19edo: Difference between revisions

← 18edo 19edo 20edo →
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

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

19edo is the second edo, after 12edo, which is able to approximate 5-limit intervals and chords with tolerable accuracy (unless you count 15edo, which has a 18-cent-sharp fifth). Having an almost just minor third and perfect fifths and major thirds about 7 cents flat, it serves as a good tuning for meantone. Unlike 12edo, where enharmonic notes are conflated, 19edo distinguishes them, and differs from 17edo in that its diatonic semitone is wider than the chromatic semitone, rather than narrower. In fact, it is nearly identical to the enharmonic scale of 1/3-comma meantone, and can be considered a closed form thereof.

It is less successful in the 7-limit as it conflates the septimal subminor third (7/6) with the septimal whole tone (8/7), but it is still better than 12edo overall.

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
Error	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)

As an approximation of other temperaments

Besides meantone, 19edo is also suitable for magic/muggles temperament, because five of its major thirds are equivalent to one of its twelfths. Its 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.

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. Finally, 19edo can be used as a tuning for sensi, though 46edo provides 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 also has the advantage of being excellent for negri, keemun, godzilla, muggles, and triton/liese. 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 the 7-odd-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 concordant 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. 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.

Adaptive 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, which has similar benefits to adaptive use, but it also works for fixed-pitch Instruments. For more on that see the section on octave stretch.

Subsets and supersets

19edo is the 8th prime edo, following 17edo and preceding 23edo. As such, it does not contain any nontrivial subset edos, though it contains 19ed4.

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.

Miscellaneous properties

19edo has the flattest possible fifth of any edo that can possibly be diamond monotone in the 15-odd-limit. Using 11\19 as the fifth, the sharpest possible mapping of 5/4 where 10/9 is no greater than 9/8 is 6\19, so the sharpest possible 15/8 is 17\19. Here 16/15 is a quarter of 4/3 (as in any negri tuning), so 15/14, 14/13, and 13/12 must all be equated with 16/15 to 2\19 for them to be monotone in size. If the fifth were any flatter, 5/4 and 15/8 would have to be flatter, and 16/15 would have to be sharper, so diamond monotone in the 15-odd-limit becomes impossible. 19edo is, in fact, diamond monotone in the 15-odd-limit, and even the 17-odd-limit (see Monotonicity limits of small EDOs). The sharpest fifth where 15-odd-limit is possible is 22\37.

Intervals

#	Cents	Note	Approximated ratios^{[note 1]}	Interval category
0	0.0	D	1/1	Unison (prime)
1	63.2	D♯	25/24, 26/25, 27/26, 28/27	Augmented unison
2	126.3	E♭	13/12, 14/13, 15/14, 16/15	Minor second
3	189.5	E	9/8, 10/9	Major second
4	252.6	E♯/F♭	7/6, 8/7, 15/13	Augmented second/ Diminished third
5	315.8	F	6/5	Minor third
6	378.9	F♯	5/4, 16/13, 56/45	Major third
7	442.1	F𝄪/G♭	9/7, 13/10, 21/16, 32/25	Augmented third/ Diminished fourth
8	505.3	G	4/3, 75/56	Perfect fourth
9	568.4	G♯	7/5, 18/13, 25/18	Augmented fourth (Small tritone)
10	631.6	A♭	10/7, 13/9, 36/25	Diminished fifth (Large tritone)
11	694.7	A	3/2, 112/75	Perfect fifth
12	757.9	A♯/B𝄫	14/9, 20/13, 25/16, 32/21	Augmented fifth/ Diminished sixth
13	821.1	B♭	8/5, 13/8, 45/28	Minor sixth
14	884.2	B	5/3	Major sixth
15	947.4	B♯/C♭	7/4, 12/7, 26/15	Augmented sixth Diminished seventh
16	1010.5	C	9/5, 16/9	Minor seventh
17	1073.7	C♯	13/7, 15/8, 24/13, 28/15	Major seventh
18	1136.8	D♭	25/13, 27/14, 48/25, 52/27	Augmented seventh
19	1200.0	D	2/1	Octave

↑ As a 2.3.5.7.13-subgroup temperament

Proposed interval names and solfèges

Table of proposed interval names and solfèges
#	Cents	Solfège	SKULO interval names
0	0.0	Do	Unison	P1
1	63.2	Di/Ro	Super unison, subminor second	S1, sm2
2	126.3	Ra	Minor second	m2
3	189.5	Re	Major second	M2
4	252.6	Ri/Ma	Supermajor second, subminor third	SM2, sm3
5	315.8	Me	Minor third	m3
6	378.9	Mi	Major third	M3
7	442.1	Mo/Fe	Supermajor third, sub fourth	SM3, s4
8	505.3	Fa	Perfect fourth	P4
9	568.4	Fi	Augmented fourth	A4
10	631.6	Se	Diminished fifth	d5
11	694.7	So	Perfect fifth	P5
12	757.9	Si/Lo	Super fifth, subminor sixth	S5, sm6
13	821.1	Le	Minor sixth	m6
14	884.2	La	Major sixth	M6
15	947.4	Li/Ta	Supermajor sixth, subminor seventh	SM6, sm7
16	1010.5	Te	Minor seventh	m7
17	1073.7	Ti	Major seventh	M7
18	1136.8	To/Da	Supermajor seventh, sub octave	SM7, s8
19	1200.0	Do	Octave	P8

Interval quality and chord names in color notation

Using Kite's 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
Minor	gu	(a, b, −1)	6/5, 9/5
Major	yo	(a, b, 1)	5/4, 5/3
Major	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 B𝄫 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.

All 19edo chords can be named using conventional methods, expanded to include augmented and diminished seconds, thirds, sixths and sevenths. Here are the zo, gu, yo and ru triads:

Color of the third	JI chord	Edosteps	Notes of C chord	Written name	Spoken name
zo (7-over)	6:7:9	0–4–11	C–E𝄫–G	Cm(♭3) or Cmin(♭3) or C(d3)	C subminor, C minor flat-three, C dim-three
gu (5-under)	10:12:15	0–5–11	C–E♭–G	Cm or Cmin	C minor
yo (5-over)	4:5:6	0–6–11	C–E–G	C or Cmaj	C, C major
ru (7-under)	14:18:21	0–7–11	C–E♯–G	C(♯3) or Cmaj(♯3) or C(A3)	C supermajor, C major sharp-three, C aug-three
yo (5-over)	4:5:6:7	0–6–11–15	C–E–G–B𝄫	Ch7 or C,d7 or Cadd(d7)	C harmonic 7, C (major) add dim-seven
gu (5-under)	1/(12:10:8:7) (1–6/5–3/2–12/7)	0–5–11–15	C–E♭–G–A♯	Cm♯6 or CmA6 or Cm(add(♯6)) or Cm(add(A6))	C minor (add) sharp-six, C minor (add) aug-six

The last two chords illustrate how the 15\19 interval can be considered as either 7/4 or 12/7, and how 19edo conflates zo and ru ratios.

For a more complete list, see 19edo chords #Ups and downs notation and Kite's ups and downs notation #Chords and chord progressions.

Notation

Standard notation

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

Any 19edo note or interval can be respelled enharmonically by adding a double-diminished second to it or subtracting one from it. Adding a dd2 is equivalent to finding the 12edo equivalent with a higher degree, then diminishing it. For example, C# becomes Db, which is diminished to become Dbb.

Notation of 19edo
#	Cents	Standard notation
#	Cents	Diatonic interval names	Note names on D
0	0.0	Perfect unison (P1)	D
1	63.2	Augmented unison (A1) Diminished second (d2)	D# Ebb
2	126.3	Doubly augmented unison (AA1) Minor second (m2)	Dx Eb
3	189.5	Major second (M2) Doubly diminished third (dd3)	E Fbb
4	252.6	Augmented second (A2) Diminished third (d3)	E# Fb
5	315.8	Doubly augmented second (AA2) Minor third (m3)	Ex F
6	378.9	Major third (M3) Doubly diminished fourth (dd4)	F# Gbb
7	442.1	Augmented third (A3) Diminished fourth (d4)	Fx Gb
8	505.3	Perfect fourth (P4)	G
9	568.4	Augmented fourth (A4) Doubly diminished fifth (dd5)	G# Abb
10	631.6	Doubly augmented fourth (AA4) Diminished fifth (d5)	Gx Ab
11	694.7	Perfect fifth (P5)	A
12	757.9	Augmented fifth (A5) Diminished sixth (d6)	A# Bbb
13	821.1	Doubly augmented fifth (AA5) Minor sixth (m6)	Ax Bb
14	884.2	Major sixth (M6) Doubly diminished seventh (dd7)	B Cbb
15	947.4	Augmented sixth (A6) Diminished seventh (d7)	B# Cb
16	1010.5	Doubly augmented sixth (AA6) Minor seventh (m7)	Bx C
17	1073.7	Major seventh (M7) Doubly diminished octave (dd8)	C# Dbb
18	1136.8	Augmented seventh (A7) Diminished octave (d8)	Cx Db
19	1200.0	Perfect octave (P8)	D

In 19edo:

Ups and downs notation is identical to standard notation;
Mixed sagittal notation is identical to standard notation, but pure sagittal notation exchanges sharps (♯) and flats (♭) for sagittal sharp () and sagittal flat () respectively.

Step offset	−2	−1	0	+1	+2
Symbol

Sagittal notation

This notation uses the same sagittal sequence as edos 5, 12, and 26, and is a subset of the notations for edos 38, 57, and 76.

Evo flavor

Because it includes no Sagittal symbols, this Evo Sagittal notation is identical to conventional notation.

Revo flavor

Dodecatonic notation

Dodecatonic notation of 19edo
#	Cents	Interval names
0	0.0	P1
1	63.2	A1, m2
2	126.3	M2, m3
3	189.5	M3
4	252.6	m4, A3
5	315.8	M4, m5
6	378.9	M5
7	442.1	A5, d6
8	505.3	P6
9	568.4	A6, m7
10	631.6	M7, d8
11	694.7	P8
12	757.9	A8, m9
13	821.1	M9, m10
14	884.2	M10
15	947.4	m11, A10
16	1010.5	M11, m12
17	1073.7	M12
18	1136.8	A12, d13
19	1200.0	P13

Approximation to JI

alt : Your browser has no SVG support. — 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

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Absolute (¢)	Relative (%)
2.3	[-30 19⟩	[⟨19 30]]	+2.277	2.277	3.612
2.3.5	81/80, 3125/3072	[⟨19 30 44]]	+2.578	1.911	3.025
2.3.5.7	49/48, 81/80, 126/125	[⟨19 30 44 53]]	+3.848	2.755	4.362
2.3.5.7.13	49/48, 65/64, 81/80, 91/90	[⟨19 30 44 53 70]]	+4.135	2.530	4.006
2.3.5.7.13.23	49/48, 65/64, 70/69, 81/80, 91/90	[⟨19 30 44 53 70 86]]	+3.319	2.936	4.649

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 equal temperaments doing better in those subgroups are 34, 31, 27e, 22, and 26, respectively.
19et is best in the 2.3.5.7.13 subgroup, and the next equal temperament that does better in this is 53.

Uniform maps

13-limit uniform maps between 18.8 and 19.2
Min. size	Max. size	Wart notation	Map
18.7816	18.9337	19e	⟨19 30 44 53 65 70]
18.9337	19.0518	19	⟨19 30 44 53 66 70]
19.0518	19.0571	19f	⟨19 30 44 53 66 71]
19.0571	19.1651	19df	⟨19 30 44 54 66 71]
19.1651	19.2228	19cdf	⟨19 30 45 54 66 71]

Commas

19et tempers out the following commas. (Note: This assumes the val ⟨19 30 44 53 66 70].)

Prime limit	Ratio^{[note 1]}	Monzo	Cents	Color name	Name
3	(20 digits)	[-30 19⟩	137.14	Trilawa	19-comma
5	16875/16384	[-14 3 4⟩	51.12	Laquadyo	Negri comma
5	(14 digits)	[-2 13 -8⟩	34.91	Laquadbigu	Unicorn comma
5	3125/3072	[-10 -1 5⟩	29.61	Laquinyo	Magic comma
5	81/80	[-4 4 -1⟩	21.51	Gu	Syntonic comma
5	78732/78125	[2 9 -7⟩	13.40	Sepgu	Sensipent comma
5	15625/15552	[-6 -5 6⟩	8.11	Tribiyo	Kleisma
5	(20 digits)	[8 14 -13⟩	5.29	Thegu	Parakleisma
5	(28 digits)	[-14 -19 19⟩	2.82	Neyo	Enneadeca
7	59049/57344	[-13 10 0 -1⟩	50.72	Laru	Harrison's comma
7	1029/1000	[-3 1 -3 3⟩	49.49	Trizogu	Keega
7	525/512	[-9 1 2 1⟩	43.41	Lazoyoyo	Avicennma
7	49/48	[-4 -1 0 2⟩	35.70	Zozo	Semaphoresma, slendro diesis
7	3645/3584	[-9 6 1 -1⟩	29.22	Laruyo	Schismean comma
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 comma
7	126/125	[1 2 -3 1⟩	13.79	Zotrigu	Starling comma
7	225/224	[-5 2 2 -1⟩	7.71	Ruyoyo	Marvel comma
7	19683/19600	[-4 9 -2 -2⟩	7.32	Labirugu	Cataharry comma
7	10976/10935	[5 -7 -1 3⟩	6.48	Satrizo-agu	Hemimage comma
7	3136/3125	[6 0 -5 2⟩	6.08	Zozoquingu	Hemimean comma
7	(12 digits)	[-11 2 7 -3⟩	1.63	Latriru-asepyo	Metric comma
7	4375/4374	[-1 -7 4 1⟩	0.40	Zoquadyo	Ragisma
11	45/44	[-2 2 1 0 -1⟩	38.91	Luyo	Undecimal fifth tone
11	56/55	[3 0 -1 1 -1⟩	31.19	Luzogu	Undecimal tritonic comma
11	100/99	[2 -2 2 0 -1⟩	17.40	Luyoyo	Ptolemisma
11	896/891	[7 -4 0 1 -1⟩	9.69	Saluzo	Pentacircle comma
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	39/38	[-1 1 0 0 0 1 0 -1⟩	44.97	Nutho	Undevicesimal two-ninth tone
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 comma, biome comma
13	676/675	[2 -3 -2 0 0 2⟩	2.56	Bithogu	Island comma
13	1001/1000	[-3 0 -3 1 1 1⟩	1.73	Tholozotrigu	Fairytale comma, sinbadma
23	2187/2116	[-2 7 0 0 0 0 0 0 -2⟩	57.14	Labitwethu	Lipsett comma
23	70/69	[1 -1 1 1 0 0 0 0 -1⟩	24.91	Twethuzoyo	Small vicesimotertial eighth tone
23	256/253	[8 0 0 0 -1 0 0 0 -1⟩	20.41	Twethulu	253rd subharmonic
23	161/160	[-5 0 -1 1 0 0 0 0 1⟩	10.79	Twethozogu	Major kirnbergisma
23	208/207	[4 -2 0 0 0 1 0 0 -1⟩	8.34	Twethutho	Vicetone comma
23	529/528	[-4 -1 0 0 -1 0 0 0 2⟩	3.28	Bitwetho-alu	Preziosisma
23	576/575	[6 2 -2 0 0 0 0 0 -1⟩	3.01	Twethugugu	Worcester comma
23	1288/1287	[3 -2 0 1 -1 -1 0 0 1⟩	1.34	Twethothuluzo	Triaphonisma

↑ Ratios longer than 10 digits are presented by placeholders with informative hints.

Linear temperaments

Since 19 is prime, all rank-2 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	Mos scales	Temperaments
1	63.16	A1, d2		Unicorn / Rhinoceros
2	126.32	m2	1L 8s, 9L 1s	Negri
3	189.47	M2	1L 5s, 6L 1s, 6L 7s	Deutone Xenial / Xenia Spell
4	252.63	A2, d3	1L 3s, 4L 1s, 5L 4s, 5L 9s	Godzilla / Helayo
5	315.79	m3	3L 1s, 4L 3s, 4L 7s, 4L 11s	Cata / keemun
6	378.95	M3	3L 1s, 3L 4s, 3L 7s, 3L 10s, 3L 13s	Magic / muggles
7	442.11	A3, d4	3L 2s, 3L 5s, 8L 3s	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	Liese Triton / pycnic

Octave stretch or compression

19edo is a promising option as a meantone tuning for pianos if split sharps are acceptable, since pianos are frequently tuned with stretched octaves due to the slight inharmonicity inherent in their strings. It also works well with harpsichords, since many have been, and are, built 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 30edt (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-odd-limit tonality diamond. The compound major and minor triads (1:5:6 and 30:6:5) are near-just as well. Another possible choice is 65zpi.

Scales

MOS scales

Main article: List of MOS scales in 19edo

Octave-equivalent mosses

Meantone pentic, 2L 3s (gen = 11\19): 3 3 5 3 5
Meantone diatonic, 5L 2s (gen = 11\19): 3 3 2 3 3 3 2
Meantone chromatic, 7L 5s (gen = 11\19): 2 1 2 1 2 2 1 2 1 2 1 2
Semaphore[5], 4L 1s (gen = 4\19): 4 4 3 4 4
Semaphore[9], 5L 4s (gen = 4\19): 3 1 3 1 3 3 1 3 1
Semaphore[14], 5L 9s (gen = 4\19): 2 1 2 1 1 2 1 1 2 1 1 2 1 1
Sensi[5], 2L 3s (gen = 7\19): 5 2 5 2 5
Sensi[8], 3L 5s (gen = 7\19): 2 3 2 2 3 2 2 3
Sensi[11], 8L 3s (gen = 7\19): 2 2 1 2 2 2 1 2 2 2 1
Negri[9], 1L 8s (gen = 2\19): 2 2 2 2 3 2 2 2 2
Negri[10], 9L 1s (gen = 2\19): 2 2 2 2 2 1 2 2 2 2
Kleismic[7], 4L 3s (gen = 5\19): 1 4 1 4 1 4 4
Kleismic[11], 4L 7s (gen = 5\19): 1 3 1 1 3 1 1 3 1 3 1
Kleismic[15], 4L 11s (gen = 5\19): 1 2 1 1 1 2 1 1 1 2 1 1 2 1 1
Magic[7], 3L 4s (gen = 6\19): 5 1 5 1 5 1 1
Magic[10], 3L 7s (gen = 6\19): 4 1 1 4 1 1 4 1 1 1
Magic[13], 3L 10s (gen = 6\19): 3 1 1 1 3 1 1 1 3 1 1 1 1
Magic[16], 3L 13s (gen = 6\19): 2 1 1 1 1 2 1 1 1 1 2 1 1 1 1 1
Liese[17], 2L 15s (gen = 9\19): 2 1 1 1 1 1 1 1 1 1 2 1 1 1 1 1 1 1 1

Other scales

Main article: 19edo modes

Meantone harmonic minor: 3 2 3 3 2 4 2
Meantone melodic minor: 3 2 3 3 3 3 2 (ascending), 3 2 3 3 2 3 3 (descending)
Meantone harmonic major: 3 3 2 3 2 4 2
Chromatic octave species – meantone / marvel double harmonic major (subset of Negri[9]): 2 4 2 3 2 4 2
Chromatic octave species (subset of Negri[9]): 2 2 4 3 2 2 4
Chromatic octave species - Sahara septatonic (subset of Negri[9]): 4 2 2 3 4 2 2
Marvel hexatonic (subset of Negri[9]): 4 2 5 2 4 2
Enharmonic pentatonic: 2 6 3 2 6
Enharmonic pentatonic: 6 2 3 6 2
Enharmonic octave species: 1 1 6 3 1 1 6
Enharmonic octave species: 6 1 1 3 6 1 1
Enharmonic octave species: 1 6 1 3 1 6 1
Pinetone major-harmonic octatonic: 3 2 3 1 2 3 2 3 (subset of Meantone[12])
Pinetone minor-harmonic octatonic: 3 2 1 3 2 3 3 2 (subset of Meantone[12])
Pinetone diminished octatonic / Porcusmine: 2 3 1 3 2 3 2 3
Pinetone harmonic diminished: 2 3 1 4 1 3 2 3
Blackville / 5-limit dipentatonic (superset of Meantone[7]): 1 2 3 2 1 2 3 2 1 2
Antipental blues: 4 4 1 2 4 4
Semiquartal 3|5 b2: 1 3 3 1 3 1 3 3 1
5-odd-limit tonality diamond: 5 1 2 3 2 1 5
7-odd-limit tonality diamond: 4 1 1 2 1 1 1 2 1 1 4
9-odd-limit tonality diamond: 3 1 1 1 1 1 1 1 1 1 1 1 1 1 3

Instruments

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]}

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.

[1] As a 2.3.5.7.13-subgroup temperament

[2] Ratios longer than 10 digits are presented by placeholders with informative hints.

[note 1]

[note 1]

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