19 equal temperament

In music, 19 equal temperament, called 19 TET, 19 EDO ("Equal Division of the Octave"), 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 192, or 63.16 cents (About this soundPlay ).

Figure 1: 19 TET on the syntonic temperament’s tuning continuum at P5= 694.737 cents, from (Milne et al. 2007).[1]
19 equal temperament keyboard, after Woolhouse (1835).[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.

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).

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

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 in effect proposed it. Salinas discussed ​13 comma meantone, in which the fifth is of size 694.786 cents. 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. 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 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 his Ph.D. thesis[5] on the properties of the 19 EDO tuning, and advocated for its use. In his thesis, he 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 the 31 tone equal temperament.[6] Mandelbaum and Joseph Yasser have written music with 19 EDO.[7] Easley Blackwood has stated that 19 EDO makes possible "a substantial enrichment of the tonal repertoire".[8]

NotationEdit

 
Usual notation, as by Easley Blackwood[9] and Wesley Woolhouse,[2] for 19 equal temperament: intervals are notated similarly to those they approximate and there are only two enharmonic equivalents without double sharps or flats (E/F and B/C).[10]  Play .

19-EDO can be represented with the traditional letter names and system of sharps and flats by treating flats and sharps as distinct notes; in 19-EDO only B is enharmonic with C, and E with F. This article will use that notation.

Interval sizeEdit

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 00 2:1 1200 00 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 ​54-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 ​23-tone 02 0126.32 13:12 0138.57 −12.26
Lesser tridecimal ​23-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

Scale diagramEdit

 
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).

ModesEdit

Ionian Mode (Major Scale)Edit

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 ModeEdit

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 ModeEdit

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 ModeEdit

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 ModeEdit

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)Edit

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 ModeEdit

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 alsoEdit

ReferencesEdit

  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.
  2. ^ a b c Woolhouse, W. S. B. (1835). Essay on Musical Intervals, Harmonics, and the Temperament of the Musical Scale, &c. London: 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. JSTOR 842948.
  7. ^ Myles Leigh Skinner (2007). Toward a Quarter-Tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky. p. 51 note 6. ISBN 9780542998478. who cites Leedy, Douglas (1991). "A Venerable Temperament Rediscovered". Perspectives of New Music. 29 (2): 205.
  8. ^ Skinner 2007, p.76.
  9. ^ Myles Leigh Skinner (2007). Toward a Quarter-Tone Syntax: Analyses of Selected Works by Blackwood, Haba, Ives, and Wyschnegradsky. p. 52. ISBN 9780542998478..
  10. ^ "19 EDO". TonalSoft.com.

Further readingEdit

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

External linksEdit