19 equal temperament explained

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, or 63.16 cents .

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 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).

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 (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 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.

The composer Joel Mandelbaum wrote on the properties of the 19 EDO tuning and advocated for its use in his Ph.D. thesis:^[1] 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 .^{[1] [2]} Mandelbaum and Joseph Yasser have written music with 19 EDO.^[3] Easley Blackwood stated that 19 EDO makes possible "a substantial enrichment of the tonal repertoire".

Notation

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

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	1200		2:1	1200		0
Septimal major seventh	18	1136.84		27:14	1137.04		-0.20
Diminished octave	18	1136.84		48:25	1129.33		+7.51
Major seventh	17	1073.68		15:8	1088.27		-14.58
Minor seventh	16	1010.53		9:5	1017.60		-7.07
Harmonic minor seventh	15	947.37		7:4	968.83		-21.46
Septimal major sixth	15	947.37		12:7	933.13		+14.24
Major sixth	14	884.21		5:3	884.36		-0.15
Minor sixth	13	821.05		8:5	813.69		+7.37
Augmented fifth	12	757.89		25:16	772.63		-14.73
Septimal minor sixth	12	757.89		14:9	764.92		-7.02
Perfect fifth	11	694.74		3:2	701.96		-7.22
Greater tridecimal tritone	10	631.58		13:9	636.62		-5.04
Greater septimal tritone, diminished fifth	10	631.58		10:7	617.49		+14.09
Lesser septimal tritone, augmented fourth	9	568.42		7:5	582.51		-14.09
Lesser tridecimal tritone	9	568.42		18:13	563.38		+5.04
Perfect fourth	8	505.26		4:3	498.04		+7.22
Augmented third	7	442.11		125:96	456.99		-14.88
Tridecimal major third	7	442.11		13:10	454.12		-10.22
Septimal major third	7	442.11		9:7	435.08		+7.03
Major third	6	378.95		5:4	386.31		-7.36
Inverted 13th harmonic	6	378.95		16:13	359.47		+19.48
Minor third	5	315.79		6:5	315.64		+0.15
Septimal minor third	4	252.63		7:6	266.87		-14.24
Tridecimal tone	4	252.63		15:13	247.74		+4.89
Septimal whole tone	4	252.63		8:7	231.17		+21.46
Whole tone, major tone	3	189.47		9:8	203.91		-14.44
Whole tone, minor tone	3	189.47		10:9	182.40		+7.07
Greater tridecimal -tone	2	126.32		13:12	138.57		-12.26
Lesser tridecimal -tone	2	126.32		14:13	128.30		-1.98
Septimal diatonic semitone	2	126.32		15:14	119.44		+6.88
Diatonic semitone, just	2	126.32		16:15	111.73		+14.59
Septimal chromatic semitone	1	63.16		21:20	84.46		-21.31
Chromatic semitone, just	1	63.16		25:24	70.67		-7.51
Septimal third-tone	1	63.16		28:27	62.96		+0.20

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

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 does not divide 12 evenly (7 is coprime to 12).

Modes

Ionian mode (major scale)

Key signature								Number of sharps		Key signature								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								Number of sharps		Key signature								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								Number of sharps		Key signature								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								Number of sharps		Key signature								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								Number of sharps		Key signature								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								Number of sharps		Key signature								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									Number of sharps		Key signature								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

Further reading

• Bucht . Saku . Huovinen . Erkki . 15–18 April 2004 . Perceived consonance of harmonic intervals in 19 tone equal temperament . Parncutt, R. . Kessler, A. . Zimmer, F. . Conference on Interdisciplinary Musicology 2004 . Proceedings of the Conference on Interdisciplinary Musicology (CIM04) . Graz, Austria . 2014-03-12 . dead . uni-graz.at . https://web.archive.org/web/20131226014305/http://www.uni-graz.at/richard.parncutt/cim04/CIM04_paper_pdf/Bucht_Huovinen_CIM04_proceedings.pdf . 2013-12-26 . dmy-all . http://gewi.uni-graz.at/~cim04/.
• Web site: Howe . Hubert S. Jr. . 19 tone theory and applications . . . Queens, NY . academic pers. site .
• Book: Levy, Kenneth J. . 1955 . Costeley's Chromatic Chanson . Annales Musicologiques: Moyen-Age et Renaissance . III . 213–261.
• Sethares . W.A. . William Sethares . April 1991 . Tunings for 19 tone equal tempered guitar . . VI . 6 . U. Wisconsin.

External links

• Web site: Darreg . Ivor . A case for nineteen . Sonic Arts . tonalsoft.com .
• Ingrid . Pearson . Graham . Hair . Dougie . McGilvray . Nick . Bailey . Amanda . Morrison . Richard . Parncutt . 2014-09-19 . 2006 . Rehearsing microtonal music: Grappling with performance and intonational problems . Bailey, Nick . Microtonalism . n-ism.org . Project summary . 2024-04-01 . dmy-all .
• Elaine . Walker . Elaine Walker (composer) . 19 TET downloadable MP3 files . music . Zia and D.D.T. . ZiaSpace.com .
• Web site: The Music of Jeff Harrington . Parnasse.com . — Jeff Harrington is a composer who has written several pieces for piano in the tuning, and there are both scores and MP3's available for download on this site.
• Chris . Vaisvil . 10 December 2009 . GR-20 Hexaphonic 19 ET guitar improvisation . music .
• Fabio . Costa . 14 October 2018 . Meditation in 19 EDO . music .

Notes and References

1. Mandelbaum . M. Joel . Joel Mandelbaum . 1961 . Multiple Division of the Octave and the Tonal Resources of 19 Tone Temperament . Mandelbaum_1961.
2. Gamer . C. . Carlton Gamer . Spring 1967 . Some combinational resources of equal-tempered systems . . 11 . 1 . 32–59 . 10.2307/842948 . 842948.
3. Leedy . Douglas . 1991 . A venerable temperament rediscovered . . 29 . 2 . 205 . 10.2307/833439 . 833439.

   cited by
   Book: Skinner, Myles Leigh . 2007 . Toward a Quarter-Tone Syntax: Analyses of selected works by Blackwood, Haba, Ives, and Wyschnegradsky . 51, footnote 6 . 9780542998478.