31 equal temperament
Encyclopedia
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO (equal division of the octave), (also known as tricesimoprimal), is the tempered scale derived by dividing the octave
into 31 equal-sized steps (equal frequency ratios). Each step represents a frequency
ratio of 21/31, or 38.71 cents
.
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 41.06 cents — was approximately a fifth
of a tone and a third of a semitone
.In 1666 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 51/4, the appeal of this method is 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
harmony, which was an advanced insight for its 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.
The remaining 10 notes can be added with, for example, five "double flat" notes and five "double sharp" notes, or by half sharp and half flats, similar to the quarter tone system.
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 and only poor fits in 19 equal temperament
. 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.
This tuning can be considered a meantone temperament
. It has the necessary property that a chain of its four fifths are equivalent to its major third (the syntonic 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.
Octave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
into 31 equal-sized steps (equal frequency ratios). Each step represents a frequency
Frequency
Frequency is the number of occurrences of a repeating event per unit time. It is also referred to as temporal frequency.The period is the duration of one cycle in a repeating event, so the period is the reciprocal of the frequency...
ratio of 21/31, or 38.71 cents
Cent (music)
The cent is a logarithmic unit of measure used for musical intervals. Twelve-tone equal temperament divides the octave into 12 semitones of 100 cents each...
.
History
Division of the octaveOctave
In music, an octave is the interval between one musical pitch and another with half or double its frequency. The octave relationship is a natural phenomenon that has been referred to as the "basic miracle of music", the use of which is "common in most musical systems"...
into 31 steps arose naturally out of Renaissance music theory
Music theory
Music theory is the study of how music works. It examines the language and notation of music. It seeks to identify patterns and structures in composers' techniques across or within genres, styles, or historical periods...
; the lesser diesis
Diesis
In classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents...
— the ratio of an octave to three major thirds, 128:125 or 41.06 cents — was approximately a fifth
Interval (music)
In music theory, an interval is a combination of two notes, or the ratio between their frequencies. Two-note combinations are also called dyads...
of a tone and a third of a semitone
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....
.In 1666 Lemme Rossi
Lemme Rossi
Lemme Rossi 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 celebri sistemi di tutti i tre generi of 1666...
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
Musical tuning
In music, there are two common meanings for tuning:* Tuning practice, the act of tuning an instrument or voice.* Tuning systems, the various systems of pitches used to tune an instrument, and their theoretical bases.-Tuning practice:...
at that time was quarter-comma meantone
Quarter-comma meantone
Quarter-comma meantone, or 1/4-comma meantone, was the most common meantone temperament in the sixteenth and seventeenth centuries, and was sometimes used later. This method is a variant of Pythagorean tuning...
, in which the fifth is tuned to 51/4, the appeal of this method is 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
Limit (music)
In music theory, limit or harmonic limit is a way of characterizing the harmony found in a piece or genre of music, or the harmonies that can be made using a particular scale. The term was introduced by Harry Partch, who used it to give an upper bound on the complexity of harmony; hence the name...
harmony, which was an advanced insight for its time. In the twentieth century, physicist, music theorist and composer Adriaan Fokker
Adriaan Fokker
Adriaan Daniël Fokker , was a Dutch physicist and musician.Fokker was born in Buitenzorg, Dutch East Indies ; he was a cousin of the aeronautical engineer Anthony 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.
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, or by half sharp and half flats, similar to the quarter tone system.
Interval size
Here are the sizes of some common intervals:| interval name | size (steps) | size (cents) | midi | just ratio | just (cents) | midi | error |
| harmonic seventh Harmonic seventh The harmonic seventh interval , also known as the septimal minor seventh, or subminor seventh, is one with an exact 7:4 ratio . This is somewhat narrower than and is "sweeter in quality" than an "ordinary" minor seventh, which has a just-intonation ratio of 9:5 , or an equal-temperament ratio of... |
25 | 967.74 | 7:4 | 968.83 | −1.09 | ||
| perfect fifth Perfect fifth In classical music from Western culture, a fifth is a musical interval encompassing five staff positions , and the perfect fifth is a fifth spanning seven semitones, or in meantone, four diatonic semitones and three chromatic semitones... |
18 | 696.77 | 3:2 | 701.96 | −5.19 | ||
| greater septimal tritone Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7... |
16 | 619.36 | 10:7 | 617.49 | +1.87 | ||
| lesser septimal tritone Septimal tritone The lesser septimal tritone is the interval with ratio 7:5 . The inverse of that interval, the greater septimal tritone, is an interval with ratio 10:7... |
15 | 580.65 | 7:5 | 582.51 | −1.86 | ||
| undecimal tritone Tritone In classical music from Western culture, the tritone |tone]]) is traditionally defined as a musical interval composed of three whole tones. In a chromatic scale, each whole tone can be further divided into two semitones... , 11th harmonic Harmonic A harmonic of a wave is a component frequency of the signal that is an integer multiple of the fundamental frequency, i.e. if the fundamental frequency is f, the harmonics have frequencies 2f, 3f, 4f, . . . etc. The harmonics have the property that they are all periodic at the fundamental... |
14 | 541.94 | 11:8 | 551.32 | −9.38 | ||
| perfect fourth Perfect fourth In classical music from Western culture, a fourth is a musical interval encompassing four staff positions , and the perfect fourth is a fourth spanning five semitones. For example, the ascending interval from C to the next F is a perfect fourth, as the note F lies five semitones above C, and there... |
13 | 503.23 | 4:3 | 498.04 | +5.19 | ||
| tridecimal major third | 12 | 464.52 | 13:10 | 454.21 | +10.31 | ||
| undecimal major third | 11 | 425.81 | 14:11 | 417.51 | +8.30 | ||
| septimal major third Septimal major third In music, the septimal major third , also called the supermajor third and sometimes Bohlen–Pierce third is the musical interval exactly or approximately equal to a just 9:7 ratio of frequencies, or alternately 14:11. It is equal to 435 cents, sharper than a just major third by the septimal... |
11 | 425.81 | 9:7 | 435.08 | −9.27 | ||
| major third Major third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the major third is one of two commonly occurring thirds. It is qualified as major because it is the largest of the two: the major third spans four semitones, the minor third three... |
10 | 387.10 | 5:4 | 386.31 | +0.79 | ||
| undecimal neutral third Neutral third A neutral third is a musical interval wider than a minor third but narrower than a major third . Three distinct intervals may be termed neutral thirds:... |
9 | 348.39 | 11:9 | 347.41 | +0.98 | ||
| minor third Minor third In classical music from Western culture, a third is a musical interval encompassing three staff positions , and the minor third is one of two commonly occurring thirds. The minor quality specification identifies it as being the smallest of the two: the minor third spans three semitones, the major... |
8 | 309.68 | 6:5 | 315.64 | −5.96 | ||
| septimal minor third Septimal minor third In music, the septimal minor third , also called the subminor third, is the musical interval exactly or approximately equal to a 7/6 ratio of frequencies. In terms of cents, it is 267 cents, a quartertone of size 36/35 flatter than a just minor third of 6/5... |
7 | 270.97 | 7:6 | 266.87 | +4.10 | ||
| septimal whole tone Septimal whole tone In music, the septimal whole tone, septimal major second, or supermajor second is the musical interval exactly or approximately equal to a 8/7 ratio of frequencies. It is about 231 cents wide in just intonation. Although 24 equal temperament does not match this interval particularly well, its... |
6 | 232.26 | 8:7 | 231.17 | +1.09 | ||
| whole tone, major tone | 5 | 193.55 | 9:8 | 203.91 | −10.36 | ||
| whole tone, minor tone | 5 | 193.55 | 10:9 | 182.40 | +11.15 | ||
| greater undecimal neutral second Neutral second A neutral second or medium second is a musical interval wider than a minor second and narrower than a major second. Three distinct intervals may be termed neutral seconds:... |
4 | 154.84 | 11:10 | 165.00 | −10.16 | ||
| lesser undecimal neutral second | 4 | 154.84 | 12:11 | 150.64 | +4.20 | ||
| septimal diatonic semitone | 3 | 116.13 | 15:14 | 119.44 | −3.31 | ||
| diatonic semitone, just Just intonation In music, just intonation is any musical tuning in which the frequencies of notes are related by ratios of small whole numbers. Any interval tuned in this way is called a just interval. The two notes in any just interval are members of the same harmonic series... |
3 | 116.13 | 16:15 | 111.73 | +4.40 | ||
| chromatic semitone, just | 2 | 77.42 | 25:24 | 70.67 | +6.75 | ||
| undecimal diesis Diesis In classical music from Western culture, a diesis is either an accidental , or a comma type of musical interval, usually defined as the difference between an octave and three justly tuned major thirds , equal to 128:125 or about 41.06 cents... |
1 | 38.71 | 45:44 | 38.91 | −0.20 | ||
| septimal diesis Septimal diesis In music, septimal diesis is an interval with the ratio of 49:48 , which is the difference between the septimal whole tone and the septimal minor third. It is about 35.7 cents wide, which is narrower than a quarter-tone but wider than the septimal comma... |
1 | 38.71 | 49:48 | 35.70 | +3.01 |
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 and only poor fits in 19 equal temperament
19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the tempered scale derived by dividing the octave into 19 equal steps . Each step represents a frequency ratio of 21/19, or 63.16 cents...
. The composer Joel Mandelbaum
Joel Mandelbaum
Joel Mandelbaum is an American music composer and teacher, best known for his use of microtonal tuning . He also has written the first Ph.D. dissertation on microtonality, in 1961. He is married to stained glass artist , and is the nephew of Abraham Edel.-Life & Music:Mandelbaum received his Ph.D...
(born 1932) used this tuning system specifically because of its good matches to the 7th and 11th partials in the harmonic series.
This tuning can be considered a meantone temperament
Meantone temperament
Meantone temperament is a musical temperament, which is a system of musical tuning. In general, a meantone is constructed the same way as Pythagorean tuning, as a stack of perfect fifths, but in meantone, each fifth is narrow compared to the ratio 27/12:1 in 12 equal temperament, the opposite of...
. It has the necessary property that a chain of its four fifths are equivalent to its major third (the syntonic comma
Syntonic comma
In music theory, the syntonic comma, also known as the chromatic diesis, the comma of Didymus, the Ptolemaic comma, or the diatonic comma is a small comma type interval between two musical notes, equal to the frequency ratio 81:80, or around 21.51 cents...
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.
External links
- The Huygens Fokker foundation for micro-tonal music, in Dutch and English
- 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
- Barbieri, Patrizio. Enharmonic instruments and music, 1470-1900. (2008) Latina, Il Levante Libreria Editrice
- M. Khramov, “Approximation to 7-Limit Just Intonation in a Scale of 31EDO,” Proceedings of the FRSM-2009 International Symposium Frontiers of Research on Speech and Music, pp. 73-82, ABV IIITM, Gwalior, 2009.