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# 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 21/19, or 63.16 cents. Because 19 is a prime number, 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).

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 theorist Francisco 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 the 31 equal temperament. [ [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] ] 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.

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 5:4 major 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 the septimal major third and septimal 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 the septimal major third and septimal minor third, and also distinguishing between this interval and the septimal whole tone. However, the 22-et does not have a close match for any whole 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 and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for meantone temperament. It is also a suitable for magic 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 a septimal 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

* [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.

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