Trienstonic clan: Difference between revisions

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The '''trienstonic clan''' of temperaments tempers out [[28/27]], the septimal third-tone or trienstonic comma.  
The '''trienstonic clan''' of temperaments tempers out [[28/27]], the septimal third-tone or trienstonic comma.  


Adding 16/15 to 28/27 leads to father temperament, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are:  
Adding 16/15 to 28/27 leads to father, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are:  
* ''[[Sharp (temperament)|Sharp]]'' → [[Dicot family #Sharp|Dicot family]]
* ''[[Sharp (temperament)|Sharp]]'' → [[Dicot family #Sharp|Dicot family]]
* ''[[Blacksmith]]'' → [[Archytas clan #Blacksmith|Archytas clan]]
* ''[[Blacksmith]]'' → [[Archytas clan #Blacksmith|Archytas clan]]
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* ''[[Opossum]]'' → [[Porcupine family #Opossum|Porcupine family]]
* ''[[Opossum]]'' → [[Porcupine family #Opossum|Porcupine family]]


== No-fives trienstonic ==
== Trienstonian ==
Subgroup: 2.3.7
[[Subgroup]]: 2.3.7


[[Comma list]]: 28/27
[[Comma list]]: 28/27


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


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


[[POTE generator]]: ~3/2 = 721.5586
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 721.5586


{{Optimal ET sequence|legend=1| 2d, 3d, 5, 78bb }}
{{Optimal ET sequence|legend=1| 2d, 3d, 5, 78bb }}


== Father ==
== Father ==
{{see also| Father family #Father }}
{{See also| Father family #Father }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 16/15, 28/27
[[Comma list]]: 16/15, 28/27


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


{{Multival|legend=1| 1 -1 3 -4 2 10 }}
{{Multival|legend=1| 1 -1 3 -4 2 10 }}


[[POTE generator]]: ~3/2 = 742.002
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 742.002


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]]: ~3/2 = {{monzo| 1/2 0 -1/4 1/4 }}
* [[7-odd-limit]]: ~3/2 = {{monzo| 1/2 0 -1/4 1/4 }}
: [[eigenmonzo]]s (unchanged-intervals): 2, 7/5
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.7/5
* [[9-odd-limit]]: ~3/2 = {{monzo| 1 -2 0 1 }} = 14/9
* [[9-odd-limit]]: ~3/2 = {{monzo| 1 -2 0 1 }} = 14/9
: eigenmonzos (unchanged-intervals): 2, 9/7
: [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.9/7


{{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
{{Optimal ET sequence|legend=1| 2d, 3d, 5, 8d, 13cd, 21bccdd }}
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== Uncle ==
== Uncle ==
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 28/27, 256/245
[[Comma list]]: 28/27, 256/245


[[Mapping]]: [{{val| 1 0 12 -2 }}, {{val| 0 1 -6 3 }}]
{{Mapping|legend=1| 1 0 12 -2 | 0 1 -6 3 }}


{{Multival|legend=1| 1 -6 3 -12 2 24 }}
{{Multival|legend=1| 1 -6 3 -12 2 24 }}


[[POTE generator]]: ~3/2 = 731.177
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 731.177


[[Minimax tuning]]:  
[[Minimax tuning]]:  
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== Wallaby ==
== Wallaby ==
{{See also|Pelogic family}}
{{See also| Pelogic family }}


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 28/27, 35/32
[[Comma list]]: 28/27, 35/32


[[Mapping]]: [{{val| 1 0 7 -2 }}, {{val| 0 1 -3 3 }}]
{{Mapping|legend=1| 1 0 7 -2 | 0 1 -3 3 }}


{{Multival|legend=1| 1 -3 3 -7 2 15 }}
{{Multival|legend=1| 1 -3 3 -7 2 15 }}


[[POTE generator]]: ~3/2 = 691.351
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 691.351


{{Optimal ET sequence|legend=1| 2d, 5c, 19ccdd }}
{{Optimal ET sequence|legend=1| 2d, 5c, 19ccdd }}
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== Octokaidecal ==
== Octokaidecal ==
=== 5-limit (supersharp) ===
=== 5-limit (supersharp) ===
Subgroup: 2.3.5
[[Subgroup]]: 2.3.5


[[Comma list]]: 800/729
[[Comma list]]: 800/729


[[Mapping]]: [{{val| 2 0 -5 }}, {{val| 0 1 3 }}]
{{Mapping|legend=1| 2 0 -5 | 0 1 3 }}


[[POTE generator]]: ~3/2 = 729.097
[[Optimal tuning]] ([[POTE]]): ~27/20 = 1\2, ~3/2 = 729.097


{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
{{Optimal ET sequence|legend=1| 8, 10, 18, 28b }}
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=== 7-limit ===
=== 7-limit ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 28/27, 50/49
[[Comma list]]: 28/27, 50/49


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


{{Multival|legend=1| 2 6 6 5 4 -2 }}
{{Multival|legend=1| 2 6 6 5 4 -2 }}


[[POTE generator]]: ~3/2 = 728.874
[[Optimal tuning]] ([[POTE]]): ~7/5 = 1\2, ~3/2 = 728.874


[[Minimax tuning]]:  
[[Minimax tuning]]:  
* [[7-odd-limit]] [[eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.5
* [[7-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5
* [[9-odd-limit]] [[eigenmonzo subgroup]] ([[unchanged-interval basis]]): 2.5
* [[9-odd-limit]] [[Eigenmonzo basis|eigenmonzo (unchanged-interval) basis]]: 2.5


{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
{{Optimal ET sequence|legend=1| 8d, 10, 18, 28b }}
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Comma list: 28/27, 50/49, 55/54
Comma list: 28/27, 50/49, 55/54


Mapping: [{{val| 2 0 -5 -4 7 }}, {{val| 0 1 3 3 0 }}]
Mapping: {{mapping| 2 0 -5 -4 7 | 0 1 3 3 0 }}


POTE generator: ~3/2 = 732.330
Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 732.330


{{Optimal ET sequence|legend=1| 8d, 10, 18e }}
{{Optimal ET sequence|legend=1| 8d, 10, 18e }}
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: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kangaroo]].''
: ''For the 5-limit version of this temperament, see [[High badness temperaments #Kangaroo]].''


Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 28/27, 1029/1000
[[Comma list]]: 28/27, 1029/1000


[[Mapping]]: [{{val| 1 0 -3 -2 }}, {{val| 0 3 10 9 }}]
{{Mapping|legend=1| 1 0 -3 -2 | 0 3 10 9 }}


{{Multival|legend=1| 3 10 9 9 6 -7 }}
{{Multival|legend=1| 3 10 9 9 6 -7 }}


[[POTE generator]]: ~7/5 = 560.328
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~7/5 = 560.328


{{Optimal ET sequence|legend=1| 2cd, 13cd, 15 }}
{{Optimal ET sequence|legend=1| 2cd, 13cd, 15 }}
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Comma list: 28/27, 77/75, 245/242
Comma list: 28/27, 77/75, 245/242


Mapping: [{{val| 1 0 -3 -2 -4 }}, {{val| 0 3 10 9 14 }}]
Mapping: {{mapping| 1 0 -3 -2 -4 | 0 3 10 9 14 }}


POTE generator: ~7/5 = 560.155
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 560.155


{{Optimal ET sequence|legend=1| 15 }}
{{Optimal ET sequence|legend=1| 15 }}
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Comma list: 28/27, 40/39, 66/65, 147/143
Comma list: 28/27, 40/39, 66/65, 147/143


Mapping: [{{val| 1 0 -3 -2 -4 0 }}, {{val| 0 3 10 9 14 7 }}]
Mapping: {{mapping| 1 0 -3 -2 -4 0 | 0 3 10 9 14 7 }}


POTE generator: ~7/5 = 559.770
Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 559.770


{{Optimal ET sequence|legend=1| 15 }}
{{Optimal ET sequence|legend=1| 15 }}
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Comma list: 28/27, 49/48, 55/54, 77/75
Comma list: 28/27, 49/48, 55/54, 77/75


Mapping: [{{val| 15 24 35 42 52 0 }}, {{val| 0 0 0 0 0 1 }}]
Mapping: {{mapping| 15 24 35 42 52 0 | 0 0 0 0 0 1 }}


POTE generator: ~13/8 = 852.924
Optimal tuning (POTE): ~22/21 = 1\15, ~13/8 = 852.924


{{Optimal ET sequence|legend=1| 15, 30, 45bc }}
{{Optimal ET sequence|legend=1| 15, 30, 45bc }}

Revision as of 13:16, 4 August 2023

The trienstonic clan of temperaments tempers out 28/27, the septimal third-tone or trienstonic comma.

Adding 16/15 to 28/27 leads to father, adding 256/245 gives uncle, adding 50/49 gives octokaidecal and adding 126/125 gives opossum. Other members of the clan discussed elsewhere are:

Trienstonian

Subgroup: 2.3.7

Comma list: 28/27

Sval mapping: [⟨1 0 -2], ⟨0 1 3]]

Gencom mapping: [⟨1 0 0 -2], ⟨0 1 0 3]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 721.5586

Optimal ET sequence: 2d, 3d, 5, 78bb

Father

See also: Father family § Father

Subgroup: 2.3.5.7

Comma list: 16/15, 28/27

Mapping: [⟨1 0 4 -2], ⟨0 1 -1 3]]

Wedgie: ⟨⟨ 1 -1 3 -4 2 10 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 742.002

Minimax tuning:

eigenmonzo (unchanged-interval) basis: 2.7/5
eigenmonzo (unchanged-interval) basis: 2.9/7

Optimal ET sequence: 2d, 3d, 5, 8d, 13cd, 21bccdd

Badness: 0.021312

Uncle

Subgroup: 2.3.5.7

Comma list: 28/27, 256/245

Mapping: [⟨1 0 12 -2], ⟨0 1 -6 3]]

Wedgie: ⟨⟨ 1 -6 3 -12 2 24 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 731.177

Minimax tuning:

Optimal ET sequence: 5, 13d, 18, 23bc, 41bcd

Badness: 0.072653

Wallaby

See also: Pelogic family

Subgroup: 2.3.5.7

Comma list: 28/27, 35/32

Mapping: [⟨1 0 7 -2], ⟨0 1 -3 3]]

Wedgie: ⟨⟨ 1 -3 3 -7 2 15 ]]

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 691.351

Optimal ET sequence: 2d, 5c, 19ccdd

Badness: 0.058468

Octokaidecal

5-limit (supersharp)

Subgroup: 2.3.5

Comma list: 800/729

Mapping: [⟨2 0 -5], ⟨0 1 3]]

Optimal tuning (POTE): ~27/20 = 1\2, ~3/2 = 729.097

Optimal ET sequence: 8, 10, 18, 28b

Badness: 0.122848

7-limit

Subgroup: 2.3.5.7

Comma list: 28/27, 50/49

Mapping: [⟨2 0 -5 -4], ⟨0 1 3 3]]

Wedgie: ⟨⟨ 2 6 6 5 4 -2 ]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 728.874

Minimax tuning:

Optimal ET sequence: 8d, 10, 18, 28b

Badness: 0.036747

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 50/49, 55/54

Mapping: [⟨2 0 -5 -4 7], ⟨0 1 3 3 0]]

Optimal tuning (POTE): ~7/5 = 1\2, ~3/2 = 732.330

Optimal ET sequence: 8d, 10, 18e

Badness: 0.030235

Kangaroo

For the 5-limit version of this temperament, see High badness temperaments #Kangaroo.

Subgroup: 2.3.5.7

Comma list: 28/27, 1029/1000

Mapping: [⟨1 0 -3 -2], ⟨0 3 10 9]]

Wedgie: ⟨⟨ 3 10 9 9 6 -7 ]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 560.328

Optimal ET sequence: 2cd, 13cd, 15

Badness: 0.077857

11-limit

Subgroup: 2.3.5.7.11

Comma list: 28/27, 77/75, 245/242

Mapping: [⟨1 0 -3 -2 -4], ⟨0 3 10 9 14]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 560.155

Optimal ET sequence: 15

Badness: 0.043195

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 40/39, 66/65, 147/143

Mapping: [⟨1 0 -3 -2 -4 0], ⟨0 3 10 9 14 7]]

Optimal tuning (POTE): ~2 = 1\1, ~7/5 = 559.770

Optimal ET sequence: 15

Badness: 0.032653

Quindecic

Subgroup: 2.3.5.7.11.13

Comma list: 28/27, 49/48, 55/54, 77/75

Mapping: [⟨15 24 35 42 52 0], ⟨0 0 0 0 0 1]]

Optimal tuning (POTE): ~22/21 = 1\15, ~13/8 = 852.924

Optimal ET sequence: 15, 30, 45bc

Badness: 0.028944

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