Rank-2 temperaments with the generator close to a fifth and period an octave

In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET (31 tone ET) or 31- EDO (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

into 31 equal-sized steps (equal frequency ratios). Each step represents a

frequency Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

ratio of , or 38.71 cents (). 31-ET is a very good approximation of

quarter-comma meantone Quarter-comma meantone, or -comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. In this system the perfect fifth is flattened by one quarter of a syntonic comma (81:80 ...

temperament. More generally, it is a

regular diatonic tuning A regular diatonic tuning is any musical scale consisting of " tones" (T) and "semitones" (S) arranged in any rotation of the sequence TTSTTTS which adds up to the octave with all the T's being the same size and all the S's the being the same s ...

in which the tempered perfect fifth is equal to 696.77 cents, as shown in Figure 1. On an

isomorphic keyboard An isomorphic keyboard is a musical input device consisting of a two-dimensional grid of note-controlling elements (such as buttons or keys) on which any given sequence and/or combination of musical intervals has the "same shape" on the keyboard wh ...

, the fingering of music composed in 31-ET is precisely the same as it is in any other syntonic tuning (such as 12-ET), so long as the notes are spelled properly — that is, with no assumption of enharmonicity.

History and use

Division of the

into 31 steps arose naturally out of Renaissance

music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the "rudiments", that are needed to understand music notation (ke ...

; 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 or a third of a

semitone A semitone, also called a half step or a half tone, is the smallest musical interval commonly used in Western tonal music, and it is considered the most dissonant when sounded harmonically. It is defined as the interval between two adjacent no ...

. In 1555,

Nicola Vicentino Nicola Vicentino (1511 – 1575 or 1576) was an Italian music theorist and composer of the Renaissance. He was one of the most progressive musicians of the age, inventing, among other things, a microtonal keyboard. Life Little is known of h ...

proposed an extended-meantone tuning of 31 tones. In 1666,

Lemme Rossi Lemme Rossi (died 1673) was an Italian music theorist who was the first to publish a discussion of 31 equal temperament, the division of the octave into 31 equal parts, in his ''Sistema musico, ouero Musica speculativa doue SI spiegano i più celeb ...

first proposed an equal temperament of this order. In 1691, having discovered it independently, scientist

Christiaan Huygens Christiaan Huygens, Lord of Zeelhem, ( , , ; also spelled Huyghens; la, Hugenius; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor, who is regarded as one of the greatest scientists of ...

wrote about it also. Since the standard system of

tuning Tuning can refer to: Common uses * Tuning, the process of tuning a tuned amplifier or other electronic component * Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice ** Guitar tunings ** Piano tuning, adjusti ...

at that time was

, in which the fifth is tuned to , 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 7-limit or septimal tunings and intervals are musical instrument tunings that have a limit of seven: the largest prime factor contained in the interval ratios between pitches is seven. Thus, for example, 50:49 is a 7-limit interval, but 14 ...

harmony. In the twentieth century, physicist, music theorist and composer

Adriaan Fokker Adriaan Daniël Fokker (; 17 August 1887 – 24 September 1972) was a Dutch physicist. He worked in the fields of special relativity and statistical mechanics. He was the inventor of the Fokker organ, a 31-tone equal-tempered (31-TET) organ. ...

, 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. Fokker designed the Fokker organ, a 31-tone equal-tempered organ, which was installed in Teyler's Museum in

Haarlem Haarlem (; predecessor of ''Harlem'' in English) is a city and municipality in the Netherlands. It is the capital of the province of North Holland. Haarlem is situated at the northern edge of the Randstad, one of the most populated metropoli ...

in 1951 and moved to

Muziekgebouw aan 't IJ Muziekgebouw aan 't IJ (English language, English: "Music Building on the IJ") is the main concert hall for contemporary classical music on the IJ (Amsterdam), IJ in Amsterdam, Netherlands. The building opened in 2005 and is located above the IJt ...

in 2010 where it has been frequently used in concerts since it moved.

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, which have no approximate fits in

12 equal temperament Twelve-tone equal temperament (12-TET) is the musical system that divides the octave into 12 parts, all of which are equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the 12th root of 2 ( ≈ 1.05946). That result ...

and only poor fits in

19 equal temperament In music, 19 Tone Equal Temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), or 19 ET, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represent ...

. The composer

Joel Mandelbaum Mayer Joel Mandelbaum (born October 12, 1932) is an American music composer and teacher, best known for his use of microtonal tuning (notably just intonation and 19 equal temperament and the 31 equal temperament). He wrote the first Ph.D. dissertat ...

(born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.Keislar, Douglas. "Six American Composers on Nonstandard Tunnings: Easley Blackwood;

John Eaton John Eaton may refer to: * John Eaton (divine) (born 1575), English divine *John Eaton (pirate) (fl. 1683–1686), English buccaneer *Sir John Craig Eaton (1876–1922), Canadian businessman *John Craig Eaton II (born 1937), Canadian businessman an ...

;

Lou Harrison Lou Silver Harrison (May 14, 1917 – February 2, 2003) was an American composer, music critic, music theorist, painter, and creator of unique musical instruments. Harrison initially wrote in a dissonant, ultramodernist style similar to his form ...

; Ben Johnston;

; William Schottstaedt", ''

Perspectives of New Music ''Perspectives of New Music'' (PNM) is a peer-reviewed academic journal specializing in music theory and analysis. It was established in 1962 by Arthur Berger and Benjamin Boretz (who were its initial editors-in-chief). ''Perspectives'' was first ...

'', vol. 29, no. 1. (Winter 1991), pp. 176–211. The tuning has poor matches to both the 9:8 and 10:9 intervals (major and minor tone in just intonation); however, it has a good match for the ''average'' of the two. Practically it is very close to quarter-comma meantone. This tuning can be considered a

meantone temperament Meantone temperament is a musical temperament, that is a tuning system, obtained by narrowing the fifths so that their ratio is slightly less than 3:2 (making them ''narrower'' than a perfect fifth), in order to push the thirds closer to pure. Me ...

. It has the necessary property that a chain of its four fifths is equivalent to its major third (the

syntonic comma In music theory, the syntonic comma, also known as the chromatic diesis, the Didymean comma, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80 (= 1.0125) ( ...

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.

Scale diagram

The following are the 31 notes in the scale: The five "double flat" notes and five "double sharp" notes may be replaced by half sharps and half flats, similar to the

quarter tone A quarter tone is a pitch halfway between the usual notes of a chromatic scale or an interval about half as wide (aurally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, a ...

system:

Chords of 31 equal temperament

Many chords of 31-ET are discussed in the article on

septimal meantone temperament 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 am ...

. Chords not discussed there include the

neutral third A neutral third is a musical interval wider than a minor third but narrower than a major third , named by Jan Pieter Land in 1880. Land makes reference to the neutral third attributed to Zalzal (8th c.), described by Al-Farabi (10th c.) as ...

s triad (), which might be written C–E–G, C–D–G or C–F–G, and the

Orwell Eric Arthur Blair (25 June 1903 – 21 January 1950), better known by his pen name George Orwell, was an English novelist, essayist, journalist, and critic. His work is characterised by lucid prose, social criticism, opposition to totalitari ...

tetrad, which is C–E–F–B. Simple I-IV-V-I isomorphic 31-TET

Usual chords like the major chord are rendered nicely in 31-ET because the third and the fifth are very well approximated. Also, it is possible to play subminor chords (where the first third is subminor) and supermajor chords (where the first third is

supermajor Big Oil is a name used to describe the world's six or seven largest publicly traded and investor-owned oil and gas companies, also known as supermajors. The term, particularly in the United States, emphasizes their economic power and influence ...

). It is also possible to render nicely the

harmonic seventh chord The harmonic seventh chord is a major triad plus the harmonic seventh interval (ratio of 7:4, about 968.826 centsBosanquet, Robert Holford Macdowall (1876). ''An elementary treatise on musical intervals and temperament'', pp. 41-42. Diapason ...

. For example on C with C–E–G–A. The seventh here is different from stacking a fifth and a minor third, which instead yields B to make a

dominant seventh In music theory, a dominant seventh chord, or major minor seventh chord, is a seventh chord, usually built on the fifth degree of the major scale, and composed of a root, major third, perfect fifth, and minor seventh. Thus it is a major triad tog ...

. This difference cannot be made in 12-ET.

References

External links