# 31 equal temperament

In music,

**31 equal temperament**, which can be abbreviated 31-tET, 31-EDO, 31-ET, is the tempered scale derived by dividing theoctave into 31 equal-sized steps. Each step represents afrequency ratio of 2^{1/31}, or 38.71 cents.Division of the

octave into 31 steps arose naturally out of Renaissancemusic theory ; the lesserdiesis — the ratio of an octave to three major thirds, 128/125 or 41.1 cents — was approximately a fifth of a tone and a third of asemitone . On this basis,Nicola Vicentino produced a 31-step keyboard instrument, theArchicembalo , in 1555, but it was not until 1666 thatLemme Rossi first proposed an equal temperament of this order. Shortly thereafter, having discovered it independently, famed scientistChristiaan Huygens wrote about it also. Since the standard system of tuning at that time wasquarter-comma meantone , in which the fifth is tuned to 5^{1/4}, the appeal of this method is immediate, as the fifth of 31-et, at 696.77 cents, is only a fifth of a cent sharper 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, which was an advanced insight for its time. In the twentieth century, physicist, music theorist and composerAdriaan 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.**cale diagram**The following are 21 of the 31 notes in the scale:

The remaining 10 notes can be added with, for example, five "double flat" notes and five "double sharp" notes.

**Interval size**Here are the sizes of some common intervals:

The 31 equal temperament has a very close fit to the 7:6, 8:7, and 7:5 ratios, ratios which do not even have approximate fits within the

12 equal temperament and which have only a poor fit with the19 equal temperament . The composerJoel Mandelbaum (born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series. [*http://links.jstor.org/sici?sici=0031-6016%28199124%2929%3A1%3C176%3ASACONT%3E2.0.CO%3B2-G "Six American Composers on Nonstandard Tunnings: Douglas Keislar; 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.*] It should be noted, however, that this tuning does not distinguish between theseptimal major third and the (14:11) ratio, neither of which is matched particularly well in this tuning.This tuning can be considered a

meantone temperament . It has the necessary property that a chain of its four fifths are equivalent to its major third (the 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.**Tempering**One property of 31-et is that it equates to the unison, or "tempers out", the

syntonic comma of 81/80. It can therefore be considered ameantone temperament . It also tempers the 5-limit intervals 393216/390625, known as the Würschmidt comma after music theoristJosé Würschmidt , and 2109375/2097152, known as the semicomma.In addition, it also tempers out 126/125, the

septimal semicomma or starling comma .Fact|date=April 2008. Because it tempers out both 81/80 and 126/125, it supportsseptimal meantone temperament . It also tempers out 1029/1024, the gamelan residue .Fact|date=April 2008, and 1728/1715, the Orwell comma .Fact|date=April 2008. Consequently it supports a wide variety oflinear temperament s .Fact|date=April 2008.31-et also tempers out 99/98.

**Chords of 31 equal temperament**Many interesting chords of 31-et are discussed in the article on

septimal meantone temperament . Chords not discussed there include the neutral thirds triad, which might be written either C-Dmusic|##-G or C-Fmusic|bb-G, and the Orwell tetrad, which is C-E-Fmusic|##-Bmusic|bb.**References****External links*** [

*http://www.xs4all.nl/~huygensf/english/index.html de Beer, Anton, "The Development of 31-tone Music"*]

* [*http://www.xs4all.nl/~huygensf/doc/fokkerorg.html Fokker, Adriaan Daniël, "Equal Temperament and the Thirty-one-keyed organ"*]

* [*http://www.xs4all.nl/~huygensf/doc/rap31.html Rapoport, Paul, "About 31-tone Equal Temperament"*]

* [*http://www.xs4all.nl/~huygensf/doc/terp31.html Terpstra, Siemen, "Toward a Theory of Meantone (and 31-et) Harmony"*]

*Wikimedia Foundation.
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### Look at other dictionaries:

**Equal temperament**— is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio. In equal temperament tunings an interval mdash; usually the octave mdash; is divided into a series of equal steps (equal… … Wikipedia**Equal temperament**— Equal E qual, a. [L. aequalis, fr. aequus even, equal; akin to Skr. ?ka, and perh. to L. unus for older oinos one, E. one.] 1. Agreeing in quantity, size, quality, degree, value, etc.; having the same magnitude, the same value, the same degree,… … The Collaborative International Dictionary of English**Equal temperament**— Temperament Tem per*a*ment, n. [L. temperamentum a mixing in due proportion, proper measure, temperament: cf. F. temp[ e]rament. See {Temper}, v. t.] 1. Internal constitution; state with respect to the relative proportion of different qualities,… … The Collaborative International Dictionary of English**equal temperament**— Music. the division of an octave into 12 equal semitones, as in the tuning of a piano. * * * ▪ music in music, a tuning system in which the octave is divided into 12 semitones of equal size. Because it enables keyboard instruments (keyboard … Universalium**equal temperament**— noun the division of the scale based on an octave that is divided into twelve exactly equal semitones equal temperament is the system commonly used in keyboard instruments • Hypernyms: ↑temperament … Useful english dictionary**53 equal temperament**— In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into fifty three equally large steps. Each step represents a frequency ratio of 21/53, or 22.6415 cents, an interval sometimes… … Wikipedia**19 equal temperament**— In music, 19 equal temperament, called 19 TET, 19 EDO, or 19 ET, is the tempered scale derived by dividing the octave into 19 equally large steps. Each step represents a frequency ratio of 21/19, or 63.16 cents. Because 19 is a prime number, one… … Wikipedia**72 equal temperament**— In music, 72 equal temperament, called twelfth tone, 72 tet, 72 edo, or 72 et, is the tempered scale derived by dividing the octave into twelfth tones, or in other words 72 equally large steps. Each step represents a frequency ratio of 21/72, or… … Wikipedia**22 equal temperament**— In music, 22 equal temperament, called 22 tet, 22 edo, or 22 et, is the tempered scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of 21/22, or 54.55 cents.The idea of dividing the octave… … Wikipedia**34 equal temperament**— In musical theory, 34 equal temperament, also referred to as 34 tet, 34 edo or 34 et, is the tempered tuning derived by dividing the octave into 34 equal sized steps. Each step represents a frequency ratio of 21/34, or 35.29 cents.Unlike… … Wikipedia