No-twos subgroup temperaments

(WIP, further entries in the catalog of 3.5.7 subgroup rank two temperaments will eventually be documented 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

Tridecimal Mintra

This temperament uses the canonical extension for prime 13 described at Tridecimal Mintaka.

Subgroup: 3.5.7.11.13

Comma list: 245/243, 275/273, 1575/1573

Subgroup-val mapping: [⟨1 5 0 1 10], ⟨0 -6 3 2 -13]]

Sval mapping generators: ~3, ~21/11

Optimal tuning (CWE): ~3 = 1\1edt, ~11/7 = 780.428

Supporting ETs: 39, 17, 22, 56, 5f, 61, 95, 100, 134, 73f, 139cf, 83cf, 173e, 178cef

Badness (Sintel): 0.373

Dubhe

Main article: 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. However, seeing the sheer complexity and size of the commas, Nekkar is much more naturally thought of as 3.5.7.11 than 3.5.7, whereupon it becomes a strong extension of Mintaka.

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

3.5.7.11.13 subgroup

This uses the Minalzidar mapping of 13.

Subgroup: 3.5.7.11.13

Comma list: 169/165, 351/343, 11011/10935

Subgroup-val mapping: [⟨1 8 3 3 6], ⟨0 -16 -3 -2 -9]]

Sval mapping generators: ~3, ~11/7

Optimal tuning (CWE): ~3 = 1\1edt, ~11/7 = 776.678

Supporting ETs: 22, 5c, 27, 49, 71f, 17cf

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

Mizar

This temperament uses a weak extension to 3.5.7.17 similar to what Dubhe does: tempering out 2025/2023 to split the 7-limit generator in half; in this case, 25/7 is split into two intervals of 17/9, which turns out to occupy the position of a macrodiatonic fifth, specifically a macro-flattone fifth.

Subgroup: 3.5.7.17

Comma list: 3125/3087, 2025/2023

Subgroup-val mapping: [⟨1 -2 -4 2], ⟨0 6 10 1]]

sval mapping generators: ~3, ~17/9

Optimal tuning (CWE): ~3 = 1\1edt, ~17/9 = 1097.800

Supporting ETs: 26, 7, 19, 45, 71, 97, 33, 123, 12d, 149, 59d, 175, 64d, 85cd

Badness (Sintel): 0.383

Full no-twos 17-limit

This exploits the Sirius tuning of the 25/21 generator being close to 13/11 (in order to split 7/5 evenly); additionally this tempers out 459/455, equating 27/13 to 35/17.

Subgroup: 3.5.7.11.13.17

Comma list: 275/273, 459/455, 1625/1617, 2025/2023

Subgroup-val mapping: [⟨1 -2 -4 12 11 2], ⟨0 6 10 -17 -15 1]]

sval mapping generators: ~3, ~17/9

Optimal tuning (CWE): ~3 = 1\1edt, ~17/9 = 1098.298

Supporting ETs: 26, 71, 45, 19, 97f, 116d

Badness (Sintel): 0.841

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

3.7.11 subgroup temperaments

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

Tridecimal Mintaka

This extension to prime 13 works in the sharper half of the Mintaka tuning range, where the most important pental extension is Mintra.

Subgroup: 3.7.11.13

Comma list: 1331/1323, 218491/216513

Sval mapping: [⟨1 0 1 10], ⟨0 3 2 -13]]

Sval mapping generators: ~3, ~21/11

Optimal tunings:

• PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~11/7 = 780.155
• CWE: ~3 = 1\1ed3, ~11/7 = 780.183

Supporting ETs: b39, b22, b17, b5f, b61, b56, b100, b139f, b95, b178ef, b83f, b134, b73f, b217ef

Minalzidar

This extension to prime 13 works in the flatter half of the Mintaka tuning range, where the most important pental extension is Nekkar.

Subgroup: 3.7.11.13

Comma list: 1331/1323, 351/343

Sval mapping: [⟨1 0 1 -3], ⟨0 3 2 9]]

Sval mapping generators: ~3, ~21/11

Optimal tunings:

• PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~11/7 = 774.432
• CWE: ~3 = 1\1ed3, ~11/7 = 774.782

Supporting ETs: b5, b27, b22, b32, b17f, b37f, b12ff, b49, b59, b42df, b76, b39ff, b86d, b71f

Mebsuta

Subgroup: 3.7.11

Comma list: 387420489/386683451

Sval mapping: [⟨1 2 2], ⟨0 -5 4]]

Sval mapping generators: ~3, ~81/77

Optimal tunings:

• PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~81/77 = 86.957
• CWE: ~3 = 1\1ed3, ~81/77 = 86.957

Supporting ETs: b22, b175, b153, b197, b131, b328, b109, b21, b219, b87, b43, b372, b65, b23

3.7.11.19 subgroup

Subgroup: 3.7.11.19

Comma list: 3971/3969, 41553/41503

Sval mapping: [⟨1 2 2 3], ⟨0 -5 4 -7]]

Sval mapping generators: ~3, ~81/77

Optimal tunings:

• PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~81/77 = 86.929
• CWE: ~3 = 1\1ed3, ~81/77 = 86.932

Supporting ETs: b22, b175, b197, b153, b131, b219, b372, b109, b328, b241, b87, b21, b65, b43

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

Keladic

Subgroup: 3.7.13

Comma list: 351/343

Sval mapping: [⟨1 1 0], ⟨0 1 3]]

Sval mapping generators: ~3, ~7/3

Optimal tunings:

• PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~7/3 = 1480.661
• CWE: ~3 = 1\1ed3, ~7/3 = 1479.487

Supporting ETs: b9, b5, b14, b13, b23, b22, b32, b6f, b31, b19f, b17f, b41, b7ff, b40

Sadalmelik

Subgroup: 3.13.17

Comma list: 85293/83521

Sval mapping: [⟨1 0 2], ⟨0 4 1]]

Sval mapping generators: ~3, ~17/9

Optimal tunings:

• CTE: ~3 = 1\1ed3, ~17/9 = 1109.689
• CWE: ~3 = 1\1ed3, ~17/9 = 1110.376

Supporting ETs: b12, b5, b7, b17, b29, b19, b41, b53, b31, b65, b22f, b9ff, b77, b43

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

See: Catalog of 3.5.7 subgroup rank two temperaments#Graphs

Projective tuning space diagrams

See: Catalog of 3.5.7 subgroup rank two temperaments#Projective tuning space diagrams