No-twos subgroup temperaments: Difference between revisions

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This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 2. Because of the absence of octaves, these are all [[nonoctave]] scales using a period of a [[tritave]], or if harmonic 3 is also excluded, [[5/1]].
This is a collection of [[subgroup temperament]]s which omit the prime harmonic of 2. Because of the absence of octaves, these are all [[nonoctave]] scales using a period of a [[tritave]], or if harmonic 3 is also excluded, [[5/1]].


== Izar ==
== Arcturus ==
[[Subgroup]]: 3.5.7
''see [[Arcturus]], [[Arcturus clan]]''
 
Subgroup: 3.5.7
 
[[Comma list]]: 15625/15309


[[Comma list]]: 13841287201/13839609375
[[Sval]] [[mapping]]: [{{val| 1 0 -7 }}, {{val| 0 1 6 }}]


{{Mapping|legend=2|1 7 5|0 -12 -7}}
Sval mapping generators: ~3, ~5


Sval mapping generators: ~3, ~16807/10125
[[POTE generator]]: ~5/3 = 878.042


[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~16807/10125 = 877.280
[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]


[[Support]]ing [[ET]]s: {{EDs|b13, b11cd, b193, b15cd, b180, b24c, b167, b37c, b154, 141, b50c, b28cd, b128, b63c|equave=t}}
== Aldebaran ==
==Canopus==
''see [[Aldebaran]]''
[[Subgroup]]: 3.5.7


[[Comma list]]: 16875/16807
[[Subgroup]]: 3.5.13


{{Mapping|legend=2|1 3 3|0 -5 -4}}
[[Comma list]]: 3159/3125


Sval mapping generators: ~3, ~7/5
[[Mapping]]: [{{val|1 0 5}}, {{val|0 1 -2}}]


[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~7/5 = 584.017
[[Support]]ing [[ET]]s: 15, 17, 13, 32, 47, 28, 11[-13], 19[+13], 43, 9[-13], 7[-13], 49[+13], 21[+13], 41[-13]


[[Support]]ing [[ET]]s: {{EDs|b13, b10, b75, b62, b23, b49, b36, b16, b7cd, 29, b42c, b19cd, b33cd, b55c|equave=t}}
[[CTE tuning|CTE generator]]: ~[[5/3]] = 887.76


== BPS ==
== BPS ==
Line 40: Line 43:


[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
[[Optimal ET sequence]]: [[4edt|b4]], [[9edt|b9]], [[13edt|b13]], [[56edt|b56]], [[69edt|b69]], [[82edt|b82]], [[95edt|b95]]
==Canopus==
[[Subgroup]]: 3.5.7


== Arcturus ==
[[Comma list]]: 16875/16807
''see [[Arcturus]], [[Arcturus clan]]''


Subgroup: 3.5.7
{{Mapping|legend=2|1 3 3|0 -5 -4}}


[[Comma list]]: 15625/15309
Sval mapping generators: ~3, ~7/5


[[Sval]] [[mapping]]: [{{val| 1 0 -7 }}, {{val| 0 1 6 }}]
[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~7/5 = 584.017


Sval mapping generators: ~3, ~5
[[Support]]ing [[ET]]s: {{EDs|b13, b10, b75, b62, b23, b49, b36, b16, b7cd, 29, b42c, b19cd, b33cd, b55c|equave=t}}


[[POTE generator]]: ~5/3 = 878.042
== Izar ==
[[Subgroup]]: 3.5.7


[[Optimal ET sequence]]: [[2edt|b2]], [[11edt|b11]], [[13edt|b13]]
[[Comma list]]: 13841287201/13839609375


== Aldebaran ==
{{Mapping|legend=2|1 7 5|0 -12 -7}}
''see [[Aldebaran]]''


[[Subgroup]]: 3.5.13
Sval mapping generators: ~3, ~16807/10125


[[Comma list]]: 3159/3125
[[Optimal tuning]] (CTE): ~3 = 1\1edt, ~16807/10125 = 877.280
 
[[Mapping]]: [{{val|1 0 5}}, {{val|0 1 -2}}]
 
[[Support]]ing [[ET]]s: 15, 17, 13, 32, 47, 28, 11[-13], 19[+13], 43, 9[-13], 7[-13], 49[+13], 21[+13], 41[-13]
 
[[CTE tuning|CTE generator]]: ~[[5/3]] = 887.76


[[Support]]ing [[ET]]s: {{EDs|b13, b11cd, b193, b15cd, b180, b24c, b167, b37c, b154, 141, b50c, b28cd, b128, b63c|equave=t}}
== No-twos-or-threes subgroup temperaments ==
== No-twos-or-threes subgroup temperaments ==
=== Juggernaut ===
=== Juggernaut ===

Revision as of 09:15, 3 January 2024

(WIP, catalog of 3.5.7 subgroup rank two temperaments will eventually be redirected here)

This is a collection of subgroup temperaments which omit the prime harmonic of 2. Because of the absence of octaves, these are all nonoctave scales using a period of a tritave, or if harmonic 3 is also excluded, 5/1.

Arcturus

see Arcturus, Arcturus clan

Subgroup: 3.5.7

Comma list: 15625/15309

Sval mapping: [⟨1 0 -7], ⟨0 1 6]]

Sval mapping generators: ~3, ~5

POTE generator: ~5/3 = 878.042

Optimal ET sequence: b2, b11, b13

Aldebaran

see Aldebaran

Subgroup: 3.5.13

Comma list: 3159/3125

Mapping: [⟨1 0 5], ⟨0 1 -2]]

Supporting ETs: 15, 17, 13, 32, 47, 28, 11[-13], 19[+13], 43, 9[-13], 7[-13], 49[+13], 21[+13], 41[-13]

CTE generator: ~5/3 = 887.76

BPS

Subgroup: 3.5.7

Comma list: 245/243

Sval mapping: [⟨1 1 2], ⟨0 -2 1]]

Sval mapping generators: ~3, ~9/7

Optimal tuning (POTE): ~3 = 1\1edt, ~9/7 = 440.4881

Optimal ET sequence: b4, b9, b13, b56, b69, b82, b95

Canopus

Subgroup: 3.5.7

Comma list: 16875/16807

Subgroup-val mapping: [⟨1 3 3], ⟨0 -5 -4]]

Sval mapping generators: ~3, ~7/5

Optimal tuning (CTE): ~3 = 1\1edt, ~7/5 = 584.017

Supporting ETs: b13, b10, b75, b62, b23, b49, b36, b16, b7cd, 29, b42c, b19cd, b33cd, b55c

Izar

Subgroup: 3.5.7

Comma list: 13841287201/13839609375

Subgroup-val mapping: [⟨1 7 5], ⟨0 -12 -7]]

Sval mapping generators: ~3, ~16807/10125

Optimal tuning (CTE): ~3 = 1\1edt, ~16807/10125 = 877.280

Supporting ETs: b13, b11cd, b193, b15cd, b180, b24c, b167, b37c, b154, 141, b50c, b28cd, b128, b63c

No-twos-or-threes subgroup temperaments

Juggernaut

Subgroup: 5.7.11

Comma list: 125/121

Subgroup-val mapping: [⟨2 4 3], ⟨0 1 0]]

Mapping generators: ~11/5, ~7/25

Optimal tuning (CTE): ~11/5 = 1\2ed5, ~7/5 = 582.512

Supporting ETs: c14, c10, c6, c18, c24, c22, c32, c16, c38, c8d, c34, c26d, c46, c52e

No-twos-or-threes 13-limit

Subgroup: 5.7.11.13

Comma list: 125/121, 637/625

Subgroup-val mapping: [⟨2 4 3 0], ⟨0 1 0 -2]]

Mapping generators: ~11/5, ~7/25

Optimal tuning (CTE): ~11/5 = 1\2ed5, ~7/5 = 582.512

Supporting ETs: c10, c14, c6, c24, c34, c16f, c44, c18f, c38, c26f, c54, c64

Antipyth

Subgroup: 5.7.11

Comma list: 859375/823543

Subgroup-val mapping: [⟨1 2 7], ⟨0 1 7]]

Mapping generators: ~5, ~7/25

Optimal tuning (CTE): ~5 = 1\1ed5, ~7/5 = 592.728

Supporting ETs: c14, c5, c19, c33, c47, c9e, c61, c75, c23e, c24e, c52e, c80e, c89e, c37e

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