No-twos subgroup temperaments
(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
Sval 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
Sval 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
Antipyth
see Antipyth
Subgroup: 5.7.11
Comma list: 859375/823543
Sval 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
Juggernaut
Subgroup: 5.7.11
Comma list: 125/121
Sval 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
Sval 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
Graphs
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| Complexity vs. damage plot. z<1 corresponds to the "Middle Path" inclusion criterion. |
Projective tuning space diagrams
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| Temperaments with smaller commas, labeled by name |
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| Temperaments with smaller commas, labeled by comma |
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| Temperaments with larger commas, labeled by name |
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| Temperaments with larger commas, labeled by comma |
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| Both sets, labeled by name |
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| Both sets, labeled by comma |
| 11.675 |