19edo

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Prime factorization 19 (prime)
Step size 63.1579¢ 
Fifth 11\19 (694.737¢)
Semitones (A1:m2) 1:2 (63.16¢ : 126.3¢)
Consistency limit 9
Distinct consistency limit 5
Special properties
English Wikipedia has an article on:

19 equal divisions of the octave (abbreviated 19edo or 19ed2), also called 19-tone equal temperament (19tet) or 19 equal temperament (19et) when viewed under a regular temperament perspective, is the tuning system that divides the octave into 19 equal parts of about 63.2 ¢ each. Each step represents a frequency ratio of 21/19, or the 19th root of 2.

Theory

History

Interest in this tuning system goes back to the sixteenth 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, which he defined as dividing the just major second into three approximately equal parts. Costeley had other compositions that made use of intervals, such as the diminished third, which have a meaningful context in 19edo, but not in other tuning systems contemporary with the work.

In 1577 music theorist Francisco de Salinas proposed 13-comma meantone, in which the fifth is 694.786 cents; the fifth of 19edo is 694.737, which is only a twentieth of a cent flatter. Salinas suggested tuning nineteen tones to the octave to this tuning, which comes within less than one cent of closing exactly, so that his suggestion is effectively 19edo.

In 1835, mathematician and music theorist Wesley Woolhouse proposed it as a more practical alternative to meantone tunings he regarded as better, such as 50 equal temperament (summary of Woolhouse's essay).

As an approximation of other temperaments

19edo's most salient characteristic is that, having an almost just minor third and perfect fifths and major thirds about seven cents flat, it serves as a good tuning for meantone. It is also suitable for magic/muggles temperament, because five of its major thirds are equivalent to one of its twelfths. For all of these there are more optimal tunings: the fifth of 19edo is flatter than the usual for meantone, and 31edo is more optimal. Similarly, the generating interval of magic temperament is a major third, and again 19edo's is flatter; 41edo more closely matches it. It does make for a good tuning for muggles, which in 19edo is the same as magic. 19edo's 7-step supermajor third can be used for sensi, whose generator is a very sharp major third, two of which make an approximate 5/3 minor sixth, though 46edo is a better sensi tuning.

However, for all of these 19edo has the practical advantage of requiring fewer pitches, which makes it easier to implement in physical instruments, and many 19edo instruments have been built. 19et is in fact the second equal temperament, after 12et which is able to approximate 5-limit intervals and chords with tolerable accuracy, and is the fifth (after 12) zeta integral edo. It is less successful in the 7-limit (but still better than 12et), as it conflates the septimal subminor third (7/6) with the septimal whole tone (8/7). 19edo also has the advantage of being excellent for negri, keemun, godzilla, magic/muggles, and triton/liese, and fairly decent for sensi. Keemun and negri are of particular note for being very simple 7-limit temperaments, with their mos scales in 19edo offering a great abundance of septimal tetrads. The Graham complexity of a 7-limit tetrad is 6 for keemun, 7 for negri, 8 for godzilla, 10 for meantone, 11 for triton, 12 for magic/muggles and 13 for sensi.

Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. Practically 19edo can be used adaptively on instruments which allow you to bend notes up: by different amounts, the 3rd, 5th, 7th, and 13th harmonics are all tuned flat. This is in contrast to 12edo, where this is not possible since the 5 and 7 are not only much farther from just than they are in 19, but fairly sharp already. 19edo's negri, sensi and semaphore scales have many 13-limit chords. (You can think of the sensi[8] 3L 5s mos scale as 19edo's answer to the diminished scale. Both are made of two diminished seventh chords, but sensi[8] gives you additional ratios of 7 and 13.)

Another option would be to employ octave stretching; the closest local zeta peak to 19 occurs at 18.9481, which makes the octaves 1203.29 cents, and a step size of between 63.2 and 63.4 cents would be preferable in theory. Pianos are frequently tuned with stretched octaves anyway due to the slight inharmonicity inherent in their strings, which makes 19edo a promising option for pianos with split sharps. Octave stretching also means that an out-of-tune interval can be replaced with a compounded or inverted version of it which is near-just. For example, if we are using 49ed6 or 30ed3 (which tune 6:1 and 3:1 just and have octaves stretched by 2.8 and 4.57 cents, respectively), then we have near-just minor thirds (6:5), compound major thirds (as 5:1), and compound fifths (as 6:1), giving us versions of everything in the 5-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. A more extreme option would be 11edf, which has octaves stretched by 12.47 cents.

As a means of extending harmony

Because 19edo allows for more blended, consonant harmonies than 12edo does, it can be a much better candidate for using alternate forms of harmony such as quartal, secundal, and poly chords. William Lynch suggests the use of seventh chords of various types to be the fundamental sonorities with a triad deemed as incomplete. Higher extensions involving the 7th harmonic as well as other non diatonic chord extensions which tend to clash in 12edo blend much better in 19edo.

19edo's diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3 cents off 23/16.

In addition, Joseph Yasser talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19edo becomes akin to the pentatonic of western music; as it would sound to a future generation, ambiguous and not tonally fortified. As paraphrased "A system in which the undeniable laws of tonal gravity exist, yet in a much more complex tonal universe." Yasser believed that music would eventually move to a 19-tone system with a 12-note supra diatonic scale would become the standard. While this has yet to happen, Yasser's concept of supra-diatonicity is intriguing and worth exploring for those wanting to extend tonality without sounding too alien.

19edo also closely approximates most of the intervals of Bozuji tuning (a 21st century tuning based on Gioseffo Zarlino's approach to just intonation). with most of the adjacent diatonic diminished and augmented intervals of Bozuji tuning represented enharmonically by one interval in 19edo.

Due to the narrow whole tones and wide diatonic semitones, 19edo's diatonic scale tends to sound somewhat dull compared to 12edo, but the pentatonic scale is said by many to sound much more expressive owing to the significantly larger contrast between the narrow whole tone and wide minor third. While 12edo has an expressive diatonic and dull pentatonic, the reverse is true in 19. Pentatonicism thus becomes more important in 19edo, and one option is to use the pentatonic scale as a sort of "super-chord", with "chord progressions" being modulations between pentatonic subsets of the superdiatonic scale.

Prime harmonics

Approximation of prime harmonics in 19edo
Harmonic 2 3 5 7 11 13 17 19 23 29 31
Error Absolute (¢) +0.0 -7.2 -7.4 -21.5 +17.1 -19.5 +21.4 +18.3 +3.3 -19.1 -8.2
Relative (%) +0.0 -11.4 -11.7 -34.0 +27.1 -30.8 +33.8 +28.9 +5.2 -30.2 -13.0
Steps
(reduced)
19
(0)
30
(11)
44
(6)
53
(15)
66
(9)
70
(13)
78
(2)
81
(5)
86
(10)
92
(16)
94
(18)

Subsets and supersets

19edo is the 8th prime edo, following 17edo and preceding 23edo.

38edo, which doubles 19edo, provides an approximation of harmonic 11 that works well with the flat tendency of its 5-limit mapping. See undevigintone. 57edo effectively corrects the harmonic 7 to just, although it is 76edo that fits the best. See meanmag.

Intervals

Degree Cents Interval Region Approximated JI Intervals* Solfege Dodecatonic Notation SKULO Interval
0 0.00 Unison (prime) 1/1 Do P1 unison P1
1 63.16 Augmented unison 25/24, 28/27, 26/25 Di/Ro A1, m2 super unison, subminor second S1, sm2
2 126.32 Minor second 15/14, 16/15, 13/12, 14/13 Ra M2, m3 minor second m2
3 189.47 Major second 9/8, 10/9 Re M3 major second M2
4 252.63 Diminished third 7/6, 8/7, 15/13 Ri/Ma m4, A3 supermajor second, subminor third SM2, sm3
5 315.79 Minor third 6/5 Me M4, m5 minor third m3
6 378.95 Major third 5/4, 16/13, 56/45 Mi M5 major third M3
7 442.11 Augmented third 32/25, 9/7, 13/10 Mo/Fe A5, d6 supermajor third, sub fourth SM3, s4
8 505.26 Perfect fourth 4/3, 75/56 Fa P6 perfect fourth P4
9 568.42 Augmented fourth
(Small tritone)
25/18, 7/5, 18/13, 11/8 Fi A6, m7 augmented fourth A4
10 631.58 Diminished fifth
(Large tritone)
36/25, 10/7, 13/9, 16/11 Se M7, d8 diminished fifth d5
11 694.74 Perfect fifth 3/2, 112/75 So P8 perfect fifth P5
12 757.89 Augmented fifth 25/16, 14/9, 20/13 Si/Lo A8, m9 super fifth, subminor 6th S5, sm6
13 821.05 Minor sixth 8/5, 13/8, 45/28 Le M9, m10 minor sixth m6
14 884.21 Major sixth 5/3 La M10 major sixth M6
15 947.37 Diminished seventh 7/4, 12/7, 26/15 Li/Ta m11, A10 supermajor sixth, subminor seventh SM6, sm7
16 1010.53 Minor seventh 9/5, 16/9 Te M11, m12 minor seventh m7
17 1073.68 Major seventh 15/8, 13/7, 28/15, 24/13 Ti M12 major seventh M7
18 1136.84 Augmented seventh 48/25, 27/14, 25/13 To/Da A12, d13 supermajor seventh, sub octave SM7, s8
19 1200.00 Octave 2/1 Do P13 octave P8

* based on treating 19edo as a 2.3.5.7.13 subgroup temperament; other approaches are possible.

Interval quality and chord names in color notation

Using color notation, qualities can be loosely associated with colors:

Quality Color Name Monzo Format Examples
diminished zo (a, b, 0, 1) 7/6, 7/4
minor fourthward wa (a, b), b < -1 32/27, 16/9
gu (a, b, -1) 6/5, 9/5
major yo (a, b, 1) 5/4, 5/3
fifthward wa (a, b), b > 1 9/8, 27/16
augmented ru (a, b, 0, -1) 9/7, 12/7

Key signatures are the same, but with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of B𝄫 would have double-flats on B and E, and flats on C, D, F, G, and A. Thinking of rewriting this key as A♯ might seem better, but then the key signature would contain double-sharps on C, F, and G, and sharps on A, B, D, and E, which is actually worse.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths. Here are the zo, gu, yo and ru triads:

Color of the 3rd JI Chord Edosteps Notes of C Chord Written Name Spoken Name
zo 6:7:9 0–4–11 C–E𝄫–G Cm(♭3), Cmin(♭3), C(d3) C subminor, C minor flat-three, C diminished-three
gu 10:12:15 0–5–11 C–E♭–G Cm, Cmin C minor
yo 4:5:6 0–6–11 C–E–G C, Cmaj C, C major
ru 14:18:21 0–7–11 C–E♯–G C(♯3), Cmaj(♯3), C(A3) C supermajor, C major sharp-three, C augmented-three
4:5:6:7 0–6–11–15 C–E–G–B𝄫 C(h7), Cadd(d7), Cmaj(add(d7)) C harmonic 7, C (major) add dim-seven
1/(4:5:6:7)
= 1:6/5:3/2:12/7
0–5–11–15 C–E♭–G–A♯ Cm(♯6), Cm(A6), Cm(add(♯6)), Cm(add(A6)) C minor (add) sharp-six, C minor (add) aug-six

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo tends to conflate zo and ru ratios.

For a more complete list, see 19edo Chord Names and Ups and downs notation #Chords and Chord Progressions.

Notation

Standard 12edo notation can be used, whether it is staff notation (with five lines), letter chain-of-fifths notation (with standard accidentals), solfege, or sargam. Note that D# and Eb are two different notes.

Notation of 19edo
Degree Cents Standard Notation
Diatonic Interval Names Note Names
on D
0 0.00 Perfect unison (P1) D
1 63.16 Augmented unison (A1)
Diminished second (d2)
D#
Ebb
2 126.32 Doubly augmented unison (AA1)
Minor second (m2)
Dx
Eb
3 189.47 Major second (M2)
Doubly diminished third (dd3)
E
Fbb
4 252.63 Augmented second (A2)
Diminished third (d3)
E#
Fb
5 315.79 Doubly augmented second (AA2)
Minor third (m3)
Ex
F
6 378.95 Major third (M3)
Doubly diminished fourth (dd4)
F#
Gbb
7 442.11 Augmented third (A3)
Diminished fourth (d4)
Fx
Gb
8 505.26 Perfect fourth (P4) G
9 568.42 Augmented fourth (A4)
Doubly diminished fifth (dd5)
G#
Abb
10 631.58 Doubly augmented fourth (AA4)
Diminished fifth (d5)
Gx
Ab
11 694.74 Perfect fifth (P5) A
12 757.89 Augmented fifth (A5)
Diminished sixth (d6)
A#
Bbb
13 821.05 Doubly augmented fifth (AA5)
Minor sixth (m6)
Ax
Bb
14 884.21 Major sixth (M6)
Doubly diminished seventh (dd7)
B
Cbb
15 947.37 Augmented sixth (A6)
Diminished seventh (d7)
B#
Cb
16 1010.53 Doubly augmented sixth (AA6)
Minor seventh (m7)
Bx
C
17 1073.68 Major seventh (M7)
Doubly diminished octave (dd8)
C#
Dbb
18 1136.84 Augmented seventh (A7)
Diminished octave (d8)
Cx
Db
19 1200.00 Perfect octave (P8) D

In 19edo:

  • ups and downs notation is identical to standard notation;
  • mixed sagittal notation is identical to standard notation, but pure sagittal notation exchanges sharps (#) and flats (b) for sagittal sharp (Sagittal sharp.png) and sagittal flat (Sagittal flat.png) respectively.

Approximation to JI

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Selected 19-limit intervals approximated in 19edo

Interval mappings

The following tables show how 15-odd-limit intervals are represented in 19edo. Prime harmonics are in bold; inconsistent intervals are in italics.

15-odd-limit intervals in 19edo (direct approximation, even if inconsistent)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
5/3, 6/5 0.148 0.2
13/7, 14/13 1.982 3.1
15/13, 26/15 4.891 7.7
13/9, 18/13 5.039 8.0
15/14, 28/15 6.873 10.9
9/7, 14/9 7.021 11.1
9/5, 10/9 7.070 11.2
3/2, 4/3 7.218 11.4
5/4, 8/5 7.366 11.7
13/10, 20/13 12.109 19.2
13/12, 24/13 12.257 19.4
7/5, 10/7 14.091 22.3
7/6, 12/7 14.239 22.5
9/8, 16/9 14.436 22.9
15/8, 16/15 14.585 23.1
11/8, 16/11 17.103 27.1
13/8, 16/13 19.475 30.8
7/4, 8/7 21.457 34.0
11/6, 12/11 24.321 38.5
11/10, 20/11 24.469 38.7
11/7, 14/11 24.597 38.9
13/11, 22/13 26.580 42.1
15/11, 22/15 31.470 49.8
11/9, 18/11 31.539 49.9
15-odd-limit intervals in 19edo (patent val mapping)
Interval and complement Error (abs, ¢) Error (rel, %)
1/1, 2/1 0.000 0.0
5/3, 6/5 0.148 0.2
13/7, 14/13 1.982 3.1
15/13, 26/15 4.891 7.7
13/9, 18/13 5.039 8.0
15/14, 28/15 6.873 10.9
9/7, 14/9 7.021 11.1
9/5, 10/9 7.070 11.2
3/2, 4/3 7.218 11.4
5/4, 8/5 7.366 11.7
13/10, 20/13 12.109 19.2
13/12, 24/13 12.257 19.4
7/5, 10/7 14.091 22.3
7/6, 12/7 14.239 22.5
9/8, 16/9 14.436 22.9
15/8, 16/15 14.585 23.1
11/8, 16/11 17.103 27.1
13/8, 16/13 19.475 30.8
7/4, 8/7 21.457 34.0
11/6, 12/11 24.321 38.5
11/10, 20/11 24.469 38.7
11/9, 18/11 31.539 49.9
15/11, 22/15 31.688 50.2
13/11, 22/13 36.578 57.9
11/7, 14/11 38.561 61.1

Regular temperament properties

Subgroup Comma List Mapping Optimal
8ve Stretch (¢)
Tuning Error
Absolute (¢) Relative (%)
2.3 [-30 19 [19 30]] +2.28 2.28 3.61
2.3.5 81/80, 3125/3072 [19 30 44]] +2.58 1.91 3.02
2.3.5.7 49/48, 81/80, 126/125 [19 30 44 53]] +3.85 2.76 4.35
2.3.5.7.13 49/48, 65/64, 81/80, 91/90 [19 30 44 53 70]] +4.14 2.53 3.99
2.3.5.7.13.23 49/48, 65/64, 70/69, 81/80, 91/90 [19 30 44 53 70 86]] +3.32 2.93 4.64

19et is lower in relative error than any previous equal temperaments in the 5-, 7-, 13-, 17-, and 19-limit – both 19 and 19e val achieve this in the case of 13-limit, 19eg val in the 17-limit, and 19egh val in the 19-limit. The next equal temperaments doing better in those subgroups are 34, 31, 27e, 22, and 26, respectively.

19et is prominent in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is 53.

Uniform maps

13-limit uniform maps between 18.5 and 19.5
Min. size Max. size Wart notation Map
18.5000 18.5113 19bbccddeeeffff 19 29 43 52 64 68]
18.5113 18.6124 19bbccddeeeff 19 29 43 52 64 69]
18.6124 18.6447 19ccddeeeff 19 30 43 52 64 69]
18.6447 18.7009 19ccddeff 19 30 43 52 65 69]
18.7009 18.7344 19cceff 19 30 43 53 65 69]
18.7344 18.7816 19eff 19 30 44 53 65 69]
18.7816 18.9337 19e 19 30 44 53 65 70]
18.9337 19.0518 19 19 30 44 53 66 70]
19.0518 19.0571 19f 19 30 44 53 66 71]
19.0571 19.1651 19df 19 30 44 54 66 71]
19.1651 19.2228 19cdf 19 30 45 54 66 71]
19.2228 19.2434 19cdeef 19 30 45 54 67 71]
19.2434 19.3220 19bcdeef 19 31 45 54 67 71]
19.3220 19.4133 19bcdeefff 19 31 45 54 67 72]
19.4133 19.5000 19bcdddeefff 19 31 45 55 67 72]

Commas

19et tempers out the following commas. (Note: This assumes the val 19 30 44 53 66 70].)

Prime
Limit
Ratio[1] Monzo Cents Color Name Name
3 (20 digits) [-30 19 137.14 Trilawa 19-comma
5 16875/16384 [-14 3 4 51.12 Laquadyo Negri comma
5 (14 digits) [-2 13 -8 34.91 Laquadbigu Unicorn comma
5 3125/3072 [-10 -1 5 29.61 Laquinyo Magic comma
5 81/80 [-4 4 -1 21.51 Gu Syntonic comma
5 78732/78125 [2 9 -7 13.40 Sepgu Sensipent comma
5 15625/15552 [-6 -5 6 8.11 Tribiyo Kleisma
5 (20 digits) [8 14 -13 5.29 Thegu Parakleisma
5 (28 digits) [-14 -19 19 2.82 Neyo Enneadeca
7 59049/57344 [-13 10 0 -1 50.72 Laru Harrison's comma
7 1029/1000 [-3 1 -3 3 49.49 Trizogu Keega
7 525/512 [-9 1 2 1 43.41 Lazoyoyo Avicennma
7 49/48 [-4 -1 0 2 35.70 Zozo Slendro diesis
7 3645/3584 [-9 6 1 -1 29.22 Laruyo Schismean comma
7 686/675 [1 -3 -2 3 27.99 Trizo-agugu Senga
7 875/864 [-5 -3 3 1 21.90 Zotrigu Keema
7 245/243 [0 -5 1 2 14.19 Zozoyo Sensamagic comma
7 126/125 [1 2 -3 1 13.79 Zotrigu Starling comma
7 225/224 [-5 2 2 -1 7.71 Ruyoyo Marvel comma
7 19683/19600 [-4 9 -2 -2 7.32 Labirugu Cataharry comma
7 10976/10935 [5 -7 -1 3 6.48 Satrizo-agu Hemimage comma
7 3136/3125 [6 0 -5 2 6.08 Zozoquingu Hemimean comma
7 (12 digits) [-11 2 7 -3 1.63 Latriru-asepyo Meter comma
7 4375/4374 [-1 -7 4 1 0.40 Zoquadyo Ragisma
11 45/44 [-2 2 1 0 -1 38.91 Luyo Undecimal fifth tone
11 56/55 [3 0 -1 1 -1 31.19 Luzogu Undecimal tritonic comma
11 100/99 [2 -2 2 0 -1 17.40 Luyoyo Ptolemisma
11 896/891 [7 -4 0 1 -1 9.69 Saluzo Pentacircle comma
11 65536/65219 [16 0 0 -2 -3 8.39 Satrilu-aruru Orgonisma
11 385/384 [-7 -1 1 1 1 4.50 Lozoyo Keenanisma
11 540/539 [2 3 1 -2 -1 3.21 Lururuyo Swetisma
13 39/38 [-1 1 0 0 0 1 0 -1 44.97 Nutho Undevicesimal two-ninth tone
13 65/64 [-6 0 1 0 0 1 26.84 Thoyo Wilsorma
13 343/338 [-1 0 0 3 0 -2 25.42 Thuthutrizo
13 91/90 [-1 -2 -1 1 0 1 19.13 Thozogu Superleap comma, biome comma
13 676/675 [2 -3 -2 0 0 2 2.56 Bithogu Island comma
13 1001/1000 [-3 0 -3 1 1 1 1.73 Tholozotrigu Fairytale comma, sinbadma
23 2187/2116 [-2 7 0 0 0 0 0 0 -2 57.14 Labitwethu Lipsett comma
23 70/69 [1 -1 1 1 0 0 0 0 - 24.91 Twethuzoyo Small vicesimotertial eighth tone
23 256/253 [8 0 0 0 -1 0 0 0 -1 20.41 Twethulu 253rd subharmonic
23 161/160 [-5 0 -1 1 0 0 0 0 1 10.79 Twethozogu Major kirnbergisma
23 208/207 [4 -2 0 0 0 1 0 0 -1 8.34 Twethutho Vicetone comma
23 529/528 [-4 -1 0 0 -1 0 0 0 2 3.28 Bitwetho-alu Preziosisma
23 576/575 [6 2 -2 0 0 0 0 0 -1 3.01 Twethugugu Worcester comma
23 1288/1287 [3 -2 0 1 -1 -1 0 0 1 1.34 Twethothuluzo Triaphonisma
  1. Ratios longer than 10 digits are presented by placeholders with informative hints

Linear temperaments

Since 19 is prime, all rank-2 temperaments in 19edo have one period per octave (i.e. are linear). Therefore you can make a correspondence between intervals and the linear temperaments they generate.

Degree Cents Interval MOSes Temperaments
1 63.16 A1, d2 Unicorn / rhinocerus
2 126.32 m2 1L 8s, 9L 1s Negri
3 189.47 M2 1L 5s, 6L 1s, 6L 7s Deutone
Spell
4 252.63 A2, d3 1L 3s, 4L 1s,
5L 4s, 5L 9s
Godzilla
5 315.79 m3 3L 1s, 4L 3s,
4L 7s, 4L 11s
Cata / keemun
6 378.95 M3 3L 1s, 3L 4s, 3L 7s,
3L 10s, 3L 13s
Magic / muggles
7 442.11 A3, d4 3L 2s, 3L 5s, 8L 3s Sensi
8 505.26 P4 2L 3s, 5L 2s, 7L 5s Meantone / flattone
9 568.42 A4 2L 3s, 2L 5s, 2L 7s,
2L 9s, 2L 11s, 2L 13s,
2L 15s
Liese / pycnic
Triton

Scales

MOS scales

Octave-equivalent mosses

Other scales

Instruments

19 note per octave Ibanez conversion by Brad Smith (Indianapolis)
19edo 5 string Bass 34"-37" scale length
19edo bass conversion by Ron Sword

Music

See also: Category:19edo tracks
XA 19-ET Index
A number of compositions that were perfomed at the midwestmicrofest concert in 2007[dead link]

See also

References

  • Bucht, Saku and Huovinen, Erkki, Perceived consonance of harmonic intervals in 19-tone equal temperament, CIM04_proceedings.
  • Levy, Kenneth J., Costeley's Chromatic Chanson, Annales Musicologues: Moyen-Age et Renaissance, Tome III (1955), pp. 213-261.

Further reading

External links