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.

Overview by mapping of 5

Classified by focusing on the mapping of 5th harmonic, similar to Rank-2 temperaments by mapping of 3.

• Arcturus, Aldebaran and Polaris are with 3/1 period and ~5/3 generator. There is one-to-one correspondence between 3.5 subgroup and mapped intervals.
• BPS is with a ~9/7 generator, two of which give the ~5/3.
• Sirius is with a ~25/21 generator, three of which give the ~5/3.
• Deneb is with a ~11/9 generator, three of which give the ~9/5.
• Canopus is with a ~7/5 generator, five of which give the ~27/5 (9/5 up a tritave).
• Alnilam is with a ~81/55 generator, ten of which give the ~243/5 (9/5 up three tritaves).
• Izar is with a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 up five tritaves).
• Mintaka does not include the 5th harmonic, and has an ~11/7 generator, two of which give the ~27/11, and three of which give the ~27/7 (9/7 and a tritave).

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

see Bohlen-Pierce-Stearns

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

Mintaka

see Mintaka

Subgroup: 3.7.11

Comma list: 1331/1323

Sval mapping: [⟨1 0 1], ⟨0 3 2]]

Sval mapping generators: ~3, ~21/11

Optimal tunings:

• PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~11/7 = 778.961
• CWE: ~3 = 1\1ed3, ~11/7 = 778.803

Supporting ETs: b22, b5, b17, b39, b12, b61, b27, b7, b83, b49, b56, b32, b29, b100

No-twos-or-threes subgroup temperaments

Antipyth

see 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

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

Graphs

Complexity vs. damage plot. z<1 corresponds to the "Middle Path" inclusion criterion.

Projective tuning space diagrams

Temperaments with smaller commas, labeled by name

Temperaments with smaller commas, labeled by comma

Temperaments with larger commas, labeled by name

Temperaments with larger commas, labeled by comma

Both sets, labeled by name

Both sets, labeled by comma

11.675