# 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 equally large steps. Each step represents a frequency ratio of 2^{1/19}, or 63.16 cents. Because 19 is aprime number , one can use any interval from this tuning system to cycle through all possible notes (as one may cycle through 12-et on thecircle of fifths ).**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, 1296/1250, 62.6 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 used it in his chanson "Seigneur Dieu ta pitié" of 1558. Costeley understood and desired the circulating aspect of this tuning; in 1577 music theoristFrancisco de Salinas 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 only a twentieth of a cent flatter. 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 proposed it as a more practical alternative to meantone tunings he regarded as better, such as 50 equal temperament.The composer

Joel Mandelbaum 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 the31 equal temperament . [*[*] Mandelbaum has written music with both the 19-et and 31-et tunings.*http://links.jstor.org/sici?sici=0022-2909%28196721%2911%3A1%3C32%3ASCROES%3E2.0.CO%3B2-A C. Gamer, "Some Combinational Resources of Equal-Tempered Systems". Journal of Music Theory, Vol. 11, No. 1 (Spring, 1967), pp. 32-59*]People have built instruments (such as guitars) and recorded music using the 19-et tuning, but the tuning has not come into widespread use.

**cale 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 Bmusic|sharp with Cmusic|flat and Emusic|sharp with Fmusic|flat. With this interpretation, the 19 notes in the scale become:

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 1.96 cents, and the difference from the major third is 13.69 cents.Compared to 12-et, this system has a slightly poorer fit to the 3:2 ratio

perfect fifth but a closer fit for the 5:4major third . There are no equal temperaments between 12 and 19 that achieve a better fit for both intervals. Unlike 12-et, 19-et utilizes the seventh harmonic, matching it fairly well for three intervals. The 19-et distinguishes between the normal thirds and the two intervals of theseptimal major third andseptimal minor third . This allows the construction of septimal triadic chords, which correspond to the harmonics 7:9:14. The seventh harmonic is also utilized in the two tritones. The 19-et does not match intervals containing the 11th harmonic.The

22 equal temperament offers a similarly close fit for most intervals, improving the fit in particular for theseptimal major third andseptimal minor third , and also distinguishing between this interval and theseptimal whole tone . However, the 22-et does not have a close match for anywhole tone which makes it less suitable for playing diatonic music.This tuning is considered a meantone temperament. It has the necessary property that a chain of its four fifths are equivalent to its major third (the 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.

**As an approximation of other temperaments**The most salient characteristic of 19-et is that, having an almost just

minor third andperfect fifth s andmajor third s about seven cents narrow, it serves as a good tuning formeantone temperament . It is also a suitable formagic temperament , because five of its major thirds are equivalent to one of its "twelfths".For both of these there are more optimal tunings, however. The generating interval for meantone is a fifth, and the fifth of 19-et is flatter than the usual for meantone; a more accurate approximation is

31 equal temperament . Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter;41 equal temperament more closely matches it.However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with 5-limit music in a tolerable manner. It is less successful with 7-limit (but still better than 12-et), as it eliminates the distinction between a

septimal minor third (7/6), and aseptimal whole tone (8/7).**References*** Levy, Kenneth J.,"Costeley's Chromatic Chanson", Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

* W. S. B. Woolhouse [*http://books.google.com/books?id=4VjsjvqMcZgC "Essay on Musical Intervals, Harmonics, and the Temperament of the Musical Scale, &c."*] J. Souter, London, 1835**External links*** [

*http://gewi.uni-graz.at/~cim04/CIM04_paper_pdf/Bucht_Huovinen_CIM04_proceedings.pdf Bucht, Saku and Huovinen, Erkki, "Perceived consonance of harmonic intervals in 19-tone equal temperament"*]

* [*http://sonic-arts.org/darreg/CASE.HTM Darreg, Ivor, "A Case for Nineteen"*]

* [*http://qcpages.qc.edu/~howe/articles/19-Tone%20Theory.html Howe, Hubert S. Jr., "19-Tone Theory and Applications"*]

* [*http://eceserv0.ece.wisc.edu/~sethares/tet19/guitarchords19.html Sethares, William A., "Tunings for 19 Tone Equal Tempered Guitar"*]

* [*http://www.n-ism.org/Projects/microtonalism.php Hair, Bailey, Morrison, Pearson and Parncutt, "Rehearsing Microtonal Music: Grappling with Performance and Intonational Problems" (project summary)*]

* [*http://www.ziaspace.com/ZIA/sections/music.html 19tet downloadable mp3s by Elaine Walker of Zia and D.D.T.*]

* [*http://www.parnasse.com/jh/blog/ The Music of Jeff Harrington*] - Jeff Harrington is a composer who has written several pieces for piano in the 19-TET tuning, and there are both scores and MP3's available for download on this site.

*Wikimedia Foundation.
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### Look at other dictionaries:

**Equal temperament**— is a musical temperament, or a system of tuning in which every pair of adjacent notes has an identical frequency ratio. In equal temperament tunings an interval mdash; usually the octave mdash; is divided into a series of equal steps (equal… … Wikipedia**Equal temperament**— Equal E qual, a. [L. aequalis, fr. aequus even, equal; akin to Skr. ?ka, and perh. to L. unus for older oinos one, E. one.] 1. Agreeing in quantity, size, quality, degree, value, etc.; having the same magnitude, the same value, the same degree,… … The Collaborative International Dictionary of English**Equal temperament**— Temperament Tem per*a*ment, n. [L. temperamentum a mixing in due proportion, proper measure, temperament: cf. F. temp[ e]rament. See {Temper}, v. t.] 1. Internal constitution; state with respect to the relative proportion of different qualities,… … The Collaborative International Dictionary of English**equal temperament**— Music. the division of an octave into 12 equal semitones, as in the tuning of a piano. * * * ▪ music in music, a tuning system in which the octave is divided into 12 semitones of equal size. Because it enables keyboard instruments (keyboard … Universalium**equal temperament**— noun the division of the scale based on an octave that is divided into twelve exactly equal semitones equal temperament is the system commonly used in keyboard instruments • Hypernyms: ↑temperament … Useful english dictionary**53 equal temperament**— In music, 53 equal temperament, called 53 TET, 53 EDO, or 53 ET, is the tempered scale derived by dividing the octave into fifty three equally large steps. Each step represents a frequency ratio of 21/53, or 22.6415 cents, an interval sometimes… … Wikipedia**31 equal temperament**— In music, 31 equal temperament, which can be abbreviated 31 tET, 31 EDO, 31 ET, is the tempered scale derived by dividing the octave into 31 equal sized steps. Each step represents a frequency ratio of 21/31, or 38.71 cents.Division of the octave … Wikipedia**72 equal temperament**— In music, 72 equal temperament, called twelfth tone, 72 tet, 72 edo, or 72 et, is the tempered scale derived by dividing the octave into twelfth tones, or in other words 72 equally large steps. Each step represents a frequency ratio of 21/72, or… … Wikipedia**22 equal temperament**— In music, 22 equal temperament, called 22 tet, 22 edo, or 22 et, is the tempered scale derived by dividing the octave into 22 equally large steps. Each step represents a frequency ratio of 21/22, or 54.55 cents.The idea of dividing the octave… … Wikipedia**34 equal temperament**— In musical theory, 34 equal temperament, also referred to as 34 tet, 34 edo or 34 et, is the tempered tuning derived by dividing the octave into 34 equal sized steps. Each step represents a frequency ratio of 21/34, or 35.29 cents.Unlike… … Wikipedia