Valinorsmic clan: Difference between revisions

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"optimal GPV sequence" → "optimal ET sequence", per Talk:Optimal_ET_sequence
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Discussed elsewhere are:  
Discussed elsewhere are:  
* ''[[Nickel]]'', {36/35, 45/44} → [[Mint family #Nickel|Mint family]]
* ''[[Nickel]]'' (+36/35 or 45/44) → [[Mint family #Nickel|Mint family]]
* [[Ares]], {64/63, 100/99} → [[Archytas family #Ares|Archytas family]]
* [[Ares]] (+64/63 or 100/99) → [[Archytas family #Ares|Archytas family]]
* ''[[Clio]]'', {81/80, 176/175} → [[Didymus rank three family #Clio|Didymus rank-3 family]]
* ''[[Clio]]'' (+81/80) → [[Didymus rank three family #Clio|Didymus rank-3 family]]
* ''[[Minerva]]'', {99/98, 176/175} → [[Marvel family #Minerva|Marvel family]]
* ''[[Minerva]]'' (+99/98) → [[Marvel family #Minerva|Marvel family]]
* ''[[Zeus]]'', {121/120, 176/175} → [[Porwell family #Zeus|Porwell family]]
* ''[[Zeus]]'' (+121/120) → [[Porwell family #Zeus|Porwell family]]
* ''[[Thrush]]'', {126/125, 176/175} → [[Starling family #Thrush|Starling family]]
* ''[[Thrush]]'' (+126/125) → [[Starling family #Thrush|Starling family]]
* ''[[Shrusus]]'', {176/175, 245/243} → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Shrusus]]'' (+245/243) → [[Sensamagic family #Shrusus|Sensamagic family]]
* ''[[Guanyin]]'', {176/175, 540/539} → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[Guanyin]]'' (+540/539) → [[Orwellismic family #Guanyin|Orwellismic family]]
* ''[[Julius]]'' or ''[[varda]]'', {176/175, 896/891} → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
* ''[[Julius]]'' or ''[[varda]]'' (+896/891) → [[Diaschismic rank three family #Julius aka varda|Diaschismic rank-3 family]]
* ''[[Lono]]'', {176/175, 5120/5103} → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[Lono]]'' (+5120/5103) → [[Hemifamity family #Lono|Hemifamity family]]
* ''[[Draco]]'', {176/175, 19683/19600} → [[Cataharry family #Draco|Cataharry family]]
* ''[[Draco]]'' (+19683/19600) → [[Cataharry family #Draco|Cataharry family]]


Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament, but we can begin with the rank-4 temperament.  
Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament, but we can begin with the rank-4 temperament.  
== Valinorsmic ==
Subgroup: 2.3.5.7.11
[[Comma list]]: [[176/175]]
[[Mapping]]: [{{val| 1 0 0 0 -4 }}, {{val| 0 1 0 0 0 }}, {{val| 0 0 1 0 2 }}, {{val| 0 0 0 1 1 }}]
Mapping generators: ~2, ~3, ~5, ~7
[[POTE generator]]s: ~3/2 = 703.0449, ~5/4 = 389.7641, ~7/4 = 972.1113
{{Optimal ET sequence|legend=1| 22, 31, 46, 53, 58, 80, 111 }}
[[Badness]]: 0.0186 × 10<sup>-6</sup>


== No-3 valinorsmic ==
== No-3 valinorsmic ==
Subgroup: 2.5.7.11
[[Subgroup]]: 2.5.7.11


[[Comma list]]: 176/175
[[Comma list]]: 176/175


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


Sval mapping generators: ~2, ~5, ~7
: sval mapping generators: ~2, ~5, ~7


[[POTE generator]]s: ~5/4 = 389.7641, ~7/4 = 972.1113
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~5/4 = 389.7641, ~7/4 = 972.1113


{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}
{{Optimal ET sequence|legend=1| 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd }}
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== Manwe ==
== Manwe ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 1331/1323
[[Comma list]]: 176/175, 1331/1323


[[Mapping]]: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}]
{{Mapping|legend=1| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}


{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }}
{{Optimal ET sequence|legend=1| 15, 31, 46, 77, 80, 111 }}
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Comma list: 176/175, 1331/1323, 351/350
Comma list: 176/175, 1331/1323, 351/350


Mapping: [{{val| 1 0 0 12 8 }}, {{val| 0 1 0 3 3 }}, {{val| 0 0 1 -6 -4 }}]
Mapping: {{mapping| 1 0 0 12 8 | 0 1 0 3 3 | 0 0 1 -6 -4 }}


{{Optimal ET sequence|legend=1| 31, 46, 77, 80, 111 }}
{{Optimal ET sequence|legend=1| 31, 46, 77, 80, 111 }}
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== Mandos ==
== Mandos ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 243/242
[[Comma list]]: 176/175, 243/242


[[Mapping]]: [{{val| 1 1 0 6 2 }}, {{val| 0 2 0 5 5 }}, {{val| 0 0 1 -2 0 }}]
{{Mapping|legend=1| 1 1 0 6 2 | 0 2 0 5 5 | 0 0 1 -2 0 }}


{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}
{{Optimal ET sequence|legend=1| 7, 24, 31, 58, 89 }}
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Comma list: 144/143, 176/175, 243/242
Comma list: 144/143, 176/175, 243/242


Mapping: [{{val| 1 1 0 6 2 4 }}, {{val| 0 2 0 5 5 -1 }}, {{val| 0 0 1 -2 0 0 }}]
Mapping: {{mapping| 1 1 0 6 2 4 | 0 2 0 5 5 -1 | 0 0 1 -2 0 0 }}


{{Optimal ET sequence|legend=1| 7, 24, 31, 58 }}
{{Optimal ET sequence|legend=1| 7, 24, 31, 58 }}
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== Ulmo ==
== Ulmo ==
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 176/175, 2200/2187
[[Comma list]]: 176/175, 2200/2187


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


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd }}
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Comma list: 169/168, 176/175, 325/324
Comma list: 169/168, 176/175, 325/324


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


{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}
{{Optimal ET sequence|legend=1| 7, 39, 46, 53, 80, 133d, 179, 312cdf }}
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Valinorsmic clan| ]] <!-- main article -->
[[Category:Valinorsmic clan| ]] <!-- main article -->
[[Category:Valinorsmic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 10:56, 9 September 2023

The valinorsmic clan of rank-3 temperaments tempers out the valinorsma, 176/175, which equates 48/35 with 15/11 (rather than 11/8, which is what the keenanisma 385/384 does).

Discussed elsewhere are:

Considered below are manwe, mandos and ulmo besides the no-3 subgroup temperament, but we can begin with the rank-4 temperament.

No-3 valinorsmic

Subgroup: 2.5.7.11

Comma list: 176/175

Sval mapping: [⟨1 0 0 -4], ⟨0 1 0 2], ⟨0 0 1 1]]

sval mapping generators: ~2, ~5, ~7

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 389.7641, ~7/4 = 972.1113

Optimal ET sequence: 6, 15, 21, 22, 25, 28, 31, 37, 163c, 200cd, 237cd, 274cd

Badness: 1.73 × 10-6

Manwe

Subgroup: 2.3.5.7.11

Comma list: 176/175, 1331/1323

Mapping: [⟨1 0 0 12 8], ⟨0 1 0 3 3], ⟨0 0 1 -6 -4]]

Optimal ET sequence: 15, 31, 46, 77, 80, 111

Badness: 0.679 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 176/175, 1331/1323, 351/350

Mapping: [⟨1 0 0 12 8], ⟨0 1 0 3 3], ⟨0 0 1 -6 -4]]

Optimal ET sequence: 31, 46, 77, 80, 111

Badness: 1.147 × 10-3

Mandos

Subgroup: 2.3.5.7.11

Comma list: 176/175, 243/242

Mapping: [⟨1 1 0 6 2], ⟨0 2 0 5 5], ⟨0 0 1 -2 0]]

Optimal ET sequence: 7, 24, 31, 58, 89

Badness: 0.751 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 144/143, 176/175, 243/242

Mapping: [⟨1 1 0 6 2 4], ⟨0 2 0 5 5 -1], ⟨0 0 1 -2 0 0]]

Optimal ET sequence: 7, 24, 31, 58

Badness: 0.923 × 10-3

Ulmo

Subgroup: 2.3.5.7.11

Comma list: 176/175, 2200/2187

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

Optimal ET sequence: 7, 39, 46, 53, 80, 126, 179, 259cd, 385bcd

Badness: 1.23 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 169/168, 176/175, 325/324

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

Optimal ET sequence: 7, 39, 46, 53, 80, 133d, 179, 312cdf

Badness: 0.932 × 10-3

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