Alphatricot family: Difference between revisions

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== Tricot ==
== Tricot ==
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: {{monzo | 39 -29 3 }} = 68719476736000/68630377364883
[[Comma list]]: {{monzo| 39 -29 3 }} = 68719476736000/68630377364883


[[Mapping]]: [{{val| 1 0 -13 }}, {{val| 0 3 29 }}]
{{Mapping|legend=1| 1 0 -13 | 0 3 29 }}


Mapping generators: ~2, ~59049/40960
: mapping generators: ~2, ~59049/40960


{{Multival|legend=1| 3 29 39 }}
{{Multival|legend=1| 3 29 39 }}


[[POTE generator]]: ~59049/40960 = 634.012
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~59049/40960 = 634.012


{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
{{Optimal ET sequence|legend=1| 53, 229, 282, 335, 388, 441, 1376, 1817, 2258 }}
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[[Comma list]]: 2430/2401, 5120/5103
[[Comma list]]: 2430/2401, 5120/5103


[[Mapping]]: [{{val| 1 0 -13 -3 }}, {{val| 0 3 29 11 }}]
{{Mapping|legend=1| 1 0 -13 -3 | 0 3 29 11 }}


{{Multival|legend=1| 3 29 11 39 9 -56 }}
{{Multival|legend=1| 3 29 11 39 9 -56 }}
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Comma list: 99/98, 121/120, 5120/5103
Comma list: 99/98, 121/120, 5120/5103


Mapping: [{{val| 1 0 -13 -3 -5 }}, {{val| 0 3 29 11 16 }}]
Mapping: {{mapping| 1 0 -13 -3 -5 | 0 3 29 11 16 }}


Optimal tuning (POTE): ~2 = 1\1, ~63/44 = 634.0273
Optimal tuning (POTE): ~2 = 1\1, ~63/44 = 634.0273
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Comma list: 99/98, 121/120, 169/168, 352/351
Comma list: 99/98, 121/120, 169/168, 352/351


Mapping: [{{val| 1 0 -13 -3 -5 0 }}, {{val| 0 3 29 11 16 7 }}]
Mapping: {{mapping| 1 0 -13 -3 -5 0 | 0 3 29 11 16 7 }}


Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0115
Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0115
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== Trident ==
== Trident ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 6144/6125, 14348907/14336000
[[Comma list]]: 6144/6125, 14348907/14336000


[[Mapping]]: [{{val| 1 0 -13 25 }}, {{val| 0 3 29 -42 }}]
{{Mapping|legend=1| 1 0 -13 25 | 0 3 29 -42 }}


{{Multival|legend=1| 3 29 -42 39 -75 -179 }}
{{Multival|legend=1| 3 29 -42 39 -75 -179 }}


[[POTE generator]]: ~4096/2835 = 634.0480
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~4096/2835 = 634.0480


{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511 }}
{{Optimal ET sequence|legend=1| 53, 176, 229, 282, 511 }}
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Comma list: 3388/3375, 6144/6125, 8019/8000
Comma list: 3388/3375, 6144/6125, 8019/8000


Mapping: [{{val| 1 0 -13 25 -33 }}, {{val| 0 3 29 -42 69 }}]
Mapping: {{mapping| 1 0 -13 25 -33 | 0 3 29 -42 69 }}


POTE generator: ~231/160 = 634.0669
Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0669


{{Optimal ET sequence|legend=1| 53, 176, 229 }}
{{Optimal ET sequence|legend=1| 53, 176, 229 }}
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Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125
Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125


Mapping: [{{val| 1 0 -13 25 -33 0 }}, {{val| 0 3 29 -42 69 7 }}]
Mapping: {{mapping| 1 0 -13 25 -33 0 | 0 3 29 -42 69 7 }}


POTE generator: ~13/9 = 634.0652
Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0652


{{Optimal ET sequence|legend=1| 53, 176, 229 }}
{{Optimal ET sequence|legend=1| 53, 176, 229 }}
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== Trillium ==
== Trillium ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 4375/4374, 1099511627776/1098337086315
[[Comma list]]: 4375/4374, 1099511627776/1098337086315


[[Mapping]]: [{{val| 1 0 -13 53 }}, {{val| 0 3 29 -95 }}]
{{Mapping|legend=1| 1 0 -13 53 | 0 3 29 -95 }}


{{Multival|legend=1| 3 29 -95 39 -159 -302 }}
{{Multival|legend=1| 3 29 -95 39 -159 -302 }}


[[POTE generator]]: ~23625/16384 = 634.0118
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~23625/16384 = 634.0118


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1376, 3193, 4569 }}
{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1376, 3193, 4569 }}
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Comma list: 4375/4374, 131072/130977, 759375/758912
Comma list: 4375/4374, 131072/130977, 759375/758912


Mapping: [{{val| 1 0 -13 53 -89 }}, {{val| 0 3 29 -95 175 }}]
Mapping: {{mapping| 1 0 -13 53 -89 | 0 3 29 -95 175 }}


POTE generator: ~3888/2695 = 634.0094
Optimal tuning (POTE): ~2 = 1\1, ~3888/2695 = 634.0094


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
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Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078
Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078


Mapping: [{{val| 1 0 -13 53 -89 -28 }}, {{val| 0 3 29 -95 175 60 }}]
Mapping: {{mapping| 1 0 -13 53 -89 -28 | 0 3 29 -95 175 60 }}


POTE generator: ~75/52 = 634.0095
Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0095


{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
{{Optimal ET sequence|legend=1| 53, 441, 494, 935, 1429 }}
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Comma list: 4375/4374, 5632/5625, 4108797/4096000
Comma list: 4375/4374, 5632/5625, 4108797/4096000


Mapping: [{{val| 1 0 -13 53 -61 }}, {{val| 0 3 29 -95 122 }}]
Mapping: {{mapping| 1 0 -13 53 -61 | 0 3 29 -95 122 }}


POTE generator: ~231/160 = 634.0190
Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0190


{{Optimal ET sequence|legend=1| 53, 335, 388 }}
{{Optimal ET sequence|legend=1| 53, 335, 388 }}
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Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374
Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374


Mapping: [{{val| 1 0 -13 53 -61 -28 }}, {{val| 0 3 29 -95 122 60 }}]
Mapping: {{mapping| 1 0 -13 53 -61 -28 | 0 3 29 -95 122 60 }}


POTE generator: ~75/52 = 634.0181
Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0181


{{Optimal ET sequence|legend=1| 53, 335, 388 }}
{{Optimal ET sequence|legend=1| 53, 335, 388 }}
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== Tritricot ==
== Tritricot ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 250047/250000, 11785390260224/11767897353375
[[Comma list]]: 250047/250000, 11785390260224/11767897353375


[[Mapping]]: [{{val| 3 6 19 30 }}, {{val| 0 -3 -29 -52 }}]
{{Mapping|legend=1| 3 6 19 30 | 0 -3 -29 -52 }}


{{Multival|legend=1| 9 87 156 117 222 118 }}
{{Multival|legend=1| 9 87 156 117 222 118 }}


[[POTE generator]]: ~100352/91125 = 165.9837
[[Optimal tuning]] ([[POTE]]): ~63/50 = 1\3, ~100352/91125 = 165.9837


{{Optimal ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }}
{{Optimal ET sequence|legend=1| 159, 282, 441, 2487, 2928, 3369 }}
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Comma list: 4000/3993, 166698/166375, 200704/200475
Comma list: 4000/3993, 166698/166375, 200704/200475


Mapping: [{{val| 3 6 19 30 22 }}, {{val| 0 -3 -29 -52 -28 }}]
Mapping: {{mapping| 3 6 19 30 22 | 0 -3 -29 -52 -28 }}


POTE generator: ~11/10 = 165.9835
Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9835


{{Optimal ET sequence|legend=1| 159, 282, 441 }}
{{Optimal ET sequence|legend=1| 159, 282, 441 }}
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Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875
Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875


Mapping: [{{val| 3 6 19 30 22 36 }}, {{val| 0 -3 -29 -52 -28 -60 }}]
Mapping: {{mapping| 3 6 19 30 22 36 | 0 -3 -29 -52 -28 -60 }}


POTE generator: ~11/10 = 165.9842
Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9842


{{Optimal ET sequence|legend=1| 159, 282, 441 }}
{{Optimal ET sequence|legend=1| 159, 282, 441 }}
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Mapping: [{{val| 3 6 19 30 22 36 16 }}, {{val| 0 -3 -29 -52 -28 -60 -9 }}]
Mapping: [{{val| 3 6 19 30 22 36 16 }}, {{val| 0 -3 -29 -52 -28 -60 -9 }}]


POTE generator: ~11/10 = 165.9805
Optimal tuning (POTE): ~34/27 = 1\3, ~11/10 = 165.9805


{{Optimal ET sequence|legend=1| 159, 282, 441 }}
{{Optimal ET sequence|legend=1| 159, 282, 441 }}
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[[Category:Temperament families]]
[[Category:Temperament families]]
[[Category:Tricot family| ]] <!-- main article -->
[[Category:Tricot family| ]] <!-- main article -->
[[Category:Tricot| ]] <!-- key article -->
[[Category:Rank 2]]
[[Category:Rank 2]]
[[Category:Tricot]]

Revision as of 11:43, 13 September 2023

The generator for tricot temperament is the real cube root of third harmonic, 31/3, tuned between 63/44 and 13/9. Tricot temperament can be described as 53&70 temperament, tempering out the tricot comma, [39 -29 3⟩ in the 5-limit. There are some mappings for 7-limit extension of this temperament: septimal tricot (53&70, also called as "trimot"), trident (53&229) and trillium (53&441). Tempering out hemifamity comma (5120/5103) leads to septimal tricot, porwell comma (6144/6125) leads to trident, and ragisma (4375/4374) leads to trillium.

Tricot

Subgroup: 2.3.5

Comma list: [39 -29 3⟩ = 68719476736000/68630377364883

Mapping: [⟨1 0 -13], ⟨0 3 29]]

mapping generators: ~2, ~59049/40960

Wedgie: ⟨⟨ 3 29 39 ]]

Optimal tuning (POTE): ~2 = 1\1, ~59049/40960 = 634.012

Optimal ET sequence: 53, 229, 282, 335, 388, 441, 1376, 1817, 2258

Badness: 0.046093

Septimal tricot aka trimot

Subgroup: 2.3.5.7

Comma list: 2430/2401, 5120/5103

Mapping: [⟨1 0 -13 -3], ⟨0 3 29 11]]

Wedgie: ⟨⟨ 3 29 11 39 9 -56 ]]

Optimal tuning (POTE): ~2 = 1\1, ~81/56 = 634.0259

Optimal ET sequence: 17c, 36c, 53, 70, 229dd, 282dd

Badness: 0.100127

11-limit

Subgroup: 2.3.5.7.11

Comma list: 99/98, 121/120, 5120/5103

Mapping: [⟨1 0 -13 -3 -5], ⟨0 3 29 11 16]]

Optimal tuning (POTE): ~2 = 1\1, ~63/44 = 634.0273

Optimal ET sequence: 17c, 36ce, 53, 70, 123de

Badness: 0.056134

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 99/98, 121/120, 169/168, 352/351

Mapping: [⟨1 0 -13 -3 -5 0], ⟨0 3 29 11 16 7]]

Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0115

Optimal ET sequence: 17c, 36ce, 53, 70, 123de

Badness: 0.032102

Trident

Subgroup: 2.3.5.7

Comma list: 6144/6125, 14348907/14336000

Mapping: [⟨1 0 -13 25], ⟨0 3 29 -42]]

Wedgie: ⟨⟨ 3 29 -42 39 -75 -179 ]]

Optimal tuning (POTE): ~2 = 1\1, ~4096/2835 = 634.0480

Optimal ET sequence: 53, 176, 229, 282, 511

Badness: 0.101694

11-limit

Subgroup: 2.3.5.7.11

Comma list: 3388/3375, 6144/6125, 8019/8000

Mapping: [⟨1 0 -13 25 -33], ⟨0 3 29 -42 69]]

Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0669

Optimal ET sequence: 53, 176, 229

Badness: 0.074272

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 351/350, 2080/2079, 2197/2187, 3146/3125

Mapping: [⟨1 0 -13 25 -33 0], ⟨0 3 29 -42 69 7]]

Optimal tuning (POTE): ~2 = 1\1, ~13/9 = 634.0652

Optimal ET sequence: 53, 176, 229

Badness: 0.046593

Trillium

Subgroup: 2.3.5.7

Comma list: 4375/4374, 1099511627776/1098337086315

Mapping: [⟨1 0 -13 53], ⟨0 3 29 -95]]

Wedgie: ⟨⟨ 3 29 -95 39 -159 -302 ]]

Optimal tuning (POTE): ~2 = 1\1, ~23625/16384 = 634.0118

Optimal ET sequence: 53, 441, 494, 935, 1376, 3193, 4569

Badness: 0.030852

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 131072/130977, 759375/758912

Mapping: [⟨1 0 -13 53 -89], ⟨0 3 29 -95 175]]

Optimal tuning (POTE): ~2 = 1\1, ~3888/2695 = 634.0094

Optimal ET sequence: 53, 441, 494, 935, 1429

Badness: 0.046758

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 2080/2079, 4096/4095, 4375/4374, 78125/78078

Mapping: [⟨1 0 -13 53 -89 -28], ⟨0 3 29 -95 175 60]]

Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0095

Optimal ET sequence: 53, 441, 494, 935, 1429

Badness: 0.019393

Pseudotrillium

Subgroup: 2.3.5.7.11

Comma list: 4375/4374, 5632/5625, 4108797/4096000

Mapping: [⟨1 0 -13 53 -61], ⟨0 3 29 -95 122]]

Optimal tuning (POTE): ~2 = 1\1, ~231/160 = 634.0190

Optimal ET sequence: 53, 335, 388

Badness: 0.111931

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 847/845, 1001/1000, 4096/4095, 4375/4374

Mapping: [⟨1 0 -13 53 -61 -28], ⟨0 3 29 -95 122 60]]

Optimal tuning (POTE): ~2 = 1\1, ~75/52 = 634.0181

Optimal ET sequence: 53, 335, 388

Badness: 0.054837

Tritricot

Subgroup: 2.3.5.7

Comma list: 250047/250000, 11785390260224/11767897353375

Mapping: [⟨3 6 19 30], ⟨0 -3 -29 -52]]

Wedgie: ⟨⟨ 9 87 156 117 222 118 ]]

Optimal tuning (POTE): ~63/50 = 1\3, ~100352/91125 = 165.9837

Optimal ET sequence: 159, 282, 441, 2487, 2928, 3369

Badness: 0.086081

11-limit

Subgroup: 2.3.5.7.11

Comma list: 4000/3993, 166698/166375, 200704/200475

Mapping: [⟨3 6 19 30 22], ⟨0 -3 -29 -52 -28]]

Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9835

Optimal ET sequence: 159, 282, 441

Badness: 0.074002

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 1575/1573, 2080/2079, 34398/34375, 43904/43875

Mapping: [⟨3 6 19 30 22 36], ⟨0 -3 -29 -52 -28 -60]]

Optimal tuning (POTE): ~63/50 = 1\3, ~11/10 = 165.9842

Optimal ET sequence: 159, 282, 441

Badness: 0.035641

17-limit

Subgroup: 2.3.5.7.11.13.17

Comma list: 936/935, 1575/1573, 1701/1700, 2025/2023, 8624/8619

Mapping: [⟨3 6 19 30 22 36 16], ⟨0 -3 -29 -52 -28 -60 -9]]

Optimal tuning (POTE): ~34/27 = 1\3, ~11/10 = 165.9805

Optimal ET sequence: 159, 282, 441

Badness: 0.025972

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