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In music,19 equal temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), 19-ED2 ("Equal Division of 2:1) or 19ET, is thetempered scale derived by dividing the octave into 19 equal steps (equal frequency ratios). Each step represents a frequency ratio of¹⁹√2, or 63.16 cents (Play^ⓘ).

The fact that traditional western music maps unambiguously onto this scale (unless it presupposes 12-EDO enharmonic equivalences) makes it easier to perform such music in this tuning than in many other tunings.

19 EDO is the tuning of thesyntonic 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 anisomorphic keyboard, the fingering of music composed in 19 EDO is precisely the same as it is in any other syntonic tuning (such as12 EDO), so long as the notes are "spelled properly" – that is, with no assumption that the sharp below matches the flat immediately above it (enharmonicity).

Division of the octave into 19 equal-width steps arose naturally out of Renaissance music theory. The ratio of four minor thirds to an octave (⁠ 648 / 625⁠ or 62.565 cents – the"greater" diesis) was almost exactly a nineteenth of an octave. Interest in such a tuning system goes back to the 16th century, when composerGuillaume Costeley used it in his chansonSeigneur Dieu ta pitié of 1558. Costeley understood and desired the circulating aspect of this tuning.

In 1577, music theoristFrancisco de Salinas discussed⁠ 1 / 3⁠commameantone, in which the tempered perfect fifth is 694.786 cents. Salinas proposed tuning nineteen tones to the octave to this fifth, which falls within one cent of closing. The fifth of 19 EDO is 694.737 cents, which is less than a twentieth of a cent narrower, imperceptible and less than tuning error, so Salinas' suggestion is, for purposes relating to human hearing, functionally identical to 19 EDO.

In the 19th century, mathematician and music theoristWesley Woolhouse proposed it as a more practical alternative tomeantone temperaments he regarded as better, such as 50 EDO.^[2]

The composerJoel Mandelbaum wrote on the properties of the 19 EDO tuning and advocated for its use in his Ph.D. thesis:^[5]Mandelbaum 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 is31TET.^[5][6]Mandelbaum andJoseph Yasser have written music with 19 EDO.^[7]Easley Blackwood stated that 19 EDO makes possible "a substantial enrichment of the tonal repertoire".^[8]

19-EDO can be represented with the traditional letter names and system of sharps and flats simply by treating flats and sharps as distinct notes, as usual in standard musical practice; however, in 19-EDO the distinction is a real pitch difference, rather than a notational fiction. In 19-EDO only B♯ isenharmonic with C♭, and E♯ with F♭.

This article uses that re-adapted standard notation: Simply using conventionally enharmonic sharps and flats as distinct notes "as usual".

Here are the sizes of some common intervals and comparison with the ratios arising in theharmonic 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 used12 TET is 1.955 cents flat, the difference from the major third is 13.686 cents sharp, the minor third is 15.643 cents flat, and the (lost) harmonic minor seventh is 31.174 cents sharp.

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

Interval name	Size (steps)	Size (cents)	Midi	Just ratio	Just (cents)	Midi	Error (cents)
Octave	19	120000		2:1	120000		00
Septimal major seventh	18	1136.84		27:14	1137.04		−00.20
Diminished octave	18	1136.84		48:25	1129.33	Playⓘ	+07.51
Major seventh	17	1073.68		15:8	1088.27	Playⓘ	−14.58
Minor seventh	16	1010.53		9:5	1017.60	Playⓘ	−07.07
Harmonic minor seventh	15	0947.37		7:4	0968.83	Playⓘ	−21.46
Septimal major sixth	15	0947.37		12:7	0933.13	Playⓘ	+14.24
Major sixth	14	0884.21		5:3	0884.36	Playⓘ	−00.15
Minor sixth	13	0821.05		8:5	0813.69	Playⓘ	+07.37
Augmented fifth	12	0757.89		25:16	0772.63	Playⓘ	−14.73
Septimal minor sixth	12	0757.89		14:9	0764.92		−07.02
Perfect fifth	11	0694.74	Playⓘ	3:2	0701.96	Playⓘ	−07.22
Greater tridecimal tritone	10	0631.58		13:90	0636.62		−05.04
Greaterseptimal tritone, diminished fifth	10	0631.58	Playⓘ	10:70	0617.49	Playⓘ	+14.09
Lesserseptimal tritone, augmented fourth	09	0568.42	Playⓘ	7:5	0582.51		−14.09
Lesser tridecimal tritone	09	0568.42		18:13	0563.38		+05.04
Perfect fourth	08	0505.26	Playⓘ	4:3	0498.04	Playⓘ	+07.22
Augmented third	07	0442.11		125:96	0456.99	Playⓘ	−14.88
Tridecimal major third	07	0442.11		13:10	0454.12		−10.22
Septimal major third	07	0442.11	Playⓘ	9:7	0435.08	Playⓘ	+07.03
Major third	06	0378.95	Playⓘ	5:4	0386.31	Playⓘ	−07.36
Inverted 13th harmonic	06	0378.95		16:13	0359.47		+19.48
Minor third	05	0315.79	Playⓘ	6:5	0315.64	Playⓘ	+00.15
Septimal minor third	04	0252.63		7:6	0266.87	Playⓘ	−14.24
Tridecimal⁠ 5 / 4⁠ tone	04	0252.63		15:13	0247.74		+04.89
Septimal whole tone	04	0252.63	Playⓘ	8:7	0231.17	Playⓘ	+21.46
Whole tone,major tone	03	0189.47		9:8	0203.91	Playⓘ	−14.44
Whole tone,minor tone	03	0189.47	Playⓘ	10:90	0182.40	Playⓘ	+07.07
Greater tridecimal⁠ 2 / 3⁠-tone	02	0126.32		13:12	0138.57		−12.26
Lesser tridecimal⁠ 2 / 3⁠-tone	02	0126.32		14:13	0128.30		−01.98
Septimal diatonic semitone	02	0126.32		15:14	0119.44	Playⓘ	+06.88
Diatonic semitone,just	02	0126.32		16:15	0111.73	Playⓘ	+14.59
Septimal chromatic semitone	01	0063.16	Playⓘ	21:20	0084.46		−21.31
Chromatic semitone, just	01	0063.16		25:24	0070.67	Playⓘ	−07.51
Septimal third-tone	01	0063.16	Playⓘ	28:27	0062.96		+00.20

A possible variant of 19-ED2 is 93-ED30, i.e. the division of 30:1 in 93 equal steps, corresponding to a stretching of the octave by 27.58¢, which improves the approximation of most natural ratios.

Because 19 is aprime number, repeating any fixed interval in this tuning system cycles through all possible notes; just as one may cycle through 12-EDO on thecircle of fifths, since afifth is 7 semitones, and number 7 iscoprime to 12.

• Archicembalo, instrument with a double keyboard layout consisting of a 19 tone system close to 19tet in pitch with an additional 12 tone keyboard that is tuned approximately aquartertone in between the white keys of the 19 tone keyboard.
• Beta scale
• Elaine Walker (composer)
• Meantone temperament
• Musical temperament
• 23 tone equal temperament
• 31 tone equal temperament

• Bucht, Saku; Huovinen, Erkki (15–18 April 2004)."Perceived consonance of harmonic intervals in 19 tone equal temperament"(PDF). In Parncutt, R.; Kessler, A.; Zimmer, F. (eds.).Proceedings of the Conference on Interdisciplinary Musicology (CIM04).Conference on Interdisciplinary Musicology 2004. Graz, Austria. Archived fromthe original(PDF) on 26 December 2013. Retrieved12 March 2014 – via uni-graz.at.
• Howe, Hubert S. Jr."19 tone theory and applications" (online journal).Aaron Copland School of Music. Queens, NY:Queens College /City University of New York.
• Levy, Kenneth J. (1955).Costeley's Chromatic Chanson. Annales Musicologiques: Moyen-Age et Renaissance. Vol. III. pp. 213–261.
• Sethares, W.A. (April 1991)."Tunings for 19 tone equal tempered guitar".Experimental Musical Instruments. 6. Vol. VI – viaU. Wisconsin.

• Darreg, Ivor."A case for nineteen".tonalsoft.com. Sonic Arts.
• Pearson, Ingrid; Hair, Graham; McGilvray, Dougie; Bailey, Nick; Morrison, Amanda; Parncutt, Richard (19 September 2014) [2006]. Bailey, Nick (ed.).Rehearsing microtonal music: Grappling with performance and intonational problems. Project summary.n-ism.org (Report). Microtonalism. Retrieved1 April 2024.
• Walker, Elaine.19 TET downloadable MP3 files.ZiaSpace.com (music). Zia and D.D.T.
• "The Music of Jeff Harrington".Parnasse.com. — Jeff Harrington is a composer who has written several pieces for piano in the19TET tuning, and there are both scores and MP3's available for download on this site.
• Vaisvil, Chris (10 December 2009).GR-20 Hexaphonic 19 ET guitar improvisation (music).
• Costa, Fabio (14 October 2018).Meditation in 19 EDO (music).

• Equal temperaments
• Microtonality

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