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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

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19 equal temperament

Main page

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 2^1/19, or the 19th root of 2.

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 694.786 ¢; 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).

Theory

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, but in 19edo it 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 major 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. 19edo is in fact the second edo, after 12edo which is able to approximate 5-limit intervals and chords with tolerable accuracy (unless you count 15edo), and is the fifth zeta integral edo, after 12edo. It is less successful in the 7-limit (but still better than 12edo), 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.

As a means of extending harmony

Because 19edo's 5-limit chords are more blended and consonant than those of 12edo, 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.

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. 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	37
Error	Absolute (¢)	+0.0	-7.2	-7.4	-21.5	+17.1	-19.5	+21.4	+18.3	+3.3	-19.1	-8.2	+1.3
	Relative (%)	+0.0	-11.4	-11.7	-34.0	+27.1	-30.8	+33.8	+28.9	+5.2	-30.2	-13.0	+2.0
Steps (reduced)		19 (0)	30 (11)	44 (6)	53 (15)	66 (9)	70 (13)	78 (2)	81 (5)	86 (10)	92 (16)	94 (18)	99 (4)

Adaptive tuning and octave stretch

Being a zeta integral tuning, the no-11's 13-limit is represented relatively well and consistently. 19edo's negri, sensi and godzilla 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.) Its diminished fifth is also a very accurate approximation of the 23rd harmonic, being only 3.3 ¢ off 23/16.

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 5th and 7th harmonics are not only much farther from just than they are in 19edo, but fairly sharp already.

Another option would be to use octave stretching; the closest local zeta peak to 19 occurs at 18.9481, which makes the octaves 1203.29 ¢, and a step size of between 63.2–63.4 ¢ 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 ¢, 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. The most extreme of these options would be 11edf, which has octaves stretched by 12.47 ¢.

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.

Instruments

19 note per octave Ibanez conversion by Brad Smith (Indianapolis)

19edo bass conversion by Ron Sword

Music

Main article: 19edo/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

• 19edo modes
• 19edo chords
• Strictly proper 19edo scales
• How to tune a 19edo guitar by ear
• Primer for 19edo
• Mason Green's New Common Practice Notation
• Extraclassical tonality
• Lumatone mapping for 19edo
• List of 19et rank two temperaments by badness
• List of 19et rank two temperaments by complexity
• List of edo-distinct 19et rank two temperaments
• Syntonic–kleismic equivalence continuum

Further reading

• Darreg, Ivor. A Case for Nineteen. 1982.
• Darreg, Ivor. Nineteen for the Nineties^{[dead link]}. (Unknown date of publication).
• Howe, Hubert S., Jr. 19-Tone Theory and Applications. c. 2004.
• Sethares, William A. Tunings for 19 Tone Equal Tempered Guitar. 1991.
• Sword, Ron. Enneadecaphonic Scales for Guitar: A Repository of Scales, Chord-Scales, Notations and Techniques for Nineteen Equal Divisions of the Octave. 2010.
• Yasser, Joseph. Theory of Evolving Tonality. 1932.

External links

• 19-tone equal-temperament and 1/3-comma meantone / 19-edo / 19-ed2 on the Tonalsoft Encyclopedia
• Microtonalism by Ingrid Pearson, Graham Hair, Dougie McGilvray, Nick Bailey, Amanda Morrison and Richard Parncutt (from n-ISM, the Network for Interdisciplinary Studies in Science, Technology, and Music)
• Forum Discussion with some 19-EDO xenharmonic scales Hanson (Keemun), Liese, Negri, Magic, Semaphore, Sensi played on guitar.
• Bostjan Zupancic's 19-EDO pages
• 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.

Intervals (delisted page)

Degrees

Degree	Cents	Interval Region	Approximated JI Intervals	Solfege
0	0.00	Unison (prime)	1/1	Do
1	63.16	Augmented unison	25/24, 26/25, 28/27	Di/Ro
2	126.32	Minor second	13/12, 14/13, 15/14, 16/15	Ra
3	189.47	Major second	9/8, 10/9	Re
4	252.63	Augmented second Diminished third	7/6, 8/7, 15/13	Ri/Ma
5	315.79	Minor third	6/5	Me
6	378.95	Major third	5/4, 16/13, 56/45	Mi
7	442.11	Augmented third	9/7, 13/10, 32/25	Mo/Fe
8	505.26	Perfect fourth	4/3, 75/56	Fa
9	568.42	Augmented fourth (Small tritone)	7/5, 18/13, 25/18	Fi
10	631.58	Diminished fifth (Large tritone)	10/7, 13/9, 36/25	Se
11	694.74	Perfect fifth	3/2, 112/75	So
12	757.89	Augmented fifth	14/9, 20/13, 25/16	Si/Lo
13	821.05	Minor sixth	8/5, 13/8, 45/28	Le
14	884.21	Major sixth	5/3	La
15	947.37	Augmented sixth Diminished seventh	7/4, 12/7, 26/15	Li/Ta
16	1010.53	Minor seventh	9/5, 16/9	Te
17	1073.68	Major seventh	13/7, 15/8, 24/13, 28/15	Ti
18	1136.84	Augmented seventh	25/13, 27/14, 48/25	To/Da
19	1200.00	Octave	2/1	Do

Approximation to JI

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