTE generator]]: ~5/4 = 331.916
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 331.916


{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
{{Optimal ET sequence|legend=1| 3, 4, 7d, 11cd, 18bcddd }}
Line 96: Line 100:
Comma list: 21/20, 25/24, 33/32
Comma list: 21/20, 25/24, 33/32


Mapping: [{{val|1 1 2 3 4}}, {{val|0 2 1 -1 -2}}]
Mapping: {{mapping| 1 1 2 3 4 | 0 2 1 -1 -2 }}


POTE generator: ~5/4 = 337.532
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 337.532


{{Optimal ET sequence|legend=1| 3, 4, 7d }}
{{Optimal ET sequence|legend=1| 3, 4, 7d }}
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Comma list: 14/13, 21/20, 25/24, 33/32
Comma list: 14/13, 21/20, 25/24, 33/32


Mapping: [{{val|1 1 2 3 4 4}}, {{val|0 2 1 -1 -2 -1}}]
Mapping: {{mapping| 1 1 2 3 4 4 | 0 2 1 -1 -2 -1 }}


POTE generator: ~5/4 = 341.023
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 341.023


{{Optimal ET sequence|legend=1| 3, 4, 7d }}
{{Optimal ET sequence|legend=1| 3, 4, 7d }}
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== Sharp ==
== Sharp ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 25/24, 28/27
[[Comma list]]: 25/24, 28/27


[[Mapping]]: [{{val|1 1 2 1}}, {{val|0 2 1 6}}]
{{Mapping|legend=1| 1 1 2 1 | 0 2 1 6 }}


[[POTE generator]]: ~5/4 = 357.938
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 357.938


{{Multival|legend=1|2 1 6 -3 4 11}}
{{Multival|legend=1| 2 1 6 -3 4 11 }}


{{Optimal ET sequence|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
{{Optimal ET sequence|legend=1| 3d, 7d, 10, 37cd, 47bccd, 57bccdd }}
Line 137: Line 141:
Comma list: 25/24, 28/27, 35/33
Comma list: 25/24, 28/27, 35/33


Mapping: [{{val|1 1 2 1 2}}, {{val|0 2 1 6 5}}]
Mapping: {{mapping| 1 1 2 1 2 | 0 2 1 6 5 }}


POTE generator: ~5/4 = 356.106
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.106


{{Optimal ET sequence|legend=1| 3de, 7d, 10, 17d, 27cde }}
{{Optimal ET sequence|legend=1| 3de, 7d, 10, 17d, 27cde }}
Line 146: Line 150:


== Decimal ==
== Decimal ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 25/24, 49/48
[[Comma list]]: 25/24, 49/48


[[Mapping]]: [{{val|2 0 3 4}}, {{val|0 2 1 1}}]
{{Mapping|legend=1| 2 0 3 4 | 0 2 1 1 }}


{{Multival|legend=1|4 2 2 -6 -8 -1}}
{{Multival|legend=1| 4 2 2 -6 -8 -1 }}


[[POTE generator]]: ~7/6 = 251.557
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~7/6 = 251.557


{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
{{Optimal ET sequence|legend=1| 4, 10, 14c, 24c, 38ccd, 62cccdd }}
Line 165: Line 169:
Comma list: 25/24, 45/44, 49/48
Comma list: 25/24, 45/44, 49/48


Mapping: [{{val|2 0 3 4 -1}}, {{val|0 2 1 1 5}}]
Mapping: {{mapping| 2 0 3 4 -1 | 0 2 1 1 5 }}


POTE generator: ~7/6 = 253.493
Optimal tuning (POTE): ~7/5 = 1\2, ~7/6 = 253.493


{{Optimal ET sequence|legend=1| 10, 14c, 24c, 38ccd, 52cccde }}
{{Optimal ET sequence|legend=1| 10, 14c, 24c, 38ccd, 52cccde }}
Line 178: Line 182:
Comma list: 25/24, 33/32, 49/48
Comma list: 25/24, 33/32, 49/48


Mapping: [{{val|2 0 3 4 10}}, {{val|0 2 1 1 -2}}]
Mapping: {{mapping| 2 0 3 4 10 | 0 2 1 1 -2 }}


POTE generator: ~7/6 = 255.066
Optimal tuning (POTE): ~7/5 = 1\2, ~7/6 = 255.066


{{Optimal ET sequence|legend=1| 4, 10e, 14c }}
{{Optimal ET sequence|legend=1| 4, 10e, 14c }}
Line 191: Line 195:
Comma list: 25/24, 35/33, 49/48
Comma list: 25/24, 35/33, 49/48


Mapping: [{{val|2 0 3 4 7}}, {{val|0 2 1 1 0}}]
Mapping: {{mapping| 2 0 3 4 7 | 0 2 1 1 0 }}


POTE generator: ~8/7 = 243.493
Optimal tuning (POTE): ~7/5 = 1\2, ~8/7 = 243.493


{{Optimal ET sequence|legend=1| 4, 6, 10 }}
{{Optimal ET sequence|legend=1| 4, 6, 10 }}
Line 200: Line 204:


== Dichotic ==
== Dichotic ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 25/24, 64/63
[[Comma list]]: 25/24, 64/63


[[Mapping]]: [{{val|1 1 2 4}}, {{val|0 2 1 -4}}]
{{Mapping|legend=1| 1 1 2 4 | 0 2 1 -4 }}


{{Multival|legend=1|2 1 -4 -3 -12 -12}}
{{Multival|legend=1| 2 1 -4 -3 -12 -12 }}


[[POTE generator]]: ~5/4 = 356.264
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 356.264


{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
{{Optimal ET sequence|legend=1| 3, 7, 10, 17, 27c, 37c, 64bccc }}
Line 219: Line 223:
Comma list: 25/24, 45/44, 64/63
Comma list: 25/24, 45/44, 64/63


Mapping: [{{val|1 1 2 4 2}}, {{val|0 2 1 -4 5}}]
Mapping: {{mapping| 1 1 2 4 2 | 0 2 1 -4 5 }}


POTE generator: ~5/4 = 354.262
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.262


{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
Line 232: Line 236:
Comma list: 25/24, 40/39, 45/44, 64/63
Comma list: 25/24, 40/39, 45/44, 64/63


Mapping: [{{val|1 1 2 4 2 4}}, {{val|0 2 1 -4 5 -1}}]
Mapping: {{mapping| 1 1 2 4 2 4 | 0 2 1 -4 5 -1 }}


POTE generator: ~5/4 = 354.365
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.365


{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
{{Optimal ET sequence|legend=1| 7, 10, 17, 27ce, 44cce }}
Line 245: Line 249:
Comma list: 22/21, 25/24, 33/32
Comma list: 22/21, 25/24, 33/32


Mapping: [{{val|1 1 2 4 4}}, {{val|0 2 1 -4 -2}}]
Mapping: {{mapping| 1 1 2 4 4 | 0 2 1 -4 -2 }}


POTE generator: ~5/4 = 354.073
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.073


{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
Line 258: Line 262:
Comma list: 22/21, 25/24, 33/32, 40/39
Comma list: 22/21, 25/24, 33/32, 40/39


Mapping: [{{val|1 1 2 4 4 4}}, {{val|0 2 1 -4 -2 -1}}]
Mapping: {{mapping| 1 1 2 4 4 4 | 0 2 1 -4 -2 -1 }}


POTE generator: ~5/4 = 354.313
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.313


{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
{{Optimal ET sequence|legend=1| 3, 7, 10e, 17e }}
Line 271: Line 275:
Comma list: 25/24, 35/33, 64/63
Comma list: 25/24, 35/33, 64/63


Mapping: [{{val|1 1 2 4 5}}, {{val|0 2 1 -4 -5}}]
Mapping: {{mapping| 1 1 2 4 5 | 0 2 1 -4 -5 }}


POTE generator: ~5/4 = 360.659
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.659


{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
Line 284: Line 288:
Comma list: 25/24, 35/33, 40/39, 64/63
Comma list: 25/24, 35/33, 40/39, 64/63


Mapping: [{{val|1 1 2 4 5 4}}, {{val|0 2 1 -4 -5 -1}}]
Mapping: {{mapping| 1 1 2 4 5 4 | 0 2 1 -4 -5 -1 }}


POTE generator: ~5/4 = 360.646
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.646


{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
{{Optimal ET sequence|legend=1| 3, 7e, 10 }}
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== Jamesbond ==
== Jamesbond ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 25/24, 81/80
[[Comma list]]: 25/24, 81/80


[[Mapping]]: [{{val|7 11 16 0}}, {{val|0 0 0 1}}]
{{Mapping|legend=1| 7 11 16 0 | 0 0 0 1 }}


{{Multival|legend=1|0 0 7 0 11 16}}
{{Multival|legend=1| 0 0 7 0 11 16 }}


[[POTE generator]]: ~8/7 = 258.139
[[Optimal tuning]] ([[POTE]]): ~10/9 = 1\7, ~8/7 = 258.139


{{Optimal ET sequence|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 7, 14c }}
Line 312: Line 316:
Comma list: 25/24, 33/32, 45/44
Comma list: 25/24, 33/32, 45/44


Mapping: [{{val|7 11 16 0 24}}, {{val|0 0 0 1 0}}]
Mapping: {{mapping| 7 11 16 0 24 | 0 0 0 1 0 }}


POTE generator: ~8/7 = 258.910
Optimal tuning (POTE): ~10/9 = 1\7, ~8/7 = 258.910


{{Optimal ET sequence|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 7, 14c }}
Line 325: Line 329:
Comma list: 25/24, 27/26, 33/32, 40/39
Comma list: 25/24, 27/26, 33/32, 40/39


Mapping: [{{val|7 11 16 0 24 26}}, {{val|0 0 0 1 0 0}}]
Mapping: {{mapping| 7 11 16 0 24 26 | 0 0 0 1 0 0 }}


POTE generator: ~8/7 = 250.764
Optimal tuning (POTE): ~10/9 = 1\7, ~8/7 = 250.764


{{Optimal ET sequence|legend=1| 7, 14c }}
{{Optimal ET sequence|legend=1| 7, 14c }}
Line 338: Line 342:
Comma list: 25/24, 33/32, 45/44, 65/63
Comma list: 25/24, 33/32, 45/44, 65/63


Mapping: [{{val|7 11 16 0 24 6}}, {{val|0 0 0 1 0 1}}]
Mapping: {{mapping| 7 11 16 0 24 6 | 0 0 0 1 0 1 }}


POTE generator: ~8/7 = 247.445
Optimal tuning (POTE): ~10/9 = 1\7, ~8/7 = 247.445


{{Optimal ET sequence|legend=1| 7, 14cf }}
{{Optimal ET sequence|legend=1| 7, 14cf }}
Line 347: Line 351:


== Sidi ==
== Sidi ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 25/24, 245/243
[[Comma list]]: 25/24, 245/243


[[Mapping]]: [{{val|1 3 3 6}}, {{val|0 -4 -2 -9}}]
{{Mapping|legend=1| 1 3 3 6 | 0 -4 -2 -9 }}


{{Multival|legend=1|4 2 9 -12 3 15}}
{{Multival|legend=1| 4 2 9 -12 3 15 }}


[[POTE generator]]: ~9/7 = 427.208
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~9/7 = 427.208


{{Optimal ET sequence|legend=1| 3d, 14c, 45cc, 59bcccd }}
{{Optimal ET sequence|legend=1| 3d, 14c, 45cc, 59bcccd }}
Line 366: Line 370:
Comma list: 25/24, 45/44, 99/98
Comma list: 25/24, 45/44, 99/98


Mapping: [{{val|1 3 3 6 7}}, {{val|0 -4 -2 -9 -10}}]
Mapping: {{mapping| 1 3 3 6 7 | 0 -4 -2 -9 -10 }}


POTE generator: ~9/7 = 427.273
POTE generator: ~9/7 = 427.273
Line 375: Line 379:


== Quad ==
== Quad ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 9/8, 25/24
[[Comma list]]: 9/8, 25/24


[[Mapping]]: [{{val|4 6 9 0}}, {{val|0 0 0 1}}]
{{Mapping|legend=1| 4 6 9 0 | 0 0 0 1 }}


{{Multival|legend=1|0 0 4 0 6 9}}
{{Multival|legend=1|0 0 4 0 6 9}}


[[POTE generator]]: ~8/7 = 324.482
[[Optimal tuning]] ([[POTE]]): ~6/5 = 1\4, ~8/7 = 324.482


{{Optimal ET sequence|legend=1| 4 }}
{{Optimal ET sequence|legend=1| 4 }}
Line 391: Line 395:
[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot family| ]] <!-- main article -->
[[Category:Dicot]]
[[Category:Dicot| ]] <!-- key article -->
[[Category:Rank-2]]

Revision as of 13:08, 6 August 2023

The 5-limit parent comma for the dicot family is 25/24, the classical chromatic semitone. Its monzo is [-3 -1 2⟩, and flipping that yields ⟨⟨ 2 1 -3 ]] for the wedgie. This tells us the generator is a classical third (major and minor mean the same thing), and that two such thirds give a fifth. In fact, (5/4)2 = (3/2)(25/24).

Possible tunings for dicot are 7edo, 17edo, 24edo using the val ⟨24 38 55] (24c) and 31edo using the val ⟨31 49 71] (31c). In a sense, what dicot is all about is using neutral thirds and pretending that is 5-limit, and like any temperament which seems to involve pretending, dicot is at the edge of what can sensibly be called a temperament at all. In other words, it is an exotemperament.

Dicot

Subgroup: 2.3.5

Comma list: 25/24

Mapping: [⟨1 1 2], ⟨0 2 1]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 348.594

Tuning ranges:

  • 5-odd-limit diamond monotone: ~5/4 = [300.000, 400.000] (1\4 to 1\3)
  • 5-odd-limit diamond tradeoff: ~5/4 = [315.641, 386.314] (full comma to untempered)
  • 5-odd-limit diamond monotone and tradeoff: ~5/4 = [315.641, 386.314]

Optimal ET sequence: 3, 4, 7, 17, 24c, 31c

Badness: 0.013028

Overview to extensions

The second comma of the normal comma list defines which 7-limit family member we are looking at. Septimal dicot, with wedgie ⟨⟨ 2 1 3 -3 -1 4 ]] adds 36/35, sharp with wedgie ⟨⟨ 2 1 6 -3 4 11 ]] adds 28/27, and dichotic with wedgie ⟨⟨ 2 1 -4 -3 -12 -12 ]] adds 64/63, all retaining the same period and generator.

Decimal with wedgie ⟨⟨ 4 2 2 -6 -8 -1 ]] adds 49/48, sidi with wedgie ⟨⟨ 4 2 9 -3 6 15 ]] adds 245/243, and jamesbond with wedgie ⟨⟨ 0 0 7 0 11 16 ]] adds 81/80. Here decimal divides the period to 1/2 octave, and sidi uses 9/7 as a generator, with two of them making up the combined 5/3 and 8/5 neutral sixth. Jamesbond has a period of 1/7 octave, and uses an approximate 15/14 as generator.

Septimal dicot

Subgroup: 2.3.5.7

Comma list: 15/14, 25/24

Mapping: [⟨1 1 2 2], ⟨0 2 1 3]]

Wedgie: ⟨⟨ 2 1 3 -3 -1 4 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.381

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness: 0.019935

11-limit

Subgroup: 2.3.5.7.11

Comma list: 15/14, 22/21, 25/24

Mapping: [⟨1 1 2 2 2], ⟨0 2 1 3 5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 342.125

Optimal ET sequence: 3de, 4e, 7

Badness: 0.019854

Eudicot

Subgroup: 2.3.5.7.11

Comma list: 15/14, 25/24, 33/32

Mapping: [⟨1 1 2 2 4], ⟨0 2 1 3 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 336.051

Optimal ET sequence: 3d, 4, 7, 18bc, 25bccd

Badness: 0.027114

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 15/14, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 2 4 4], ⟨0 2 1 3 -2 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 338.846

Optimal ET sequence: 3d, 4, 7, 25bccd, 32bccddef, 39bcccdddef

Badness: 0.023828

Flat

Subgroup: 2.3.5.7

Comma list: 21/20, 25/24

Mapping: [⟨1 1 2 3], ⟨0 2 1 -1]]

Wedgie: ⟨⟨ 2 1 -1 -3 -7 -5 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 331.916

Optimal ET sequence: 3, 4, 7d, 11cd, 18bcddd

Badness: 0.025381

11-limit

Subgroup: 2.3.5.7.11

Comma list: 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4], ⟨0 2 1 -1 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 337.532

Optimal ET sequence: 3, 4, 7d

Badness: 0.024988

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 14/13, 21/20, 25/24, 33/32

Mapping: [⟨1 1 2 3 4 4], ⟨0 2 1 -1 -2 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 341.023

Optimal ET sequence: 3, 4, 7d

Badness: 0.023420

Sharp

Subgroup: 2.3.5.7

Comma list: 25/24, 28/27

Mapping: [⟨1 1 2 1], ⟨0 2 1 6]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 357.938

Wedgie: ⟨⟨ 2 1 6 -3 4 11 ]]

Optimal ET sequence: 3d, 7d, 10, 37cd, 47bccd, 57bccdd

Badness: 0.028942

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 28/27, 35/33

Mapping: [⟨1 1 2 1 2], ⟨0 2 1 6 5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.106

Optimal ET sequence: 3de, 7d, 10, 17d, 27cde

Badness: 0.022366

Decimal

Subgroup: 2.3.5.7

Comma list: 25/24, 49/48

Mapping: [⟨2 0 3 4], ⟨0 2 1 1]]

Wedgie: ⟨⟨ 4 2 2 -6 -8 -1 ]]

Optimal tuning (POTE): ~7/5 = 1\2, ~7/6 = 251.557

Optimal ET sequence: 4, 10, 14c, 24c, 38ccd, 62cccdd

Badness: 0.028334

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 49/48

Mapping: [⟨2 0 3 4 -1], ⟨0 2 1 1 5]]

Optimal tuning (POTE): ~7/5 = 1\2, ~7/6 = 253.493

Optimal ET sequence: 10, 14c, 24c, 38ccd, 52cccde

Badness: 0.026712

Decimated

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 49/48

Mapping: [⟨2 0 3 4 10], ⟨0 2 1 1 -2]]

Optimal tuning (POTE): ~7/5 = 1\2, ~7/6 = 255.066

Optimal ET sequence: 4, 10e, 14c

Badness: 0.031456

Decibel

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 49/48

Mapping: [⟨2 0 3 4 7], ⟨0 2 1 1 0]]

Optimal tuning (POTE): ~7/5 = 1\2, ~8/7 = 243.493

Optimal ET sequence: 4, 6, 10

Badness: 0.032385

Dichotic

Subgroup: 2.3.5.7

Comma list: 25/24, 64/63

Mapping: [⟨1 1 2 4], ⟨0 2 1 -4]]

Wedgie: ⟨⟨ 2 1 -4 -3 -12 -12 ]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 356.264

Optimal ET sequence: 3, 7, 10, 17, 27c, 37c, 64bccc

Badness: 0.037565

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 64/63

Mapping: [⟨1 1 2 4 2], ⟨0 2 1 -4 5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.262

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness: 0.030680

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 40/39, 45/44, 64/63

Mapping: [⟨1 1 2 4 2 4], ⟨0 2 1 -4 5 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.365

Optimal ET sequence: 7, 10, 17, 27ce, 44cce

Badness: 0.021674

Dichotomic

Subgroup: 2.3.5.7.11

Comma list: 22/21, 25/24, 33/32

Mapping: [⟨1 1 2 4 4], ⟨0 2 1 -4 -2]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.073

Optimal ET sequence: 3, 7, 10e, 17e

Badness: 0.031719

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 22/21, 25/24, 33/32, 40/39

Mapping: [⟨1 1 2 4 4 4], ⟨0 2 1 -4 -2 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 354.313

Optimal ET sequence: 3, 7, 10e, 17e

Badness: 0.022741

Dichosis

Subgroup: 2.3.5.7.11

Comma list: 25/24, 35/33, 64/63

Mapping: [⟨1 1 2 4 5], ⟨0 2 1 -4 -5]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.659

Optimal ET sequence: 3, 7e, 10

Badness: 0.041361

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 35/33, 40/39, 64/63

Mapping: [⟨1 1 2 4 5 4], ⟨0 2 1 -4 -5 -1]]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 360.646

Optimal ET sequence: 3, 7e, 10

Badness: 0.027938

Jamesbond

Subgroup: 2.3.5.7

Comma list: 25/24, 81/80

Mapping: [⟨7 11 16 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨ 0 0 7 0 11 16 ]]

Optimal tuning (POTE): ~10/9 = 1\7, ~8/7 = 258.139

Optimal ET sequence: 7, 14c

Badness: 0.041714

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 33/32, 45/44

Mapping: [⟨7 11 16 0 24], ⟨0 0 0 1 0]]

Optimal tuning (POTE): ~10/9 = 1\7, ~8/7 = 258.910

Optimal ET sequence: 7, 14c

Badness: 0.023524

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 27/26, 33/32, 40/39

Mapping: [⟨7 11 16 0 24 26], ⟨0 0 0 1 0 0]]

Optimal tuning (POTE): ~10/9 = 1\7, ~8/7 = 250.764

Optimal ET sequence: 7, 14c

Badness: 0.023003

Septimal

Subgroup: 2.3.5.7.11.13

Comma list: 25/24, 33/32, 45/44, 65/63

Mapping: [⟨7 11 16 0 24 6], ⟨0 0 0 1 0 1]]

Optimal tuning (POTE): ~10/9 = 1\7, ~8/7 = 247.445

Optimal ET sequence: 7, 14cf

Badness: 0.022569

Sidi

Subgroup: 2.3.5.7

Comma list: 25/24, 245/243

Mapping: [⟨1 3 3 6], ⟨0 -4 -2 -9]]

Wedgie: ⟨⟨ 4 2 9 -12 3 15 ]]

Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 427.208

Optimal ET sequence: 3d, 14c, 45cc, 59bcccd

Badness: 0.056586

11-limit

Subgroup: 2.3.5.7.11

Comma list: 25/24, 45/44, 99/98

Mapping: [⟨1 3 3 6 7], ⟨0 -4 -2 -9 -10]]

POTE generator: ~9/7 = 427.273

Optimal ET sequence: 3de, 14c, 45cce, 59bcccdee

Badness: 0.032957

Quad

Subgroup: 2.3.5.7

Comma list: 9/8, 25/24

Mapping: [⟨4 6 9 0], ⟨0 0 0 1]]

Wedgie: ⟨⟨ 0 0 4 0 6 9 ]]

Optimal tuning (POTE): ~6/5 = 1\4, ~8/7 = 324.482

Optimal ET sequence: 4

Badness: 0.045911

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