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In music, 31 equal temperament, which can also be abbreviated (31 tone ) or (equal division of the octave), also known as tricesimoprimal, is the tempered scale derived by dividing the
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
into 31 equally-proportioned steps (equal frequency ratios). Each step represents a
frequency Frequency is the number of occurrences of a repeating event per unit of time. Frequency is an important parameter used in science and engineering to specify the rate of oscillatory and vibratory phenomena, such as mechanical vibrations, audio ...
ratio of , or 38.71  cents (). 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 with ...
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 ...
, the fingering of music composed in is precisely the same as it is in any other syntonic tuning (such as so long as the notes are spelled properly—that is, with no assumption of enharmonicity.


History and use

Division of the
octave In music, an octave (: eighth) or perfect octave (sometimes called the diapason) is an interval between two notes, one having twice the frequency of vibration of the other. The octave relationship is a natural phenomenon that has been referr ...
into 31 steps arose naturally out of Renaissance
music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first is the "Elements of music, ...
; the lesser
diesis In classical music from Western culture, a diesis ( or enharmonic diesis, plural dieses ( , or "difference"; Greek: "leak" or "escape" is either an accidental (see sharp), or a very small musical interval, usually defined as the differe ...
– the ratio of an octave to three major thirds, 128:125 or 41.06 cents – was approximately one-fifth of a
tone Tone may refer to: Visual arts and color-related * Tone (color theory), a mix of tint and shade, in painting and color theory * Tone (color), the lightness or brightness (as well as darkness) of a color * Toning (coin), color change in coins * ...
or two-fifths of a
semitone A semitone, also called a minor second, 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 ...
. In 1555,
Nicola Vicentino Nicola Vicentino (1511 – 1575 or 1576) was an Italian music theory, music theorist and composer of the Renaissance music, Renaissance. He was one of the most progressive musicians of the age, inventing, among other things, a microtonal keyb ...
proposed an extended-meantone tuning of 31 tones. In 1666, Lemme Rossi first proposed an equal temperament of this order. In 1691, having discovered it independently, scientist
Christiaan Huygens Christiaan Huygens, Halen, Lord of Zeelhem, ( , ; ; also spelled Huyghens; ; 14 April 1629 – 8 July 1695) was a Dutch mathematician, physicist, engineer, astronomer, and inventor who is regarded as a key figure in the Scientific Revolution ...
wrote about it also. Since the standard system of tuning at that time was
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 with ...
, in which the fifth is tuned to , the appeal of this method was immediate, as the fifth of , 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 provides an excellent approximation of septimal, or 7 limit 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 equal temperament, 31-tone equal-temp ...
, 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 Teylers Museum () is an art, natural history, and science museum in Haarlem, Netherlands. Established in 1778, Teylers Museum was founded as a centre for contemporary art and science. The historic centre of the museum is the neoclassical Oval Ro ...
in
Haarlem Haarlem (; predecessor of ''Harlem'' in English language, English) is a List of cities in the Netherlands by province, city and Municipalities of the Netherlands, municipality in the Netherlands. It is the capital of the Provinces of the Nether ...
in 1951 and moved to
Muziekgebouw aan 't IJ Muziekgebouw aan 't IJ ( English: "Music Building on the IJ") is the main concert hall for contemporary classical music on the IJ in Amsterdam, Netherlands. The building opened in 2005 and is located above the IJtunnel, a ten-minute walk from ...
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 12 equal temperament (12-ET) is the musical system that divides the octave into 12 parts, all of which are Equal temperament, equally tempered (equally spaced) on a logarithmic scale, with a ratio equal to the Twelfth root of two, 12th root of 2 ...
and only poor fits in
19 equal temperament In music, 19 equal temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), 19-ED2 ("Equal Division of 2:1) or 19 Equal temperament, ET, is the musical temperament, tempered scale derived by dividing the octave into 19 equal steps ...
. The composer Joel Mandelbaum (born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series. 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 temperaments are musical temperaments; that is, a variety of Musical tuning#Tuning systems, tuning systems constructed, similarly to Pythagorean tuning, as a sequence of equal fifths, both rising and descending, scaled to remain within th ...
. It has the necessary property that a chain of its four fifths is equivalent to its major third (the
syntonic comma In music theory Music theory is the study of theoretical frameworks for understanding the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory": The first i ...
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 (orally, or logarithmically) as a semitone, which itself is half a whole tone. Quarter tones divide the octave by 50 cents each, a ...
system:


Modes Mode ( meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * MO''D''E (magazine), a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is the setting fo ...


Ionian mode The Ionian mode is a Mode (music), musical mode or, in modern usage, a diatonic scale also called the major scale. It is named after the Ionians, Ionian Greeks. It is the name assigned by Heinrich Glarean in 1547 to his new Gregorian mode#Authent ...
(
major scale The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at doubl ...
)


Chords of 31 equal temperament

Many chords of 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 c ...
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) was an English novelist, poet, essayist, journalist, and critic who wrote under the pen name of George Orwell. His work is characterised by lucid prose, social criticism, opposition to a ...
tetrad, which is C–E–F𝄪–B𝄫. Usual chords like the major chord are rendered nicely in 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). 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 P ...
. For example on tonic C, with The seventh here is different from stacking a fifth and a minor third, which instead yields B to make a
dominant seventh Domination or dominant may refer to: Society * World domination, structure where one dominant power governs the planet * Colonialism in which one group (usually a nation) invades another region for material gain or to eliminate competition * Ch ...
. This difference cannot be made in .


Footnotes


See also

*
Archicembalo The archicembalo (or arcicembalo, ) was a musical instrument described by Nicola Vicentino in 1555. This was a harpsichord built with many extra keys and strings, enabling experimentation in microtonality and just intonation. Construction The ...
, alternate keyboard instrument with 36 keys per octave that was sometimes tuned as


References


External links

* * * * * * * {{Musical tuning Equal temperaments Microtonality