19 equal temperament

# 19 equal temperament

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In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET
Equal temperament
An equal temperament is a musical temperament, or a system of tuning, in which every pair of adjacent notes has an identical frequency ratio. As pitch is perceived roughly as the logarithm of frequency, this means that the perceived "distance" from every note to its nearest neighbor is the same for...

, is the tempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represents a frequency ratio of 21/19, or 63.16 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...

. Because 19 is a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...

, one can use any interval from this tuning system to cycle through all possible notes (as one may cycle through 12-et on the circle of fifths
Circle of fifths
In music theory, the circle of fifths shows the relationships among the 12 tones of the chromatic scale, their corresponding key signatures, and the associated major and minor keys...

).

## History

Division of the octave into 19 steps arose naturally out of Renaissance music theory: the greater diesis, the ratio of four minor thirds to an octave (648:625 or 62.565 cents) was almost exactly a 19th of an octave. Interest in such a tuning system goes back to the sixteenth century, when composer Guillaume Costeley
Guillaume Costeley
Guillaume Costeley was a French composer of the Renaissance. He was the court organist to Charles IX of France and famous for his numerous chansons, which were representative of the late development of the form; his work in this regard was part of the early development of the style known as...

used it in his chanson Seigneur Dieu ta pitié of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theorist Francisco de Salinas
Francisco de Salinas
Francisco de Salinas was a Spanish music theorist and organist, noted as among the first to describe meantone temperament in mathematically precise terms, and one of the first to describe, in effect, 19 equal temperament. In his De musica libri septem of 1577 he discusses 1/3-, 1/4- and 2/7-comma...

in effect proposed it. Salinas discussed 1/3-comma meantone, in which the fifth is of size 694.786 cents; the fifth of 19-tet is 694.737, which is less than a twentieth of a cent narrower, imperceptible and less than tuning error. Salinas suggested tuning nineteen tones to the octave to this tuning, which fails to close by less than a cent, so that his suggestion is effectively 19-tet. In the nineteenth century mathematician and music theorist Wesley Woolhouse
Wesley S. B. Woolhouse
Wesley Stoker Barker Woolhouse was an actuary with diverse interests in music theory, the design of steam locomotives, measurements, and many other fields, publishing books in all these fields....

proposed it as a more practical alternative to meantone tunings he regarded as better, such as 50 equal temperament.

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...

wrote his Ph.D. thesis (1961) on the properties of the 19-et tuning, and advocated for its use. In his thesis he demonstrated why he believed that this system represents the only viable system with a number of divisions between 12 and 22, and furthermore that the next smallest number of divisions resulting in a significant improvement in match to natural intervals is the 31 equal temperament
31 equal temperament
In music, 31 equal temperament, 31-ET, which can also be abbreviated 31-TET, 31-EDO , , is the tempered scale derived by dividing the octave into 31 equal-sized steps...

. Mandelbaum has written music with both the 19-et and 31-et tunings.

People have built instruments (such as guitars) and recorded music using the 19-et tuning, but the tuning has not come into widespread use.

## Scale diagram

The 19-tone system can be represented with the traditional letter names and system of sharps and flats by treating flats and sharps as distinct notes, but identifying B as enharmonic
Enharmonic
In modern musical notation and tuning, an enharmonic equivalent is a note , interval , or key signature which is equivalent to some other note, interval, or key signature, but "spelled", or named, differently...

with C and E with F. With this interpretation, the 19 notes in the scale become:
 Step (cents) 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 63 Note name A A B B B/C C C D D D E E E/F F F G G G A A Interval (cents) 0 63 126 189 253 316 379 442 505 568 632 695 758 821 884 947 1011 1074 1137 1200

The fact that traditional western music maps unambiguously onto this scale makes it easier to perform such music in this tuning than in many other tunings.

## Interval size

Here are the sizes of some common intervals and comparison with the ratios arising in the harmonic series; the difference column measures in cents the distance from an exact fit to these ratios. For reference, the difference from the perfect fifth in the widely used 12 equal temperament is 1.955 cents, and the difference from the major third is 13.686 cents.
Interval Name Size (steps) Size (cents) Midi Just Ratio Just (cents) Midi Error (cents)
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...

11 694.74 3:2 701.96 −7.22
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...

, augmented fourth
10 631.58 10:7 617.49 +14.09
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...

, diminished fifth
9 568.42 7:5 582.51 −14.09
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...

8 505.26 4:3 498.04 +7.22
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...

7 442.11 9:7 435.08 +7.03
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...

6 378.95 5:4 386.31 −7.36
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...

5 315.79 6:5 315.64 +0.15
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...

4 252.63 7:6 266.87 −14.24
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...

4 252.63 8:7 231.17 +21.46
Whole tone, major tone 3 189.47 9:8 203.91 −14.44
Whole tone, minor tone 3 189.47 10:9 182.40 +7.07
Septimal diatonic semitone 2 126.32 15:14 119.44 +6.88
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...

2 126.32 16:15 111.73 +14.59
Septimal chromatic semitone
Septimal chromatic semitone
In music, a septimal chromatic semitone or minor semitone is the interval 21:20 . It is about 84.47 cents. The septimal chromatic semitone may be derived from the harmonic series as the interval between the twentieth and twenty-first harmonics....

1 63.16 21:20 84.46 −21.31
Chromatic semitone, just 1 63.16 25:24 70.67 −7.51
Septimal third-tone
Septimal third-tone
A septimal 1/3-tone is an interval with the ratio of 28:27, which is the difference between the perfect fourth and the supermajor third. It is about 62.96 cents wide. The septimal 1/3-tone can be viewed either as a musical interval in its own right, or as a comma; if it is tempered out in a given...

1 63.16 28:27 62.96 +0.20