User:VIxen/Cemetery
Shaka
Two commas that split 2/1 in half, corresponding to convergents to sqrt(2), are the shaftesburisma S29/S41 and the kalisma S99, prompting to temper out {S29, S41, S99}, approximating /29 and /41 primodal chords well.
Subgroup: 2.3.35.11.29.41
Comma list: 841/840, 1189/1188, 1681/1680
Sval mapping: [⟨2 2 6 5 7 8], ⟨0 1 1 -1 1 1], ⟨0 0 2 2 1 1]]
Optimal tuning (CTE): ~41/29 = 1\2, ~3/2 = 702.031, ~41/24 = 926.693
Supporting ETs: 22, 26, 36, 48, 70, 96, 106, 118, 140, 154, 176, 188, 224, 272, 294, 342
Scale: Shaka10
Spog
This temperament produces superpelog-like semiquartal scales while being more accurate (see rational approximations to their intervals).
Subgroup: 2.15.55
Comma list: 100663296/100656875
Sval mapping: [⟨1 0 5], ⟨0 5 1]]
Optimal tuning (subgroup CTE): ~55/32 = 937.655
Optimal ET sequence: 5, 9, 23, 32, 151, 183, 215, 247, 956, 1203, 1450, 3147, 4597
2.15.55.325
Subgroup: 2.15.55.325
Comma list: 4225/4224, 6656/6655
Sval mapping: [⟨1 0 5 6], ⟨0 5 1 3]]
Optimal tuning (subgroup CTE): ~55/32 = 937.647
Supporting ETs: 5, 9, 13[-15], 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
2.15.189.55.325
Related temperament: lux
Subgroup: 2.15.189.55.325
Comma list: 2080/2079, 3025/3024, 4096/4095
Sval mapping: [⟨1 0 6 5 6], ⟨0 5 2 1 3]]
Optimal tuning (subgroup CTE): ~55/32 = 937.677
Supporting ETs: 5, 9, 14, 23, 32, 37, 41, 50, 55, 64, 73, 78, 87, 96, 101, 105, 119, 128, 151, 183, 206, 311
2.15.189.55.325.725
Subgroup: 2.15.189.55.325.725
Comma list: 1625/1624, 2080/2079, 3025/3024, 4096/4095
Sval mapping: [⟨1 0 6 5 6 -3], ⟨0 5 2 1 3 16]]
Optimal tuning (subgroup CTE): ~55/32 = 937.649
Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 142[+725], 151, 183, 206[+725], 311
2.15.189.55.325.725.279
Here are rational approximations to the intervals of the semiquartal scale.
Sharp: 12/11, 25/21, 33/26, 18/13, 31/21 ~ 65/44 ~ 96/65, 50/31 ~ 29/18, 55/32, 15/8.
Flat: 16/15, 64/55, 31/25 ~ 36/29, 42/31 ~ 65/48 ~ 88/65, 13/9, 52/33, 42/25, 11/6.
Subgroup: 2.15.189.55.325.725.279
Comma list: 1625/1624, 2016/2015, 2080/2079, 3025/3024, 4096/4095
Sval mapping: [⟨1 0 6 5 6 -3 5], ⟨0 5 2 1 3 16 4]]
Optimal tuning (subgroup CTE): ~55/32 = 937.638
Supporting ETs: 9[-725], 14[+725], 23, 32, 41[-725], 55, 73[-725], 87, 105[-725], 119, 151, 183, 206[+725], 311
Poggers
Related temperaments: pogo, supers
Subgroup: 2.9.7.15/11.13
Comma list: 540/539, 1716/1715, 2080/2079
Sval mapping: [⟨1 1 1 -1 -1], ⟨0 6 5 4 13]]
Optimal tuning (subgroup CTE): ~9/7 = 433.888
Supporting ETs: 8[+9, +7, +13], 11, 14[-13], 19[+9, +7, ++13], 25[-13], 36, 47, 58, 61[-13], 69[+13], 80[+13], 83, 91[+9, +7, +13], 105
Hnoss
To the wizma [-6 -8 2 5⟩ = 420175/419904, the kalisma is a natural complement, as their product is the argyria, [-9 -4 0 3 2⟩ = 41503/41472. VIxen named it and gersemi after Odr's and Freya's daughters since in 11-limit, they both temper out one comma of odin (9801/9800) and one of freya (41503/41472). There is also a word play on "semi-" that hints at the split of the octave in half.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 41503/41472
Mapping: [⟨2 0 1 2 6], ⟨0 1 4 0 2], ⟨0 0 -5 2 -3]]
- mapping generators: ~99/70, ~3, ~144/77
Optimal ET sequence: 22, 50, 72, 166, 176, 198, 248, 270, 342, 612, 954, 1566, 4086dee, 5652cddeee
Badness: 0.368 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 1716/1715, 2080/2079, 17303/17280
Mapping: [⟨2 0 1 2 6 -3], ⟨0 1 4 0 2 1], ⟨0 0 -5 2 -3 4]]
Optimal ET sequence: 22f, 32cf, 54cff, 72, 166, 198, 270, 634, 904, 1174, 1880ef
Badness: 0.867 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 715/714, 1089/1088, 1225/1224, 2025/2023
Mapping: [⟨2 0 1 2 6 -3 0], ⟨0 1 4 0 2 1 6], ⟨0 0 -5 2 -3 4 -6]]
Optimal ET sequence: 22f, 54cffgg, 72, 166g, 198g, 270, 364, 436, 634g, 706f
Badness: 0.862 × 10-3
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 4200/4199
Mapping: [⟨2 0 1 2 6 -3 0 13], ⟨0 1 4 0 2 1 6 2], ⟨0 0 -5 2 -3 4 -6 -6]]
Optimal ET sequence: 72, 94, 166g, 198g, 270, 436, 634g, 706f
Badness: 0.901 × 10-3
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 715/714, 1225/1224, 1540/1539, 2080/2079, 2530/2527, 2737/2736
Mapping: [⟨2 0 1 2 6 -3 0 13 19], ⟨0 1 4 0 2 1 6 2 -2], ⟨0 0 -5 2 -3 4 -6 -6 -2]]
Optimal ET sequence: 72, 94, 166g, 270, 342f, 436, 706fi
Badness: 1.14 × 10-3
Gersemi
The extension to 13-limit with 4225/4224 is weak but facilitates the use of 18/7 as the equave. Fokker blocks of 128 notes are available for the latter, corresponding to 94edo. 18/7 is split into 4 parts that become ~19/15 in 19-limit. Also, (18/7)3 ~ 17/1 via the chlorisma. However, the tones 9/8 and (19/15)/(9/8) = 152/135 have distinct mappings.
Subgroup: 2.3.5.7.11.13
Comma list: 4225/4224, 9801/9800, 41503/41472
Mapping: [⟨2 0 1 2 6 9], ⟨0 1 9 -2 5 -6], ⟨0 0 -10 4 -6 7]]
- mapping generators: ~99/70, ~3, ~154/65
Optimal ET sequence: 44, 50, 94, 144, 176, 220, 270, 590, 684, 954
Badness: 1.06 × 10-3
17-limit
Subgroup: 2.3.5.7.11.13.17
Comma list: 1089/1088, 1225/1224, 2025/2023, 4225/4224
Mapping: [⟨2 0 1 2 6 9 0], ⟨0 1 9 -2 5 -6 12], ⟨0 0 -10 4 -6 7 -12]]
Optimal ET sequence: 44, 50, 94, 144g, 176g, 220g, 270, 364, 414, 634g, 684
Badness: 1.46 × 10-3
19-limit
Subgroup: 2.3.5.7.11.13.17.19
Comma list: 1089/1088, 1225/1224, 1729/1728, 2926/2925, 3762/3757
Mapping: [⟨2 0 1 2 6 9 0 1], ⟨0 1 9 -2 5 -6 12 11], ⟨0 0 -10 4 -6 7 -12 -11]]
Optimal ET sequence: 44, 50, 94, 144gh, 176g, 220g, 270, 414h, 590, 634g, 684h
Badness: 1.11 × 10-3
23-limit
Subgroup: 2.3.5.7.11.13.17.19.23
Comma list: 897/896, 1089/1088, 1225/1224, 1729/1728, 2737/2736, 2926/2925
Mapping: [⟨2 0 1 2 6 9 0 1 7], ⟨0 1 9 -2 5 -6 12 11 3], ⟨0 0 -10 4 -6 7 -12 -11 -3]]
Optimal ET sequence: 44, 50, 94, 144gh, 176g, 220g, 226, 270, 320i, 364i, 414hi
Badness: 1.23 × 10-3
Rishi
The 7-limit comma [65 -84 10 16⟩ ~ 0.13¢ has the ratio of the exponents of 3 and 2 that is close to the one in 81/8. The square root of the latter is close to 35/11. This suggests tempering out (81/8)(35/11)-2, which is the kalisma.
Apart from 35/11, 35/33, and the equivalents of their squares, 81/8 and 9/8, another equave that comes to mind is 3/2, especially after tempering out the chalmersia. When 3/2 is chosen as the equave, Fokker blocks of 34 notes can be used that are close to 34edf and 58edo.
When VIxen was naming this temperament, sruti was its only named rank-2 subtemperament. Rishis (sages) are believed to have created shruti texts.
Subgroup: 2.3.5.7.11
Comma list: 9801/9800, 572145834917888/571919811374025
Mapping: [⟨2 0 3 -10 -4], ⟨0 1 2 4 4], ⟨0 0 8 -5 3]]
- mapping generators: ~99/70, ~3, ~17364375/14172488
Optimal ET sequence: 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 7414, 9874, 12828e
Badness: 2.10 × 10-3
13-limit
Subgroup: 2.3.5.7.11.13
Comma list: 9801/9800, 10648/10647, 371293/371250
Mapping: [⟨2 0 3 -10 -4 2], ⟨0 1 2 4 4 3], ⟨0 0 8 -5 3 7]]
Optimal ET sequence: 24, 34d, 58, …, 436, 460, 494, 954, 1448, 1506, 2460, 2954, 5414, 6920, 7414, 9874, 12828e
Badness: 0.505 × 10-3