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:

• Terrain9 – 9-tone albitonic scale
• Terrain15 – 15-tone chromatic scale

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:

• Petrtri5 – 5-tone haplotonic scale
• Petrtri8 – 8-tone albitonic scale
• Petrtri13 – 13-tone chromatic scale

Barton

"Barton" redirects here. For Jacob Barton's scale, see Barton (scale).

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.

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:

• Barton5 – 5-tone haplotonic scale
• Barton7 – 7-tone albitonic scale
• Barton9 – 9-tone mega-albitonic scale
• Barton11 – 11-tone chromatic scale
• Barton13 – 13-tone mega-chromatic scale

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 tuning:

• POTE: ~2 = 1\1, ~16/13 = 357.677
• POTT: ~2 = 1\1, ~16/13 = 357.794 (1200 - 300 log₂(7))

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

Scales:

• Tridec8 – 8-tone albitonic scale
• Tridec13 – 13-tone chromatic scale

Nestoria

See also: Schismatic family § Nestoria (2.3.5.19)

Scales:

• Nestoria7 – 7-tone albitonic scale
• Nestoria12 – 12-tone chromatic scale
• Nestoria17 – 17-tone mega chromatic scale

Orgone

Main article: Orgone

See also: Orgonia

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:

• Orgone7 – 7-tone albitonic scale
• Orgone11 – 11-tone chromatic scale
• Orgone15 – 15-tone mega-chromatic scale

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

Main article: Hanson and cata

See also: Kleismic family § Cata

Scales:

• Cata7 – 7-tone albitonic scale
• Cata11 – 11-tone mega-albitonic scale
• Cata15 – 15-tone chromatic scale
• Cata19 – 19-tone mega-chromatic scale

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 (2011) by Chris Vaisvil – blog | play – in Bridgetown[14] for flute, bassoons, and contra-bassoon

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:

• Haumea5 – 5-tone haplotonic scale
• Haumea9 – 9-tone albitonic scale
• Haumea14 – 14-tone chromatic scale
• Haumea19 – 19-tone mega-chromatic scale

Neutral

See also: Rastmic clan § 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

Main article: Slendric

See also: Gamelismic clan #Slendric and No-fives subgroup temperaments #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:

• Tobago6 – 6-tone haplotonic scale
• Tobago10 – 10-tone albitonic scale
• Tobago14 – 14-tone chromatic scale
• Tobago24 – 24-tone quatertone scale

Pepperoni

See also: Parapyth

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:

• Wizz6 – 6-tone haplotonic scale
• Wizz10 – 10-tone albitonic scale
• Wizz16 – 16-tone chromatic scale

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:

• Bossier8 – 8-tone albitonic scale
• Bossier11 – 11-tone chromatic scale
• Bossier14 – 14-tone mega-chromatic scale

Penta a.k.a. mechanism

See also: Subgroup temperaments § Penta a.k.a. mechanism

Scales:

• Penta5 – 5-tone haplotonic scale
• Penta8 – 8-tone albitonic scale
• Penta11 – 11-tone chromatic scale
• Penta19 – 19-tone mega-chromatic scale

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 tunings:

• subgroup CTE: ~2 = 1\1, ~3/2 = 703.4398
• subgroup POTE: ~2 = 1\1, ~3/2 = 703.414

Optimal ET sequence: 12, 17, 29

RMS error: 0.5102 cents

Scales:

• Edson5 – 5-tone haplotonic scale
• Edson7 – 7-tone albitonic scale
• Edson12 – 12-tone chromatic scale
• Edson17 – 17-tone mega-chromatic scale

Photia

See also: Schismatic family § Photia (2.3.5.17)

Scales:

• Photia7 – 7-tone albitonic scale
• Photia12 – 12-tone chromatic scale

Shoe a.k.a. llywelyn

See also: Llywelynsmic clan § Llywelyn a.k.a. shoe

Scales:

• Shoe5 – 5-tone mini-haplotonic scale
• Shoe6 – 6-tone haplotonic scale
• Shoe11 – 11-tone albitonic scale
• Shoe16 – 16-tone chromatic scale

Roulette

See also: Hemimean clan § 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:

• Silver7 – 7-tone haplotonic scale
• Silver10 – 10-tone albitonic scale
• Silver17 – 17-tone chromatic scale

Baldy

See also: Subgroup temperaments § 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

• Baldy6 – 6-tone haplotonic scale
• Baldy11 – 11-tone albitonic scale
• Baldy17 – 17-tone chromatic scale

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

See also: Gamelismic clan § Radon

• Radon5 – 5-tone haplotonic scale
• Radon11 – 11-tone albitonic scale
• Radon16 – 16-tone chromatic scale

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

See also: Rastmic clan § Namo

Scales:

• Namo7 – 7-tone haplotonic scale
• Namo10 – 10-tone albitonic scale
• Namo17 – 17-tone chromatic scale

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:

• Sentry5 – 5-tone haplotonic scale
• Sentry8 – 8-tone albitonic scale
• Sentry11 – 11-tone chromatic scale

Hemif

See also: No-fives subgroup temperaments § Hemif

Scales:

• Hemif7 – 7-tone haplotonic scale
• Hemif10 – 10-tone albitonic scale
• Hemif17 – 17-tone chromatic scale

Garibaldi

Main article: Garibaldi temperament

See also: Schismatic family § Garibaldi

Scales:

• Garibaldi12 – 12-tone albitonic scale
• Garibaldi17 – 17-tone chromatic scale

Mabilic

Given below is the no-three version of semabila, or equivalently the no-threes version of trismegistus. It is the 7 & 9 temperament in the 2.5.7 subgroup, and tempers out 1071875/1048576, the mabilisma.

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:

• Mabila7 – 7-tone haplotonic scale
• Mabila9 – 9-tone albitonic scale
• Mabila16 – 16-tone chromatic scale

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:

• Indium8 – 8-tone albitonic scale
• Indium17 – 17-tone chromatic scale

Music

• The Cry of Indium (2011) by Chris Vaisvil – blog | play – in Indium[17]

Starlingtet

Starlingtet, the 4 & 15 temperament in the 2.5/3.7/3 subgroup, is related to starling as well as to myna.

Subgroup: 2.5/3.7/3

Comma list: 126/125

Sval mapping: [⟨1 0 -1], ⟨0 1 3]]

Gencom mapping: [⟨1 -1 0 1], ⟨0 4/3 1/3 -5/3]]

gencom: [2 6/5; 126/125]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 311.154

Optimal ET sequence: 15, 19, 23, 27, 239b, 266b, 293b, 320b, 347b, 374b, 401b

RMS error: 0.8398 cents

Scales:

• Starlingtet4 – 4-tone haplotonic scale
• Starlingtet7 – 7-tone albitonic scale
• Starlingtet11 – 11-tone chromatic scale

Lovecraft

Lovecraft, the 4 & 13 temperament in the 2.11.13 subgroup, is generated by ~13/11. Two generator steps give ~11/8 and three generator steps give ~13/8.

Subgroup: 2.11.13

Comma list: 1352/1331

Sval mapping: [⟨1 3 3], ⟨0 2 3]]

Gencom mapping: [⟨1 0 0 0 3 3], ⟨0 0 0 0 2 3]]

gencom: [2 13/11; 1352/1331]

Optimal tuning (POTE): ~2 = 1\1, ~13/11 = 279.318

Optimal ET sequence: 13, 30, 43, 73, 116b

RMS error: 0.8449 cents

Scales:

• Lovecraft5 – 5-tone haplotonic scale
• Lovecraft9 – 9-tone albitonic scale
• Lovecraft13 – 13-tone chromatic scale

Blackbirds

Blackbirds is a fairly straightforward temperament. It simply equates ~13/11 to 1/4 of the octave with a generator for prime 11 or 13.

Subgroup: 2.11.13

Comma list: 29282/28561

Sval mapping: [⟨4 0 1], ⟨0 1 1]]

Gencom mapping: [⟨4 0 0 0 12 13], ⟨0 0 0 0 1 1]]

gencom: [13/11 11/8; 29282/28561]

Optimal tuning (POTE): ~13/11 = 1\4, ~11/8 = 546.660

Optimal ET sequence: 4, 16, 20, 24, 44, 68, 112c, 180bc

RMS error: 0.8685 cents

Scales:

• Blackbirds8 – 8-tone albitonic scale
• Blackbirds12 – 12-tone chromatic scale
• Blackbirds16 – 16-tone mega-chromatic scale

Laz

Laz is related to georgian as well as to winston.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 144/143, 176/175, 196/195

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

Gencom mapping: [⟨1 -5/4 3 -1/4 7/4 -1/4], ⟨0 -1/4 -3 3/4 -21/4 19/4]]

gencom: [2 7/6; 144/143 176/175 196/195]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 269.300

Optimal ET sequence: 9, 31, 40, 49, 58, 156bde, 205bde

RMS error: 0.8790 cents

Scales:

• Laz5 – 5-tone haplotonic scale
• Laz9 – 9-tone albitonic scale
• Laz13 – 13-tone chromatic scale

Myna

Main article: Myna

See also: Starling temperaments § Myna

Scales:

• Myna7 – 7-tone haplotonic scale
• Myna11 – 11-tone albitonic scale
• Myna15 – 15-tone chromatic scale

Greeley

Greeley is related to opossum as well as to nusecond.

Subgroup: 2.5/3.7/3.11/3

Comma list: 121/120, 126/125

Sval mapping: [⟨1 1 2 2], ⟨0 -2 -6 -1]]

Gencom mapping: [⟨1 -5/4 -1/4 3/4 3/4], ⟨0 9/4 1/4 -15/4 5/4]]

gencom: [2 11/10; 121/120 126/125]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~11/10 = 155.776

Optimal ET sequence: 8, 15, 23, 54, 77, 100, 131d, 208bd

RMS error: 1.034 cents

Scales:

• Greeley8 – 8-tone albitonic scale
• Greeley15 – 15-tone chromatic scale

Music

• A Greeley Steals Halloween by Chris Vaisvil (2011) – blog | play – in Greeley[15]

Semidim

Semidim can be described as the 8 & 29 temperament in the 2.5/3.7/3.11/3.13/3 subgroup.

Subgroup: 2.5/3.7/3.11/3.13/3

Comma list: 121/120, 169/168, 275/273

Sval mapping: [⟨1 3 5 3 4], ⟨0 -6 -10 -3 -5]]

Gencom mapping: [⟨1 -3 0 2 0 1], ⟨0 24/5 -6/5 -26/5 9/5 -1/5]]

gencom: [2 13/10; 121/120 169/168 275/273]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/10 = 453.242

Optimal ET sequence: 5, 8, 13, 21, 29, 37, 44, 45, 188bde

RMS error: 1.052 cents

Scales:

• Semidim8 – 8-tone albitonic scale
• Semidim13 – 13-tone chromatic scale

Mothwelltri

Mothwelltri, the 1 & 4 temperament in the 2.7/3.11 subgroup, is related to orwell.

Subgroup: 2.7/3.11

Comma list: 99/98

Sval mapping: [⟨1 0 1], ⟨0 1 2]]

Gencom mapping: [⟨1 -1/2 0 1/2 3], ⟨0 -1/2 0 1/2 2]]

gencom: [2 7/6; 99/98]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~7/6 = 273.174

Optimal ET sequence: 9, 22, 40, 49c, 58c, 67c, 76c, 79, 101b, 123bc

RMS error: 1.064 cents

Scales:

• Mothwelltri5 – 5-tone haplotonic scale
• Mothwelltri9 – 9-tone albitonic scale
• Mothwelltri13 – 13-tone chromatic scale

Sulis

Sulis is related to minerva as well as to würschmidt.

Subgroup: 2.5.9/7.11/9

Comma list: 99/98, 176/175

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

Gencom mapping: [⟨1 4/9 2 -1/9 -1/9], ⟨0 -4/3 1 -2/3 4/3]]

gencom: [2 5/4; 99/98 176/175]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~5/4 = 386.558

Optimal ET sequence: 22, 25, 28, 31, 59, 90c, 149c

RMS error: 1.074 cents

Scales:

• Sulis7 – 7-tone haplotonic scale
• Sulis10 – 10-tone albitonic scale
• Sulis13 – 13-tone chromatic scale

Skwares

Main article: Squares

See also: No-fives subgroup temperaments § Skwares

Scales:

• Skwares8 – 8-tone albitonic scale
• Skwares11 – 11-tone mega-albitonic scale
• Skwares14 – 14-tone chromatic scale

Orwell

Main article: Orwell

See also: Semicomma family § Orwell

Scales:

• Orwell5 – 5-tone haplotonic scale
• Orwell9 – 9-tone albitonic scale
• Orwell13 – 13-tone chromatic scale

Magicaltet

Magicaltet is related to supermagic, superkleismic, and magic.

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 tunings:

• subgroup CTE: ~2 = 1\1, ~5/3 = 877.3426
• subgroup POTE: ~2 = 1\1, ~5/3 = 877.351

Optimal ET sequence: 4, 7, 11, 15, 26, 67, 93*

* wart for 5/3

RMS error: 1.206 cents

Scales:

• Magicaltet7 – 7-tone albitonic scale
• Magicaltet11 – 11-tone chromatic scale
• Magicaltet15 – 15-tone mega-chromatic scale

Magic

Main article: Magic

See also: Magic family § Magic

Scales:

• Magic7 – 7-tone haplotonic scale
• Magic10 – 10-tone albitonic scale
• Magic13 – 13-tone chromatic scale
• Magic16 – 16-tone mega-chromatic scale

Kiribati

Kiribati is related to nakika as well as to octacot.

Subgroup: 2.9/5.7/3.11/9

Comma list: 100/99, 245/242

Sval mapping: [⟨1 1 1 0], ⟨0 -2 3 4]]

Gencom mapping: [⟨1 1/10 -4/5 11/10 1/5], ⟨0 -3/2 -1 3/2 1]]

gencom: [2 21/20; 100/99 245/242]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~21/20 = 87.892

Optimal ET sequence: 13, 14, 27, 41, 55, 191bd, 232bcd, 273bcd

RMS error: 1.245 cents

Scales:

• Kiribati13 – 13-tone chromatic scale
• Kiribati14 – 14-tone chromatic scale

Score

Subgroup: 2.7.11.13

Comma list: 343/338, 847/832

Sval mapping: [⟨1 1 3 1], ⟨0 4 1 6]]

Gencom mapping: [⟨1 0 0 1 3 1], ⟨0 0 0 4 1 6]]

gencom: [2 11/8; 343/338 847/832]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 540.099

Optimal ET sequence: 5, 7, 9, 11, 20

RMS error: 1.282 cents

Scales:

• Score5 – 5-tone haplotonic scale
• Score7 – 7-tone albitonic scale
• Score9 – 9-tone mega-albitonic scale
• Score11 – 11-tone chromatic scale

Sensi

Main article: Sensi

See also: Sensipent family § Sensi

Scales:

• Sensi5 – 5-tone haplotonic scale
• Sensi8 – 8-tone albitonic scale
• Sensi11 – 11-tone mega-albitonic scale
• Sensi19 – 19-tone chromatic scale

Sensation

Main article: Sensi

See also: Sensipent family § Sensi

Scales:

• Sensi5 – 5-tone haplotonic scale
• Sensi8 – 8-tone albitonic scale
• Sensi11 – 11-tone mega-albitonic scale
• Sensi19 – 19-tone chromatic scale

Meantone

Main article: Meantone

See also: Meantone family § Meantone

Scales:

• Meantone5 – 5-tone haplotonic scale
• Meantone7 – 7-tone albitonic scale
• Meantone12 – 12-tone chromatic scale

Mohaha

See also: Rastmic clan § Mohaha

Scales:

• Mohaha7 – 7-tone albitonic scale
• Mohaha10 – 10-tone chromatic scale

Music:

Billy Stiltner

• Mohaha10ping2 ^{[dead link]}

Mohoho

See also: Rastmic clan § Mohoho

Joan

Joan is related to casablanca as well as to orwell.

Subgroup: 2.9.7.11

Comma list: 99/98, 9317/9216

Sval mapping: [⟨1 0 1 3], ⟨0 7 4 1]]

Gencom mapping: [⟨1 0 0 1 3], ⟨0 7/2 0 4 1]]

gencom: [2 11/8; 99/98 9317/9216]

Optimal tuning (POTE): ~2 = 1\1, ~11/8 = 542.672 cents

Optimal ET sequence: 11, 20, 31, 42, 115bd, 157bd

RMS error: 1.424 cents

Scales:

• Joan5 – 5-tone haplotonic scale
• Joan7 – 7-tone albitonic scale
• Joan9 – 9-tone mega-albitonic scale
• Joan11 – 11-tone chromatic scale

Tutone

See also: Hemimean clan § Tutone

Scales:

• Tutone6 – 6-tone haplotonic scale
• Tutone7 – 7-tone albitonic scale
• Tutone13 – 13-tone chromatic scale

Stacks

See also: Subgroup temperaments § Stacks (a.k.a. 2magic)

Scales:

• Stacks5 – 5-tone haplotonic scale
• Stacks8 – 8-tone albitonic scale
• Stacks11 – 11-tone chromatic scale
• Stacks19 – 19-tone mega-chromatic scale

Bleu

See also: No-fives subgroup temperaments § Bleu

Scales:

• Bleu8 – 8-tone haplotonic scale
• Bleu9 – 9-tone albitonic scale
• Bleu17 – 17-tone chromatic scale

Sephiroth

Sephiroth is the no-7 restriction of muggles.

Subgroup: 2.5.11.13.17

Comma list: 65/64, 170/169, 221/220

Sval mapping: [⟨1 0 15 6 11], ⟨0 1 -5 -1 -3]]

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

gencom: [2 5/4; 65/64 170/169 221/220]

Optimal tuning (POTE): ~2 = 1\1, ~5/4 = 372.236

Optimal ET sequence: 10, 13, 16, 29

RMS error: 1.774 cents

Scales:

• Sephiroth7 – 7-tone haplotonic scale
• Sephiroth10 – 10-tone albitonic scale
• Sephiroth13 – 13-tone chromatic scale
• Sephiroth16 – 16-tone mega-chromatic scale

Archy

Main article: Superpyth

See also: Archytas clan § Archy

Scales:

• Archy5 – 5-tone haplotonic scale
• Archy7 – 7-tone albitonic scale
• Archy12 – 12-tone chromatic scale

Supra

Main article: Superpyth

See also: Archytas clan § Supra

Scales:

• Supra7 – 7-tone albitonic scale
• Supra12 – 12-tone chromatic scale

Supraphon

Main article: Superpyth

See also: Archytas clan § Supraphon

Suhajira

See also: Archytas clan § Suhajira

Scales:

• Suhajira7 – 7-tone haplotonic scale
• Suhajira10 – 10-tone albitonic scale
• Suhajira17 – 17-tone chromatic scale

Machine

See also: Subgroup temperaments § Machine

Scales:

• Machine5 – 5-tone haplotonic scale
• Machine6 – 6-tone albitonic scale
• Machine11 – 11-tone chromatic scale

Deutone

See also: Hemimean clan § Deutone

Scales:

• Deutone6 – 6-tone haplotonic scale
• Deutone7 – 7-tone albitonic scale
• Deutone13 – 13-tone chromatic scale

Negra

Main article: Negri

See also: Marvel temperaments § Negri

Porkypine

Main article: Porcupine

See also: Porcupine family § Porcupine

Scales:

• Porkypine7 – 7-tone haplotonic scale
• Porkypine8 – 8-tone albitonic scale
• Porkypine15 – 15-tone chromatic scale

Skateboard

Skateboard is related to thrasher.

Subgroup: 2.5/3.7/3.11.13/9

Comma list: 56/55, 91/90, 100/99

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

Gencom mapping: [⟨1 -3/7 4/7 11/7 4 -6/7], ⟨0 0 -1 -3 -2 2]]

gencom: [2 6/5; 56/55 91/90 100/99]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 313.842

Optimal ET sequence: 11, 15, 19, 23, 42d, 65d

RMS error: 2.396 cents

Scales:

• Skateboard7 – 7-tone haplotonic scale
• Skateboard11 – 11-tone albitonic scale
• Skateboard15 – 15-tone chromatic scale

Augment

Augment is related to augmented.

Subgroup: 2.5.7.11

Comma list: 56/55, 128/125

Sval mapping: [⟨3 7 0 2], ⟨0 0 1 1]]

Gencom mapping: [⟨3 0 7 9 11], ⟨0 0 0 -1 -1]]

gencom: [5/4 8/7; 56/55 128/125]

Optimal tuning (POTE): ~5/4 = 1\3, ~8/7 = 228.275

Optimal ET sequence: 3, 6, 9, 15, 21

RMS error: 2.422 cents

Scales:

• Augment6 – 6-tone haplotonic scale
• Augment9 – 9-tone albitonic scale
• Augment12 – 12-tone chromatic scale

Semaphore

Main article: Semaphore and godzilla

See also: Semaphoresmic clan § Semaphore

Scales:

• Semaphore5 – 5-tone haplotonic scale
• Semaphore9 – 9-tone albitonic scale
• Semaphore14 – 14-tone chromatic scale

Kryptonite

Kryptonite is related to krypton.

Subgroup: 2.5.7/3.11/3.13/3

Comma list: 56/55, 78/77, 91/90

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

Gencom mapping: [⟨1 -5/4 2 -1/4 3/4 3/4], ⟨0 -1/2 3 3/2 -3/2 1/2]]

gencom: [2 13/12; 56/55 78/77 91/90]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~13/12 = 132.428

Optimal ET sequence: 9, 63, 82bd, 91bde

RMS error: 2.545 cents

Scales:

• Kryptonite9 – 9-tone albitonic scale
• Kryptonite10 – 10-tone albitonic scale
• Kryptonite19 – 19-tone mega-chromatic scale

Oregon

Subgroup: 2.9.15.7.11

Comma list: 55/54, 64/63, 100/99

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

Gencom mapping: [⟨1 2 3 2 4], ⟨0 -3/2 -5/2 3 -2]]

gencom: [2 6/5; 55/54 64/63 100/99]

Optimal tuning (subgroup POTE): ~2 = 1\1, ~6/5 = 325.495

Optimal ET sequence: 7, 11, 37b, 48b, 59b

RMS error: 2.550 cents

Scales:

• Oregon7 – 7-tone haplotonic scale
• Oregon11 – 11-tone albitonic scale
• Oregon15 – 15-tone chromatic scale

Antikythera

See also: Jubilismic clan § Antikythera

Scales:

• Antikythera6 – 6-tone haplotonic scale
• Antikythera10 – 10-tone albitonic scale
• Antikythera16 – 16-tone chromatic scale

Glacial

Subgroup: 2.9.5.11.13

Comma list: 45/44, 65/64, 81/80

Sval mapping: [⟨1 0 -4 -6 10], ⟨0 1 2 3 -2]]

Gencom mapping: [⟨1 3/2 2 0 3 4], ⟨0 1/2 2 0 3 -2]]

gencom: [2 9/8; 45/44 65/64 81/80]

Optimal tuning (POTE): ~2 = 1\1, ~9/8 = 186.151

Optimal ET sequence: 6, 13, 45be, 58bce, 71bce, 84bce

RMS error: 2.887 cents

Scales:

• Glacial6 – 6-tone haplotonic scale
• Glacial7 – 7-tone albitonic scale
• Glacial13 – 13-tone chromatic scale

Leantone

See also: Hemimean clan § Leantone

Scales:

• Leantone6 – 6-tone haplotonic scale
• Leantone7 – 7-tone albitonic scale
• Leantone13 – 13-tone chromatic scale