31 equal temperament
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In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the octave into 31 equal-sized steps (equal frequency ratios). Play (help·info) Each step represents a frequency ratio of 21/31, or 38.71 cents ( Play (help·info)).
31-ET is a tuning of the syntonic temperament in which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On an isomorphic keyboard, the fingering of music composed in 31-edo is precisely the same as it is in any other syntonic tuning (such as 12-edo), so long as the notes are spelled properly -- that is, with no assumption of enharmonicity.
Division of the octave into 31 steps arose naturally out of Renaissance music theory; the lesser diesis — the ratio of an octave to three major thirds, 128:125 or 41.06 cents — was approximately a fifth of a tone and a third of a semitone. In 1666, Lemme Rossi first proposed an equal temperament of this order. Shortly thereafter, having discovered it independently, scientist Christiaan Huygens wrote about it also. Since the standard system of tuning at that time was quarter-comma meantone, in which the fifth is tuned to 51/4, the appeal of this method was immediate, as the fifth of 31-et, at 696.77 cents, is only 0.19 cent wider than the fifth of quarter-comma meantone. Huygens not only realized this, he went farther and noted that 31-ET provides an excellent approximation of septimal, or 7-limit harmony. In the twentieth century, physicist, music theorist and composer Adriaan Fokker, after reading Huygens's work, led a revival of interest in this system of tuning which led to a number of compositions, particularly by Dutch composers.
Scale diagram 
The following are the 31 notes in the scale:
Sometimes the five "double flat" notes and five "double sharp" notes are replaced by half sharps and half flats, similar to the quarter tone system:
Interval size 
Here are the sizes of some common intervals:
|interval name||size (steps)||size (cents)||midi||just ratio||just (cents)||midi||error|
|harmonic seventh||25||967.74||Play (help·info)||7:4||968.83||Play (help·info)||−1.09|
|perfect fifth||18||696.77||Play (help·info)||3:2||701.96||Play (help·info)||−5.19|
|greater septimal tritone||16||619.35||10:7||617.49||+1.87|
|lesser septimal tritone||15||580.65||Play (help·info)||7:5||582.51||Play (help·info)||−1.86|
|undecimal tritone, 11th harmonic||14||541.94||Play (help·info)||11:8||551.32||Play (help·info)||−9.38|
|perfect fourth||13||503.23||Play (help·info)||4:3||498.04||Play (help·info)||+5.19|
|septimal narrow fourth||12||464.52||Play (help·info)||21:16||470.78||play (help·info)||−6.26|
|tridecimal major third||12||464.52||Play (help·info)||13:10||454.21||Play (help·info)||+10.31|
|septimal major third||11||425.81||Play (help·info)||9:7||435.08||Play (help·info)||−9.27|
|undecimal major third||11||425.81||Play (help·info)||14:11||417.51||Play (help·info)||+8.30|
|major third||10||387.10||Play (help·info)||5:4||386.31||Play (help·info)||+0.79|
|tridecimal neutral third||9||348.39||Play (help·info)||16:13||359.47||play (help·info)||−11.09|
|undecimal neutral third||9||348.39||Play (help·info)||11:9||347.41||Play (help·info)||+0.98|
|minor third||8||309.68||Play (help·info)||6:5||315.64||Play (help·info)||−5.96|
|septimal minor third||7||270.97||Play (help·info)||7:6||266.87||Play (help·info)||+4.10|
|septimal whole tone||6||232.26||Play (help·info)||8:7||231.17||Play (help·info)||+1.09|
|whole tone, major tone||5||193.55||Play (help·info)||9:8||203.91||Play (help·info)||−10.36|
|whole tone, minor tone||5||193.55||Play (help·info)||10:9||182.40||Play (help·info)||+11.15|
|greater undecimal neutral second||4||154.84||Play (help·info)||11:10||165.00||−10.16|
|lesser undecimal neutral second||4||154.84||Play (help·info)||12:11||150.64||Play (help·info)||+4.20|
|septimal diatonic semitone||3||116.13||Play (help·info)||15:14||119.44||Play (help·info)||−3.31|
|diatonic semitone, just||3||116.13||Play (help·info)||16:15||111.73||Play (help·info)||+4.40|
|septimal chromatic semitone||3||77.42||Play (help·info)||21:20||84.47||Play (help·info)||−7.05|
|chromatic semitone, just||2||77.42||Play (help·info)||25:24||70.67||Play (help·info)||+6.75|
|lesser diesis||1||38.71||Play (help·info)||128:125||41.06||Play (help·info)||−2.35|
|undecimal diesis||1||38.71||Play (help·info)||45:44||38.91||Play (help·info)||−0.20|
|septimal diesis||1||38.71||Play (help·info)||49:48||35.70||Play (help·info)||+3.01|
The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, which have no approximate fits in 12 equal temperament and only poor fits in 19 equal temperament. The composer Joel Mandelbaum (born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.
This tuning can be considered a meantone temperament. It has the necessary property that a chain of its four fifths is equivalent to its major third (the syntonic comma 81:80 is tempered out), which also means that it contains a "meantone" that falls between the sizes of 10:9 and 9:8 as the combination of one of each of its chromatic and diatonic semitones.
Chords of 31 equal temperament 
Many chords of 31-et are discussed in the article on septimal meantone temperament. Chords not discussed there include the neutral thirds triad ( Play (help·info)), which might be written C-E-G, C-D-G or C-F-G, and the Orwell tetrad, which is C-E-F-B.
- Milne, A., Sethares, W.A. and Plamondon, J.,"Isomorphic Controllers and Dynamic Tuning: Invariant Fingerings Across a Tuning Continuum", Computer Music Journal, Winter 2007, Vol. 31, No. 4, Pages 15-32.
- Keislar, Douglas. "Six American Composers on Nonstandard Tunnings: Easley Blackwood; John Eaton; Lou Harrison; Ben Johnston; Joel Mandelbaum; William Schottstaedt", Perspectives of New Music, Vol. 29, No. 1. (Winter, 1991), pp. 176-211.
- Andrew Milne, William Sethares, and James Plamondon (2007). "Isomorphic Controllers and Dynamic Tuning: Invariant Fingering over a Tuning Continuum", p.29. Computer Music Journal, 31:4, pp.15–32, Winter 2007.
- The Huygens Fokker foundation for micro-tonal music, in Dutch and English
- Fokker, Adriaan Daniël, Equal Temperament and the Thirty-one-keyed organ
- Rapoport, Paul, About 31-tone Equal Temperament
- Terpstra, Siemen, Toward a Theory of Meantone (and 31-et) Harmony
- Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice
- M. Khramov, “Approximation to 7-Limit Just Intonation in a Scale of 31EDO,” Proceedings of the FRSM-2009 International Symposium Frontiers of Research on Speech and Music, pp. 73–82, ABV IIITM, Gwalior, 2009.