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Semicomma family: Difference between revisions

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{{Technical data page}}
The [[5-limit]] parent [[comma]] for the '''semicomma family''' of [[regular temperament|temperaments]] is the [[semicomma]] ({{monzo|legend=1| -21 3 7 }}, [[ratio]]: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.
The [[5-limit]] parent [[comma]] for the '''semicomma family''' of [[regular temperament|temperaments]] is the [[semicomma]] ({{monzo|legend=1| -21 3 7 }}, [[ratio]]: 2109375/2097152). This is the amount by which three pure 3/1 twelfths exceed seven pure 8/5 minor sixths.


Line 12: Line 13:
: mapping generators: ~2, ~75/64
: mapping generators: ~2, ~75/64


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~75/64 = 271.627
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.670
: [[error map]]: {{val| 0.000 -0.264 -1.324 }}
* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.627
: error map: {{val| 0.000 -0.564 -1.195 }}


[[Tuning ranges]]:
[[Tuning ranges]]:
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=== 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. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add
The second comma of the [[Normal lists #Normal interval list|normal comma list]] defines which 7-limit family member we are looking at. Adding 65625/65536 (or 225/224) leads to orwell, but we could also add
* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell) with wedgie {{multival| 21 -9 -7 -63 -70 9 }}, or
* 1029/1024, leading to the {{nowrap| 31 & 159 }} temperament (triwell), or
* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell) with wedgie {{multival| 28 -12 1 -84 -77 36 }}, or
* 2401/2400, giving the {{nowrap| 31 & 243 }} temperament (quadrawell), or
* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric) with wedgie {{multival| 7 -3 61 -21 77 150 }}.
* 4375/4374, giving the {{nowrap| 53 & 243 }} temperament (sabric).


== Orwell ==
== Orwell ==
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{{Mapping|legend=1| 1 0 3 1 | 0 7 -3 8 }}
{{Mapping|legend=1| 1 0 3 1 | 0 7 -3 8 }}


{{Multival|legend=1| 7 -3 8 -21 -7 27 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~7/6 = 271.513
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~7/6 = 271.509
: [[error map]]: {{val| 0.000 -1.364 -0.853 +3.278 }}
* [[POTE]]: ~2 = 1200.000, ~7/6 = 271.509
: error map: {{val| 0.000 -1.394 -0.840 +3.243 }}


[[Minimax tuning]]:
[[Minimax tuning]]:
* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
* [[7-odd-limit]]: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}
: {{monzo list| 1 0 0 0 | 14/11 0 -7/11 7/11 | 27/11 0 3/11 -3/11 | 27/11 0 -8/11 8/11 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.7/5
* 9-odd-limit: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
* [[9-odd-limit]]: ~7/6 = {{monzo| 3/17 2/17 -1/17 }}
: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}
: {{monzo list| 1 0 0 0 | 21/17 14/17 -7/17 0 | 42/17 -6/17 3/17 0 | 41/17 16/17 -8/17 0 }}
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/5
: [[Eigenmonzo basis|unchanged-interval (eigenmonzo) basis]]: 2.9/5


[[Tuning ranges]]:
[[Tuning ranges]]:
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Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}
Mapping: {{mapping| 1 0 3 1 3 | 0 7 -3 8 2 }}


{{Multival|legend=1| 7 -3 8 2 -21 -7 -21 27 15 -22 }}
Optimal tunings:
 
* CTE: ~2 = 1200.000, ~7/6 = 271.560
Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 271.426
* POTE: ~2 = 1200.000, ~7/6 = 271.426


Minimax tuning:
Minimax tuning:
* 11-odd-limit: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
* 11-odd-limit: ~7/6 = {{monzo| 2/11 0 -1/11 1/11 }}
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 0 }}, {{monzo| 27/11 0 3/11 -3/11 0 }}, {{monzo| 27/11 0 -8/11 8/11 0 }}, {{monzo| 37/11 0 -2/11 2/11 0 }}]
: [{{monzo| 1 0 0 0 0 }}, {{monzo| 14/11 0 -7/11 7/11 0 }}, {{monzo| 27/11 0 3/11 -3/11 0 }}, {{monzo| 27/11 0 -8/11 8/11 0 }}, {{monzo| 37/11 0 -2/11 2/11 0 }}]
: Eigenmonzo (unchanged-interval) basis: 2.7/5
: Unchanged-interval (eigenmonzo) basis: 2.7/5


Tuning ranges:
Tuning ranges:
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Mapping: {{mapping| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}
Mapping: {{mapping| 1 0 3 1 3 8 | 0 7 -3 8 2 -19 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 271.546
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.556
* POTE: ~2 = 1200.000, ~7/6 = 271.546


Tuning ranges:
Tuning ranges:
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* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 275.659]


{{Optimal ET sequence|legend=0| 22, 31, 53, 84e, 137e }}
{{Optimal ET sequence|legend=0| 22, 31, 53, 84e }}


Badness (Smith): 0.019718
Badness (Smith): 0.019718
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Mapping: {{mapping| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}
Mapping: {{mapping| 1 0 3 1 3 3 | 0 7 -3 8 2 3 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 271.301
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.747
* POTE: ~2 = 1200.000, ~7/6 = 271.301


{{Optimal ET sequence|legend=0| 9, 22, 31f }}
{{Optimal ET sequence|legend=0| 9, 22, 31f }}
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Mapping: {{mapping| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}
Mapping: {{mapping| 1 0 3 1 3 1 | 0 7 -3 8 2 12 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 271.088
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.163
* POTE: ~2 = 1200.000, ~7/6 = 271.088


Tuning ranges:
Tuning ranges:
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* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]
* 13- and 15-odd-limit diamond tradeoff: ~7/6 = [266.871, 281.691]


{{Optimal ET sequence|legend=0| 22f, 31 }}
{{Optimal ET sequence|legend=0| 9, 22f, 31 }}


Badness (Smith): 0.019931
Badness (Smith): 0.019931


==== Doublethink ====
==== Doublethink ====
Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two [[13/12]]~[[14/13]]'s by tempering out their difference, [[169/168]].
Doublethink is a weak extension of orwell to the 13-limit. It splits the generator of ~7/6 into two [[13/12]]~[[14/13]]'s by tempering out their difference, [[169/168]]. Its ploidacot is alpha-tetradecacot.  


Subgroup: 2.3.5.7.11.13
Subgroup: 2.3.5.7.11.13
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Mapping: {{mapping| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}
Mapping: {{mapping| 1 0 3 1 3 2 | 0 14 -6 16 4 15 }}


Optimal tuning (POTE): ~2 = 1200.000, ~13/12 = 135.723
Optimal tunings:
* CTE: ~2 = 1200.000, ~13/12 = 135.811
* POTE: ~2 = 1200.000, ~13/12 = 135.723


Tuning ranges:
Tuning ranges:
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* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]
* 13- and 15-odd-limit diamond tradeoff: ~13/12 = [128.298, 138.573]


{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 62, 115ef, 168eef }}
{{Optimal ET sequence|legend=0| 9, 35bd, 44, 53, 62, 115ef }}


Badness (Smith): 0.027120
Badness (Smith): 0.027120
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Mapping: {{mapping| 1 0 3 1 -4 | 0 7 -3 8 33 }}
Mapping: {{mapping| 1 0 3 1 -4 | 0 7 -3 8 33 }}


Optimal tuning (POTE): ~2 = 1200.000, ~7/6 = 271.288
Optimal tunings:
* CTE: ~2 = 1200.000, ~7/6 = 271.316
* POTE: ~2 = 1200.000, ~7/6 = 271.288


Tuning ranges:
Tuning ranges:
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* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]
* 11-odd-limit diamond tradeoff: ~7/6 = [266.871, 272.514]


{{Optimal ET sequence|legend=0| 31, 84, 115, 376b, 491bd, 606bde }}
{{Optimal ET sequence|legend=0| 22e, 31, 84, 115 }}


Badness (Smith): 0.031438
Badness (Smith): 0.031438
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Mapping: {{mapping| 1 7 0 9 17 | 0 -14 6 -16 -35 }}
Mapping: {{mapping| 1 7 0 9 17 | 0 -14 6 -16 -35 }}


Optimal tuning (POTE): ~2 = 1200.000, ~55/36 = 735.752
: mapping generators: ~2, ~72/55
 
Optimal tunings:
* CTE: ~2 = 1200.000, ~55/36 = 735.754
* POTE: ~2 = 1200.000, ~55/36 = 735.752


{{Optimal ET sequence|legend=0| 31, 106, 137, 442bd }}
{{Optimal ET sequence|legend=0| 31, 75e, 106, 137 }}


Badness (Smith): 0.038377
Badness (Smith): 0.038377
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{{Mapping|legend=1| 1 0 3 -11 | 0 7 -3 61 }}
{{Mapping|legend=1| 1 0 3 -11 | 0 7 -3 61 }}


{{Multival|legend=1| 7 -3 61 -21 77 150 }}
[[Optimal tuning]]s:
 
* [[CTE]]: ~2 = 1200.000, ~75/64 = 271.622
[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~75/64 = 271.607
: [[error map]]: {{val| 0.000 -0.599 -1.180 +0.131 }}
* [[POTE]]: ~2 = 1200.000, ~75/64 = 271.607
: error map: {{val| 0.000 -0.707 -1.134 -0.808 }}


{{Optimal ET sequence|legend=1| 53, 137d, 190, 243 }}
{{Optimal ET sequence|legend=1| 53, 137d, 190, 243, 1511bccd }}


[[Badness]] (Smith): 0.088355
[[Badness]] (Smith): 0.088355


== Triwell ==
== Triwell ==
The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the fifth harmonic.
The triwell temperament ({{nowrap| 31 & 159 }}) slices orwell major sixth ~128/75 into three generators, nine of which give the 5th harmonic.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
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{{Mapping|legend=1| 1 7 0 1 | 0 -21 9 7 }}
{{Mapping|legend=1| 1 7 0 1 | 0 -21 9 7 }}


{{Multival|legend=1| 21 -9 -7 -63 -70 9 }}
: mapping generators: ~2, ~448/375


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~448/375 = 309.472
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~448/375 = 309.456
: [[error map]]: {{val| 0.000 -0.522 -1.213 -2.637 }}
* [[POTE]]: ~2 = 1200.000, ~448/375 = 309.472
: error map: {{val| 0.000 -0.872 -1.063 -2.520 }}


{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}
{{Optimal ET sequence|legend=1| 31, 97, 128, 159, 190 }}
Line 228: Line 255:
Mapping: {{mapping| 1 7 0 1 13 | 0 -21 9 7 -37 }}
Mapping: {{mapping| 1 7 0 1 13 | 0 -21 9 7 -37 }}


Optimal tuning (POTE): ~2 = 1200.000, ~448/375 = 309.471
Optimal tunings:
* CTE: ~2 = 1200.000, ~448/375 = 309.444
* POTE: ~2 = 1200.000, ~448/375 = 309.471


{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}
{{Optimal ET sequence|legend=0| 31, 97, 128, 159, 190 }}
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== Quadrawell ==
== Quadrawell ==
The ''quadrawell'' temperament ({{nowrap| 31 & 212 }}) has an [[8/7]] generator of about 232 cents, twelve of which give the fifth harmonic.
The ''quadrawell'' temperament ({{nowrap| 31 & 212 }}) has an [[8/7]] generator of about 232 cents, twelve of which give the 5th harmonic.


[[Subgroup]]: 2.3.5.7
[[Subgroup]]: 2.3.5.7
Line 243: Line 272:
{{Mapping|legend=1| 1 7 0 3 | 0 -28 12 -1 }}
{{Mapping|legend=1| 1 7 0 3 | 0 -28 12 -1 }}


{{Multival|legend=1| 28 -12 1 -84 -77 36 }}
: mapping generators: ~2, ~8/7


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~8/7 = 232.094
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~8/7 = 232.082
: [[error map]]: {{val| 0.000 -0.255 -1.328 -0.908 }}
* [[POTE]]: ~2 = 1200.000, ~8/7 = 232.094
: error map: {{val| 0.000 -0.574 -1.191 -0.919 }}


{{Optimal ET sequence|legend=1| 31, 119, 150, 181, 212, 243, 698cd, 941cd }}
{{Optimal ET sequence|legend=1| 31, 119, 150, 181, 212, 243, 698cd, 941cd }}
Line 258: Line 291:
Mapping: {{mapping| 1 7 0 3 11 | 0 -28 12 -1 -39 }}
Mapping: {{mapping| 1 7 0 3 11 | 0 -28 12 -1 -39 }}


Optimal tuning (POTE): ~2 = 1200.000, ~8/7 = 232.083
Optimal tunings:
* CTE: ~2 = 1200.000, ~8/7 = 232.065
* POTE: ~2 = 1200.000, ~8/7 = 232.083


{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}
{{Optimal ET sequence|legend=0| 31, 119, 150, 181, 212, 455ee, 667cdee }}
Line 273: Line 308:
{{Mapping|legend=1| 1 14 -3 6 | 0 -35 15 -9 }}
{{Mapping|legend=1| 1 14 -3 6 | 0 -35 15 -9 }}


{{Multival|legend=1| 35 -15 9 -105 -84 63 }}
: mapping generators: ~2, ~2625/2048


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~2625/2048 = 425.673
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~2625/2048 = 425.666
: [[error map]]: {{val| 0.000 -0.278 -1.318 0.177 }}
* [[POTE]]: ~2 = 1200.000, ~2625/2048 = 425.673
: error map: {{val| 0.000 -0.526 -1.212 0.113 }}


{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}
{{Optimal ET sequence|legend=1| 31, 172, 203, 234, 265, 296 }}
Line 288: Line 327:
Mapping: {{mapping| 1 14 -3 6 29 | 0 -35 15 -9 -72 }}
Mapping: {{mapping| 1 14 -3 6 29 | 0 -35 15 -9 -72 }}


Optimal tuning (POTE): ~2 = 1200.000, ~2625/2048 = 425.679
Optimal tunings:
* CTE: ~2 = 1200.000, ~2625/2048 = 425.671
* POTE: ~2 = 1200.000, ~2625/2048 = 425.679


{{Optimal ET sequence|legend=0| 31, 172e, 203e, 234, 265, 296, 919bc, 1215bcc, 1511bcc }}
{{Optimal ET sequence|legend=0| 31, 234, 265, 296, 919bc }}


Badness (Smith): 0.052774
Badness (Smith): 0.052774
Line 303: Line 344:
{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}
{{Mapping|legend=1| 1 0 3 0 | 0 35 -15 62 }}


{{Multival|legend=1| 35 -15 62 -105 0 186 }}
: mapping generators: ~2, ~405/392


[[Optimal tuning]] ([[POTE]]): ~2 = 1200.000, ~405/392 = 54.324
[[Optimal tuning]]s:
* [[CTE]]: ~2 = 1200.000, ~405/392 = 54.335
: [[error map]]: {{val| 0.000 -0.233 -1.338 -0.061 }}
* [[POTE]]: ~2 = 1200.000, ~405/392 = 54.324
: error map: {{val| 0.000 -0.604 -1.178 -0.718 }}


{{Optimal ET sequence|legend=1| 22, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}
{{Optimal ET sequence|legend=1| 22, …, 199d, 221, 243, 751c, 994cd, 1237bccd, 1480bccd }}


[[Badness]] (Smith): 0.168897
[[Badness]] (Smith): 0.168897
Line 318: Line 363:
Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}
Mapping: {{mapping| 1 0 3 0 5 | 0 35 -15 62 -34 }}


Optimal tuning (POTE): ~2 = 1200.000, ~33/32 = 54.334
Optimal tunings:
* CTE: ~2 = 1200.000, ~33/32 = 54.338
* POTE: ~2 = 1200.000, ~33/32 = 54.334


{{Optimal ET sequence|legend=0| 22, 221, 243, 265, 773ce, 1038ccee, 1303ccee }}
{{Optimal ET sequence|legend=0| 22, 221, 243, 265 }}


Badness (Smith): 0.097202
Badness (Smith): 0.097202
Line 331: Line 378:
Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}
Mapping: {{mapping| 1 0 3 0 4 | 0 35 -15 62 -12 }}


Optimal tuning (POTE): ~2 = 1200.000, ~405/392 = 54.316
Optimal tunings:
* CTE: ~2 = 1200.000, ~405/392 = 54.332
* POTE: ~2 = 1200.000, ~405/392 = 54.316


{{Optimal ET sequence|legend=0| 22, 199d, 221e, 243e }}
{{Optimal ET sequence|legend=0| 22, …, 199d, 221e, 243e, 707bcdeee }}


Badness (Smith): 0.078657
Badness (Smith): 0.078657


[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Pages with mostly numerical content]]
[[Category:Semicomma family| ]] <!-- main article -->
[[Category:Semicomma family| ]] <!-- main article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Orson]]
[[Category:Orson]]
[[Category:Orwell]]
[[Category:Orwell]]
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