19 equal temperament

In music, 19 equal temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), 19-ED2 ("Equal Division of 2:1) 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 ¹⁹√2, or 63.16 cents (Play^ⓘ).

19 equal temperament keyboard^[2]

Figure 1: 19-TET on the syntonic temperament's tuning continuum at P5= 694.737 cents^[1]

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.

Joseph Yasser's 19 equal temperament keyboard layout^[3]

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 that the sharp below matches the flat immediately above it (enharmonicity).

The comparison between a standard 12 tone classical guitar and a 19 tone guitar design. This is the preliminary data that Arto Juhani Heino used to develop the "Artone 19" guitar design. The measurements are in millimeters.^[4]

History and use

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 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 discussed ⁠ 1 / 3 ⁠ comma meantone, 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 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.^[2]

The composer Joel 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 is 31 TET.^[5][6] Mandelbaum and Joseph Yasser have written music with 19 EDO.^[7] Easley Blackwood stated that 19 EDO makes possible "a substantial enrichment of the tonal repertoire".^[8]

Notation

Usual pitch notation, promoted by Easley Blackwood^[9] and Wesley Woolhouse,^[2] for 19 equal temperament: Intervals are notated similarly to the 12 TET intervals that approximate them. Aside from double sharps or double flats, only the note pairs E♯ & F♭ and B♯ & C♭ are enharmonic equivalents (modern sense).^[10]

Just intonation intervals approximated in 19 EDO

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♯ is enharmonic 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".

Interval size

play diatonic scale in 19 EDO^ⓘ, contrast with diatonic scale in 12 EDO^ⓘ, contrast with just diatonic scale^ⓘ

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 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
Greater septimal tritone, diminished fifth	10	0631.58	Playⓘ	10:70	0617.49	Playⓘ	+14.09
Lesser septimal 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.

Scale diagram

Circle of fifths in 19 tone equal temperament

Major chord on C in 19 equal temperament: All notes within 8 cents of just intonation (rather than 14 for 12 equal temperament). Play 19 ET^ⓘ, Play just^ⓘ, or Play 12 ET^ⓘ

Because 19 is a prime number, repeating any fixed interval in this tuning system cycles through all possible notes; just as one may cycle through 12-EDO on the circle of fifths, since a fifth is 7 semitones, and number 7 is coprime to 12.

Modes

Ionian mode (major scale)

Key signature	Scale							Number of sharps		Key signature	Scale							Number of flats
C major	C	D	E	F	G	A	B	0
G major	G	A	B	C	D	E	F♯	1
D major	D	E	F♯	G	A	B	C♯	2
A major	A	B	C♯	D	E	F♯	G♯	3
E major	E	F♯	G♯	A	B	C♯	D♯	4
B major	B	C♯	D♯	E	F♯	G♯	A♯	5		C major	C	D	E	F	G	A	B	14
F♯ major	F♯	G♯	A♯	B	C♯	D♯	E♯	6		G major	G	A	B	C	D	E	F♭	13
C♯ major	C♯	D♯	E♯	F♯	G♯	A♯	B♯	7		D major	D	E	F♭	G	A	B	C♭	12
G♯ major	G♯	A♯	B♯	C♯	D♯	E♯	F	8		A major	A	B	C♭	D	E	F♭	G♭	11
D♯ major	D♯	E♯	F	G♯	A♯	B♯	C	9		E major	E	F♭	G♭	A	B	C♭	D♭	10
A♯ major	A♯	B♯	C	D♯	E♯	F	G	10		B major	B	C♭	D♭	E	F♭	G♭	A♭	9
E♯ major	E♯	F	G	A♯	B♯	C	D	11		F♭ major	F♭	G♭	A♭	B	C♭	D♭	E♭	8
B♯ major	B♯	C	D	E♯	F	G	A	12		C♭ major	C♭	D♭	E♭	F♭	G♭	A♭	B♭	7
F major	F	G	A	B♯	C	D	E	13		G♭ major	G♭	A♭	B♭	C♭	D♭	E♭	F	6
C major	C	D	E	F	G	A	B	14		D♭ major	D♭	E♭	F	G♭	A♭	B♭	C	5
										A♭ major	A♭	B♭	C	D♭	E♭	F	G	4
										E♭ major	E♭	F	G	A♭	B♭	C	D	3
										B♭ major	B♭	C	D	E♭	F	G	A	2
										F major	F	G	A	B♭	C	D	E	1
										C major	C	D	E	F	G	A	B	0

Dorian mode

Key signature	Scale							Number of sharps		Key signature	Scale							Number of flats
D Dorian	D	E	F	G	A	B	C	0
A Dorian	A	B	C	D	E	F♯	G	1
E Dorian	E	F♯	G	A	B	C♯	D	2
B Dorian	B	C♯	D	E	F♯	G♯	A	3
F♯ Dorian	F♯	G♯	A	B	C♯	D♯	E	4
C♯ Dorian	C♯	D♯	E	F♯	G♯	A♯	B	5		D Dorian	D	E	F	G	A	B	C	14
G♯ Dorian	G♯	A♯	B	C♯	D♯	E♯	F♯	6		A Dorian	A	B	C	D	E	F♭	G	13
D♯ Dorian	D♯	E♯	F♯	G♯	A♯	B♯	C♯	7		E Dorian	E	F♭	G	A	B	C♭	D	12
A♯ Dorian	A♯	B♯	C♯	D♯	E♯	F	G♯	8		B Dorian	B	C♭	D	E	F♭	G♭	A	11
E♯ Dorian	E♯	F	G♯	A♯	B♯	C	D♯	9		F♭ Dorian	F♭	G♭	A	B	C♭	D♭	E	10
B♯ Dorian	B♯	C	D♯	E♯	F	G	A♯	10		C♭ Dorian	C♭	D♭	E	F♭	G♭	A♭	B	9
F Dorian	F	G	A♯	B♯	C	D	E♯	11		G♭ Dorian	G♭	A♭	B	C♭	D♭	E♭	F♭	8
C Dorian	C	D	E♯	F	G	A	B♯	12		D♭ Dorian	D♭	E♭	F♭	G♭	A♭	B♭	C♭	7
G Dorian	G	A	B♯	C	D	E	F	13		A♭ Dorian	A♭	B♭	C♭	D♭	E♭	F	G♭	6
D Dorian	D	E	F	G	A	B	C	14		E♭ Dorian	E♭	F	G♭	A♭	B♭	C	D♭	5
										B♭ Dorian	B♭	C	D♭	E♭	F	G	A♭	4
										F Dorian	F	G	A♭	B♭	C	D	E♭	3
										C Dorian	C	D	E♭	F	G	A	B♭	2
										G Dorian	G	A	B♭	C	D	E	F	1
										D Dorian	D	E	F	G	A	B	C	0

Phrygian mode

Key signature	Scale							Number of sharps		Key signature	Scale							Number of flats
E Phrygian	E	F	G	A	B	C	D	0
B Phrygian	B	C	D	E	F♯	G	A	1
F♯ Phrygian	F♯	G	A	B	C♯	D	E	2
C♯ Phrygian	C♯	D	E	F♯	G♯	A	B	3
G♯ Phrygian	G♯	A	B	C♯	D♯	E	F♯	4
D♯ Phrygian	D♯	E	F♯	G♯	A♯	B	C♯	5		E Phrygian	E	F	G	A	B	C	D	14
A♯ Phrygian	A♯	B	C♯	D♯	E♯	F♯	G♯	6		B Phrygian	B	C	D	E	F♭	G	A	13
E♯ Phrygian	E♯	F♯	G♯	A♯	B♯	C♯	D♯	7		F♭ Phrygian	F♭	G	A	B	C♭	D	E	12
B♯ Phrygian	B♯	C♯	D♯	E♯	F	G♯	A♯	8		C♭ Phrygian	C♭	D	E	F♭	G♭	A	B	11
F Phrygian	F	G♯	A♯	B♯	C	D♯	E♯	9		G♭ Phrygian	G♭	A	B	C♭	D♭	E	F♭	10
C Phrygian	C	D♯	E♯	F	G	A♯	B♯	10		D♭ Phrygian	D♭	E	F♭	G♭	A♭	B	C♭	9
G Phrygian	G	A♯	B♯	C	D	E♯	F	11		A♭ Phrygian	A♭	B	C♭	D♭	E♭	F♭	G♭	8
D Phrygian	D	E♯	F	G	A	B♯	C	12		E♭ Phrygian	E♭	F♭	G♭	A♭	B♭	C♭	D♭	7
A Phrygian	A	B♯	C	D	E	F	G	13		B♭ Phrygian	B♭	C♭	D♭	E♭	F	G♭	A♭	6
E Phrygian	E	F	G	A	B	C	D	14		F Phrygian	F	G♭	A♭	B♭	C	D♭	E♭	5
										C Phrygian	C	D♭	E♭	F	G	A♭	B♭	4
										G Phrygian	G	A♭	B♭	C	D	E♭	F	3
										D Phrygian	D	E♭	F	G	A	B♭	C	2
										A Phrygian	A	B♭	C	D	E	F	G	1
										E Phrygian	E	F	G	A	B	C	D	0

Lydian mode

Key signature	Scale							Number of sharps		Key signature	Scale							Number of flats
F Lydian	F	G	A	B	C	D	E	0
C Lydian	C	D	E	F♯	G	A	B	1
G Lydian	G	A	B	C♯	D	E	F♯	2
D Lydian	D	E	F♯	G♯	A	B	C♯	3
A Lydian	A	B	C♯	D♯	E	F♯	G♯	4
E Lydian	E	F♯	G♯	A♯	B	C♯	D♯	5		F Lydian	F	G	A	B	C	D	E	14
B Lydian	B	C♯	D♯	E♯	F♯	G♯	A♯	6		C Lydian	C	D	E	F♭	G	A	B	13
F♯ Lydian	F♯	G♯	A♯	B♯	C♯	D♯	E♯	7		G Lydian	G	A	B	C♭	D	E	F♭	12
C♯ Lydian	C♯	D♯	E♯	F	G♯	A♯	B♯	8		D Lydian	D	E	F♭	G♭	A	B	C♭	11
G♯ Lydian	G♯	A♯	B♯	C	D♯	E♯	F	9		A Lydian	A	B	C♭	D♭	E	F♭	G♭	10
D♯ Lydian	D♯	E♯	F	G	A♯	B♯	C	10		E Lydian	E	F♭	G♭	A♭	B	C♭	D♭	9
A♯ Lydian	A♯	B♯	C	D	E♯	F	G	11		B Lydian	B	C♭	D♭	E♭	F♭	G♭	A♭	8
E♯ Lydian	E♯	F	G	A	B♯	C	D	12		F♭ Lydian	F♭	G♭	A♭	B♭	C♭	D♭	E♭	7
B♯ Lydian	B♯	C	D	E	F	G	A	13		C♭ Lydian	C♭	D♭	E♭	F	G♭	A♭	B♭	6
F Lydian	F	G	A	B	C	D	E	14		G♭ Lydian	G♭	A♭	B♭	C	D♭	E♭	F	5
										D♭ Lydian	D♭	E♭	F	G	A♭	B♭	C	4
										A♭ Lydian	A♭	B♭	C	D	E♭	F	G	3
										E♭ Lydian	E♭	F	G	A	B♭	C	D	2
										B♭ Lydian	B♭	C	D	E	F	G	A	1
										F Lydian	F	G	A	B	C	D	E	0

Mixolydian mode

Key signature	Scale							Number of sharps		Key signature	Scale							Number of flats
G Mixolydian	G	A	B	C	D	E	F	0
D Mixolydian	D	E	F♯	G	A	B	C	1
A Mixolydian	A	B	C♯	D	E	F♯	G	2
E Mixolydian	E	F♯	G♯	A	B	C♯	D	3
B Mixolydian	B	C♯	D♯	E	F♯	G♯	A	4
F♯ Mixolydian	F♯	G♯	A♯	B	C♯	D♯	E	5		G Mixolydian	G	A	B	C	D	E	F	14
C♯ Mixolydian	C♯	D♯	E♯	F♯	G♯	A♯	B	6		D Mixolydian	D	E	F♭	G	A	B	C	13
G♯ Mixolydian	G♯	A♯	B♯	C♯	D♯	E♯	F♯	7		A Mixolydian	A	B	C♭	D	E	F♭	G	12
D♯ Mixolydian	D♯	E♯	F	G♯	A♯	B♯	C♯	8		E Mixolydian	E	F♭	G♭	A	B	C♭	D	11
A♯ Mixolydian	A♯	B♯	C	D♯	E♯	F	G♯	9		B Mixolydian	B	C♭	D♭	E	F♭	G♭	A	10
E♯ Mixolydian	E♯	F	G	A♯	B♯	C	D♯	10		F♭ Mixolydian	F♭	G♭	A♭	B	C♭	D♭	E	9
B♯ Mixolydian	B♯	C	D	E♯	F	G	A♯	11		C♭ Mixolydian	C♭	D♭	E♭	F♭	G♭	A♭	B	8
F Mixolydian	F	G	A	B♯	C	D	E♯	12		G♭ Mixolydian	G♭	A♭	B♭	C♭	D♭	E♭	F♭	7
C Mixolydian	C	D	E	F	G	A	B♯	13		D♭ Mixolydian	D♭	E♭	F	G♭	A♭	B♭	C♭	6
G Mixolydian	G	A	B	C	D	E	F	14		A♭ Mixolydian	A♭	B♭	C	D♭	E♭	F	G♭	5
										E♭ Mixolydian	E♭	F	G	A♭	B♭	C	D♭	4
										B♭ Mixolydian	B♭	C	D	E♭	F	G	A♭	3
										F Mixolydian	F	G	A	B♭	C	D	E♭	2
										C Mixolydian	C	D	E	F	G	A	B♭	1
										G Mixolydian	G	A	B	C	D	E	F	0

Aeolian mode (natural minor scale)

Key signature	Scale							Number of sharps		Key signature	Scale							Number of flats
A minor	A	B	C	D	E	F	G	0
E minor	E	F♯	G	A	B	C	D	1
B minor	B	C♯	D	E	F♯	G	A	2
F♯ minor	F♯	G♯	A	B	C♯	D	E	3
C♯ minor	C♯	D♯	E	F♯	G♯	A	B	4
G♯ minor	G♯	A♯	B	C♯	D♯	E	F♯	5		A minor	A	B	C	D	E	F	G	14
D♯ minor	D♯	E♯	F♯	G♯	A♯	B	C♯	6		E minor	E	F♭	G	A	B	C	D	13
A♯ minor	A♯	B♯	C♯	D♯	E♯	F♯	G♯	7		B minor	B	C♭	D	E	F♭	G	A	12
E♯ minor	E♯	F	G♯	A♯	B♯	C♯	D♯	8		F♭ minor	F♭	G♭	A	B	C♭	D	E	11
B♯ minor	B♯	C	D♯	E♯	F	G♯	A♯	9		C♭ minor	C♭	D♭	E	F♭	G♭	A	B	10
F minor	F	G	A♯	B♯	C	D♯	E♯	10		G♭ minor	G♭	A♭	B	C♭	D♭	E	F♭	9
C minor	C	D	E♯	F	G	A♯	B♯	11		D♭ minor	D♭	E♭	F♭	G♭	A♭	B	C♭	8
G minor	G	A	B♯	C	D	E♯	F	12		A♭ minor	A♭	B♭	C♭	D♭	E♭	F♭	G♭	7
D minor	D	E	F	G	A	B♯	C	13		E♭ minor	E♭	F	G♭	A♭	B♭	C♭	D♭	6
A minor	A	B	C	D	E	F	G	14		B♭ minor	B♭	C	D♭	E♭	F	G♭	A♭	5
										F minor	F	G	A♭	B♭	C	D♭	E♭	4
										C minor	C	D	E♭	F	G	A♭	B♭	3
										G minor	G	A	B♭	C	D	E♭	F	2
										D minor	D	E	F	G	A	B♭	C	1
										A minor	A	B	C	D	E	F	G	0

Locrian mode

Key signature	Scale							Number of sharps		Key signature	Scale							Number of flats
B Locrian	B	C	D	E	F	G	A	0
F♯ Locrian	F♯	G	A	B	C	D	E	1
C♯ Locrian	C♯	D	E	F♯	G	A	B	2
G♯ Locrian	G♯	A	B	C♯	D	E	F♯	3
D♯ Locrian	D♯	E	F♯	G♯	A	B	C♯	4
A♯ Locrian	A♯	B	C♯	D♯	E	F♯	G♯	5		B Locrian	B	C	D	E	F	G	A	14
E♯ Locrian	E♯	F♯	G♯	A♯	B	C♯	D♯	6		F♭ Locrian	F♭	G	A	B	C	D	E	13
B♯ Locrian	B♯	C♯	D♯	E♯	F♯	G♯	A♯	7		C♭ Locrian	C♭	D	E	F♭	G	A	B	12
F Locrian	F	G♯	A♯	B♯	C♯	D♯	E♯	8		G♭ Locrian	G♭	A	B	C♭	D	E	F♭	11
C Locrian	C	D♯	E♯	F	G♯	A♯	B♯	9		D♭ Locrian	D♭	E	F♭	G♭	A	B	C♭	10
G Locrian	G	A♯	B♯	C	D♯	E♯	F	10		A♭ Locrian	A♭	B	C♭	D♭	E	F♭	G♭	9
D Locrian	D	E♯	F	G	A♯	B♯	C	11		E♭ Locrian	E♭	F♭	G♭	A♭	B	C♭	D♭	8
A Locrian	A	B♯	C	D	E♯	F	G	12		B♭ Locrian	B♭	C♭	D♭	E♭	F♭	G♭	A♭	7
E Locrian	E	F	G	A	B♯	C	D	13		F Locrian	F	G♭	A♭	B♭	C♭	D♭	E♭	6
B Locrian	B	C	D	E	F	G	A	14		C Locrian	C	D♭	E♭	F	G♭	A♭	B♭	5
										G Locrian	G	A♭	B♭	C	D♭	E♭	F	4
										D Locrian	D	E♭	F	G	A♭	B♭	C	3
										A Locrian	A	B♭	C	D	E♭	F	G	2
										E Locrian	E	F	G	A	B♭	C	D	1
										B Locrian	B	C	D	E	F	G	A	0

See also

• 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 a quartertone 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

References

1. Milne, A.; Sethares, W. A.; Plamondon, J. (Winter 2007). "Isomorphic controllers and dynamic tuning: Invariant fingerings across a tuning continuum". Computer Music Journal. 31 (4): 15–32. doi:10.1162/comj.2007.31.4.15. S2CID 27906745.
2. Woolhouse, W.S.B. (1835). Essay on Musical Intervals, Harmonics, and the Temperament of the Musical Scale, &c. London, UK: J. Souter.
3. Joseph Yasser. "A Theory of Evolving Tonality". MusAnim.com.
4. Heino, Arto Juhani. "Artone 19 Guitar Design". Heino names the 19 note scale Parvatic.
5. Mandelbaum, M. Joel (1961). Multiple Division of the Octave and the Tonal Resources of 19 Tone Temperament (Thesis).
6. Gamer, C. (Spring 1967). "Some combinational resources of equal-tempered systems". Journal of Music Theory. 11 (1): 32–59. doi:10.2307/842948. JSTOR 842948.
7. Leedy, Douglas (1991). "A venerable temperament rediscovered". Perspectives of New Music. 29 (2): 205. doi:10.2307/833439. JSTOR 833439.

   cited by
   

   Skinner, Myles Leigh (2007). Toward a Quarter-Tone Syntax: Analyses of selected works by Blackwood, Haba, Ives, and Wyschnegradsky. p. 51, footnote 6. ISBN 9780542998478.
8. Skinner (2007), p. 76.
9. Skinner (2007), p. 52.
10. "19 EDO". TonalSoft.com.

Further reading

• 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 from the original (PDF) on 26 December 2013. Retrieved 12 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 – via U. Wisconsin.

External links

• 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. Retrieved 1 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 the 19 TET 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).