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 have a 3/1 period and ~5/3 generator. There is one-to-one correspondence between the 3.5 subgroup and mapped intervals.
• BPS has a ~9/7 generator, two of which give the ~5/3.
• Sirius has a ~25/21 generator, three of which give the ~5/3.
• Deneb has a ~11/9 generator, three of which give the ~9/5.
• Canopus has a ~7/5 generator, five of which give the ~27/5 (9/5 up a tritave).
• Alnilam has a ~81/55 generator, ten of which give the ~243/5 (9/5 up three tritaves).
• Izar has a ~16807/10125 generator, twelve of which give the ~2187/5 (9/5 up five tritaves).
• Nekkar has a ~16807/10935 generator, sixteen of which give the ~6561/5 (9/5 up six 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).
• Antipyth uses 5/1 as a period, and has a ~7/5 generator. There is one-to-one correspondence between the 5.7 subgroup and mapped intervals.
• Juggernaut uses half-pentave(~11/5) as a period, and has a ~7/5 generator.

3.5.7 subgroup temperaments

Arcturus

Main article: Arcturus

As for extensions of this temperament that include the prime 2, see opossum, crepuscular, catalan, bunya, bohpier, and superkleismic.

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

Polturus

This extension of Arcturus adds Polaris's mapping for 11/9, mapping it to 5 generators down.

Subgroup: 3.5.7.11

Comma list: 15625/15309, 177147/171875

Gencom: [3/1 5/3; 15625/15309 177147/171875]

Mapping: [⟨1 1 -1 5], ⟨0 1 6 -6]]

POTE generator: ~5/3 = 884.268

EDTs: 15, 13e, 28e, 43dee

BPS

Main article: Bohlen-Pierce-Stearns

For extensions to this temperament that include the prime 2, see Sensamagic clan. No-twos extensions will be documented below.

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

Badness (Sintel): 0.066

Mintra

See also Mintaka and Deneb.

This temperament splits 27/7 (the BPS generator up a tritave) into three by means of 11/7, and is the intersection of BPS, Deneb, and Mintaka temperaments as well as the most natural temperament satisfied in the 3.5.7.11 subgroup in 39edt.

Subgroup: 3.5.7.11

Comma list: 245/243, 1331/1323

Subgroup-val mapping: [⟨1 5 0 1], ⟨0 -6 3 2]]

Sval mapping generators: ~3, ~21/11

Optimal tuning (CWE): ~3 = 1\1edt, ~11/7 = 780.752

Supporting ETs: 39, 17, 56, 22, 5, 95, 12, 61, 73, 134, 27c, 151e, 100, 90

Badness (Sintel): 0.302

Dubhe

This temperament is a simple 3.5.7.17 weak extension of BPS that splits the generator of 9/7 into two intervals of 17/15. The name was suggested by MidnightBlue after Dubhe, a bright double star (the ninth brightest) and similarities to the word "double".

Subgroup: 3.5.7.17

Comma list: 245/243, 2025/2023

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

Optimal tuning (CWE): ~3 = 1\1edt, ~17/15 = 220.142

Supporting ETs: 26, 9, 17, 43, 69, 8, 35, 95, 61, 60, 121, 25g, 112, 44

Badness (Sintel): 0.177

Canopus

Main article: Canopus

For extensions to this temperament that include the prime 2, see Mirkwai clan. No-twos extensions will be documented below.

Subgroup: 3.5.7

Comma list: 16875/16807

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

Sval mapping generators: ~3, ~7/5

Optimal tunings:

• CTE: ~3 = 1\1edt, ~7/5 = 584.017
• PETE: ~3 = 1\1edt, ~7/5 = 583.9584

Optimal ET sequence: b13, b62, b75, b88, b101, b114, b355, b469, b583, b697

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

Nekkar

This temperament is the no-twos restriction of squares, and as such is named after a star that belonged to the obsolete constellation of Quadrans Muralis, whose name has to do with squares.

Subgroup: 3.5.7

Comma list: 35303692060125/33232930569601

Subgroup-val mapping: [⟨1 8 3], ⟨0 -16 -3]]

Sval mapping generators: ~3, ~16807/10935

Optimal tuning (CWE): ~3 = 1\1edt, ~16807/10935 = 776.767

Supporting ETs: 22, 49, 5c, 71, 27, 17c, 120, 93, 76c, 32cc, 169d, 115, 191d, 164d

3.5.7.11 subgroup

See also Mintaka.

This continues the canonical 11-limit extension of squares.

Subgroup: 3.5.7.11

Comma list: 1331/1323, 120285/117649

Subgroup-val mapping: [⟨1 8 3 3], ⟨0 -16 -3 -2]]

Sval mapping generators: ~3, ~11/7

Optimal tuning (CWE): ~3 = 1\1edt, ~11/7 = 776.781

Supporting ETs: 22, 49, 71, 5c, 27, 120, 93, 17c, 76c, 169d, 191d, 115, 164d, 125cd

Sirius

Main article: Sirius

For an overview of extensions to this temperament that include prime 2, see Gariboh clan#Overview to extensions.

Subgroup: 3.5.7

Comma list: 3125/3087

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

sval mapping generators: ~3, ~25/21

Optimal tuning (POTE): ~3 = 1\1edt, ~25/21 = 293.740

Optimal ET sequence: b6, b7, b13, b71, b84, b97, b110, b123, b136

3.5.11 subgroup temperaments

Polaris

Main article: Polaris

Polaris tempers out the comma 177147/171875, and thus equates 7 5/3's with 15/11, or equivalently 7 9/5's with 11/9.

Subgroup: 3.5.11

Comma list: 177147/171875

Gencom: [3/1 5/3; 177147/171875]

Sval mapping: [⟨1 2 1], ⟨0 1 -6]]

POTE generator: ~5/3 = 892.6

EDTs: 17, 15, 32, 49, 13[+11], 47, 19, 11[+11], 81, 66, 79[+11], 62[+11], 28[+11], 21[-11]

Deneb

Main article: Deneb

Subgroup: 3.5.11

Comma list: 6655/6561

Gencom: [3/1 11/9; 6655/6561]

Sval mapping: [⟨1 2 2], ⟨0 -3 1]]

POTE generator: ~11/9 = 340.242

EDTs: 28, 11, 17, 6, 39, 5, 67, 45, 50, 16, 23, 73, 61, 62

Alnilam

Effectively a microtemperament, Alnilam takes a generator of an 81/55 flat fifth and equates 9 of them with 11/9. The name was given by CompactStar to continue with the theme of naming no-twos temperaments after proper star names, but also to indirectly reference mavila.

Subgroup: 3.5.11

Comma list: [0 -35 9 0 10⟩

Gencom: [3/1 81/55; [0 -35 9 0 10⟩]

Sval mapping: [⟨1 5 -1], ⟨0 -10 9]]

CTE generator: ~81/55 = 672.410

EDTs: 99, 17, 82, 116, 181, 65, 14[-5], 280, 48, 215, 31, 133, 314, 263

Other tritave-based subgroups

Aldebaran

Main article: Aldebaran

Subgroup: 3.5.13

Comma list: 3159/3125

Sval 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

Mintaka

Main article: 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

Main article: 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

Main article: 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

Tridecimal juggernaut

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