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.
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-et 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-et. 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 (summary of Woolhouse's essay).

Contents

Scale diagram

Interval size

As an approximation of other temperaments

External links

References

Scale diagram

The following are the 19 notes in the scale:

'Interval (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
'Note (cents)'	0		63		126		189		253		316		379		442		505		568		632		695		758		821		884		947		1011		1074		1137		1200

Note that B♯ equals C♭ and E♯ equals F♭.

Interval size

Here are the sizes of some common intervals:

interval name	size (steps)	size (cents)	just ratio	just (cents)
perfect fifth	11	695	3:2	702
perfect fourth	8	505	4:3	498
major third	6	379	5:4	386
minor third	5	316	6:5	316
whole tone, major tone	3	189	9:8	203
whole tone, minor tone	3	189	10:9	182
diatonic semitone, just	2	126	16:15	112
chromatic semitone, just	1	63	25:24	71

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).

External links

★ Bucht, Saku and Huovinen, Erkki, ''Perceived consonance of harmonic intervals in 19-tone equal temperament''

★ Darreg, Ivor, ''A Case for Nineteen''

★ Howe, Hubert S. Jr., ''19-Tone Theory and Applications''

★ Sethares, William A., ''Tunings for 19 Tone Equal Tempered Guitar''

★ Hair, Bailey, Morrison, Pearson and Parncutt, ''Rehearsing Microtonal Music: Grappling with Performance and Intonational Problems'' (project summary)

★ 19tet downloadable mp3s by Elaine Walker of Zia and D.D.T.

References

Levy, Kenneth J.,''Costeley's Chromatic Chanson'', Annales Musicologues:
Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.