User:Dummy index/Chromatic pairs and how we define haplotonic

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Proposal to simplify the definition

  • In order to distinguish between the fixed category of Category:Haplotonic scales, etc., and the reciprocal relationship (haplotonic) + (albitonic) = (chromatic), the latter will be called, for example, triplet. Every 3 successive generations of MOS that satisfies the above relationship will be called a triplet.
  • If the number of notes in the scale is important, consider using the Pentatonic and Heptatonic pages.
  • Based on the above equation and the nature of MOS, the chromatic must be a direct descendant of the albitonic. So I don't like the existence of mega-albitonic between albitonic and chromatic, even though the meanings of relational and categorical usages don't have to be exactly the same. (I will also withdraw the mini-albitonic that I said.)
  • Categorical haplotonic scales include the last MOS that s ≥ 125 ¢ and some subset of such scales. in [[Chromatic pairs]], such subsets all may be labelled mini-haplotonic (even if there are more than one).
  • Categorical albitonic scale is the first MOS that s < 125 ¢.
  • Categorical chromatic scale include direct descendant of albitonic (by adding corresponding haplotonic), and further descendant (by adding same haplotonic). in [[Chromatic pairs]], such further descendants all may be labelled mega-chromatic (even if there are more than one).

Memos

I couldn't find a comprehensive explanation of haplotonic, so I'm going to consider the current situation in my own way.

haplotonic is something similar to the black keys of a piano. hmm...

  1. Number of notes. about 5-notes. here I tentatively state the best is 4–6 notes.
  2. (haplotonic)+(albitonic)=(chromatic). strong. (strict, but local)
  3. Semitone-free MOS. some authors might consider this property. All labelled scales are s > 125 ¢.

In fact, it can be seen from condition 3 that the haplotonic scale is up to 9 notes.

By the way, many of the scale-lists listed on [[Chromatic pairs]] are actually "chain of albitonic-chromatic pairs". Moreover, the part that is a arithmetic sequence is composed by repeatedly adding the same strong haplotonic scale. And the strong haplotonic is direct parent MOS of first albitonic of the arithmetic chain.

  • Nestoria[7], Nestoria[12], Nestoria[17] — adding Nestoria[5] repeatedly
  • (memo: strong haplotonic is "L will be new s if cutting s from L anymore" so soft-of-basic)
  • What I mean of albitonic-chromatic pairs is strict operation of equation of condition 2. And with MOS tree, albitonic-chromatic relation usually results in direct parent-daughter. Exception is (1L xs)+(1L ys)=(1a (1+x+y)b) (i dunno another case). Or adopt his strict definition. You can then use n*(haplotonic)+(albitonic)=(...) to move a few steps at once (possibly multiple copies of one of them).

~~ Exercise ~~

  • I would like to exclude 1L xs from the albitonic-chromatic pair. To reason this... excluding 1L from strong haplotonic.
  • First two strong haplotonic of Barton are Barton[2] and Barton[11]. Barton[2] is too few notes. Barton[11] is too many notes. Barton[3] and all descendants aren't semitone-free. Strong haplotonic can be obtained for any rank-2, but not always any haplotonic (in condition 1 and 3) can be.
  • Seni[5] and Sensi[8] both are labelled Category:Haplotonic scales. Only Sensi[8] is strong haplotonic, but Sensi[5] is shorter genchain, obviously more haplo than Sensi[8]. both haplotonic? mini-haplotonic and haplotonic? haplotonic and strong-haplotonic? Well, shouldn't both be haplotonic? Isn't it just that we want to differentiate it because we're trying to include it in the scale-list?
    • If there are multiple haplotonic scale in condition 1 and 3, it may be nice to name most-notes scale (must be strong haplotonic!) "strong haplotonic" or "maximum haplotonic" or "the end of haplotonic".
      • Baldy[5] and Baldy[6] both are strong haplotonic, so we need to use different terms to distinguish them. e.g. Baldy4/5/6 are haplotonic and Baldy6 is max-haplotonic. Until Baldy5 become listed, Baldy6 is simply called haplotonic.
  • Fractional-octave?
  • Are they undoubtedly haplotonic? neutral third 3L 4s, mavila 2L 5s, ...


Category:Haplotonic scales (November 2024)

Caption text
Name Size of s (¢) # of notes
1–3: −, 4–6: ☆, 7–9: △		 Strong Remarks
Baldy6 180 ☆ Yes Baldy5 is also strong haplotonic for Baldy6-Baldy11 pair
Bleu8 140 △ Yes
Roulette6 195 ☆ Yes
Sensi5 130 ☆ No direct parent of Sensi8
Sensi8 130 △ Yes
Silver7 128 △ Yes
Tutone6 194 ☆ Yes

Chromatic pairs (November 2024)

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

terrain_3558edo.png

Scales:

Music

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

petrtri_385edo.png

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:

  • 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

tridec_235edo.png

Scales:

Nestoria

See also: Schismatic family § Nestoria (2.3.5.19)
nestoria_171edo.png

Scales:

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

orgone_308edo.jpg

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

marveltwintri_321edo.png

Scales:

Cata

Main article: Hanson and cata
See also: Kleismic family § Cata
cata_246edo.jpg

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

bridgetown_29edo.png

Scales:

Music

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

See also: Rastmic clan § Neutral
neutral_380edo.png

Scales:

Music

Slendric

Main article: Slendric
See also: Gamelismic clan #Slendric and No-fives subgroup temperaments #Slendric
slendric_190edo.jpg

Scales:

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

huntington_400edo.jpg

Scales:

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

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

pepperoni_271edo.png

Scales:

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

bossier_225edo.jpg

Scales:

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:

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:

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:

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)
photia_65edo.png

Scales:

Shoe a.k.a. llywelyn

See also: Llywelynsmic clan § Llywelyn a.k.a. shoe
shoe_37edo.jpg

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
roulette_37edo.jpg

Scales:

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

silver_487edo.jpg

Scales:

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

baldy_147edo.jpg

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:

Radon

See also: Gamelismic clan § Radon
radon_128edo.jpg
  • 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:

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:

Hemif

See also: No-fives subgroup temperaments § Hemif
hemif_58edo.jpg

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
garibaldi_94edo.png

Scales:

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

mabila_66edo.jpg

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

indium_253edo.png

Scales:

Music

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:

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

lovecraft_116edo.png

Scales:

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:

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
myna_89edo.jpg

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

greeley_100edo.png

Scales:

Music

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

semidim_770edo.png

Scales:

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:

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
skwares_206edo.png

Scales:

Orwell

Main article: Orwell
See also: Semicomma family § Orwell
orwell_84edo.png

Scales:

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:

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

* wart for 5/3

RMS error: 1.206 cents

Scales:

Magic

Main article: Magic
See also: Magic family § Magic
magic_145edo.jpg

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:

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

score_20edo.jpg

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
sensi_46edo.jpg

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
meantone_31edo.jpg

Scales:

Mohaha

See also: Rastmic clan § Mohaha
mohaha_131edo.png

Scales:

Music:

Billy Stiltner

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
tutone_99edo.png

Scales:

Stacks

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

Scales:

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:

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
supra_56edo.png

Scales:

Supraphon

Main article: Superpyth
See also: Archytas clan § Supraphon

Suhajira

See also: Archytas clan § Suhajira

Scales:

Machine

See also: Subgroup temperaments § Machine
machine_28edo.png

Scales:

Deutone

See also: Hemimean clan § Deutone
deutone_44edo.png

Scales:

Negra

Main article: Negri
See also: Marvel temperaments § Negri

Porkypine

Main article: Porcupine
See also: Porcupine family § Porcupine

Scales:

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:

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

augment_21edo.png

Scales:

Semaphore

Main article: Semaphore and godzilla
See also: Semaphoresmic clan § Semaphore
semaphore_24edo.png

Scales:

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:

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:

Antikythera

See also: Jubilismic clan § Antikythera
antikythera_28edo.png

Scales:

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:

Leantone

See also: Hemimean clan § Leantone
leantone_81edo.png

Scales:

Notes

  1. ↑ Yahoo! Tuning Group | "father" variant?
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