31 EQUAL TEMPERAMENT

In music, '31 equal temperament', called 31-tet, 31-edo, 31-et, or tricesimoprimal temperament, is the tempered scale derived by dividing the octave into 31 equal-sized steps. Each step represents a frequency ratio of 2^1/31, or 38.71 cents.
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 41.1 cents, was approximately a fifth of a tone and a third of a semitone. On this basis Nicola Vicentino produced a 31-step keyboard instrument, the Archicembalo, in 1555, but it was not until 1666 that Lemme Rossi first proposed an equal temperament of this order. Shortly thereafter, having discovered it independently, famed 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 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 that, he went farther and noted that 31-et provides an excellent approximation of septimal, or 7-limit harmony, which was a very advanced insight for the time. 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.

Contents

Scale diagram

Interval size

Theoretical properties

Chords of 31 equal temperament

External links

Scale diagram

The following are 21 of the 31 notes in the scale:

'Interval (cents)'		77		39		77		39		39		39		77		39		77		77		39		77		39		39		39		77		39		77		77		39		77
'Note name'	A		A♯		B♭		B		C♭		B♯		C		C♯		D♭		D		D♯		E♭		E		F♭		E♯		F		F♯		G♭		G		G♯		A♭		A
'Note (cents)'	0		77		116		194		232		271		310		387		426		503		581		619		697		735		774		813		890		929		1006		1084		1123		1200

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:

interval name	size (steps)	size (cents)	just ratio	just (cents)
perfect fifth	18	697	3:2	702
perfect fourth	13	503	4:3	498
major third	10	387	5:4	386
minor third	8	310	6:5	316
whole tone, major tone	5	194	9:8	203
whole tone, minor tone	5	194	10:9	182
diatonic semitone, just	3	116	16:15	112
chromatic semitone, just	2	77	25:24	71

This tuning is 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.

Theoretical properties

The single most important fact about 31-et is that it equates to the unison, or ''tempers out'', the syntonic comma of 81/80. It is therefore a meantone temperament. It also tempers the 5-limit intervals 393216/390625, known as the Würschmidt comma after music theorist José Würschmidt, and 2109375/2097152, known as the semicomma.
More significantly, perhaps, it tempers out 126/125, the septimal semicomma or starling comma. Because it tempers out both 81/80 and 126/125, it supports septimal meantone temperament. It also tempers out 1029/1024, the gamelan residue, and 1728/1715, the Orwell comma. Consequently it supports a wide variety of linear temperaments.

Chords of 31 equal temperament

Many of the most 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-Dx-G or C-Fbb-G, and the Orwell tetrad, which is C-E-Fx-Bbb.

External links

★ de Beer, Anton, ''The Development of 31-tone Music''

★ 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''