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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), | '''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), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in 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 pump]]s 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 detemperings 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, 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 interval qualities supported by 159edo are many, 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 - resulting in [[slendric]] temperament and hence 159edo's distinction from 53 in the 7-limit - 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]] temperament. 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. | ||
On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. Once can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on. | On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us [[nexus]] temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. Once can also split the Ptolemaic minor third into three instances of 17/16, giving us [[archagall]] temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to [[parimic]] temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to [[fidesmic]] temperament, which acts like a more accurate rendition of [[archy]] temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to [[quartismic]] temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing [[phaotismic]] temperament, and the list goes on. | ||
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* The kleisma (the difference between a stack of three 25/24 intervals and 9/8) | * 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) | * The semicomma (the difference between a stack of three 75/64 intervals and 8/5) | ||
* | * The gamelisma (the difference between a stack of three 8/7 intervals and 3/2) | ||
* 385/384 (the difference between 77/64 and 6/5) | * 385/384 (the difference between 77/64 and 6/5) | ||
* | * The pine comma (the difference between a stack of three 11/10 intervals and 4/3) | ||
* 625/624 (the difference between 25/24 and 26/25) | * 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) | * 676/675 (the difference between a stack of two 15/13 intervals and the perfect fourth) | ||
* | * The twosquare comma (the difference between a stack of two 33/32 intervals and 17/16) | ||
==== 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. | 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; regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. 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}} | ||
| Line 111: | Line 111: | ||
|Lesser Supraminor | |Lesser Supraminor | ||
|Greater Supraminor | |Greater Supraminor | ||
|Artoneutral | |Artoneutral | ||
|Tendoneutral | |Tendoneutral | ||
|Lesser Submajor | |Lesser Submajor | ||
Revision as of 00:55, 29 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), however it includes near-just approximations of the 11th and 17th harmonics, and a slightly more accurate 7th harmonic, resulting in 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 detemperings 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, 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 - resulting in slendric temperament and hence 159edo's distinction from 53 in the 7-limit - 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 temperament. 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.
On top of all that, there are a number of microtemperament-based structures also supported by 159edo, each of which provides some decent, unexpected melodic possibilities. For instance, one can split the Pythagorean minor third into three instances of 128/121 which gives us nexus temperament- a temperament which also happens to split the Pythagorean diatonic semitone into two, and the octave into three. Once can also split the Ptolemaic minor third into three instances of 17/16, giving us archagall temperament, which is named for certain tunings found in other temperaments producing fractal-like acoustics. There's also the ability to split the Ptolemaic major sixth into six instances of 11/9, leading to parimic temperament. In addition, there's also the ability to split the septimal supermajor third into two instances of 17/15, leading to fidesmic temperament, which acts like a more accurate rendition of archy temperament in a different subgroup, and this can be exploited for modulation purposes. As if that weren't enough, there's the possibility of splitting the septimal subminor third into five instances of 33/32, leading to quartismic temperament. Furthermore, there's the possibility of splitting the greater tridecimal neutral tenth into three instances of 27/20, producing phaotismic temperament, and the list goes on.
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)
- The gamelisma (the difference between a stack of three 8/7 intervals and 3/2)
- 385/384 (the difference between 77/64 and 6/5)
- The pine comma (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)
- The twosquare comma (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; regardless, the inconsistency remains less than 10 cents even when either interval is stacked three times, and 13/7 or 14/13 is tuned almost perfectly. 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) | |
Currently, the ADIN system used for interval naming elsewhere on this site fails for 159edo, so another set of interval names will be used here, though the ADIN names will be referenced in places. Furthermore, because there are so many thirds, what is usually a single chart will be split into five.
| Quality | Narrow Inframinor | Inframinor | Wide Inframinor | Narrow Subminor | Lesser Subminor (Septiminor) | Greater Subminor | Wide Subminor |
|---|---|---|---|---|---|---|---|
| Cents | 234 | 242 | 249 | 257 | 264 | 272 | 279 |
| Just interpretation | 340/297 | 1024/891 | 15/13 | 51/44 | 7/6 | 117/100 | 20/17 |
| Steps | 31 | 32 | 33 | 34 | 35 | 36 | 37 |
| Quality | Narrow Minor (Gothminor) | Pythagorean Minor (Triminor) | Artomean Minor | Tendomean Minor | Ptolemaic Minor (Pentaminor) | Wide Minor |
|---|---|---|---|---|---|---|
| Cents | 287 | 294 | 302 | 309 | 317 | 325 |
| Just interpretation | 33/28, 13/11 | 32/27 | 25/21 | 153/128 | 6/5 | 135/112 |
| Steps | 38 | 39 | 40 | 41 | 42 | 43 |
| Quality | Lesser Supraminor | Greater Supraminor | Artoneutral | Tendoneutral | Lesser Submajor | Greater Submajor |
|---|---|---|---|---|---|---|
| Cents | 332 | 340 | 347 | 355 | 362 | 370 |
| Just interpretation | 40/33, 63/52 | 39/32, 17/14 | 11/9 | 27/22 | 16/13, 21/17 | 99/80, 26/21 |
| Steps | 44 | 45 | 46 | 47 | 48 | 49 |
| Quality | Narrow Major | Ptolemaic Major (Pentamajor) | Artomean Major | Tendomean Major | Pythagorean Major (Trimajor) | Wide Major (Gothmajor) |
|---|---|---|---|---|---|---|
| Cents | 377 | 385 | 392 | 400 | 408 | 415 |
| Just interpretation | 56/45 | 5/4 | 64/51 | 63/50 | 81/64 | 14/11, 33/26 |
| Steps | 50 | 51 | 52 | 53 | 54 | 55 |
| Quality | Narrow Supermajor | Lesser Supermajor | Greater Supermajor (Septimajor) | Wide Supermajor | Narrow Ultramajor | Ultramajor | Wide Ultramajor |
|---|---|---|---|---|---|---|---|
| Cents | 423 | 430 | 438 | 445 | 453 | 460 | 468 |
| Just interpretation | 51/40 | 50/39 | 9/7 | 22/17 | 13/10 | 2673/2048 | 891/680 |
| Steps | 56 | 57 | 58 | 59 | 60 | 61 | 62 |
Regular diatonic thirds are bolded.
Chords
159edo has a vast array of triads at its disposal, both fifth-bounded and fourth-bounded. However, the JI interpretation will tend to inform the usage of the various triads offered. The main exceptions to this rule involve chords that serve as detemperings of other, smaller tuning systems.
