19edo
Prime factorization | 19 (prime) |
Step size | 63.158¢ |
Fifth | 11\19 = 694.737¢ |
Major 2nd | 3\19 = 189¢ |
Minor 2nd | 2\19 = 126¢ |
Augmented 1sn | 1\19 = 63¢ |
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.
Theory
prime 2 | prime 3 | prime 5 | prime 7 | prime 11 | prime 13 | prime 23 | ||
---|---|---|---|---|---|---|---|---|
Error | absolute (¢) | 0.0 | -7.2 | -7.4 | -21.5 | +17.1 | -19.5 | +3.3 |
relative (%) | 0 | -11 | -12 | -34 | +27 | -31 | +5 | |
nearest edomapping | 19 | 11 | 6 | 15 | 9 | 13 | 10 | |
fifthspan | 0 | +1 | +4 | -9 | +6 | -4 | -6 |
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.
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.
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 | 0.0000 | unison | P1 | D | do | P1 | 1/1 |
1 | 63.1579 | aug 1sn, dim 2nd | A1, d2 | D#, Ebb | di | A1, m2 | 25/24, 28/27, 26/25 |
2 | 126.3157 | minor 2nd | m2 | Eb | ra | M2, m3 | 15/14, 16/15, 13/12, 14/13 |
3 | 189.4737 | major 2nd | M2 | E | re | M3 | 9/8, 10/9, 6272/5625 |
4 | 252.6316 | aug 2nd, dim 3rd | A2, d3 | E#, Fb | ri/ma | m4, a3 | 7/6, 8/7, 15/13 |
5 | 315.7895 | minor 3rd | m3 | F | me | M4, m5 | 6/5 |
6 | 378.9474 | major 3rd | M3 | F# | mi | M5 | 5/4, 16/13, 56/45 |
7 | 442.1053 | aug 3rd, dim 4th | A3, d4 | Fx, Gb | mo | A5, d6 | 32/25, 9/7, 13/10 |
8 | 505.2632 | perfect 4th | P4 | G | fa | P6 | 4/3, 75/56 |
9 | 568.4211 | aug 4th | A4 | G# | fi | A6, m7 | 25/18, 7/5, 18/13 |
10 | 631.5789 | dim 5th | d5 | Ab | se | M7, d8 | 36/25, 10/7, 13/9 |
11 | 694.7368 | perfect 5th | P5 | A | sol | P8 | 3/2, 112/75 |
12 | 757.8947 | aug 5th, dim 6th | A5, d6 | A#, Bbb | lo | A8, m9 | 25/16, 14/9, 20/13 |
13 | 821.0526 | minor 6th | m6 | Bb | le | M9, m10 | 8/5, 13/8, 45/28 |
14 | 884.2105 | major 6th | M6 | B | la | M10 | 5/3 |
15 | 947.3684 | aug 6th, dim 7th | A6, d7 | B#, Cb | li/ta | m11, A10 | 7/4, 12/7, 26/15 |
16 | 1010.5263 | minor 7th | m7 | C | te | M11, m12 | 9/5, 16/9, 5625/3136 |
17 | 1073.6843 | major 7th | M7 | C# | ti | M12 | 15/8, 13/7, 28/15, 24/13 |
18 | 1136.8421 | aug 7th, dim 8ve | A7, d8 | Cx, Db | da | A12, d13 | 48/25, 27/14, 25/13 |
19 | 1200.0000 | 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 | notes as 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,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 zo and ru ratios.
For a more complete list, see Ups and Downs Notation - Chords and Chord Progressions.
Just approximation
Selected just intervals by error
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.
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 |
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
Temperament measures
The following table shows TE temperament measures (RMS normalized by the rank) of 19et.
3-limit | 5-limit | 7-limit | 2.3.5.7.13 | ||
---|---|---|---|---|---|
Octave stretch (¢) | +2.28 | +2.58 | +3.85 | +4.14 | |
Error | absolute (¢) | 2.28 | 1.91 | 2.76 | 2.53 |
relative (%) | 3.61 | 3.02 | 4.35 | 3.99 |
- 19et has a lower relative error than any previous ETs in the 5- and 7-limit. The next ET that does better in these subgroups is 34 and 31, respectively.
- 19et is most 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 | |
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 |
- ↑ Ratios longer than 10 digits are presented by placeholders with informative hints
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
Important MOSes include:
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 | Generator for |
---|---|---|---|
1 | 63.1579 | A1, d2 | Unicorn/rhinocerus |
2 | 126.3157 | m2 | Negri |
3 | 189.4737 | M2 | Deutone (2-meantone) / spell |
4 | 252.6316 | A2, d3 | Godzilla |
5 | 315.7895 | m3 | Kleismic (Hanson, Keemun, catakleismic) |
6 | 378.9474 | M3 | Magic/charisma/glamour |
7 | 442.1053 | A3, d4 | Sensi |
8 | 505.2632 | P4 | Meantone/Flattone/meanenneadecal/Meanpop |
9 | 568.4211 | A4 | Liese/Triton/lisa |
Instruments
Music
- XA 19-ET Index
- WEIGEL FAMILY CHRISTMAS (xenharmonic chocolate), an album of xenharmonic Christmas covers played by Stephen Weigel, many are in 19edo
- 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
- Fugue in 19ET (2020) - SoundCloud by Aaron Andrew Hunt
- Citified Notions and
- Limp Off to School by John Starrett
- The Light Of My Betelgeuse by Mykhaylo Khramov
- Undines by Jon Lyle Smith
- Sylphs by Jon Lyle Smith
- Gnomes by Jon Lyle Smith
- 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 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 rendered by Claudi Meneghin
- 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
- Bright Objects - 17-tone fifths chain in 19EDO by Cam Taylor
- Minor Thirds, Minor Sevens - 17-tone fifths chain in 19EDO by Cam Taylor
- Wurly Minors and Enharmonic Tetrachords - 17-tone fifths chain in 19EDO by Cam Taylor
- Brain for Breakfast - 19-EDO piece (first movement uses 19 tone serialism) and
- Psychoclowns - 19-EDO pseudo-rondo by Bostjan Zupancic
See also
- 19edo Modes
- Strictly proper 19edo scales
- How to tune a 19edo guitar by ear
- Primer for 19edo
- Mason Green's New Common Practice Notation
- Arto and Tendo Theory
Articles
- A Case for Nineteen by Ivor Darreg Permalink
- Nineteen for the Nineties by Ivor Darreg
- 19-Tone Theory and Applications by Hubert S. Howe Jr. Permalink
- Tunings for 19 Tone Equal Tempered Guitar by William A. Sethares Permalink
- Microtonalism by Bailey, Morrison, Pearson and Parncutt Permalink
- 19-tone equal-temperament and 1/3-comma meantone - Encyclopedia of Microtonal Music Theory Permalink
- Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar.
- Enneadecaphonic Scales for Guitar by Ron Sword
- Bostjan Zupancic's 19-EDO pages
- Here is a catalog of all 19edo heptatonic scales.
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.