19edo

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19edo
Prime factorization 19 is a prime
Step size 63.15789¢
Fifth 11\19 ≈ 695¢
Major 2nd 3\19 ≈ 189¢
Minor 2nd 2\19 ≈ 126¢
Augmented 1sn 1\19 ≈ 63¢

19 equal divisions of the octave (19-EDO), or 19(-tone) equal temperament (19-TET, 19-ET) when viewed from a regular temperament perspective, is the tuning system derived by dividing the octave into 19 equally large steps. Each step represents a frequency ratio of the 19th root of 2, or about 63.2 cents. It is the 8th prime EDO, following 17edo and preceding 23edo.

Theory

Approximation of odd harmonics in 19edo
Odd harmonic 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Error absolute (¢) -7.2 -7.4 -21.5 -14.4 +17.1 -19.5 -14.6 +21.4 +18.3 -28.7 +3.3 -14.7 -21.7 -19.1 -8.2
relative (%) -11 -12 -34 -23 +27 -31 -23 +34 +29 -45 +5 -23 -34 -30 -13
Steps (reduced) 30 (11) 44 (6) 53 (15) 60 (3) 66 (9) 70 (13) 74 (17) 78 (2) 81 (5) 83 (7) 86 (10) 88 (12) 90 (14) 92 (16) 94 (18)

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 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 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

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 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 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. 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 27edo and 46edo are better sensi tunings for the 13-limit approximations of sensi.

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, though only the 2.3.5.7.13 subgroup is represented consistently. 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. 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 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 tonality without sounding too alien.

19-EDO 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 19-EDO.

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.

Differences between distributionally-even scales and smaller edos

N L-Nedo s-Nedo
2 31.579¢ -31.579¢
3 42.105¢ -21.053¢
4 15.7895¢ -47.895¢
5 12.632¢ -51.526¢
6 52.632¢ -10.526¢
7 18.045¢ -45.103¢
8 39.474¢ -23.684¢
9 56.14¢ -7.018¢
10 6.316¢ -56.842¢
11 17.225¢ -46.091¢
12 26.316¢ -36.842¢
13 34.008¢ -29.15¢
14 40.6015¢ -22.556¢
15 46.316¢ -16.842¢
16 51.316¢ -11.842¢
17 55.728¢ -7.43¢
18 59.649¢ -3.509¢

Intervals

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

Degree Cents Interval Solfege Dodecatonic notation Ratios*
0 0.00 unison P1 D do P1 1/1
1 63.16 aug 1sn, dim 2nd A1, d2 D#, Ebb di A1, m2 25/24, 28/27, 26/25
2 126.32 minor 2nd m2 Eb ra M2, m3 15/14, 16/15, 13/12, 14/13
3 189.47 major 2nd M2 E re M3 9/8, 10/9, 125/112
4 252.63 aug 2nd, dim 3rd A2, d3 E#, Fb ri/ma m4, a3 7/6, 8/7, 15/13
5 315.79 minor 3rd m3 F me M4, m5 6/5
6 378.95 major 3rd M3 F# mi M5 5/4, 16/13, 56/45
7 442.11 aug 3rd, dim 4th A3, d4 Fx, Gb mo A5, d6 32/25, 9/7, 13/10
8 505.26 perfect 4th P4 G fa P6 4/3, 75/56
9 568.42 aug 4th A4 G# fi A6, m7 25/18, 7/5, 18/13
10 631.58 dim 5th d5 Ab se M7, d8 36/25, 10/7, 13/9
11 694.74 perfect 5th P5 A sol P8 3/2, 112/75
12 757.89 aug 5th, dim 6th A5, d6 A#, Bbb lo A8, m9 25/16, 14/9, 20/13
13 821.05 minor 6th m6 Bb le M9, m10 8/5, 13/8, 45/28
14 884.21 major 6th M6 B la M10 5/3
15 947.37 aug 6th, dim 7th A6, d7 B#, Cb li/ta m11, A10 7/4, 12/7, 26/15
16 1010.53 minor 7th m7 C te M11, m12 9/5, 16/9, 224/125
17 1073.68 major 7th M7 C# ti M12 15/8, 13/7, 28/15, 24/13
18 1136.84 aug 7th, dim 8ve A7, d8 Cx, Db da A12, d13 48/25, 27/14, 25/13
19 1200.00 perfect 8ve P8 D do P13 2/1

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

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 Bbb 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.

Chord names

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

0-6-11-15 = C E G Bbb = C,d7 = C add dim-seven or C major dim-seven = 4:5:6:7

0-5-11-15 = C Eb G A# is Cm#6 or CmA6 = 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 zo and ru ratios.

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

JI approximation

15-odd-limit interval mappings

The following table shows how 15-odd-limit intervals are represented in 19edo. Prime harmonics are in bold; inconsistent intervals are in italic.

Direct mapping (even if inconsistent)
Interval, complement Error (abs, ¢)
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
Patent val mapping
Interval, complement Error (abs, ¢)
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
11/9, 18/11 31.539
15/11, 22/15 31.688
13/11, 22/13 36.578
14/11, 11/7 38.561

Selected 17-limit intervals

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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

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 ETs 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 ET that does better in this is 53.

Commas

19-EDO 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(s)
5 16875/16384 [-14 3 4⟩ 51.12 Laquadyo Negri comma, double augmentation diesis
5 3125/3072 [-10 -1 5⟩ 29.61 Laquinyo Small diesis, magic comma
5 81/80 [-4 4 -1⟩ 21.51 Gu Syntonic comma, Didymus comma, meantone comma
5 78732/78125 [2 9 -7⟩ 13.40 Sepgu Medium semicomma, sensipent comma
5 15625/15552 [-6 -5 6⟩ 8.11 Tribiyo Kleisma, semicomma majeur
5 (20 digits) [8 14 -13⟩ 5.29 Thegu Parakleisma
5 (28 digits) [-14 -19 19⟩ 2.82 Neyo Enneadeca, 19-tone-comma
7 1029/1000 [-3 1 -3 3⟩ 49.49 Trizogu Keega
7 525/512 [-9 1 2 1⟩ 43.41 Lazoyoyo Avicennma, Avicenna's enharmonic diesis
7 49/48 [-4 -1 0 2⟩ 35.70 Zozo Slendro diesis
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
7 126/125 [1 2 -3 1⟩ 13.79 Zotrigu Septimal semicomma, starling comma
7 225/224 [-5 2 2 -1⟩ 7.71 Ruyoyo Septimal kleisma, marvel comma
7 19683/19600 [-4 9 -2 -2⟩ 7.32 Labirugu Cataharry
7 10976/10935 [5 -7 -1 3⟩ 6.48 Satrizo-agu Hemimage
7 3136/3125 [6 0 -5 2⟩ 6.08 Zozoquingu Hemimean
7 (12 digits) [-11 2 7 -3⟩ 1.63 Latriru-asepyo Meter
7 4375/4374 [-1 -7 4 1⟩ 0.40 Zoquadyo Ragisma
11 100/99 [2 -2 2 0 -1⟩ 17.40 Luyoyo Ptolemisma
11 896/891 [7 -4 0 1 -1⟩ 9.69 Saluzo Pentacircle
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 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
13 676/675 [2 -3 -2 0 0 2⟩ 2.56 Bithogu Parizeksma
  1. ↑ Ratios longer than 10 digits are presented by placeholders with informative hints

Linear temperaments

Since 19 is prime, all rank two 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 Generator for
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 		Semaphore / godzilla
5 315.79 m3 3L 1s, 4L 3s,
4L 7s, 4L 11s 		Hanson / keemun / catakleismic
6 378.95 M3 3L 1s, 3L 4s, 3L 7s,
3L 10s, 3L 13s 		Magic
7 442.11 A3, d4 3L 2s, 3L 5s, 8L 3s Sensipent / BPS / 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 		Lee / liese / pycnic
Triton

Scales

Important MOSes include:

  • meantone diatonic 5*3-2*2 (11\19, 1\1)
  • meantone chromatic 7*2-5*1 (11\19, 1\1)
  • semaphore 5*3-4*1 (4\19, 1\1)
  • semaphore 5*2-9*1 (4\19, 1\1)
  • sensi 3*3-5*2 (7\19, 1\1)
  • sensi 8*2-3*1 (7\19, 1\1)
  • Negri 3*1-2*8 (2\19, 1\1)
  • Negri 2*9-1*1 (2\19, 1\1)
  • Kleismic diatonic 4*4-3*1 (5\19, 1\1)
  • Kleismic superdiatonic 4*3-7*1 (5\19, 1\1)
  • Kleismic chromatic 4*2-11*1 (5\19, 1\1)
  • Magic diatonic 3*5-4*1 (6\19, 1\1)
  • Magic superdiatonic 3*4-7*1 (6\19, 1\1)
  • Magic chromatic 3*3-10*1 (6\19, 1\1)
  • Pathological Magic mega chromatic 3*2-13*1 (6\19, 1\1)
  • Pathological Liese 2*2-15*1 (9\19, 1\1)

Instruments

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

Music

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.
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