Biyatismic clan: Difference between revisions

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Temperaments discussed elsewhere are:  
Temperaments discussed elsewhere are:  
* ''[[Sonic]]'', {55/54, 100/99} → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
* ''[[Sonic]]'' (+55/54 or 100/99) → [[Porcupine rank three family #Sonic|Porcupine rank-3 family]]
* ''[[Urania]]'', {81/80, 121/120} → [[Didymus rank three family #Urania|Didymus rank-3 family]]
* ''[[Urania]]'' (+81/80) → [[Didymus rank three family #Urania|Didymus rank-3 family]]
* ''[[Big brother]]'', {99/98, 121/120} → [[Nuwell family #big Brother|Nuwell family]]
* ''[[Big brother]]'' (+99/98) → [[Nuwell family #big Brother|Nuwell family]]
* ''[[Oxpecker]]'', {121/120, 126/125} → [[Starling family #Oxpecker|Starling family]]
* ''[[Oxpecker]]'' (+126/125) → [[Starling family #Oxpecker|Starling family]]
* [[Zeus]], {121/120, 176/175} → [[Porwell family #Zeus|Porwell family]]
* ''[[Artemis]]'' (+225/224) → [[Marvel family #Artemis|Marvel family]]
* ''[[Artemis]]'', {121/120, 225/224} → [[Marvel family #Artemis|Marvel family]]
* ''[[Bisector]]'' (+245/243) → [[Sensamagic family #Bisector|Sensamagic family]]
* ''[[Bisector]]'', {121/120, 245/243} → [[Sensamagic family #Bisector|Sensamagic family]]


Considered below are aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see [[Rank-4 temperament #Biyatismic (121/120)]].  
Considered below are zeus, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see [[Rank-4 temperament #Biyatismic (121/120)]].  


== Protomere ==
== Protomere ==
Subgroup: 2.3.5.11
[[Subgroup]]: 2.3.5.11


[[Comma list]]: 121/120
[[Comma list]]: 121/120


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


Sval mapping generators: ~2, ~3, ~11/10
: sval mapping generators: ~2, ~3, ~11/10


[[POTE generator]]s: ~3/2 = 701.4578, ~11/10 = 157.7466
[[Optimal tuning]] ([[POTE]]): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466


{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
{{Optimal ET sequence|legend=1| 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee }}
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=== 7-limit (squalentine) ===
=== 7-limit (squalentine) ===
Subgroup: 2.3.5.7
[[Subgroup]]: 2.3.5.7


[[Comma list]]: 64827/64000
[[Comma list]]: 64827/64000


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


Mapping generators: ~2, ~3, ~21/20
: mapping generators: ~2, ~3, ~21/20


{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d }}
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=== 11-limit ===
=== 11-limit ===
Subgroup: 2.3.5.7.11
[[Subgroup]]: 2.3.5.7.11


[[Comma list]]: 121/120, 441/440
[[Comma list]]: 121/120, 441/440


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


Mapping generators: ~2, ~3, ~22/21
: mapping generators: ~2, ~3, ~22/21


{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
{{Optimal ET sequence|legend=1| 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee }}
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Comma list: 121/120, 351/350, 441/440
Comma list: 121/120, 351/350, 441/440


Mapping: [{{val| 1 0 1 3 2 6 }}, {{val| 0 1 1 0 1 -1 }}, {{val| 0 0 4 3 2 11 }}]
Mapping: {{mapping| 1 0 1 3 2 6 | 0 1 1 0 1 -1 | 0 0 4 3 2 11 }}


{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
{{Optimal ET sequence|legend=1| 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef }}
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Comma list: 121/120, 196/195, 352/351
Comma list: 121/120, 196/195, 352/351


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


{{Optimal ET sequence|legend=1| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
{{Optimal ET sequence|legend=1| 17c, 29, 31, 46, 60e, 77, 106de, 183dee }}
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Comma list: 105/104, 121/120, 275/273
Comma list: 105/104, 121/120, 275/273


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


{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
{{Optimal ET sequence|legend=1| 14cf, 15, 29, 31, 45ef, 60e }}
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Comma list: 91/90, 121/120, 441/440
Comma list: 91/90, 121/120, 441/440


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


{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
{{Optimal ET sequence|legend=1| 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def }}
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[[Category:Temperament clans]]
[[Category:Temperament clans]]
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic clan| ]] <!-- main article -->
[[Category:Biyatismic| ]] <!-- key article -->
[[Category:Rank 3]]
[[Category:Rank 3]]

Revision as of 03:38, 8 September 2023

The biyatismic clan of rank-3 temperaments tempers out the biyatisma, 121/120 = [-3 -1 -1 0 2⟩.

Temperaments discussed elsewhere are:

Considered below are zeus, aphrodite, and the no-7 subgroup temperament, protomere. For the rank-4 biyatismic temperament, see Rank-4 temperament #Biyatismic (121/120).

Protomere

Subgroup: 2.3.5.11

Comma list: 121/120

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

sval mapping generators: ~2, ~3, ~11/10

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.4578, ~11/10 = 157.7466

Optimal ET sequence: 7, 15, 22, 31, 46, 53, 137e, 183ee, 190ee

Badness: 0.0297 × 10-3

Zeus

See also: Porwell family § Zeus

Subgroup: 2.3.5.7.11

Comma list: 121/120, 176/175

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

Mapping to lattice: [⟨0 1 -1 2 0], ⟨0 1 1 -1 1]]

Lattice basis:

11/10, 11/8
Angle (11/10, 11/8) = 87.464 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 702.1530, ~11/10 = 157.0881

Minimax tuning:

[[1 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0⟩, [22/9 2/9 1/3 -4/9 0⟩, [22/9 2/9 -2/3 5/9 0⟩, [10/3 2/3 0 -1/3 0⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5.9/7

Optimal ET sequence: 15, 22, 31, 46, 53, 68, 77, 99, 130e

Badness: 0.400 × 10-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 to 2.3.11/5

Zeus11[22] hobbit transversal

33/32, 16/15, 11/10, 8/7, 64/55, 77/64, 5/4, 14/11, 4/3,
11/8, 45/32, 16/11, 3/2, 11/7, 8/5, 5/3, 55/32, 7/4,
11/6, 15/8, 64/33, 2

Zeus11[24] hobbit transversal

33/32, 16/15, 11/10, 9/8, 8/7, 77/64, 11/9, 5/4, 21/16, 4/3,
11/8, 45/32, 16/11, 3/2, 32/21, 8/5, 18/11, 5/3, 7/4, 16/9,
11/6, 15/8, 64/33, 2

Scales:

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 176/175, 351/350

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

Mapping to lattice: [⟨0 1 -1 2 0 -3], ⟨0 1 1 -1 1 -2]]

Lattice basis:

11/10 length = 0.7898, 11/8 length = 1.002
Angle (11/10, 11/8) = 106.7439 degrees

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 701.8679, ~11/10 = 156.9582

Minimax tuning:

  • 13-odd-limit
[[1 0 0 0 0 0⟩, [11/9 10/9 -1/3 -2/9 0 0⟩, [22/9 2/9 1/3 -4/9 0 0⟩, [22/9 2/9 -2/3 5/9 0 0⟩, [10/3 2/3 0 -1/3 0 0⟩, [14/3 -8/3 1 1/3 0 0⟩]
eigenmonzo (unchanged-interval) basis: 2.9/5.9/7
  • 15-odd-limit
[[1 0 0 0 0 0⟩, [0 1 0 0 0 0⟩, [11/5 1/5 2/5 -2/5 0 0⟩, [11/5 1/5 -3/5 3/5 0 0⟩, [13/5 3/5 1/5 -1/5 0 0⟩, [38/5 -12/5 1/5 -1/5 0 0⟩]
eigenmonzo (unchanged-interval) basis: 2.3.7/5

Optimal ET sequence: 15, 22, 31, 46, 53, 77, 99, 130e

Badness: 0.934 × 10-3

Projection pairs: 5 600/121 7 2662/375 11 120/11 13 1280/99 to 2.3.11/5

Zeus13[22] hobbit transversal

260/243, 88/81, 11/10, 44/39, 162/143, 11/9, 16/13, 320/243, 4/3, 1040/729, 13/9, 729/520, 3/2, 99/65, 44/27, 18/11, 1280/729, 16/9, 11/6, 24/13, 243/130, 2

Tinia

Subgroup: 2.3.5.7.11.13

Comma list: 66/65, 121/120, 176/175

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

Optimal tuning (POTE): ~2 = 1\1, ~3/2 = 699.3420, ~11/10 = 155.3666

Optimal ET sequence: 7, 9, 15, 22f, 24, 31

Badness: 0.808 × 10-3

Aphrodite

Aphrodite tempers out the squalentine comma, 64827/64000, in the 7-limit. Its generators can be taken to be 2, 3, and 21/20, and it equates (21/20)3 with 8/7.

7-limit (squalentine)

Subgroup: 2.3.5.7

Comma list: 64827/64000

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

mapping generators: ~2, ~3, ~21/20

Optimal ET sequence: 14c, 15, 29, 31, 46, 60, 77, 91, 122, 137d, 168d

Badness: 0.943 × 10-3

Projection pairs: 5 320000/64827 7 64000/9261 to 2.3.7/5

11-limit

Subgroup: 2.3.5.7.11

Comma list: 121/120, 441/440

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

mapping generators: ~2, ~3, ~22/21

Optimal ET sequence: 14c, 15, 29, 31, 46, 60e, 77, 91e, 137de, 168dee

Badness: 0.583 × 10-3

13-limit

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 351/350, 441/440

Mapping: [⟨1 0 1 3 2 6], ⟨0 1 1 0 1 -1], ⟨0 0 4 3 2 11]]

Optimal ET sequence: 14cf, 31, 45ef, 46, 77, 122ee, 137def, 168deef

Badness: 1.456 × 10-3

Eros

Subgroup: 2.3.5.7.11.13

Comma list: 121/120, 196/195, 352/351

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

Optimal ET sequence: 17c, 29, 31, 46, 60e, 77, 106de, 183dee

Badness: 1.150 × 10-3

Inanna

Subgroup: 2.3.5.7.11.13

Comma list: 105/104, 121/120, 275/273

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

Optimal ET sequence: 14cf, 15, 29, 31, 45ef, 60e

Badness: 1.077 × 10-3

Ishtar

Subgroup: 2.3.5.7.11.13

Comma list: 91/90, 121/120, 441/440

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

Optimal ET sequence: 14cf, 15, 17c, 29, 31f, 46, 106deff, 121def

Badness: 1.151 × 10-3

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