No-twos subgroup temperaments

This is a list showing technical temperament data. For an explanation of what information is shown here, you may look at the technical data guide for regular temperaments.

Todo: 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

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

Subgroup-val mapping: [⟨1 0 -7], ⟨0 1 6]]

mapping generators: ~3, ~5

Optimal tunings:

• WE: ~3 = 1903.863 ¢, ~5/3 = 878.923 ¢
• CWE: ~3 = 1901.955 ¢, ~5/3 = 878.291 ¢

Optimal ET sequence: b2, b11, b13

Badness (Sintel): 0.535

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

Subgroup-val mapping: 1 1 -1 5 | 0 1 6 -6 }}

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

Optimal tuning (POTE): ~5/3 = 884.268 ¢

Optimal ET sequence: b13e, b15, b28e, b43dee

Badness (Sintel): 2.507

BPS

Main article: BPS

For extensions to this temperament that include the octave, see sensamagic clan. Non-octave extensions will be documented below.

Subgroup: 3.5.7

Comma list: 245/243

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

mapping generators: ~3, ~9/7

Optimal tunings:

• WE: ~3 = 1903.7398 ¢, ~9/7 = 440.9014 ¢

error map: ⟨+1.785 -0.771 -2.248]

• CWE: ~3 = 1901.9550 ¢, ~9/7 = 440.6646 ¢

error map: ⟨0.000 -3.030 -5.580]

Optimal ET sequence: b4, b9, b13, b56, b69, b82, b95, b367cdd, b462cdd

Badness (Sintel): 0.0659

Alhena

Todo: complete section Catalog the intermediate extension (3.5.7.11/2).

This is a strong extension to BPS in the subgroup 3.5.7.11/2.13/4 that equates the "semitone" of 27/25~49/45 to 13/12, and then three of these intervals to 14/11.

Subgroup: 3.5.7.11/2.13/4

Comma list: 196/195, 325/324, 1001/1000

Subgroup-val mapping: [⟨1 1 2 -1 2], ⟨0 2 -1 11 -4]]

mapping generators: ~3, ~7/3

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~9/7 = 441.025 ¢

Supporting ETs: b13, b69, b56, b82, b43, b125, b30, b151, b95, b17é, b194d, b99, b181d, b108é *

* é is used as the wart for 11/2.

Badness (Sintel): 0.187

Mintra

See also: Mintaka and Deneb.

This temperament splits 27/7 (the BPS generator up a tritave) into three by means of 11/7 or, equivalently, 7/1 in three by means of 21/11, 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]] mapping generators: ~3, ~21/11

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~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]]

mapping generators: ~3, ~21/11

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~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 5]]

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~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 Canopic clan. No-twos extensions will be documented below.

Subgroup: 3.5.7

Comma list: 16875/16807

Subgroup-val mapping: [⟨1 -2 -1], ⟨0 5 4]]

mapping generators: ~3, ~15/7

Optimal tunings:

• WE: ~3 = 1901.7826 ¢, ~15/7 = 1317.8771 ¢

error map: ⟨+1.785 -0.771 -2.248]

• CWE: ~3 = 1901.9550 ¢, ~15/7 = 1317.9686 ¢

error map: ⟨0.000 -0.381 +1.093]

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

Badness (Sintel): 0.0996

Suhail

Todo: complete section Catalog the intermediate extension (3.5.7.11).

Tempering out the 3.13-subgroup threedie splits the tritave into three, meeting 11/1 at seven generators after tempering out the sopreisma.

Subgroup: 3.5.7.11.13

Comma list: 1575/1573, 1625/1617, 4459/4455

Subgroup-val mapping: [⟨3 4 5 6 7], ⟨0 5 4 7 0]]

mapping generators: ~13/9, ~65/63

Optimal tunings:

• WE: ~13/9 = 634.144 ¢, ~65/63 = 49.695 ¢
• CWE: ~13/9 = 633.985 ¢, ~65/63 = 49.733 ¢

Optimal ET sequence: b39, b114, b153, b498cf, b651cf

Badness (Sintel): 0.330

Izar

Subgroup: 3.5.7

Comma list: 13841287201/13839609375

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

mapping generators: ~3, ~16807/10125

Optimal tunings:

• WE: ~3 = 1901.958 ¢, ~16807/10125 = 877.283 ¢
• CWE: ~3 = 1901.955 ¢, ~16807/10125 = 877.281 ¢

Optimal ET sequence: b13, b141, b154, …, b258, b271, b800, b1071, b1342, b1613, b4568, b6181

Badness (Sintel): 0.017

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]]

mapping generators: ~3, ~16807/10935

Optimal tunings:

• WE: ~3 = 1900.155 ¢, ~16807/10935 = 775.963 ¢
• CWE: ~3 = 1901.955 ¢, ~16807/10935 = 776.767 ¢

Optimal ET sequence: b22, b49, b71, b120, b191d

Badness (Sintel): 17.120

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]]

mapping generators: ~3, ~11/7

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~11/7 = 776.781 ¢

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

Badness (Sintel): 1.375

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]]

mapping generators: ~3, ~11/7

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~11/7 = 776.678 ¢

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

Badness (Sintel): 1.723

Procyon

This tempers out the Don Page comma between 7/5 and 9/7, allowing an accurate representation of the 5:7:9 chord, similar to the 3:5:7 in Sirius.

Subgroup: 3.5.7

Comma list: 823543/820125

Subgroup-val mapping: [⟨1 2 2], ⟨0 -7 -3]]

mapping generators: ~3, ~17/9

Optimal tunings:

• WE: ~3 = 1902.198 ¢, ~49/45 = 145.412 ¢
• CWE: ~3 = 1901.955 ¢, ~49/45 = 145.368 ¢

Supporting ETs: b13, b157, b144, b170, b131, b183, b118, b14, b105, b12c, b196, b92, b27, b79

Badness (Sintel): 0.200

Erigone

Todo: complete section Catalog the intermediate extension (3.5.7.11).

Erigone splits the (tritave-augmented) generator of procyon into three, allowing for an accurate representation of 11/9 at -19 generators and 13/9 at -13 generators.

Subgroup: 3.5.7.11.13

Comma list: 847/845, 1575/1573, 4459/4455

Subgroup-val mapping: [⟨1 9 5 9 7], ⟨0 -21 -9 -19 -13]]

mapping generators: ~3, ~49/33

Optimal tunings:

• WE: ~3 = 1901.9699 ¢, ~49/33 = 682.4486 ¢
• CWE: ~3 = 1901.9550 ¢, ~49/33 = 682.4427 ¢

Optimal ET sequence: b25ce, b39, b92, b131, b170, b301, b471

Badness (Sintel): 0.21396

Hemigone

Todo: complete section Catalog the intermediate extension (3.5.7.11.13.17).

By tempering out 3971/3969, erigone's tritave-augmented generator (49/11) is split into two 19/9s. Then, 17/1 is approximated at 39/35 below 19/1 (tempering out 665/663).

Subgroup: 3.5.7.11.13.17.19

Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 4459/4455

Subgroup-val mapping: [⟨1 30 14 28 20 25 2], ⟨0 -42 -18 -38 -26 -33 1]]

mapping generators: ~3, ~19/9

Optimal tunings:

• WE: ~3 = 1902.0918 ¢, ~19/9 = 1292.3032 ¢
• CWE: ~3 = 1901.9550 ¢, ~19/9 = 1292.2083 ¢

Optimal ET sequence: b25ce, b53, b78, b131, b209, b340

Badness (Sintel): 0.45479

3.5.7.11.13.17.19.23 subgroup

2277/2275 may be used in the same way to extend the simpler erigone to the 3.5.7.11.13.23 subgroup.

Subgroup: 3.5.7.11.13.17.19.23

Comma list: 665/663, 847/845, 1575/1573, 1617/1615, 2277/2275, 4459/4455

Subgroup-val mapping: [⟨1 30 14 28 20 25 2 64], ⟨0 -42 -18 -38 -26 -33 1 -90]]

mapping generators: ~3, ~19/9

Optimal tunings:

• WE: ~3 = 1902.0149 ¢, ~19/9 = 1292.2401 ¢
• CWE: ~3 = 1901.9550 ¢, ~19/9 = 1292.1988 ¢

Optimal ET sequence: b53i, b78i, b131, b340, b471

Badness (Sintel): 0.54174

Sirius

Main article: Sirius

This tempers out the Don Page comma between 5/3 and 7/5, allowing an accurate representation of the 3:5:7 chord, similar to the 5:7:9 in Procyon.

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]]

mapping generators: ~3, ~25/21

Optimal tunings:

• WE: ~3 = 1902.4455 ¢, ~25/21 = 293.7393 ¢
• CWE: ~3 = 1901.9550 ¢, ~25/21 = 293.7594 ¢

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

Badness (Sintel): 0.213

Remus

Todo: complete section Catalog the intermediate extension (3.5.7.11).

By splitting the generator of sirius into three, remus efficiently represents the no-2's 13-limit with mos scales of 18, 25, 32, or 39 steps.

This is essentially electra but with prime 7, or more accurately, electra is the no-sevens restriction of this temperament.

Subgroup: 3.5.7.11.13

Comma list: 275/273, 1625/1617, 1575/1573

Subgroup-val mapping: [⟨1 4 6 5 6], ⟨0 -9 -15 -10 -13]]

mapping generators: ~3, ~15/11

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~15/11 = 536.090 ¢

Supporting ETs: 39, 7, 32, 71, 110, 46, 149, 188, 181

Badness (Sintel): 0.286

Mizar

Todo: complete section Catalog the intermediate extensions (3.5.7.11, 3.5.7.11.13) instead.

This temperament uses a weak extension to the 3.5.7.17 subgroup 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: 2025/2023, 3125/3087

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

mapping generators: ~3, ~17/9

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~17/9 = 1097.800 ¢

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

Badness (Sintel): 0.383

3.5.7.11.13.17 subgroup

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 17/13 to 35/27.

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]]

mapping generators: ~3, ~17/9

Optimal tuning (CWE): ~3 = 1901.955 ¢, ~17/9 = 1098.298

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

Badness (Sintel): 0.841

Bohlenic

This temperament is identical to 13edt (equal-tempered Bohlen–Pierce scale), but has an independent generator for 11.

Subgroup: 3.5.7.11

Comma list: 245/243, 3125/3087

Subgroup-val mapping: ((mapping| 13 19 23 0 | 0 0 0 1 }}

mapping generators: ~27/25, ~11

Optimal tunings:

• CTE: ~27/25 = 146.304 ¢, ~11 = 4151.318 ¢
• CWE: ~27/25 = 146.304 ¢, ~11 = 4147.705 ¢

Optimal ET sequence: b13, b26, b39

Badness (Sintel): 0.499

3.5.7.11.13 subgroup

Subgroup: 3.5.7.11.13

Comma list: 245/243, 275/273, 847/845

Subgroup-val mapping: ((mapping| 13 19 23 0 2 | 0 0 0 1 1 }}

Optimal tunings:

• CTE: ~27/25 = 146.304 ¢, ~11 = 4149.733 ¢
• CWE: ~27/25 = 146.304 ¢, ~11 = 4146.033 ¢

Optimal ET sequence: b13, b26, b39

Badness (Sintel): 0.365

Tuning diagrams

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

Temperaments supported by 13edt, labelled by name

Temperaments not supported by 13edt, labelled by name

Both sets, labelled by name

Temperaments supported by 13edt, labelled by comma

Temperaments not supported by 13edt, labelled by comma

Both sets, labeled by comma

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

Fomalhaut

Fomalhaut is an extension of Deneb to higher limits that splits the interval of 11/3 in three.

The 23-limit version of Fomalhaut was created first, as an attempt to approximate the no-2s, no-7s 23-limit as accurately as possible using 25 to 35 notes per equave, defined as the b28 & b33 temperament in this limit. Then the lower limit versions were created by simply extrapolating the temperament downwards.

Fomalhaut follows the convention of naming no-twos temperaments after stars.

Subgroup: 3.5.11.13

Comma list: 6655/6561, 274625/264627

Gencom: [3/1 99/65; 6655/6561 274625/264627]

Sval mapping: [⟨1 5 1 -2], ⟨0 -9 3 11]]

POTE generator: ~99/65 = 748.0156

EDTs: b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ff

• Complexity: 1.561892
• Adjusted Error: 6.495941 cents
• TE Error: 1.755451 cents/octave

3.5.11.13.17

Subgroup: 3.5.11.13.17

Comma list: 1105/1089, 4225/4131, 6655/6561

Gencom: [3/1 99/65; 1105/1089 4225/4131 6655/6561]

Sval mapping: [⟨1 5 1 -2 1], ⟨0 -9 3 11 4]]

POTE generator: ~17/11 = 748.0236

EDTs: b28, b5, b33, b23f, b61, b56f, b38c, b10cf, b66c, b51ffg

• Complexity: 1.418914
• Adjusted Error: 6.431616 cents
• TE Error: 1.573498 cents/octave

3.5.11.13.17.19

Subgroup: 3.5.11.13.17.19

Comma list: 247/243, 325/323, 1105/1089, 4675/4617

Gencom: [3/1 99/65; 247/243 325/323 1105/1089 4675/4617]

Sval mapping: [⟨1 5 1 -2 1 7], ⟨0 -9 3 11 4 -11]]

POTE generator: ~17/11 = 747.9960

EDTs: b28, b33, b5, b61, b56f, b23f, b38ch, b66ch, b89fgh, b10cfh

• Complexity: 1.449992
• Adjusted Error: 6.125446 cents
• TE Error: 1.441985 cents/octave

3.5.11.13.17.19.23

Subgroup: 3.5.11.13.17.19.23

Comma list: 209/207, 247/243, 255/253, 325/323, 4675/4617

Gencom: [3/1 99/65; 209/207 247/243 255/253 325/323 4675/4617]

Sval mapping: [⟨1 5 1 -2 1 7 6], ⟨0 -9 3 11 4 -11 -8]]

POTE generator: ~17/11 = 748.0874

EDTs: b28, b5, b33, b23f, b61, b56f, b38ch, b10cfhi, b66ch, b51ffg

• Complexity: 1.382541
• Adjusted Error: 7.087107 cents
• TE Error: 1.566709 cents/octave

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

Extensions to prime 5 are covered at Mintra and Nekkar.

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

Mebsuta is a microtemperament in the 3.7.11 subgroup that sets the relative sizes of 9/7 and 11/9 to be in the ratio of 5:4; its generator is identifiable as the ratio between these intervals, 81/77. It produces a 21L 1s MOS scale against the tritave, which serves as a well-temperament of 22edt; that scale's chroma is identified with 1331/1323.

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

Mebsuta naturally extends itself with prime 19, identifying the two-generator interval as 21/19, since its square differs from 11/9 (the four-generator interval) by the small comma 3971/3969.

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

3.5.7.11.19 subgroup

Tempering out 12005/11979, the unisquary comma, sets the chroma 1331/1323 equal to 245/243, producing an accurate if complex mapping for prime 5 at 32 generators up; it is notable that this sets eight 11/9s equal to 5/1, which is the 3.5.11 restriction of mohaha.

Subgroup: 3.5.7.11.19

Comma list: 3971/3969, 12005/11979, 41553/41503

Sval mapping: [⟨1 0 2 2 3], ⟨0 32 -5 4 -7]]

Sval mapping generators: ~3, ~81/77

Optimal tunings:

• PEWE (Pure-Equaves WE): ~3 = 1\1ed3, ~81/77 = 87.065
• CWE: ~3 = 1\1ed3, ~81/77 = 87.066

Supporting ETs: b131, b22, b153, b284, b415, b109, b437, b175, b546, b87c, b699, b240, b590, b721

Adhara

Adhara cleaves the step of Mebsuta in three to produce a remarkable Don Page temperament for the chord 7:9:11:13:17 (that is, setting 13/11 to two-thirds of 9/7, and 17/13 to four-thirds of 11/9). It can be extended to even higher subgroups fairly naturally, and encompasses several prominent tunings within its structure (such as 65edt~41edo, 131edt, and 197edt).

Subgroup: 3.7.11.13.17

Comma list: 14161/14157, 107811/107653, 1108809/1108723

Sval mapping: [⟨1 2 2 2 2], ⟨0 -15 12 22 38]]

Sval mapping generators: ~3, ~119/117

Optimal tunings:

• PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~119/117 = 28.979
• CTE: ~3 = 1\1ed3, ~119/117 = 28.970

Optimal ET sequence: b65, b66, b131, b197, b328, b525, b722, b1247f, b3216defff

3.7.11.13.17.19 subgroup

This includes the natural extension of Mebsuta to prime 19.

Subgroup: 3.7.11.13.17.19

Comma list: 3213/3211, 3971/3969, 14161/14157, 41553/41503

Sval mapping: [⟨1 2 2 2 2 3], ⟨0 -15 12 22 38 -21]]

Sval mapping generators: ~3, ~119/117

Optimal tunings:

• PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~119/117 = 28.973
• CTE: ~3 = 1\1ed3, ~119/117 = 28.970

Optimal ET sequence: b65, b66, b131, b197, b525, b722, b919, b2035df

3.7.8.11.13.17.19 subgroup

This sets two-thirds of 11/9 to 8/7.

Subgroup: 3.7.8.11.13.17.19

Comma list: 513/512, 729/728, 833/832, 969/968, 3971/3969

Sval mapping: [⟨1 2 2 2 2 2 3], ⟨0 -15 -7 12 22 38 -21]]

Sval mapping generators: ~3, ~64/63

Optimal tunings:

• PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~64/63 = 28.978
• CTE: ~3 = 1\1ed3, ~64/63 = 28.975

Optimal ET sequence: b65, b66, b131, b197, b328, b525, b722, b1247âf, b1969ââf (â is the wart for 8.)

3.5.7.8.11.13.17.19.23 subgroup

At the cost of lower accuracy, Procyon can be added to the Adhara structure, thereby spanning the entire triple-octave 23-limit.

Subgroup: 3.5.7.8.11.13.17.19.23

Comma list: 361/360, 441/440, 513/512, 729/728, 833/832, 969/968, 1127/1125

Sval mapping: [⟨1 2 2 2 2 2 2 3 4], ⟨0 -35 -15 -7 12 22 38 -21 -75]]

Sval mapping generators: ~3, ~64/63

Optimal tunings:

• PETE (Pure-Equaves TE): ~3 = 1\1ed3, ~64/63 = 29.032
• CTE: ~3 = 1\1ed3, ~64/63 = 29.041

Optimal ET sequence: b65i, b66i, b131

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