Chromatic pairs
A chromatic pair is a pair of mos scales in some rank-2 temperament (either a full p-limit temperament or a subgroup temperament) that are analogous to the familiar diatonic and chromatic scales of standard tuning. The smaller scale is an albitonic scale (from Latin albus, "white"), named such since the diatonic scale can be played using the white keys of a piano, and the larger of the two is a chromatic scale, analogous to the chromatic scale of standard tuning. Other related scales can be associated to chromatic pairs, namely a haplotonic scale (from Greek haplóos, "single, simple"), analogous to the pentatonic scale of standard tuning and played using the black keys of a piano.
In the sections below, the pair temperaments are listed in order of increasing RMS error, which is an appropriate error measure for subgroups as well as full p-limit groups.
Terrain
- "Terrain" redirects here. For the scale, see Terrain (scale).
Terrain, the 6 & 21 temperament in the 2.9/5.9/7 subgroup, is related to domain. It is a remarkable temperament, in that while its complexity is low, it has no discernible error. The 1–7/5–9/5 and 1–9/7–9/5 chords are characteristic.
Subgroup: 2.9/5.9/7
Comma list: 250047/250000
Sval mapping: [⟨3 0 -4], ⟨0 1 2]]
Gencom mapping: [⟨3 10/9 -7/9 2/9], ⟨0 -2/3 -1/3 2/3]]
- gencom: [63/50 10/9; 250047/250000]
Optimal tuning (subgroup POTE): ~63/50 = 1\3, ~10/9 = 182.461
Optimal ET sequence: 6, 21, 27, 33, 105, 138, 171, 1848, 2019, 2190, 2361, 2532, 2703, 2874, 3045, 3216, 3387, 3558
RMS error: 0.00844 cents
Scales:
- Music
- 5 5 1 mode of 33 equal (2013) by Chris Vaisvil – blog | YouTube | play
Petrtri
Petrtri can be described as 3 & 5 temperament in the 2.11/5.13/5 subgroup.
Subgroup: 2.11/5.13/5
Comma list: 2200/2197
Sval mapping: [⟨1 0 1], ⟨0 3 1]]
Gencom mapping: [⟨1 0 -1/3 0 -1/3 2/3], ⟨0 0 -4/3 0 5/3 -1/3]]
- gencom: [2 13/10; 2200/2197]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 455.012
Optimal ET sequence: 21, 29, 153, 182, 211, 240, 269, 298, 327, 356, 385, 509, 741c, 1126c
RMS error: 0.0749 cents
Scales:
Barton
Barton may be described as the 11 & 13 temperament in the 2.5.11.13 subgroup. It was named after Jacob Barton by Gene Ward Smith and Carl Lumma in 2006[1].
Subgroup: 2.5.11.13
Comma list: 2200/2197, 6656/6655
Sval mapping: [⟨1 6 3 6], ⟨0 -8 1 -5]]
Gencom mapping: [⟨1 0 6 0 3 6], ⟨0 0 -8 0 1 -5]]
- gencom: [2 11/8; 2200/2197 6656/6655]
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 551.699
Optimal ET sequence: 11, 13, 24, 37, 50, 87, 298, 385, 472, 559, 1590cd
RMS error: 0.0822 cents
Scales:
Voltage
Voltage is the 3 & 7 temperament in the 2.7.13 subgroup.
Subgroup: 2.7.13
Comma list: 28672/28561
Sval mapping: [⟨1 4 4], ⟨0 -4 -1]]
Gencom mapping: [⟨1 0 0 4 0 4], ⟨0 0 0 -4 0 -1]]
- gencom: [2, 16/13; 28672/28561]
Optimal ET sequence: 3, 7, 10, 27, 37, 47, 57, 104
RMS error: 0.1423
Tridec
Tridec, the 5 & 8 temperament in the 2.7/5.11/5.13/5 subgroup, extends petrtri.
Subgroup: 2.7/5.11/5.13/5
Comma list: 847/845, 1001/1000
Sval mapping: [⟨1 2 0 1], ⟨0 -4 3 1]]
Gencom mapping: [⟨1 0 -3/4 5/4 -3/4 1/4], ⟨0 0 0 -4 3 1]]
- gencom: [2 13/10; 847/845 1001/1000]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 454.556
Optimal ET sequence: 5, 8, 21, 29, 37, 66, 169, 235, 404c, 639c, 953bc
RMS error: 0.1613 cents
Nestoria
Scales:
- nestoria7 – 7-tone albitonic scale
- nestoria12 – 12-tone chromatic scale
- nestoria17 – 17-tone mega chromatic scale
Orgone
Orgone, the 4 & 7 temperament in the 2.7.11 subgroup, is related to superkleismic.
Subgroup: 2.7.11
Comma list: 65536/65219
Sval mapping: [⟨1 2 4], ⟨0 3 -2]]
Gencom mapping: [⟨1 0 0 2 4], ⟨0 0 0 3 -2]]
- gencom: [2 77/64; 65536/65219]
Optimal tuning (POTE): ~2 = 1\1, ~77/64 = 323.372
Optimal ET sequence: 4, 7, 11, 15, 26, 89, 115, 141, 167, 308, 475bc, 783bc
RMS error: 0.1888 cents
Scales:
Marveltwintri
Marveltwintri can be described as the 3 & 4 temperament in the 2.5/3.13/9 subgroup.
Subgroup: 2.5/3.13/9
Comma list: 325/324
Sval mapping: [⟨1 0 2], ⟨0 1 -2]]
Gencom mapping: [⟨1 -1/6 5/6 0 0 -1/3], ⟨0 -1/2 -3/2 0 0 1]]
- gencom: [2 6/5; 325/324]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 317.139
Optimal ET sequence: 11, 15, 19, 34, 53, 87, 140, 185, 219, 253, 287b, 321b
RMS error: 0.2444 cents
Scales:
- marveltwintri7 – 7-tone albitonic scale
- marveltwintri11 – 11-tone chromatic scale
- marveltwintri15 – 15-tone mega chromatic scale
Cata
Scales:
Bridgetown
Bridgetown, the 5 & 24 temperament in the 2.3.11/5.13/5 subgroup, is related to haumea and barbados.
Subgroup: 2.3.11/5.13/5
Comma list: 352/351, 676/675
Sval mapping: [⟨1 0 -6 -1], ⟨0 2 9 3]]
Gencom mapping: [⟨1 2 -5/3 0 4/3 1/3], ⟨0 -2 4 0 -5 1]]
- gencom: [2 15/13; 352/351 676/675]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.399
Optimal ET sequence: 5, 9, 14, 19, 24, 29, 169, 198, 227, 256, 285, 314
RMS error: 0.2513 cents
Scales:
- bridgetown9 – 9-tone albitonic scale
- bridgetown14 – 14-tone chromatic scale
- Music
- Paint in the Water 29 (archived 2020) by Igliashon Jones
- Route 14 in Bridgetown (archived 2020) by Chris Vaisvil
Haumea
Related temperaments include bridgetown, namaka, hemigari, barbados, and parizekmic.
Subgroup: 2.3.7/5.11/5.13/5
Comma list: 352/351, 676/675, 847/845
Sval mapping: [⟨1 0 10 -6 -1], ⟨0 2 -12 9 3]]
Gencom mapping: [⟨1 2 -3/4 -11/4 9/4 5/4], ⟨0 -2 0 12 -9 -3]]
- gencom: [2 15/13; 352/351 676/675 847/845]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~15/13 = 248.491
Optimal ET sequence: 24, 29, 111, 140, 169, 198, 565d, 763bd, 961bd
RMS error: 0.2668 cents
Scales:
Neutral
Scales:
- neutral7 – 7-tone albitonic scale
- neutral10 – 10-tone chromatic scale
- neutral17 – 17-tone mega-chromatic scale
- Music
- Mo - Happy - Happy play by Jake Freivald in Neutral[7], 24edo tuning
Slendric
Scales:
- Slendric5 – 5-tone haplotonic scale
- Slendric6 – 6-tone haplotonic scale
- Slendric11 – 11-tone albitonic scale
- Slendric16 – 16-tone chromatic scale
Huntington
Huntington may be described as the 10 & 27 temperament in the 2.5.7.13 subgroup.
Subgroup: 2.5.7.13
Comma list: 640/637, 10985/10976
Sval mapping: [⟨1 5 4 4], ⟨0 -9 -4 -1]]
Gencom mapping: [⟨1 0 5 4 0 4], ⟨0 0 -9 -4 0 -1]]
- gencom: [2 16/13; 640/637 10985/10976]
Optimal tuning (POTE): ~2 = 1\1, ~16/13 = 357.002
Optimal ET sequence: 7, 10, 17, 27, 37, 84, 121, 279cd, 400cd
RMS error: 0.3452 cents
Scales:
- huntington7 – 7-tone haplotonic scale
- huntington10 – 10-tone albitonic scale
- huntington17 – 17-tone chromatic scale
Tobago
Tobago, the 10 & 14 temperament in the 2.3.11.13/5 subgroup, extends neutral and barbados.
Subgroup: 2.3.11.13/5
Comma list: 243/242, 676/675
Sval mapping: [⟨2 0 -1 -2], ⟨0 2 5 3]]
Gencom mapping: [⟨2 4 -2 0 9 2], ⟨0 -2 3/2 0 -5 -3/2]]
- gencom: [55/39 15/13; 243/242 676/675]
Optimal tuning (subgroup POTE): ~55/39 = 1\2, ~15/13 = 249.312
Optimal ET sequence: 10, 14, 24, 58, 82, 130
RMS error: 0.3533 cents
Scales:
Pepperoni
Pepperoni is generated by a fifth and can be described as the 5 & 12 temperament in the 2.3.11/7.13/7. It is the single-chain retraction of parapyth. The Pepper fifth, which is (40200 + 600 sqrt(5))/59 = 704.096 cents, is a good pepperoni generator, hence the name.
Subgroup: 2.3.11/7.13/7
Comma list: 352/351, 364/363
Sval mapping: [⟨1 0 7 12], ⟨0 1 -4 -7]]
Gencom mapping: [⟨1 1 0 -8/3 1/3 7/3], ⟨0 1 0 11/3 -1/3 -10/3]]
- gencom: [2 3/2; 352/351 364/363]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~3/2 = 703.856
Optimal ET sequence: 5, 7, 12, 17, 29, 46, 58, 75, 80, 87, 104, 121, 167, 196, 208, 271, 595bcd
RMS error: 0.3789 cents
Scales:
- pepperoni7 – 7-tone albitonic scale
- pepperoni12 – 12-tone chromatic scale
Wizz
Wizz, the 6 & 16 temperament in the 2.5.11 subgroup, is related to wizard.
Subgroup: 2.5.11
Comma list: 15625/15488
Sval mapping: [⟨2 0 -7], ⟨0 1 3]]
Gencom mapping: [⟨2 0 4 0 5], ⟨0 0 1 0 3]]
- gencom: [125/88 5/4; 15625/15488]
Optimal tuning (POTE): ~125/88 = 1\2, ~5/4 = 383.768
Optimal ET sequence: 6, 16, 22, 28, 50, 122, 172, 222
RMS error: 0.3997
Scales:
Bossier
Bossier can be described as the 3 & 17 in the 2.7.11.13 subgroup.
Subgroup: 2.7.11.13
Comma list: 1573/1568, 15488/15379
Sval mapping: [⟨1 0 1 3], ⟨0 8 7 2]]
Gencom mapping: [⟨1 0 0 0 1 3], ⟨0 0 0 8 7 2]]
- gencom: [2 14/11; 1573/1568 15488/15379]
Optimal tuning (POTE): ~2 = 1\1, ~14/11 = 421.309
Optimal ET sequence: 17, 20, 37, 57, 94, 225, 319cd, 413bcd
RMS error: 0.4043 cents
Scales:
Penta a.k.a. mechanism
Scales:
Bluebirds
Subgroup: 2.11.13
Comma list: 265837/262144
Sval mapping: [⟨1 0 6], ⟨0 3 -2]]
Gencom mapping: [⟨1 0 0 0 3 4], ⟨0 0 0 0 3 -2]]
- gencom: [2 143/128; 265837/262144]
Optimal tuning (POTE): ~2 = 1\1, ~143/128 = 182.368
Optimal ET sequence: 6, 7, 13, 33, 46, 79, 125c, 204bc, 329bc
RMS error: 0.4444 cents
Scales:
- bluebirds6 – 6-tone haplotonic scale
- bluebirds7 – 7-tone albitonic scale
- bluebirds13 – 13-tone chromatic scale
Marveltri
Marveltri, the 3 & 13 temperament in the 2.5.9/7 subgroup, is related to marvel, magic, and the unnamed 22 & 47 temperament.
Subgroup: 2.5.9/7
Comma list: 225/224
Sval mapping: [⟨1 0 5], ⟨0 1 -2]]
Gencom mapping: [⟨1 2/5 2 -1/5], ⟨0 -4/5 1 2/5]]
- gencom: [2 5/4; 225/224]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~5/4 = 383.638
Optimal ET sequence: 13, 16, 19, 22, 25, 47, 69, 72, 97, 122, 269bc, 660bc
RMS error: 0.4801 cents
Scales:
- marveltri7 – 7-tone albitonic scale
- marveltri10 – 10-tone chromatic scale
- marveltri13 – 13-tone mega-chromatic scale
Edson
Edson is related to pele and andromeda.
Subgroup: 2.3.7/5.11/5.13/5
Comma list: 196/195 = [2 -1 2 0 -1⟩, 352/351 = [5 -3 0 1 -1⟩, 364/363 = [2 -1 1 -2 1⟩
Sval mapping: [⟨1 0 10 17 22], ⟨0 1 -6 -10 -13]]
- mapping generators: ~2, ~3
Gencom mapping: [⟨1 1 -5 -1 2 4], ⟨0 1 29/4 5/4 -11/4 -23/4]]
- gencom: [2 3/2; 196/195, 352/351, 364/363]
Optimal ET sequence: 12, 17, 29
RMS error: 0.5102 cents
Scales:
Photia
Scales:
Shoe a.k.a. llywelyn
Scales:
Roulette
Scales:
- roulette6 – 6-tone haplotonic scale
- roulette7 – 7-tone albitonic scale
- roulette13 – 13-tone chromatic scale
- roulette19 – 19-tone mega-chromatic scale
Silver
Silver can be described as the 10 & 27 temperament in the 2.5.7.13.17 subgroup.
Subgroup: 2.5.7.13.17
Comma list: 170/169, 640/637, 5525/5488
Sval mapping: [⟨1 5 4 4 2], ⟨0 -9 -4 -1 7]]
Gencom mapping: [⟨1 0 -4 0 0 3 9], ⟨0 0 9 4 0 1 -7]]
- gencom: [2 13/8; 170/169 640/637 5525/5488]
Optimal tuning (POTE): ~2 = 1\1, ~13/8 = 842.711
Optimal ET sequence: 7, 10, 17, 27, 37, 47, 84, 131, 178e, 309cde, 487bcdee
RMS error: 0.5886 cents
Scales:
Baldy
Baldy is every other step of garibaldi, without the mapping of prime 11. It can be described as the 6 & 35 temperament.
Subgroup: 2.9.5.7.13
Comma list: 225/224, 325/324, 640/637
Sval mapping: [⟨1 0 15 25 -28], ⟨0 1 -4 -7 10]]
Gencom mapping: [⟨1 3/2 3 4 0 2], ⟨0 1/2 -4 -7 0 10]]
- gencom: [2 9/8; 225/224 325/324 640/637]
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 204.090
Optimal ET sequence: 6, 11, 17, 23, 29, 35, 41, 47, 100, 147, 488cd, 635cd
RMS error: 0.5999 cents
Scales
Guanyintet
Guanyintet, the 4 & 9 temperament in the 2.5.7/3.11/3 subgroup, is related to guanyin as well as to orwell.
Subgroup: 2.5.7/3.11/3
Comma list: 176/175, 540/539
Sval mapping: [⟨1 0 2 -2], ⟨0 3 -1 5]]
Gencom mapping: [⟨1 -4/3 3 -1/3 5/3], ⟨0 4/3 -3 7/3 -11/3]]
- gencom: [2 7/6; 176/175 540/539]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 270.093
Optimal ET sequence: 9, 31, 40, 49, 89, 191bc, 227bc, 231bc, 271bc, 311bc, 316bcd
RMS error: 0.6028 cents
Scales:
- guanyintet5 – 5-tone haplotonic scale
- guanyintet9 – 9-tone albitonic scale
- guanyintet13 – 13-tone chromatic scale
Radon
Gariberttet
Gariberttet can be described as the 4 & 29 temperament in the 2.5/3.7/3.13/11 subgroup.
Subgroup: 2.5/3.7/3.13/11
Comma list: 275/273, 847/845
Sval mapping: [⟨1 0 0 0], ⟨0 3 5 1]]
Gencom mapping: [⟨1 0 0 0 0 0], ⟨0 -8/3 1/3 7/3 -1/2 1/2]]
- gencom: [2 13/11; 275/273 847/845]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/11 = 293.679
Optimal ET sequence: 29, 33, 37, 41, 45, 49, 78, 94, 143d, 237bd, 380bd
RMS error: 0.6914 cents
Scales:
- gariberttet5 – 5-tone haplotonic scale
- gariberttet9 – 9-tone albitonic scale
- gariberttet13 – 13-tone chromatic scale
Namo
Scales:
Sentry
Sentry, the 3 & 5 temperament in the 2.5/3.9/7 subgroup, is related to sensi.
Subgroup: 2.5/3.9/7
Comma list: 245/243
Sval mapping: [⟨1 0 0], ⟨0 2 1]]
Gencom mapping: [⟨1 0 0 0], ⟨0 0 2 -1]]
- gencom: [2 9/7; 245/243]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~9/7 = 440.902
Optimal ET sequence: 8, 11, 19, 30, 41, 49, 52, 145b, 166c, 197bc, 215c, 264bc
RMS error: 0.7105 cents
Scales:
Hemif
Scales:
Garibaldi
Scales:
- Garibaldi12 – 12-tone albitonic scale
- Garibaldi17 – 17-tone chromatic scale
Mabila
Given below is the no-3 version of mabila, the 7 & 9 temperament.
Subgroup: 2.5.7
Comma list: 1071875/1048576
Sval mapping: [⟨1 1 5], ⟨0 3 -5]]
Gencom mapping: [⟨1 0 1 5], ⟨0 0 3 -5]]
- gencom: [2 175/128; 1071875/1048576]
Optimal tuning (POTE): ~2 = 1\1, ~175/128 = 527.236
Optimal ET sequence: 7, 9, 16, 25, 41, 66, 305bc
RMS error: 0.7729 cents
Scales:
Indium
Indium can be described as the 8 & 33 temperament in the 2.5/3.7/3.11/3 subgroup.
Subgroup: 2.5/3.7/3.11/3
Comma list: 3025/3024, 3125/3087
Sval mapping: [⟨1 0 0 2], ⟨0 6 10 -1]]
Gencom mapping: [⟨1 -1/2 -1/2 -1/2 3/2], ⟨0 -15/4 9/4 25/4 -19/4]]
- gencom: [2 12/11; 3025/3024 3125/3087]
Optimal tuning (subgroup POTE): ~2 = 1\1, ~12/11 = 147.010
Optimal ET sequence: 33, 41, 49, 57, 106, 204, 253
RMS error: 0.7788 cents
Scales:
- Music
- The Cry of Indium (2011) by Chris Vaisvil blog | play – in Indium[17]
Starlingtet
Svalname: 4&15
Related temperaments: starling; becomes myna
Subgroup: 2.5/3.7/3
Comma list: 126/125
Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 311.154
Gencom: [2 6/5; 126/125]
Gencom mapping: [<1 -1 0 1|, <0 4/3 1/3 -5/3|]
Mapping: [<1 0 -1|, <0 1 3|]
Optimal ET sequence: 15, 19, 23, 27, 239b, 266b, 293b, 320b, 347b, 374b, 401b
RMS error: 0.8398 cents
Lovecraft
Subgroup: 2.11.13
Comma list: 1352/1331
Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 279.318
Gencom: [2 13/11; 1352/1331]
Gencom mapping: [<1 0 0 0 3 3|, <0 0 0 0 2 3|]
Mapping: [<1 1 0|, <0 2 3|]
Optimal ET sequence: 13, 30, 43, 73, 116b
RMS error: 0.8449 cents
Blackbirds
Subgroup: 2.11.13
Comma list: 29282/28561
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 546.660
Gencom: [13/11 11/8; 29282/28561]
Gencom mapping: [<4 0 0 0 12 13|, <0 0 0 0 1 1|]
Mapping: [<4 0 1|, <0 1 1|]
Optimal ET sequence: 4, 16, 20, 24, 44, 68, 112c, 180bc
RMS error: 0.8685 cents
Laz
Related temperaments: georgian, winston
Subgroup: 2.5.7/3.11/3.13/3
Comma list: 144/143, 176/175, 196/195
Optimal tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 269.300
Gencom: [2 7/6; 144/143 176/175 196/195]
Gencom mapping: [<1 -5/4 3 -1/4 7/4 -1/4|, <0 -1/4 -3 3/4 -21/4 19/4|]
Mapping: [<1 0 2 -2 6|, <0 3 -1 5 -5|]
Optimal ET sequence: 9, 31, 40, 49, 58, 156bde, 205bde
RMS error: 0.8790 cents
Myna
Scales:
- Myna7 – 7-tone haplotonic scale
- Myna11 – 11-tone albitonic scale
- Myna15 – 15-tone chromatic scale
Greeley
Related temperament: Opossum, Nusecond
Subgroup: 2.5/3.7/3.11/3
Comma list: 121/120, 126/125
Optimal tuning (subgroup POTE): ~2 = 1\1, ~11/10 = 155.776
Gencom: [2 11/10; 121/120 126/125]
Gencom mapping: <1 -5/4 -1/4 3/4 3/4|, <0 9/4 1/4 -15/4 5/4|]
Mapping: [<1 1 2 2|, <0 -2 -6 -1|]
Optimal ET sequence: 8, 15, 23, 54, 77, 100, 131d, 208bd
RMS error: 1.034 cents
Music
A Greeley Steals Halloween by Chris Vaisvil in Greeley 15
Semidim
Sval name: 8&29
Subgroup: 2.5/3.7/3.11/3.13/3
Comma list: 121/120, 169/168, 275/273
Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 453.242
Gencom: [2 13/10; 121/120 169/168 275/273]
Gencom mapping: [<1 -3 0 2 0 1|, <0 24/5 -6/5 -26/5 9/5 -1/5|]
Mapping: [<1 3 5 3 4|, <0 -6 -10 -3 -5|]
Optimal ET sequence: 5, 8, 13, 21, 29, 37, 44, 45, 188bde
RMS error: 1.052 cents
Mothwelltri
Svalname: 1&4
Related temperament: Orwell
Subgroup: 2.7/3.11
Comma list: 99/98
Optimal tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 273.174
Gencom: [2 7/6; 99/98]
Gencom mapping: [<1 -1/2 0 1/2 3|, <0 -1/2 0 1/2 2|]
Mapping: [<1 0 1|, <0 1 2|]
Optimal ET sequence: 9, 22, 40, 49c, 58c, 67c, 76c, 79, 101b, 123bc
RMS error: 1.064 cents
Sulis
Related temperament: minerva, würschmidt
Subgroup: 2.5.9/7.11/9
Comma list: 99/98, 176/175
Optimal tuning (subgroup POTE): ~2 = 1\1, ~5/4 = 386.558
Gencom: [2 5/4; 99/98 176/175]
Gencom mapping: [<1 4/9 2 -1/9 -1/9|, <0 -4/3 1 -2/3 4/3|]
Mapping: [<1 0 5 -9|, <0 1 -2 4|]
Optimal ET sequence: 22, 25, 28, 31, 59, 90c, 149c
RMS error: 1.074 cents
Skwares
Related temperament: squares
Subgroup: 2.3.7.11
Comma list: 99/98, 243/242
Optimal tuning (POTE): ~2 = 1\1, ~9/7 = 425.244
Gencom: [2 9/7; 99/98 243/242]
Gencom mapping: [<1 3 0 6 7|, <0 -4 0 -9 -10|]
Mapping: [<1 3 6 7|, <0 -4 -9 -10|]
Optimal ET sequence: 5, 8, 11, 14, 17, 31, 48, 79, 127, 206bcd
RMS error: 1.099 cents
Orwell
Scales:
- Orwell5 – 5-tone haplotonic scale
- Orwell9 – 9-tone albitonic scale
- Orwell13 – 13-tone chromatic scale
Magicaltet
Subgroup: 2.5/3.7.11
Comma list: 100/99 = [2 2 0 -1⟩, 385/384 = [-7 1 1 1⟩
Sval mapping: [⟨1 0 5 2], ⟨0 1 -3 2]]
- mapping generators: ~2, ~5/3
Gencom mapping: [⟨1 -1/2 1/2 2 4], ⟨0 1/2 -1/2 3 -2]]
- gencom: [2 6/5; 100/99 385/384]
Optimal ET sequence: 4, 7, 11, 15, 26, 67, 93*
* wart for 5/3
RMS error: 1.206 cents
Related temperaments: supermagic, superkleismic, magic
Scales:
- magicaltet7 – 7-tone albitonic scale
- magicaltet11 – 11-tone chromatic scale
- magicaltet15 – 15-tone mega-chromatic scale
Magic
Scales:
- Magic7 – 7-tone haplotonic scale
- Magic10 – 10-tone albitonic scale
- Magic13 – 13-tone chromatic scale
- Magic16 – 16-tone mega-chromatic scale
Kiribati
Related temperament: nakika, octacot
Subgroup: 2.9/5.7/3.11/9
Comma list: 100/99, 245/242
Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/20 = 87.892
Gencom: [2 21/20; 100/99 245/242]
Gencom mapping: [<1 1/10 -4/5 11/10 1/5|, <0 -3/2 -1 3/2 1|]
Mapping: [<1 1 1 0|, <0 -2 3 4|]
Optimal ET sequence: 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd
RMS error: 1.245 cents
Score
Subgroup: 2.7.11.13
Comma list: 343/338, 847/832
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 540.099
Gencom: [2 11/8; 343/338 847/832]
Gencom mapping: [<1 0 0 1 3 1|, <0 0 0 4 1 6|]
Mapping: [<1 1 3 1|, <0 4 1 6|]
Optimal ET sequence: 5, 7, 9, 11, 20
RMS error: 1.282 cents
Sensi
Scales:
- Sensi5 – 5-tone haplotonic scale
- Sensi8 – 8-tone albitonic scale
- Sensi11 – 11-tone mega-albitonic scale
- Sensi19 – 19-tone chromatic scale
Sensation
Scales:
- Sensi5 – 5-tone haplotonic scale
- Sensi8 – 8-tone albitonic scale
- Sensi11 – 11-tone mega-albitonic scale
- Sensi19 – 19-tone chromatic scale
Meantone
Scales:
- Meantone5 – 5-tone haplotonic scale
- Meantone7 – 7-tone albitonic scale
- Meantone12 – 12-tone chromatic scale
Mohaha
Scales:
- Mohaha7 – 7-tone albitonic scale
- Mohaha10 – 10-tone chromatic scale
Music:
Mohoho
Joan
Related temperaments: casablanca, orwell
Subgroup: 2.9.7.11
Comma list: 99/98, 9317/9216
Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 542.672 cents
Gencom: [2 11/8; 99/98 9317/9216]
Gencom mapping: [<1 0 0 1 3|, <0 7/2 0 4 1|]
Mapping: [<1 0 1 3|, <0 7 4 1|]
Optimal ET sequence: 11, 20, 31, 42, 115bd, 157bd
RMS error: 1.424 cents
Tutone
Scales:
- Tutone6 – 6-tone haplotonic scale
- Tutone7 – 7-tone albitonic scale
- Tutone13 – 13-tone chromatic scale
Stacks
Related temperament: magic
Svalname: 11&30
Subgroup: 2.9.15.7.11.13
Comma list: 100/99, 105/104, 144/143, 196/195
Optimal tuning (subgroup POTE): ~2 = 1\1, ~9/7 = 439.146
Gencom: [2 9/7; 100/99 105/104 144/143 196/195]
Gencom mapping: [<1 5/2 5/2 5 2 7|, <0 -5/2 -1/2 -6 4 -9|]
Mapping: [<1 0 2 -1 6 -2|, <0 5 3 6 -4 9|]
Optimal ET sequence: 11, 30, 41, 153cdef, 194cdef, 235cdef
RMS error: 1.540 cents
Bleu
Sval name: 9&17
Subgroup: 2.3.7.11.13
Comma list: 78/77, 99/98, 144/143
POL2 generator: ~13/12 = 139.990
Gencom: [2 13/12; 78/77 99/98 144/143]
Gencom mapping: [<1 1 0 2 3 3|, <0 5 0 7 4 6|]
Mapping: [<1 1 2 3 3|, <0 5 7 4 6|]
RMS error: 1.752 cents
bleu8 - Eight note haplotonic scale
bleu9 - Nine note albitonic scale
bleu17 - Seventeen note chromatic scale
Sephiroth
Related temperament: Muggles
Subgroup: 2.5.11.13.17
Comma list: 65/64, 170/169, 221/220
Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 372.236
Gencom: [2 5/4; 65/64 170/169 221/220]
Gencom mapping: [<1 0 2 0 5 4 5|, <0 0 1 0 -5 -1 -3|]
Mapping: [<1 0 15 6 11|, <0 1 -5 -1 -3|]
Optimal ET sequence: 10, 13, 16, 29
RMS error: 1.774 cents
Archy
Scales:
Supra
Scales:
- Supra7 – 7-tone albitonic scale
- Supra12 – 12-tone chromatic scale
Supraphon
Suhajira
Scales:
- Suhajira7 – 7-tone haplotonic scale
- Suhajira10 – 10-tone albitonic scale
- Suhajira17 – 17-tone chromatic scale
Machine
Subgroup: 2.9.7.11
Comma list: 64/63, 99/98
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 214.384
Gencom: [2 9/8; 64/63 99/98]
Gencom mapping: [<1 3/2 0 3 4|, <0 1/2 0 -1 -3|]
Mapping: [<1 0 6 13|, <0 1 -1 -3|]
Optimal ET sequence: 5, 6, 11, 17, 28
RMS error: 1.977 cents
Deutone
Related temperament: meanpop
Subgroup: 2.9.5.7.13
Comma list: 65/64, 81/80, 91/90
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 191.059
Gencom: [2 9/8; 65/64 81/80 91/90]
Gencom mapping: [<1 3/2 2 2 0 4|, <0 1/2 2 5 0 -2|]
Mapping: [<1 0 -4 -13 10|, <0 1 2 5 -2|]
Optimal ET sequence: 6, 7, 13, 19, 25e, 44de
RMS error: 2.003 cents
Negra
Porkypine
Scales:
- Porkypine7 – 7-tone haplotonic scale
- Porkypine8 – 8-tone albitonic scale
- Porkypine15 – 15-tone chromatic scale
Skateboard
Related temperament: thrasher
Subgroup: 2.5/3.7/3.11.13/9
Comma list: 56/55, 91/90, 100/99
Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 313.842
Gencom: [2 6/5; 56/55 91/90 100/99]
Gencom mapping: [<1 -3/7 4/7 11/7 4 -6/7|, <0 0 -1 -3 -2 2|]
Mapping: [<1 0 -1 2 2|, <0 1 3 2 -2|]
Optimal ET sequence: 11, 15, 19, 23, 42d, 65d
RMS error: 2.396 cents
Augment
Related temperament: augmented
Subgroup: 2.5.7.11
Comma list: 56/55, 128/125
Optimal tuning (POTE): ~5/4 = 1\3, ~8/7 = 228.275
Gencom: [5/4 8/7; 56/55 128/125]
Gencom mapping: [<3 0 7 9 11|, <0 0 0 -1 -1|]
Mapping: [<3 7 0 2|, <0 0 1 1|]
Optimal ET sequence: 3, 6, 9, 15, 21
RMS error: 2.422 cents
Semaphore
Scales:
- Semaphore5 – 5-tone haplotonic scale
- Semaphore9 – 9-tone albitonic scale
- Semaphore14 – 14-tone chromatic scale
Kryptonite
Related temperaments: krypton
Subgroup: 2.5.7/3.11/3.13/3
Comma list: 56/55, 78/77, 91/90
Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/12 = 132.428
Gencom: [2 13/12; 56/55 78/77 91/90]
Gencom mapping: [<1 -5/4 2 -1/4 3/4 3/4|, <0 -1/2 3 3/2 -3/2 1/2|]
Mapping: [<1 2 1 2 2|, <0 -3 -2 1 -1|]
Optimal ET sequence: 9, 63, 82bd, 91bde
RMS error: 2.545 cents
Oregon
Subgroup: 2.9.15.7.11
Comma list: 55/54, 64/63, 100/99
Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 325.495
Gencom: [2 6/5; 55/54 64/63 100/99]
Gencom mapping: [<1 2 3 2 4|, <0 -3/2 -5/2 3 -2|]
Mapping: [<1 1 1 5 2|, <0 3 4 -3 2|]
Optimal ET sequence: 7, 11, 37b, 48b, 59b
RMS error: 2.550 cents
Antikythera
Related temperament: pajara
Subgroup: 2.9.5.7
Comma list: 50/49, 64/63
Optimal tuning (POTE): ~7/5 = 1\2, ~8/7 = 214.095
Gencom: [7/5 8/7; 50/49 64/63]
Gencom mapping: [<2 3 5 6|, <0 1/2 -1 -1|]
Mapping: [<2 0 11 12|, <0 1 -1 -1|]
Optimal ET sequence: 4, 6, 10, 16, 22, 28
RMS error: 2.572 cents
Glacial
Subgroup: 2.9.5.11.13
Comma list: 45/44, 65/64, 81/80
[[[Optimal tuning]] (POTE): ~2 = 1\1, ~9/8 = 186.151
Gencom: [2 9/8; 45/44 65/64 81/80]
Gencom mapping: [<1 3/2 2 0 3 4|, <0 1/2 2 0 3 -2|]
Mapping: [<1 0 -4 -6 10|, <0 1 2 3 -2|]
Optimal ET sequence: 6, 13, 45be, 58bce, 71bce, 84bce
RMS error: 2.887 cents
Leantone
Related temperament: meanenneadecal
Subgroup: 2.9.5.7.11
Comma list: 45/44, 56/55, 81/80
Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 192.500
Gencom: [2 9/8; 45/44 56/55 81/80]
Gencom mapping: [<1 3/2 2 2 3|, <0 1/2 2 5 3|]
Mapping: [<1 0 -4 -13 -6|, <0 1 2 5 3|]
Optimal ET sequence: 6, 7, 13, 19, 25, 31, 56, 81
RMS error: 3.882 cents
