In music, 31 equal temperament, called 31-tet, 31-edo, or 31-et, is the 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. The cent is a logarithmic unit of measure used for musical intervals. ...

Interest in this tuning system goes back to 1666, when music theorist Lemme Rossi first proposed it. Shortly thereafter, having discovered it independently, famed scientist Christiaan Huyghens 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. Huyghens 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 Huyghen's work, led a revival of interest in this system of tuning which lead to a number of compositions, particularly by Dutch composers. Quarter-comma meantone was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. ... Just intonation tunings and scales can be described by giving an upper bound on the complexity of the harmonies admitted by the tuning or scale. ... Adriaan Daniël Fokker (1887–1972) was a Dutch physicist and musician. ...

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 wuerschmidt comma after music theorist José Würschmidt, and 2109375/2097152, known as the semicomma. The syntonic comma, also known as the comma of Didymus or Ptolemaic comma, is a small interval between two musical notes, equal to the frequency ratio 81:80, or around 21. ... Meantone temperament is a system of musical tuning. ...

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 the 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. In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal...

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. In music, septimal meantone temperament, also called standard septimal meantone or simply septimal meantone, refers to the tempering of 7-limit musical intervals by a meantone temperament tuning in the range from fifths flattened by the amount of fifths for 12 equal temperament to those as flat as 19 equal...

Musical examples

Original to 31-et

• The Liberation of Gabrovo, Harold Fortuin, excerpt, mp3 file

31 equal arrangements

• Impromptu in C# minor, op 28 no 3, Hugo Reinhold, midi file

• String quartet no 3, op 67, first movement, Brahms, midi file

• String quartet no 3, op 67, second movement, Brahms, midi file

• Prelude to Tristan und Isolde, Wagner, mp3 file

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