159edo: Difference between revisions
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'''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), but this time, you have access to near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, giving you consistency up to the 17-odd-limit. The step-size, being slightly above the melodic JND of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pumps]] as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by having a pitch hue palette that's capable of imitating the pitch-hue palettes of smaller tuning systems- in this case, you get detunings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND. | '''159edo''', or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave [[2/1]]. Like [[53edo]], 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), but this time, you have access to near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, giving you consistency up to the 17-odd-limit. The step-size, being slightly above the melodic JND of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of [[comma pumps]] as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by having a pitch hue palette that's capable of imitating the pitch-hue palettes of smaller tuning systems- in this case, you get detunings of [[10edo]], [[12edo]], [[13edo]], [[14edo]], [[17edo]], [[19edo]], [[22edo]], [[24edo]] and [[31edo]] among others with errors smaller than the melodic JND. | ||
The interval qualities supported by 159edo are many, and there are a number of microtemperament-based structures also supported by 159edo. However, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo, the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us island temperament. Dividing the perfect fourth into three instances of 11/10 gives us pine temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us sextilifourths. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the interpretation of the semilimma. | The interval qualities supported by 159edo are many, and there are a number of microtemperament-based structures also supported by 159edo. However, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the [[perfect fifth]] is really only divisible by three due to 159edo being the three-fold multiple of 53edo, the [[perfect fourth]] has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us island temperament. Dividing the perfect fourth into three instances of 11/10 gives us pine temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us sextilifourths. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma. | ||
== Theory == | == Theory == | ||
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* 289/288, the difference between 17/16 and 18/17 | * 289/288, the difference between 17/16 and 18/17 | ||
159edo tempers out the following commas: | 159edo tempers out the following commas in the 17-limit: | ||
* The schisma (the difference between 5/4 and the Pythagorean diminished fourth) | * The schisma (the difference between 5/4 and the Pythagorean diminished fourth) | ||
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==== JI approximation ==== | ==== JI approximation ==== | ||
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, marveltwin, nestoria, minor semivicemic and brunisimic temperaments. | Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the [[slendric]], [[marveltwin]], [[nestoria]], [[minor semivicemic]] and [[brunisimic]] temperaments. | ||
{{Harmonics in ED|159|31|0}} | {{Harmonics in ED|159|31|0}} | ||
Revision as of 21:00, 28 March 2026
159edo, or 159 equal divisions of the octave, is the equal tuning featuring steps of (1200/159) ~= 7.55 cents, 159 of which stack to the perfect octave 2/1. Like 53edo, 159edo is an excellent approximation to Pythagorean tuning (stacking pure 3/2 fifths), but this time, you have access to near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, giving you consistency up to the 17-odd-limit. The step-size, being slightly above the melodic JND of 5 cents as well as more than twice the harmonic JND of the average trained musician at 3.5 cents, enables one to perform fluid modulations by means of comma pumps as well as by step substitutions. Furthermore, 159edo, like a number of higher edos, is characterized by having a pitch hue palette that's capable of imitating the pitch-hue palettes of smaller tuning systems- in this case, you get detunings of 10edo, 12edo, 13edo, 14edo, 17edo, 19edo, 22edo, 24edo and 31edo among others with errors smaller than the melodic JND.
The interval qualities supported by 159edo are many, and there are a number of microtemperament-based structures also supported by 159edo. However, while every step of 159edo can be interpreted harmonically or subharmonically as being a 17-limit interval or simpler, some of the intervals you get have rather complex interpretations in terms of odd-limit. While the perfect fifth is really only divisible by three due to 159edo being the three-fold multiple of 53edo, the perfect fourth has a little more to offer in terms of divisions. For starters, the perfect fourth can be divided into two instances of 15/13, giving us island temperament. Dividing the perfect fourth into three instances of 11/10 gives us pine temperament. Dividing the perfect fourth into six instances of an interval which can be interpreted as 21/20 and 22/21 tempered together gives us sextilifourths. The perfect fourth can also be cut into eleven intervals which, individually, are half of a Pythagorean limma, giving us a number of temperaments based on the exact interpretation of the semilimma.
Theory
159edo was first used for maqams by Ozan Yarman. It was later put to use by Aura for its ability to handle near-just quartertones derived from the 2.3.11 subgroup on top of the 5-limit foundation provided by 53edo.
Edostep interpretations
159edo's edostep has the following interpretations in the 2.3.5.11.17 subgroup:
- 243/242, the difference between the 11-limit artoneutral third 11/9, and the 11-limit tendoneutral third 27/22
- 256/255, the difference between 16/15 and 17/16
- 289/288, the difference between 17/16 and 18/17
159edo tempers out the following commas in the 17-limit:
- The schisma (the difference between 5/4 and the Pythagorean diminished fourth)
- The vulture comma (the difference between four 320/243 intervals and the tritave)
- The amiton (the difference between a stack of five 10/9 intervals and 27/16)
- The kleisma (the difference between a stack of three 25/24 intervals and 9/8)
- The semicomma (the difference between a stack of three 75/64 intervals and 8/5)
- 1029/1024 (the difference between a stack of three 8/7 intervals and 3/2)
- 385/384 (the difference between 77/64 and 6/5)
- 4000/3993 (the difference between a stack of three 11/10 intervals and 4/3)
- 625/624 (the difference between 25/24 and 26/25)
- 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth)
- 1089/1088 (the difference between a stack of two 33/32 intervals and 17/16)
JI approximation
Although 159edo inherits its approximations of the 5-limit from 53edo, the 5th harmonic can nonetheless be stacked twice without accumulating too much error, rendering it sufficient for Western Classical usage. While the 7th harmonic is technically more accurate in terms of absolute error than in 53edo, the relative error doesn't allow one to stack more than one instance of 7/4 without excessive error accumulation, and the same is true with 13/8. As a whole, 159edo is characterized by its combination of accuracy in the 2.3.5.11.17 subgroup, and a series of compromises in the 7.13.19.23.29 subgroup- among the compromises are the slendric, marveltwin, nestoria, minor semivicemic and brunisimic temperaments.
| Harmonic | 2 | 3 | 5 | 7 | 11 | 13 | 17 | 19 | 23 | 29 | 31 | |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Error | Absolute (¢) | 0.0 | -0.1 | -1.4 | -2.8 | -0.4 | -2.8 | +0.7 | -3.2 | -1.9 | -3.2 | +2.1 |
| Relative (%) | 0.0 | -0.9 | -18.7 | -36.9 | -5.0 | -37.0 | +9.3 | -42.0 | -24.6 | -41.9 | +28.3 | |
| Steps
(reduced) |
159
(0) |
252
(93) |
369
(51) |
446
(128) |
550
(73) |
588
(111) |
650
(14) |
675
(39) |
719
(83) |
772
(136) |
788
(152) | |
