User:Hkm/19edo

Harmonic		2	3	5	7	11	13	17	19	23
Error	Absolute	+0.0	-7.2	-7.4	-21.5	+17.1	-19.5	+21.4	+18.3	+3.3
	Relative	you	get	the	gist
Steps	Expanded	i'm	not	filling	the	entire	table	manu	ally
	Reduced

Degree	Cents	Interval	Approximated JI	Solfege
0	0.00	P1	1/1	Do
1	63.16	A1	25/24, 26/25, 28/27	Di/Ro
2	126.32	m2	13/12, 14/13, 15/14, 16/15	Ra
3	189.47	M2	9/8, 10/9	Re
4	252.63	A2/d3	7/6, 8/7, 15/13	Ri/Ma
5	315.79	m3	6/5	Me
6	378.95	M3	5/4, 16/13, 56/45	Mi
7	442.11	A3/d4	9/7, 13/10, 32/25	Mo/Fe
8	505.26	P4	4/3, 75/56	Fa
9	568.42	A4	7/5, 18/13, 25/18	Fi
10	631.58	d5	10/7, 13/9, 36/25	Se
11	694.74	P5	3/2, 112/75	So
12	757.89	A5	14/9, 20/13, 25/16	Si/Lo
13	821.05	m6	8/5, 13/8, 45/28	Le
14	884.21	M6	5/3	La
15	947.37	A6/d7	7/4, 12/7, 26/15	Li/Ta
16	1010.53	m7	9/5, 16/9	Te
17	1073.68	M7	13/7, 15/8, 24/13, 28/15	Ti
18	1136.84	A7/d8	25/13, 27/14, 48/25	To/Da
19	1200.00	P8	2/1	Do

Step offset −2 −1 0 +1 +2 Symbol

Because it includes no Sagittal symbols, this Evo Sagittal notation is also a conventional notation.

Approximation to JI

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

Zeta peak index

Tuning			Strength				Octave (cents)		Integer limit
ZPI	Steps per 8ve	Step size (cents)	Height		Integral	Gap	Size	Stretch	Consistent	Distinct
			Tempered	Pure
65zpi	18.948	63.33	5.98	5.214	1.313	16.699	1203.287	3.287	10	7

Regular temperament properties

Subgroup	Comma list	Mapping	Optimal 8ve stretch (¢)	Tuning error
				Absolute (¢)	Relative (%)
2.3	[-30 19⟩	[⟨19 30]]	+2.28	2.28	3.61
2.3.5	81/80, 3125/3072	[⟨19 30 44]]	+2.58	1.91	3.02
2.3.5.7	49/48, 81/80, 126/125	[⟨19 30 44 53]]	+3.85	2.76	4.35
2.3.5.7.13	49/48, 65/64, 81/80, 91/90	[⟨19 30 44 53 70]]	+4.14	2.53	3.99
2.3.5.7.13.23	49/48, 65/64, 70/69, 81/80, 91/90	[⟨19 30 44 53 70 86]]	+3.32	2.93	4.64

• 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	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 -⟩	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

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
• Syntonic–kleismic equivalence continuum

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 Spell
4	252.63	A2, d3	1L 3s, 4L 1s, 5L 4s, 5L 9s	Godzilla
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 / pycnic Triton

Scales

MOS scales

Main article: List of MOS scales in Hkm/19edo

Octave-equivalent mosses

• meantone pentatonic, 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

• Meantone harmonic minor: 3 2 3 3 2 4 2
• Meantone melodic minor: 3 2 3 3 3 3 2
• 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

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

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.