Wikipedia

19 equal temperament

(Redirected from 19-tone 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ⓘ).

Figure 1: 19-TET on the syntonic temperament's tuning continuum at P5= 694.737 cents[1]
19 equal temperament keyboard[2]

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

edit

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

edit
 
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

edit
 
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

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

Modes

edit

Ionian mode (major scale)

edit
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

edit
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

edit
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

edit
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

edit
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)

edit
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

edit
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

edit

References

edit
  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. ^ a b c 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. ^ a b 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

edit
edit
Retrieved from "https://en.wikipedia.org/w/index.php?title=19_equal_temperament&oldid=1252959240"