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'''19edo''' is an equal division of 2/1 into 19 steps of 1200c/19 | '''19edo''' is an equal division of 2/1 into 19 steps of 1200c/19 ~= 63.2c each. | ||
19edo is interesting as a flatter [[Meantone]] system; it is in fact very close to 1/3-comma Meantone (i.e. Meantone with exact 6/5). It has [[interordinal]]s and supports [[Semaphore]], which means that it treats half of its perfect fourth as a septimal (8/7) major second, leading to a 9-note MOS semiquartal (5L4s) that can be created from stacking its 8/7. 19edo is also a tuning of the following temperaments which provide non-12edo [[comma pump|comma loop]] progressions: | |||
* [[Kleismic]] temperament (six 6/5's = 3/1) but with lower accuracy due to its 6/5 being barely sharpened at all | |||
* [[Magic]] temperament (five 5/4's = 3/1) with the generator 5/4 on the flat end for Magic | |||
* [[Negri]], equating four 16/15's to one 4/3. | |||
An early use of 19edo was in the mid-1500s by French composer Guillaume Costeley in his chanson "Seigneur Dieu ta pitie", in which he uses mostly the standard consonances from the diatonic scale (perfect fifth, major third, etc.) as contrapuntal consonances, but incorporates motions by a third of a tone that are outside the diatonic scale. A keyboard rendition of the chanson can be heard on [https://www.youtube.com/watch?v=wT6-Ndx1EbM YouTube]. | |||
== Basic theory == | |||
=== Intervals and notation === | |||
19edo can be notated entirely with standard [[diatonic notation]], with #/b = 1\19 and x/bb = 2\19, and equivalences E# = Fb and B# = Cb. | |||
=== Prime harmonic approximations === | |||
{{Harmonics in ED|19|23|0}} | |||
{{Cat|Edos}}19edo has a reasonable approximation of 9/7 and is a good 2.3.5.7.13 temperament for its size. Its actual mappings of 7 and 13 are inaccurate, however, and so it can be seen as a 2.<3.<5.<<<7.<<<13 tuning, where 28/27, 13/7, and 125/112 are approximated very accurately, and 9/7, 15/13, and 28/25 are approximated somewhat accurately. | |||
As most of 19edo's errors on primes are approximately a third of an edostep, that suggests that 57edo is a good tuning in 2.7.11.13.17.19. | |||
=== Edostep interpretations === | |||
19edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup: | |||
* 25/24 (the interval between 6/5 and 5/4) | |||
* 26/25 (the interval between 25/16 and 13/8) | |||
* 27/26 (the interval between 13/8 and 27/16) | |||
* 28/27 (the interval between 9/8 and 7/6) | |||
25/24 ~ 26/25 ~ 27/26 is an equivalence that defines 2.3.5.13 Kleismic. | |||
=== Scales === | |||
Basic or soft MOSes in 19edo include: | |||
* diatonic | |||
* checkertonic (3-2-3-2-2-3-2-2) | |||
* manual (4-4-4-4-3) | |||
* antimachinoid (3-3-3-3-3-4) | |||
* antisubneutralic (2-2-2-2-2-2-2-2-3) | |||
Due to being a meantone tuning with an interval in between 5/4 and 4/3, 19edo has a 7-limit [[omnidiatonic]] scale (3-4-1-3-4-3-1). | |||
The semiquartal (5L4s) MOS (3-1-3-1-3-1-3-1-3) is also commonly used in 19edo. Also note the Kleismic MOSes: | |||
* 4L7s, interpreted as Kleismic[11]: 1 1 3 1 1 3 1 1 3 1 4 | |||
* 4L11s, Kleismic[15]: 1 1 1 2 1 1 1 2 1 1 1 2 1 1 2 | |||
== Multiples == | |||
=== 57edo === | |||
57edo is a good no-threes no-fives 19-limit or 2.5/3.7.11.13.19 subgroup system. | |||
{{Harmonics in ED|57}} | |||
=== 171edo === | |||
Combining 19edo's accurate 28/27 and 6/5 with [[9edo]]'s accurate 7/6 results in 171edo which is a very accurate model of 7-limit JI, even more accurate than [[99edo]]. Since its 14/13 is inherited from 19edo, its 13th harmonic is acceptable too. 171edo supports [[Schismic]] and [[Ennealimmal]]. | |||
{{Harmonics in ED|171}} | |||
=== 665edo === | |||
665edo = 19 * 35 is mostly known for its absurdly accurate Pythagorean tuning, supporting the "Satanic" temperament equating a stack of 666 perfect fifths (octave reduced) to a single perfect fifth, with an accuracy within 1/1000 of a cent. It additionally shares 19edo's tunings of 6/5 and 28/27, supporting Enneadecal temperament, and is a strong no-11 23-limit tuning. | |||
{{Harmonics in ED|665}} | |||
{{Navbox EDO}} | |||
{{Cat|Edos}} | {{Cat|Edos}} | ||
Latest revision as of 23:04, 11 May 2026
19edo is an equal division of 2/1 into 19 steps of 1200c/19 ~= 63.2c each.
19edo is interesting as a flatter Meantone system; it is in fact very close to 1/3-comma Meantone (i.e. Meantone with exact 6/5). It has interordinals and supports Semaphore, which means that it treats half of its perfect fourth as a septimal (8/7) major second, leading to a 9-note MOS semiquartal (5L4s) that can be created from stacking its 8/7. 19edo is also a tuning of the following temperaments which provide non-12edo comma loop progressions:
- Kleismic temperament (six 6/5's = 3/1) but with lower accuracy due to its 6/5 being barely sharpened at all
- Magic temperament (five 5/4's = 3/1) with the generator 5/4 on the flat end for Magic
- Negri, equating four 16/15's to one 4/3.
An early use of 19edo was in the mid-1500s by French composer Guillaume Costeley in his chanson "Seigneur Dieu ta pitie", in which he uses mostly the standard consonances from the diatonic scale (perfect fifth, major third, etc.) as contrapuntal consonances, but incorporates motions by a third of a tone that are outside the diatonic scale. A keyboard rendition of the chanson can be heard on YouTube.
Basic theory
Intervals and notation
19edo can be notated entirely with standard diatonic notation, with #/b = 1\19 and x/bb = 2\19, and equivalences E# = Fb and B# = Cb.
Prime harmonic approximations
| 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 (%) | 0.0 | -11.4 | -11.7 | -34.0 | +27.1 | -30.8 | +33.8 | +28.9 | +5.2 | |
| Steps
(reduced) |
19
(0) |
30
(11) |
44
(6) |
53
(15) |
66
(9) |
70
(13) |
78
(2) |
81
(5) |
86
(10) | |
19edo has a reasonable approximation of 9/7 and is a good 2.3.5.7.13 temperament for its size. Its actual mappings of 7 and 13 are inaccurate, however, and so it can be seen as a 2.<3.<5.<<<7.<<<13 tuning, where 28/27, 13/7, and 125/112 are approximated very accurately, and 9/7, 15/13, and 28/25 are approximated somewhat accurately.
As most of 19edo's errors on primes are approximately a third of an edostep, that suggests that 57edo is a good tuning in 2.7.11.13.17.19.
Edostep interpretations
19edo's edostep has the following interpretations in the 2.3.5.7.13 subgroup:
- 25/24 (the interval between 6/5 and 5/4)
- 26/25 (the interval between 25/16 and 13/8)
- 27/26 (the interval between 13/8 and 27/16)
- 28/27 (the interval between 9/8 and 7/6)
25/24 ~ 26/25 ~ 27/26 is an equivalence that defines 2.3.5.13 Kleismic.
Scales
Basic or soft MOSes in 19edo include:
- diatonic
- checkertonic (3-2-3-2-2-3-2-2)
- manual (4-4-4-4-3)
- antimachinoid (3-3-3-3-3-4)
- antisubneutralic (2-2-2-2-2-2-2-2-3)
Due to being a meantone tuning with an interval in between 5/4 and 4/3, 19edo has a 7-limit omnidiatonic scale (3-4-1-3-4-3-1).
The semiquartal (5L4s) MOS (3-1-3-1-3-1-3-1-3) is also commonly used in 19edo. Also note the Kleismic MOSes:
- 4L7s, interpreted as Kleismic[11]: 1 1 3 1 1 3 1 1 3 1 4
- 4L11s, Kleismic[15]: 1 1 1 2 1 1 1 2 1 1 1 2 1 1 2
Multiples
57edo
57edo is a good no-threes no-fives 19-limit or 2.5/3.7.11.13.19 subgroup system.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -7.2 | -7.4 | -0.4 | -3.9 | +1.6 | +0.3 | -2.8 | +3.3 | +2.0 | -8.2 |
| Relative (%) | 0.0 | -34.3 | -35.0 | -1.9 | -18.8 | +7.5 | +1.5 | -13.2 | +15.7 | +9.5 | -38.9 | |
| Steps
(reduced) |
57
(0) |
90
(33) |
132
(18) |
160
(46) |
197
(26) |
211
(40) |
233
(5) |
242
(14) |
258
(30) |
277
(49) |
282
(54) | |
171edo
Combining 19edo's accurate 28/27 and 6/5 with 9edo's accurate 7/6 results in 171edo which is a very accurate model of 7-limit JI, even more accurate than 99edo. Since its 14/13 is inherited from 19edo, its 13th harmonic is acceptable too. 171edo supports Schismic and Ennealimmal.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -0.2 | -0.3 | -0.4 | +3.1 | +1.6 | +0.3 | -2.8 | +3.3 | +2.0 | -1.2 |
| Relative (%) | 0.0 | -2.9 | -5.0 | -5.8 | +43.7 | +22.5 | +4.4 | -39.6 | +47.1 | +28.5 | -16.8 | |
| Steps
(reduced) |
171
(0) |
271
(100) |
397
(55) |
480
(138) |
592
(79) |
633
(120) |
699
(15) |
726
(42) |
774
(90) |
831
(147) |
847
(163) | |
665edo
665edo = 19 * 35 is mostly known for its absurdly accurate Pythagorean tuning, supporting the "Satanic" temperament equating a stack of 666 perfect fifths (octave reduced) to a single perfect fifth, with an accuracy within 1/1000 of a cent. It additionally shares 19edo's tunings of 6/5 and 28/27, supporting Enneadecal temperament, and is a strong no-11 23-limit tuning.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -0.0 | -0.1 | +0.2 | +0.9 | +0.4 | -0.3 | +0.2 | -0.3 | +0.8 | +0.8 |
| Relative (%) | 0.0 | -0.0 | -8.2 | +10.9 | +47.8 | +20.8 | -16.3 | +12.8 | -16.9 | +44.3 | +45.9 | |
| Steps
(reduced) |
665
(0) |
1054
(389) |
1544
(214) |
1867
(537) |
2301
(306) |
2461
(466) |
2718
(58) |
2825
(165) |
3008
(348) |
3231
(571) |
3295
(635) | |
