19edo

**19edo** is a reasonable alternative to 12edo with reasonably accurate tuning. In the meantone circle of fifths, it could be naturally considered an extension.

Compact notation based on meantone notation:

<inline>

```
C  C# Db D  D# Eb E  E# F  F# Gb G  G# Ab A  A# Bb B  Cb
```
</inline>
To round midi note a (0—127, with 60 being C4 and +12 being octave) with pitch wheel b (0—16383, with 8192 being center) to 19edo, the following formula could be used:

```
((a*4096L+b+32768L)*19+24575L)/49152L-16
```
. ((a\*4096L+b is the conversion of a and b to a uniform 49152edo space in a long integer, +32768L) is the subtraction of (-10\*4096L+8192) to center the space on D note, \*19+24575L)/49152L is the round to nearest to 19edo (with one less than half of 49152 being used to get G♯ note by default instead of A♭) and -16 adds the D-2 offset in 19edo to restore C−1 note on 0. The default 12edo range of 0—127 gets converted to 0—201 range (1536/1, or nine octaves and one tritave, in 12edo is 12×9+19=127 and in 19edo is 19×9+30=201), therefore extending 128 notes to 202 notes for C−1 to G9 range. Including the pitch wheel (−2 to 2 semitones of 12edo), the output range is −3—204, which is B♭−2 to A9 range of 208 notes.

# 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.

\* 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:

## 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:

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 - 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).

### Patent val mapping

File:19ed2-001.svg

19ed2-001.svg

## Commas

19 EDO tempers out the following commas. (Note: This assumes the val < 19 30 44 53 66 70 |.)

# Photos

 File:Vaisvil-19edo-guitar-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

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

# See also

* 19edo Modes
* Strictly proper 19edo scales
* How to tune a 19edo guitar by ear
* Chord progressions in 19edo-family scales

## 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

## 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.