31 equal temperament

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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 equally large steps. Each step represents a frequency ratio of 2^1/31, or 38.71 cents.

Nicola Vicentino had very nearly produced 31-equal temperament on his extended keyboard instrument, the Archicembalo, in 1555, but it was not until 1666 that Lemme Rossi first proposed it theoretically. 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.

1 Scale diagram
2 Theoretical properties
3 Chords of 31 equal temperament
4 External links

[edit] 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.

[edit] 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.

[edit] 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.

[edit] External links

Tunings

edit

Pythagorean · Just intonation · Harry Partch's 43-tone scale

Regular temperaments

Equal temperaments :	12-tone · 19-tone · 22-tone · 24-tone · 31-tone · 53-tone · 72-tone
Non-equal temperaments :	Meantone (Quarter-comma; Lucy tuning; Septimal) · Schismatic · Miracle