Catalog of Linear Temperaments

Mappings show the odd-prime axes in terms of the generator. The mapping uniquely defines a temperament, although all temperaments have more than one mapping because you could take eg a fourth or a fifth as the generator. ETs may not be consistent with the mappings given. I don't care.

This is a complement to Monzo's list of equal temperaments.


Has its own page. Mapping (1, 4, 10). Covers 12,19,31,43,50,55-equal Very efficient, long history, works well in the 7-limit.


Has its own page. Mapping (-1, 8, 14). Covers 17, 29, 41, 53, 65-equal. Can be very accurate in the 5-limit, still works well in the 9-limit. Implied by Safi ad-Din al-Urmawi (d.1294) and Qutb al-Din al-Shirazi (1236-1311). They both define 5-limit scales on a Pythagorean meta-scale, and more are schismically correct than not. Also covered by either Helmholtz. The 7-approximation was made explicit by Erv Wilson in 1975. 53-equal was covered by Ching Fang (45 BC) and Philolaus, a "disciple of Pythagoras".


Could also be called "double positive" by a slight bending of Bosanquet's terminology. Has its own page. Mapping (1, -2), period is a half-octave. Covered by 22, 34, 46-equal. May have relevance to Indian music. Covered in Paul Erlich's 22-equal generalized diatonic paper. Three different 7-approxmiations worthy of attention.

Neutral third scales

Has its own page. Two neutral thirds make up a perfect fifth. The neutral thirds can be called 11:9. The "wolf third" to make up a 7-note scale can be identified with 6:5 or 7:6 or neither. Hence no consistent mapping. Covered by 17, 24, 31, 38, 41-equal. Al-Farabi's (d.950) Rast looks this family. 24-notation (not equal temperament) is very important for Arabic music. I don't know who stated the rational approximations first. It could be called "septimally double-positive" in my version of Wilson's version of Bosanquet's terminology.


Has its own page, better covered in Monzo's tuning dictionary. Mapping (6, -7, -2, 15). Covered by 31, 41, 72-equal. Discovered by George Secor 1975, rediscovered by Paul Erlich and Dave Keenan, named for "multiple integer ratios approximated consistently, linearly and evenly" or some such.


Mapping (5, 1, 12). Covered by 19, 22, 41-equal. Major third generator, around 380 cents. Comes top of my temperament list for the 9-limit. Not much explored or documented yet, but looks pretty nifty. History unknown, but it's the kind of thing Erv Wilson probably has tucked away somewhere, and would have been on one of Dave Keenan's spreadsheets. It has multiple approximations generated iteratively and consistently.

Double diatonic

Mapping (1, 4, 4), half-octave period. Covers 14, 26, 38, 50-equal, although the 7-limit doesn't work with 50. Like meantone in the 5-limit, but splits the tone for 7-limit intervals. Interesting because it fits a Halberstadt keyboard well: tune the white notes to meantone and each black note to be exactly in the middle of the neighbouring white notes. Not that closely linked to Bosanquet's double negative temperaments. Paul Erlich came up with injera for 22222221 222221 from 26.


Mapping (1, -3). Covers 7, 9, 16, 23-equal. The complementary 7 note 5-limit mapping to meantone. Has some relationship with the Indonesian pelogs. Simple chords, but not very accurate. The best you can do is a 521.1 cent generator, with a minimax of 23 cents.


Mapping (0, 1, 1, 1, 1), 1/29-octave period. Would make sense as two keyboards each tuned to 29-equal. I found it using my temperament program in the 15-limit list.


Mapping (6, 5). Minor third generator, around 317 cents. From Dave Keenan: "See Scala's intnam.par : 15552:15625 (8.11 c) or my `A kleisma is the difference between a just fifth (2:3) and an octave reduced chain of 6 just minor thirds (5:6)'" The basis for Larry Hanson's keyboard. Covers 15, 19, 34, 53-equal.


Mapping (7, -3, 8, 2). A subminor third generator (about 272 cents). Covers 22, 31 and 53-EDO. Gene has a page about this


Mapping (2, -4, 7, -1), half octave period. Quartertone generator (around 52 cents). Covers 22 and 46-EDO. From Paul Erlich and Dave Keenan. Intended for a guitar fretting, relates to the Indian shruti scale giving 11-limit approximations. Double-diaschismic, in that the diaschismic generator is divided into two equal parts for the 5-limit.


Mapping (-2, -3, -2), 1/9-octave period. Quartertone generator (around 49 cents). Covers 27, 45 and 80-EDO. Came up on my 7-limit microtemperament list, but first noticed and named by Gene Ward Smith.


Mapping (2, 3, 2, 1), 1/18-octave period. Comma generator, around 17.6 cents. Consistent with 72-, 126, 198- and 270-EDO. Discovered by Gene Ward Smith, gets the 11-limit to around a third of a cent.


Mapping (-3, -5, 6, -4). Neutral second generator (around 164 cents). Covers 15, 22, 37 and 59-EDO. Named after Herman Miller's "Mizarian Porcupine Overture"


Mapping (-4, 3, -2, 14, -3) Generator around 125 cents. Covers 9, 10, 19, 28, 29-equal. Comes from Negri, John. "The Nineteen-Tone System as Ten Plus Nine", Interval vol. 5 no. 3, winter 1986-1987, pp. 11-13. Simple in the limit, if not that accurate.


Mapping (8, 1) Generator around 388 cents. Named after Würschmidt's comma. Covers 3, 28, 31-equal.

Tiny Diesic

Mapping (7, 9) Generator around 443 cents. Covers 8, 19, 27-equal.

Minimal Diesic

Mapping (4, 9) Generator around 176 cents. Covers 7, 27-equal.

Acute Minor Third

Mapping (-5, -13) Generator around 339 cents.


Mapping (7, 3) Generator around 12 cents.


Mapping (6, 5, 22, -21), minor third generator (around 317 cents). Kleismic variant, covers 19, 72 and 91-EDO. Unison vectors 225:224, 385:384 and 4375:4374. Found and named by Gene Ward Smith.


Mapping (-13, -14) Generator around 315 cents.


Mapping (-15, 2) Generator around 193 cents.


Mapping (-3, -4, -5, -3), 1/8-octave period. Comma generator (around 16 cents). Covers 72, 80, 152, 224 and 296-EDO. Another one of Gene Ward Smith's.


Mapping (1, 1, 4), 1/4-octave period. Semitone generator (around 90 cents). Unison vectors 648:625 and 225:224. Covers 12, 28, 40, 52, 64-EDO. Originally called octatonic. See this page of Herman Miller's.


Mapping (-1, 0), 1/3-octave period. Semitone generator (around 91 cents). Unison vector 128:125. Covers 3, 9, 12, 15, 18, 27-EDO.

Blackwood decatonic

Mapping (0, -1), 1/5-octave period. Semitone generator (around 85 cents). Covers 5, 10, 15-EDO.

Cassandra 1

Mapping (-1, 8, 14, -23, -20). Perfect fourth generator (about 498 c). Schismic variant, covers 41, 53 and 94-EDO. Suggested by Erv Wilson, in pp 7-8. See also my 13-limit keyboard mappings

Cassandra 2

Generator (-1, 8, 14, 18, 21). Perfect fourth generator (about 497 c). Schismic variant, covers 29, 41-EDO. Suggested by Erv Wilson as a keyboard mapping, see p.6, and the alternative 13-limit mapping on p.8. See also my 15-limit keyboard mappings

Bohlen-Pierce scale

Has its own website. The 9:7 generator was proposed by Carol Krumhansl in a 1987 conference paper. Dan Stearns also thought of the linear temperament bit. Here's his mapping:

2       7
3       0
5       2
7      -1

See also this post and this other one.


Mapping (1, -2, -2), half octave period. Diaschismic variant. Also called "twintone" or "paultone", it's behind Paul Erlich's decatonic scales. Consistent with 10, 12, 22-equal.

13-limit Diaschismic

Mapping (1, -2, -8, -12, -15). Half-octave period. Diaschismic variant, covers 46- and 58-EDO. I suggested this on the Tuning List as an alternative to Wilson's. It's currently second place in my 13-limit keyboard mappings. It should work better on a Bosanquet keyboard than multiple-29 and will have a more compact diamond, if you think that's important.


Also known as the wonder tuning, proposed by Margo Schulter. Based on the primes 2, 3, and 7 -- no 5. Mapping (3, x, -1)., tempers out 1029:1024 (set x=7 if you want to bring 5 back). Supermajor second generator (around 233 cents). Works with 5, 16, 21-equal.

Golden non-Meantone

From article by Walter O'Connell in Xenharmonikon 15 (1993), but written in the 60s. Period golden ratio of 1.618.., generator: 2:1 ratio. Works with 7, 9, 11 and 14 note scales. Intended for a timbre consisting of golden-section partials.

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