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.

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 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

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.

Degrees of 19edo	Solfege	Diatonic Category	Dodecatonic category	Cents	pions	7mus	Ratios*	Generator for
0	do	P1	P1	0			1/1
1	di	A1, d2	A1, m2	63.1579	66.9474	80.8421 (50.D794₁₆)	25/24, 28/27, 26/25, 27/26	Unicorn/rhinocerus
2	ra	m2	M2, m3	126.3157	133.8947	161.6842 (A1.AF29₁₆)	15/14, 16/15, 13/12, 14/13	Negri
3	re	M2	M3	189.4737	200.8421	242.5263 (F2.86BD₁₆)	9/8, 10/9	Deutone (2-meantone) / spell
4	ri/ma	A2, d3	m4, a3	252.6316	267.7895	323.3684 (143.5E51₁₆)	7/6, 8/7, 15/13	Godzilla
5	me	m3	M4, m5	315.7895	334.7368	404.2105 (194.35E5₁₆)	6/5	Kleismic (hanson, keemun, catakleismic)
6	mi	M3	M5	378.9474	401.6843	485.0526 (1E5.0D79₁₆)	5/4, 16/13, 26/21	Magic/charisma/glamour
7	mo	A3, d4	A5, d6	442.1053	468.6316	565.8947 (235.E50E₁₆)	32/25, 9/7, 13/10	Sensi
8	fa	P4	P6	505.2632	535.57895	646.7368 (286.BCA2₁₆)	4/3	Meantone/flattone/meanenneadecal/meanpop
9	fi	A4	A6, m7	568.42105	602.5263	727.57895 (2D7.9436₁₆)	25/18, 7/5, 18/13	Liese/triton/lisa
10	se	d5	M7, d8	631.57895	669.4737	808.42105 (328.6BCA₁₆)	36/25, 10/7, 13/9	Liese/triton/lisa
11	sol	P5	P8	694.7368	736.42105	889.2632 (379.435E₁₆)	3/2	Meantone
12	lo	A5, d6	A8, m9	757.8947	803.3684	970.1053 (3CA.1AF3₁₆)	25/16, 14/9, 20/13	Sensi
13	le	m6	M9, m10	821.0526	870.3157	1050.9474 (41A.F287₁₆)	8/5, 13/8, 21/13	Magic
14	la	M6	M10	884.2105	937.2632	1131.7895 (46B.CA1B₁₆)	5/3	Kleismic (hanson, keemun, catakleismic)
15	li/ta	A6, d7	m11, A10	947.3684	1004.2105	1212.6316 (4BC.A1AF₁₆)	7/4, 12/7, 26/15	Godzilla
16	te	m7	M11, m12	1010.5263	1071.1579	1293.4737 (50D.7943₁₆)	9/5, 16/9	Deutone / spell
17	ti	M7	M12	1073.6843	1138.1053	1374.3157 (55E.50D8₁₆)	15/8, 13/7, 28/15, 24/13	Negri
18	da	A7, d8	A12, d13	1136.8421	1205.0526	1455.1579 (5AF.286C₁₆)	48/25, 40/21, 27/14, 25/13, 52/27	Unicorn/rhinocerus
19	do	P8	P13	1200	1272	1536 (600₁₆)	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	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

Notation

Looking at 19edo as an extension of 12edo, standard notation can be used, whether it is staff notation (with five lines), letter notation (with standard accidentals), solfege, or sargam. Notes with enharmonic equivalents are different than they are in 12edo, though.

Letter Notation (anglophonic standard)

Using the letters A-G and "accidentals" b to lower a tone and # to raise a tone, with also bb to lower a tone two degrees and x to raise a tone two degrees, the notes and enharmonic equivalents are shown in the table below:

A Basic Look at Letter Notation in 19edo
Degree	Interval with A	Alternative Interval with A	Name in the key of A	Letter(s)	Enharmonic Equivalents
1	Unison		Tonic	A
2		Diminished Second		A#	Bbb
3	Minor Second			Bb (*)	Ax
4	Major Second		Supertonic	B (*)	Cbb
5		Augmented Second, Diminished Third		B# or Cb
6	Minor Third			C	Bx
7	Major Third		Mediant	C#	Dbb
8		Augmented Third, Diminished Fourth		Db	Cx
9	Perfect Fourth		Subdominant	D
10	Augmented Fourth			D#	Ebb
11	Diminished Fifth			Eb	Dx
12	Perfect Fifth		Dominant	E	Fbb
13	Augmented Fifth			E# or Fb
14	Minor Sixth			F	Ex
15	Major Sixth		Submediant	F#	Gbb
16	Diminished Seventh	Augmented Sixth		Gb	Fx
17	Minor Seventh			G
18	Major Seventh		Subtonic	G#	Abb
19		Augmented Seventh		Ab	Gx

*Some cultures use letter notation, but there is a common variation to replace Bb from the table with B and then replace B from the table with H.

Chords would follow the same spelling as with standard 12edo notation, just be careful with spelling. For example, Bb chord would be spelled Bb D F, and A# chord would be A# Cx E#; but the two are different chords, one degree apart from each other.

Key signatures are the same, but again, with the extra notes and different enharmonic equivalents, some key signatures can get messy. For example, the key of Bbb would have bb's on B and E, and b's on C, D, F, G, and A. Thinking of rewriting this key as A# might seem better, but then the key signature would contain x's on C, F, and G, and #'s 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	14:18:27	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 - 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).

Best direct mapping, even if inconsistent

Interval, complement	Error (abs., in cents)
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., in cents)
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

19ed2-001.svg

Commas

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

Comma	Monzo	Value (Cents)	Name(s)
16875/16384	\| -14 3 4 >	51.12	Negri Comma, Double Augmentation Diesis
3125/3072	\| -10 -1 5 >	29.61	Small Diesis, Magic Comma
81/80	\| -4 4 -1 >	21.51	Syntonic Comma, Didymos Comma, Meantone Comma
78732/78125	\| 2 9 -7 >	13.40	Medium Semicomma, Sensipent Comma
15625/15552	\| -6 -5 6 >	8.11	Kleisma, Semicomma Majeur
	\| 8 14 -13 >	5.29	Parakleisma
	\| -14 -19 19 >	2.82	Enneadeca, 19-Tone-Comma
1029/1000	\| -3 1 -3 3 >	49.49	Keega
525/512	\| -9 1 2 1 >	43.41	Avicenna, Avicenna's Enharmonic Diesis
49/48	\| -4 -1 0 2 >	35.70	Slendro Diesis
686/675	\| 1 -3 -2 3 >	27.99	Senga
875/864	\| -5 -3 3 1 >	21.90	Keema
245/243	\| 0 -5 1 2 >	14.19	Sensamagic
126/125	\| 1 2 -3 1 >	13.79	Septimal Semicomma, Starling Comma
225/224	\| -5 2 2 -1 >	7.71	Septimal Kleisma, Marvel Comma
19683/19600	\| -4 9 -2 -2 >	7.32	Cataharry
10976/10935	\| 5 -7 -1 3 >	6.48	Hemimage
3136/3125	\| 6 0 -5 2 >	6.08	Hemimean
	\| -11 2 7 -3 >	1.63	Meter
4375/4374	\| -1 -7 4 1 >	0.40	Ragisma
100/99	\| 2 -2 2 0 -1 >	17.40	Ptolemisma
896/891	\| 7 -4 0 1 -1 >	9.69	Pentacircle
65536/65219	\| 16 0 0 -2 -3 >	8.39	Orgonisma
385/384	\| -7 -1 1 1 1 >	4.50	Keenanisma
540/539	\| 2 3 1 -2 -1 >	3.21	Swetisma
91/90	\| -1 -2 -1 1 0 1 >	19.13	Superleap
676/675	\| 2 -3 -2 0 0 2 >	2.56	Parizeksma