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Theory


In music, 19 equal temperament, called 19-TET, 19-EDO, or 19-ET, is the scale derived by dividing the octave into 19 equally large steps. Each step represents a frequency ratio of the 19th root of 2, or 63.16 cents. It is the 8th prime edo, following 17edo and coming before 23edo.

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; in 1577 music theorist Francisco de Salinas in effect proposed it. Salinas discussed 1/3-comma meantone, in which the fifth is of size 694.786 cents; the fifth of 19-et is 694.737, which is only a twentieth of a cent flatter. 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-et. In the nineteenth century 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


The most salient characteristic of 19-et is that, having an almost just minor third and perfect fifths and major thirds about seven cents narrow, it serves as a good tuning for meantone temperament. It is also a suitable for magic/muggles temperament, because five of its major thirds are equivalent to one of its twelfths. For both of these there are more optimal tunings: the fifth of 19-et is flatter than the usual for meantone, and a more accurate approximation is 31 equal temperament. Similarly, the generating interval of magic temperament is a major third, and again 19-et's is flatter; 41 equal temperament more closely matches it. It does make for a good tuning for muggles, which in 19et is the same as magic.

However, for all of these 19-et has the practical advantage of requiring fewer pitches, which makes physical realizations of it easier to build. (Many 19-et instruments have been built.) 19-et is in fact the second equal temperament, after 12-et which is able to deal with 5-limit music in a tolerable manner, and is the fifth (after 12) zeta integral edo. It is less successful with 7-limit (but still better than 12-et), as it eliminates the distinction between a septimal minor third (7/6), and a septimal whole tone (8/7). 19-EDO 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 19-EDO 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 13-limit is represented relatively well, and practically 19-edo 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. The same cannot be said of 12edo, in which the 5th and 7th are - not only farther than they are in 19, but fairly sharp already.

Another option would be to use a stretched octave; the zeta function-optimal tuning has an octave of roughly 1203 cents. Stringed instruments, in particular the piano, are frequently tuned with stretched octaves anyway due to the inharmonicity inherent in strings, which makes 19edo a promising option for them. 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 instance, if we're using 93ed30 (a variant of 19edo in which 30:1 is just), 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.

As a means of extending harmony

Because 19 EDO allows for more blended, consonant harmonies than 12 EDO 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 12 EDO blend much better in 19 EDO.

In addition, Joseph Yasser talks about the idea of a 12 tone supra diatonic scale where the 7 tone major scale in 19 EDO 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 tonlaity without sounding too alien.

The narrow whole tones and wide diatonic semitones of 19edo give the diatonic scale a somewhat duller quality, but has the opposite effect on the pentatonic scale, which becomes much more expressive owing to the 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.


Intervals and linear temperaments

List of 19et rank two temperaments by badness
List of 19et rank two temperaments by complexity
List of edo-distinct 19et rank two temperaments

Since 19 is prime, all rank two temperaments in 19edo have one period per octave. Therefore you can make a correspondence between intervals and the linear temperaments they generate.
Degrees of 19edo
Solfege
Diatonic Category
Dodecatonic category
Cents
Ratios*
Generator for
0
do
P1
P1
0
1/1

1
di
A1, d2
A1, m2
63.1579
25/24, 21/20, 28/27, 26/25, 27/26
Unicorn/rhinocerus
2
ra
m2
M2, m3
126.326
15/14, 16/15, 13/12, 14/13
Negri
3
re
M2
M3
189.474
9/8, 10/9
Deutone (2-meantone) / spell
4
ri/ma
A2, d3
m4, a3
252.632
7/6, 8/7, 15/13
Godzilla
5
me
m3
M4, m5
315.789
6/5, 25/21
Kleismic (hanson, keemun, catakleismic)
6
mi
M3
M5
378.947
5/4, 16/13, 26/21
Magic/charisma/glamour
7
mo
A3, d4
A5, d6
442.105
32/25, 9/7, 13/10
Sensi
8
fa
P4
P6
505.263
4/3
Meantone/flattone/meanenneadecal/meanpop
9
fi
A4
A6, m7
568.421
25/18, 7/5, 18/13
Liese/triton/lisa
10
se
d5
M7, d8
631.579
36/25, 10/7, 13/9
Liese/triton/lisa
11
sol
P5
P8
694.737
3/2
Meantone
12
lo
A5, d6
A8, m9
757.895
25/16, 14/9, 20/13
Sensi
13
le
m6
M9, m10
821.053
8/5, 13/8, 21/13
Magic
14
la
M6
M10
884.210
5/3, 42/25
Kleismic (hanson, keemun, catakleismic)
15
li/ta
A6, d7
m11, A10
947.368
7/4, 12/7, 26/15
Godzilla
16
te
m7
M11, m12
1010.53
9/5, 16/9
Deutone / spell
17
ti
M7
M12
1073.68
15/8, 13/7, 28/15, 24/13
Negri
18
da
A7, d8
A12, d13
1136.84
48/25, 40/21, 27/14, 25/13, 52/27
Unicorn/rhinocerus
19
do
P8
P13
1200
2/1


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

Chord Names


Chords can be named using conventional methods, expanded to include augmented and diminished 2nd, 3rds, 6ths and 7ths.

0-6-11 = C E G = C = C major (approximate 4:5:6)
0-5-11 = C Eb G = Cm = C minor (approximate 10:12:15)
0-4-11 = C Ebb G = C(b3) or C(d3) = C flat-three or C dim-three (6:7:9)
0-7-11 = C E# G = C(#3) or C(A3) = C sharp-three or C aug-three (14:18:21)

0-6-11-15 = C E G Bbb = C(bb7) or C(d7) = C double-flat-seven or C major dim-seven or C add dim-seven = 4:5:6:7
0-5-11-15 = C Eb G A# is Cm(#6) or Cm(A6) = C minor sharp-six or C minor aug-six = 1/(4:5:6:7) = 1/1 - 6/5 - 3/2 - 12/7

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 otonal and utonal septimal ratios.

For a more complete list, see Ups and Downs Notation - Chord names in other EDOs.

Selected just intervals by error

The following table shows how some prominent just intervals are represented within 19edo (ordered by absolute error).
Interval, complement
Error (abs., in cents)
6/5, 5/3
0.148
14/13, 13/7
1.982
15/13, 26/15
4.891
18/13, 13/9
5.039
15/14, 28/15
6.873
9/7, 14/9
7.021
10/9, 9/5
7.070
4/3, 3/2
7.218
5/4, 8/5
7.366
13/10, 20/13
12.109
13/12, 24/13
12.257
7/5, 10/7
14.091
7/6, 12/7
14.239
9/8, 16/9
14.436
16/15, 15/8
14.585
11/8, 16/11
17.103
16/13, 13/8
19.475
8/7, 7/4
21.457
12/11, 11/6
24.321
11/10, 20/11
24.469
14/11, 11/7
24.597
13/11, 22/13
26.580
15/11, 22/15
31.470
11/9, 18/11
31.539

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Commas

19 EDO tempers out the following commas. (Note: This assumes the val < 19 30 44 53 66 70 |.)
Comma
Monzo
Value (Cents)
Name(s)
16875/16384
| -14 3 4 >
51.12
Negri Comma, Double Augmentation Diesis
3125/3072
| -10 -1 5 >
29.61
Small Diesis, Magic Comma
81/80
| -4 4 -1 >
21.51
Syntonic Comma, Didymos Comma, Meantone Comma
78732/78125
| 2 9 -7 >
13.40
Medium Semicomma, Sensipent Comma
15625/15552
| -6 -5 6 >
8.11
Kleisma, Semicomma Majeur

| 8 14 -13 >
5.29
Parakleisma

| -14 -19 19 >
2.82
Enneadeca, 19-Tone-Comma
1029/1000
| -3 1 -3 3 >
49.49
Keega
525/512
| -9 1 2 1 >
43.41
Avicennma, Avicenna's Enharmonic Diesis
49/48
| -4 -1 0 2 >
35.70
Slendro Diesis
686/675
| 1 -3 -2 3 >
27.99
Senga
875/864
| -5 -3 3 1 >
21.90
Keema
245/243
| 0 -5 1 2 >
14.19
Sensamagic
126/125
| 1 2 -3 1 >
13.79
Septimal Semicomma, Starling Comma
225/224
| -5 2 2 -1 >
7.71
Septimal Kleisma, Marvel Comma
19683/19600
| -4 9 -2 -2 >
7.32
Cataharry
10976/10935
| 5 -7 -1 3 >
6.48
Hemimage
3136/3125
| 6 0 -5 2 >
6.08
Hemimean

| -11 2 7 -3 >
1.63
Meter
4375/4374
| -1 -7 4 1 >
0.40
Ragisma
100/99
| 2 -2 2 0 -1 >
17.40
Ptolemisma
896/891
| 7 -4 0 1 -1 >
9.69
Pentacircle
65536/65219
| 16 0 0 -2 -3 >
8.39
Orgonisma
385/384
| -7 -1 1 1 1 >
4.50
Keenanisma
540/539
| 2 3 1 -2 -1 >
3.21
Swetisma
91/90
| -1 -2 -1 1 0 1 >
19.13
Superleap
676/675
| 2 -3 -2 0 0 2 >
2.56
Parizeksma


See also


Articles


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.

Photos

19 note per octave Ibanez conversion by Brad Smith (Indianapolis)
external image Media-Card_BlackBerry_pictures_IMG00145-1024x768.jpg


Compositions

XA 19-ET Index

The Juggler by Aaron Krister Johnson
Foum play by Jacob Barton
Sand by Christopher Bailey
Walking Down the Hillside at Cortona, and Seeing its Towers Rise Before Me by Christopher Bailey
Ditty by Christopher Bailey
Seigneur Dieu ta pitié by Guillaume Costeley
Prelude 2 for 19 tone guitar by Ivor Darreg
Sympathetic Metaphor play by William Sethares Permalink
Truth on a bus play by William Sethares Permalink
Rondo in 19ET by Aaron Andrew Hunt
Citified Notions and
Limp Off to School by John Starrett.
The Light Of My Betelgeuse by Mykhaylo Khramov
Undines, Sylphs, Gnomes, and Salamanders by Jon Lyle Smith
Another Aire For Lute by Jon Lyle Smith
A number of compositions that were perfomed at the midwestmicrofest concert in 2007
Fanfare in 19-note Equal Tuning by Easley Blackwood
Zvíře by Milan Guštar
19tet downloadable mp3s by ZIA, Elaine Walker and D.D.T.
Comets Over Flatland 14 by Randy Winchester
Forgetting Even Her Beauty blog play Forgetting Even Her Beauty by Chris Vaisvil
19 Black Hawks for Osama blog play video for 19 Black Hawks for Osama by Chris Vaisvil
Summer Song blog play Summer Song by Trevor (The TwoRegs) and Norm Harris and Chris Vaisvil
19 ImprovFridays blog play video of performance of 19 ImprovFridays by Chris Vaisvil
The World has Changed blog play The World has Changed by Chris Vaisvil
jjj play by Chris Vaisvil
Now listen! Pitch! play by Omega9
Cordas (19-edo version) play by Omega9
A Piece in 19edo by Omega9
A Piece in 19edo (ver.3) play by Omega9
Bach’s Prelude number 24 from Well Tempered Clavier, Book II
Bach’s Fugue number 24 from Well Tempered Clavier, Book II
rendered by Claudi Meneghin
Movi-Nove by Roncevaux (Löis Lancaster)
Psychedelic Delt by Rewarrp