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19 equal temperament
In music, 19 equal temperament, called 19-TET, 19-EDO ("Equal Division of the Octave"), or 19-ET, 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 (). Because 19 is a prime number, one can use any interval from this tuning system to cycle through all possible notes; just as one may cycle through 12-edo on the circle of fifths, the number 7 (of semitones in a fifth in 12-edo) being coprime to 12.
19-edo is the tuning of the syntonic temperament in which the tempered perfect fifth is equal to 694.737 cents, as shown in Figure 1 (look for the label "19-TET"). On an isomorphic keyboard, the fingering of music composed in 19-edo is precisely the same as it is in any other syntonic tuning (such as 12-edo), so long as the notes are spelled properly—that is, with no assumption of enharmonicity.
==History==
Division of the octave into 19 equal-width 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 nineteenth of an octave. Interest in such a tuning system goes back to the 16th 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-edo 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-edo. In the 19th century, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone temperaments he regarded as better, such as 50-edo.〔
The composer Joel Mandelbaum wrote his Ph.D. thesis (1961) on the properties of the 19-edo tuning, and advocated for its use. In his thesis, he argued that it is 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 approximating just intervals is the 31 equal temperament.〔(C. Gamer, ''Some Combinational Resources of Equal-Tempered Systems''. Journal of Music Theory, Vol. 11, No. 1 (Spring, 1967), pp. 32–59 )〕 Mandelbaum and Joseph Yasser have written music with 19-edo.〔Myles Leigh Skinner (2007). ''Toward a Quarter-tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky'', p.51n6. ISBN 9780542998478. Cites Leedy, Douglas (1991). "A Venerable Temperament Rediscovered", ''Persepctives of New Music'' 29/2, p.205.〕 Easley Blackwood has stated that 19-edo makes possible, "a substantial enrichment of the tonal repertoire."〔Skinner 2007, p.76.〕
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