List of regular temperaments

From Xenharmonic Reference
Main article: Regular temperament

Rank-2

2.3.5.x families

Rank-2 Temperaments
Form Family Name Generator Ploidacot Usual Scale Type ETs Description Subgroup Commas Accuracy (Vector)
7 Syntonic Meantone 692-697c monocot softer diatonic, m-chromatic 7, 12 Common historical temperament for 5-limit diatonic harmony. 2.3.5 81/80 4 Medium
Septimal Meantone 695-697c monocot softer diatonic, m-chromatic 19, 31 Canonical extension of the above to 2...7. 2.3.5.7 81/80, 225/224 4 Medium
Flattone 692-694c monocot softer diatonic, m-chromatic 19, 26 Less accurate extension, more melodic intuition, easily extends to higher limits. 7-form version of Meantone. 2.3.5.7 525/512, 81/80 6 Low
12 Injera 92-96c diploid monocot thalassic 12, 26 Adds a 600c tritone representing 7/5 to meantone. 2.3.5.7 81/80, 50/49 5 Medium-low
7 Porcupine Porcupine 161-166c omega-tricot onyx, pine 15, 22 Moderate-accuracy 2.3.5.11 temperament with a ~160c generator and a heptatonic MOS. 2.3.5.11 250/243, 100/99 5 Medium-low
Septimal Porcupine 161-163c omega-tricot onyx, pine 15, 22 Extension of the above to the full 11-limit. 2.3.5.7.11 250/243, 100/99, 64/63 5 Medium-low
Interclassical Interclassical, Dicot 670-680c, 720-730c dicot mosh, dicoid 7, 10 5-limit exotemperament equating 5/4 and 6/5 to the same interval. 2.3.5 25/24 7 Very low
Tetracot Tetracot 175-180c tetracot archeotonic, 7L6s 34, 41 Interprets (3/2)^(1/4) as 10/9. 2.3.5.11 100/99, 243/242 2 High (2.3.5), Medium (extensions)
12 Diminished Diminished 685-700c tetraploid monocot tetrawood, 4L 8s 12, 16 Sets 6/5 to 300 cents; a step up from 600 cents is 3/2. 2.3.5 648/625 5 Medium-low
Dimisept 685-700c tetraploid monocot tetrawood, 4L 8s 12, 16 Exotempered extension of the above that sets 7/6 to 300 cents. 2.3.5.7 36/35, 50/49 7 Very low
Augmented Augmented 705-715c triploid monocot triwood, tcherepnin 12, 15 Sets 5/4 to 400 cents, a step down from 800c is 3/2 and 2 steps up is 7/4. 2.3.5.7 128/125 6 Low
Schismic Schismic 701-702c monocot harder diatonic, p-chromatic 41, 53 5-limit interpretation of Pythagorean tuning, best tuned when the fifth is flattened by a fraction of a cent. 2.3.5 schisma 1 Very high
Garibaldi 702-703c monocot harder diatonic, p-chromatic 41, 53 7-limit interpretation of Pythagorean tuning. It is most accurate when the fifth is tuned slightly sharp. 2.3.5.7 schisma, 225/224 3 Medium-high
Misty Misty 701-708c triploid monocot 12, 51 5/4 is 4 times the difference between 3/2 and 800c. 2.3.5 misty comma 3 Medium-high
Diaschismic Diaschismic 100-111c diploid monocot jaric, 10L 2s 12, 34 Temperament characterized by a perfect semioctave and a sharpened fifth or semitone generator. Two generators down reaches 5/4. 2.3.5.17 diaschisma, 136/135 4 Medium
Septimal Diaschismic 103-104c diploid monocot jaric, 10L 2s 12, 34 Rather complex 7-limit extension of the above. 2.3.5.7.17 diaschisma, 126/125, 136/135 4 Medium
10 Pajara 109-111c diploid monocot jaric, 10L 2s 12, 22 Jubilic archytas diaschismic temperament. Contains jubilic chord structure and is strongly associated with 22edo. 2.3.5.7.17 diaschisma, 50/49, 136/135 5 Medium-low
3 Magic Magic 378-382c alpha-pentacot mosh, sephiroid 19, 22 Stacks five flattened major thirds to form a perfect twelfth. 5/4-generated analog of Meantone. 2.3.5 magisma 4 Medium
Wurschmidt Wurschmidt 386-389c beta-octacot - 31, 34 Eight 5/4s stack to 3/2 and three 25/24s stack to two 16/15s. 5/4-generated analog of Schismic. Bad for MOSes. 2.3.5.11.23 576/575, 12167/12150 3 Medium-high
8 Father Father 720-800c monocot antipentic 3, 5 Extremely inaccurate exotemperament which equates 5/4 with 4/3. 2.3.5 16/15 8 Extremely low
4 Kleismic Kleismic, Cata 317c alpha-hexacot smitonic, 4L7s, 4L11s 19, 34 A highly accurate temperament equating a stack of six slightly sharp 6/5's to one 3/1 and three 6/5's to one 26/15. 2.3.5.13 kleisma, 325/324 2 High
10 Negri Negri 124-128c omega-tetracot 10, 19 Generator is a sharp minor second of 16/15 which stacks 3 times to 5/4, and can be seen as a counterpart of Porcupine. 2.3.5 16875/16384 5 Medium-low
Semibuzzard 124-128c taric 10, 28 Weak Jubilismic or Buzzard extension of Negri allowing intervals of 7 to be reached with a 600-cent offset. 2.3.5.7.11 16875/16384, 50/49, 243/242 6 Low
Negrisept 124-128c 10, 19 Semaphore extension of negri. 2.3.5.7 16875/16384, 49/48 7 Very low

2.3.7.x families

Form Family Name Generator Ploidacot Usual Scale Type ETs Description Subgroup Commas Accuracy
5 Archy Archy 709-720c monocot soft pentic, harder diatonic, p-chromatic 5, 22 2.3.7 counterpart of Meantone, which sharpens the fifth. 2.3.7 64/63 5 Medium-low
Superpyth 709-711c monocot soft pentic, harder diatonic, p-chromatic 22, 27 Extension of the above to 2...7, favoring flatter tunings. 2.3.5.7 64/63, 245/243 5 Medium-low
Gamelic Slendric, Wonder 231-234c tricot 1L 4s, machinoid, 5L 6s 5, 31 Splits the fifth in 3 parts, each of which is 8/7. Little relation to actual Slendro tuning. 2.3.7 gamelisma 3 Medium-high
Mothra 231c tricot 1L 4s, machinoid, 5L 6s 26, 31 Meantone extension of the above. 2.3.5.7 81/80, gamelisma 4 Medium
Rodan 234c tricot 1L 4s, machinoid, 5L 6s 41, 46 More accurate extension of the above. 2.3.5.7 245/243, gamelisma 4 Medium
Miracle 117c hexacot antisinatonic, 10L 1s 31, 41 Generated by a 15/14~16/15 semitone, two of which reach a slendric 8/7. 2.3.5.7.11 225/224, 243/242, gamelisma 3 Medium-high
Valentine 78c enneacot 15L 1s, Carlos Alpha 15, 16 Scale with small steps strongly associated with Carlos Alpha. 2.3.5.7 126/125, gamelisma 3 Medium-high
Buzzard Buzzard 474-478c alpha-tetracot 53, 58 Sharpens the 21/16 so that four of them stacks to the 3/1. 2.3.5.7.13 buzzardsma 4 Medium
Interseptimal Interseptimal, Semaphore 240-250c alpha-dicot 4L 1s, semiquartal 5, 19 Equipentatonic, inaccurate 7-limit temperament. 2.3.7 49/48 6 Low
13 Squares Squares 424-426c beta-tetracot 3L 5s, 3L 8s, 3L 11s 14, 17 No-fives temperament generated by a flattened 9/7 equated with 14/11. 2.3.7.11 99/98, 243/242 4 Medium

2.3.x families

Form Family Name Generator Ploidacot Usual Scale Type ETs Description Subgroup Commas Accuracy
7 Rastmatic Rastmatic 345-355c dicot mosh, dicoid 7, 10 Maps 11/9 and its fifth complement to a perfect neutral third. 2.3.11 243/242 3 Medium-high
Mohajira 347-350c dicot mosh, dicoid 24, 31 Meantone extension of the above. Can optionally be extended to set 7/4 equal to the semiflat minor seventh, or via a strong meantone extension. 2.3.5.11 243/242, 81/80 5 Medium-low
Intratridecimal Intratridecimal 350-360c dicot mosh, dicoid 27, 10 Maps 16/13 and its fifth complement to a perfect neutral third. 2.3.13 512/507 4 Medium
(To be named) 355-360c dicot mosh, dicoid 27, 10 Archy extension of the above. 2.3.7.13 512/507, 64/63 5 Medium-low

2.3.5.7.x families

Form Family Name Generator Ploidacot Usual Scale Type ETs Description Commas Subgroup Accuracy
7 Amity Amity 338-340c gamma-pentacot 7L 18s, 7L 25s 46, 53 Sets the spacing of the thirds to 7/6-#-6/5-#-#-5/4-#-9/7; mostly appears as a structural element of EDOs rather than an independent structure. 4375/4374, 5120/5103 2.3.5.7 2 High
12 Compton Compton 385c dodecaploid acot dodecawood 12, 60 Acts as a closed circle of 12 fifths (see 12edo), but with 5/4 flattened by a diesis from 400c, and 7/4 flattened by twice that amount from 1000c. pythagorean comma 2.3.5.7 3 Medium-high
4 Doublewide Doublewide 325c 22, 48 Sets 6/5 and 7/6 the same distance from the 300c period, so that four ~25c generators stack to the semitone separating 3/2 from 600c. 50/49, 875/864 2.3.5.7 5 Medium-low
Myna Myna 309-311c beta-decacot - 27, 31 Sets 25/24 equal to twice 36/35. 126/125, 1728/1715 2.3.5.7 3 Medium-high
9 Orwell Orwell 270-273c alpha-heptacot gramitonic 22, 31 Generator is a sharpened 7/6. Interval chain finds 7/6, 11/8, 8/5, 15/8, 11/10, 9/7, 3/2. 99/98, 121/120, 176/175 2.3.5.7.11 3 Medium-high
Ennealimmal Ennealimmal 44-53c enneaploid dicot enneawood 27, 45 Divides the octave into nine equal parts representing 27/25 and half of 7/6. 2401/2400, 4375/4374 2.3.5.7 1 Very high
8 Nusecond Nusecond 154-155c onyx, pine 31, 70 Generator is a neutral second, but places primes at high complexity, preferring ratios between them. 126/125, 2430/2401 2.3.5.7.11 4 Medium

No-threes families

Form Family Name Generator Ploidacot Usual Scale Type ETs Description Commas Subgroup Error
7 Mabilic Mabilic 668-680c alpha-triseph[a] antidiatonic, armotonic, 9L 7s 7, 9 Basic antidiatonic temperament with no 3. mabilisma 2.5.7 4 Medium
Trismegistus 672-675c alpha-triseph antidiatonic, armotonic, 9L 7s 16, 25 High-accuracy but high complexity extension of prime 3. gamelisma, magisma 2.3.5.7 4 Medium
Semabila 668-672c alpha-triseph antidiatonic, armotonic, 9L 7s 9, 25 Combination of Mabilic and Semaphore. 49/48, 28672/28125 2.3.5.7 6 Low
Mavila 675-680c monocot antidiatonic, armotonic, 7L 9s 7, 9 Exotemperament serving as an antidiatonic analog of meantone. 36/35, 135/128 2.3.5.7 7 Very low
11 Orgonismic Orgone, Orgonic 320-325c trimech[b] 4L7s 15, 26 A high-accuracy rank-2 temperament generated by a tempered 77/64. 65536/65219 2.7.11 3 Medium-high
Superkleismic 320-325c trimech 4L7s 15, 26 The main (albeit less accurate) extension of orgone to the full 11-limit. 100/99, 245/242, 385/384 2...11 4 Medium
6 Hemimean Didacus 192-196c diseph 1L 5s, 6L 1s 6, 25 Every other step of septimal meantone. 3136/3125 2.5.7 2 High

Non-octave families

Rank-2 Temperaments
Form Family Name Generator Ploidacot Usual Scale Type ETs Description Commas Subgroup Accuracy
b13 Sensamagic Sensamagic 435-440c monogem[c] lambda b4, b9[d] Basic tritave temperament that stacks 9/7 twice to reach 5/3. Generates the lambda (4L5s⟨3/1⟩) MOS scale, or can be used with octaves as Sensamagic.2. 245/243 3.5.7 3 Medium-high
8 Sensi 440-445c beta-heptacot 3L 2s, checkertonic 19, 27 Very sharp extension of Sensamagic, which finds the octave at 125/63. 91/90, 126/125, 169/168 2.3.5.7.13 4 Medium

[a] seph = divisions of 5/4

[b] mech = divisions of 7/4

[c] gem = divisions of 7/3 in a perfect twelfth (tritave) equivalent context

[d] A "b" prefixed to an equal temperament indicates the equal division of 3/1.

Rank-3

Name Commas Subgroup ETs Description Generators
Marvel 225/224, 385/384 2.3.5.7.11 19, 22, 31 16/15 and 15/14 are equated, or equivalently 32/25 and 9/7 are equated. ~3/2, ~81/80
Hemifamity / Aberschismic 5120/5103 2.3.5.7 41, 46, 53 81/80 and 64/63 are equated. Sometimes used in aberrismic theory to interpret ternary scales as 2.3.5.7, and associated with argent tuning. ~3/2, ~81/80
Parapyth(ic) 352/351, 896/891 2.3.7.11.13 41, 46, 63 Based on Margo Schulter's regular tuning construct called "parapyth". ~3/2, ~28/27

See also

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