159edo
The 159 equal divisions of the octave (159edo), or the 159(-tone) equal temperament (159tet, 159et) when viewed from a regular temperament perspective, divides the octave into 159 equal parts of about 7.55 cents each.
Theory
As the step size of 159edo is simultaneously above the average peak JND of human pitch perception and small enough to be well within the margin of error between Just 5-limit intervals and their 12edo counterparts, 159edo offers a decent balance between allowing the possibility of seamless modulation to keys that are not in the same series of fifths, and not having a step-size so small as to have individual steps blend completely into one another. Thus, it can be said that 159edo falls in what can perhaps be considered the ideal range for a Mega-EDO in terms of possible musical functionality outside of pitch bends.
Prime harmonics
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Mappings
A salient fact about 159edo is that 159 = 3 × 53, and it shares the same 3rd, 5th and 13th harmonics with 53edo. However, compared to 53edo, the patent vals differ on the mappings for 7, 11 and 17 – in fact, this EDO has a very accurate 11 and an only slightly less accurate 17. Although 159edo is consistent up to the 17-odd-limit, it proves to be inconsistent in the 19-odd-limit, with the 19th harmonic having multiple reasonable mappings. Furthermore, 159edo demonstrates 3-to-2 telicity, as despite being contorted in the 5-limit, it is the largest EDO to temper out Mercator's comma in which said comma is less than half the size of a single EDO step. This means, among other things, that there is a perfect match between the direct mapping and the more complicated traditional mapping for an octave-reduced stack of fifty-three tempered 3/2 perfect fifths – a complete circle of fifths for this EDO. However, for intervals such as 49/32 and 128/125, these two mappings don't match. While the patent val supports cartography temperament, which is among the best 13-limit temperaments in the Mercator family, the 159d and 159e mappings support other members of this temperament family.
Commas
In the 5-limit, it tempers out the same commas as 53edo, including 15625/15552, 32805/32768, 1600000/1594323, 2109375/2097152, and 10485760000/10460353203.
In the 7-limit, it tempers out 1029/1024, 10976/10935, 117649/116640, 250047/250000, 235298/234375 and 703125/702464; this makes it among other things an excellent tuning for guiron and tritikleismic temperaments, as well as a possible tuning for metric temperament.
In the 11-limit, it tempers out not only 385/384, 441/440, 3025/3024, 4000/3993, 4375/4356, and 6250/6237, but both 1771561/1769472 and 117440512/117406179, which in turn means that 19712/19683 is tempered out as well.
In the 13-limit it tempers out 325/324, 364/363, 625/624, 676/675, 1001/1000, 1575/1573, 2080/2079, 6656/6655, 10985/10976 and 13720/13689.
In the 17-limit tempers out 273/272, 375/374, 595/594, 715/714, 833/832, 936/935, 1089/1088, 1701/1700, 8624/8619, 11271/11264, 15379/15300, and 24576/24565.
In the 19-limit, it is known to temper out 343/342 and 361/360, but since it is inconsistent in the 19-limit, there are other potential mappings available that temper out different commas.
In addition to the above, 159edo actually tempers out the 7-limit termite comma and the 13-limit chalmersia, as well as the 17-limit sparkisma, the latter of which is also tempered out by 53edo despite it having a different mapping for 17.
Notably, 159edo provides the optimal patent val for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament portending. In addition to this, it also supports yarman temperament, with a generator of 2\159 which can be taken as an approximate 105/104. 159 supplies the optimal patent val for 7, 11, 13, 17 and 19-limit yarman, so they are very closely associated. Curiously, the temperament does not temper out 1029/1024, however. Yarman temperament has MOS of 79 and 80 notes to the octave, and the 79-note MOS has been proposed by Ozan Yarman as a tuning standard for arabic/turkish/persian music.
MOSes and other scales
No less than five possible generators for the Diatonic MOS Scale are supported by 159edo. The 91\159 generator results in large and small scale steps at 23\159 and 22\159 respectively, making for a quasi-equalized scale, while the 95\159 results in large and small scale steps at 31\159 and 2\159 respectively, making for a version approaching paucitonic. The 92\159 generator results in large and small scale steps at 25\159 and 17\159 respectively, and this makes for a very meantone-like diatonic scale perfect for xenharmonic pieces that follow in the classical tradition. Conversely, the 94\159 generator results in results in large and small scale steps at 29\159 and 7\159 respectively, and this makes for a superpyth diatonic scale that is slightly harder and better than that of 22edo. Finally, the patent 93\159 generator results in the same diatonic MOS scale found in 53edo, which, despite now having competition from other possible generators, is still the go-to for those looking for something more akin to the classic Pythagorean tuning, as well as for those looking to deal with good approximations of related 5-limit scales.
In addition, 159edo has no less than four possible generators for the Oneirotonic MOS Scale, and of these, two of them are also supported by 53edo. The 60\159 generator results in large and small scale steps at 21\159 and 18\159 respectively, making for a distinctly ultra-soft scale, while the 63\159 generator results in large and small scale steps at 30\159 and 3\159 respectively, making for a distinctly ultra-hard scale. As for the remaining two generators, the 61\159 generator results in large and small scale steps at 24\159 and 13\159 respectively and comes the closest to any sort of basic form of this scale, however, the 62\159 generator is also a solid choice, and is also useful for at least one related non-MOS scale due to 62\159 approximating 21/16.
Intervals
Notation
Because of the complexity of 159edo, notation requires systems that make use of multiple extra pairs of accidentals. This is because at high EDOs, systems with only a single extra accidental pair become unwieldy due to the sheer number of such accidentals required for notating some pitches, which in turn results in high amounts of clutter on scores. So far, several notation systems addressing this problem have been proposed.
Regular temperament properties
| Subgroup | Comma list | Mapping | Optimal 8ve stretch (¢) |
Tuning error | |
|---|---|---|---|---|---|
| Absolute (¢) | Relative (%) | ||||
| 2.3.5.7 | 1029/1024, 10976/10935, 15625/15552 | [⟨159 252 369 446]] | +0.411 | 0.413 | 5.47 |
| 2.3.5.7.11 | 385/384, 441/440, 4000/3993, 10976/10935 | [⟨159 252 369 446 550]] | +0.350 | 0.389 | 5.15 |
| 2.3.5.7.11.13 | 325/324, 364/363, 385/384, 625/624, 10976/10935 | [⟨159 252 369 446 550 588]] | +0.418 | 0.385 | 5.11 |
| 2.3.5.7.11.13.17 | 273/272, 325/324, 364/363, 375/374, 385/384, 3773/3757 | [⟨159 252 369 446 550 588 650]] | +0.333 | 0.412 | 5.46 |
Rank-2 temperaments
Note: Temperaments supported by 53et are not included unless they are also supported by at least one other equal temperament.
| Periods per octave |
Generator (reduced) |
Cents (reduced) |
Associated ratio |
Temperament |
|---|---|---|---|---|
| 1 | 2\159 | 15.094 | 121/120 | Yarman I / yarman II |
| 1 | 11\159 | 83.019 | 21/20 | Sextilififths |
| 1 | 22\159 | 166.038 | 11/10 | Tertiaschis |
| 1 | 31\159 | 233.962 | 8/7 | Slendric / guiron |
| 1 | 38\159 | 286.792 | 13/11 | Gamity |
| 1 | 41\159 | 309.434 | 448/375 | Triwell |
| 1 | 64\159 | 483.019 | 160/121 | Quarterframe |
| 1 | 67\159 | 505.660 | 75/56 | Marfifths |
| 1 | 85\159 | 641.509 | 81/56 | Condor |
| 3 | 8\159 | 60.377 | 28/27 | Chromat |
| 3 | 20\159 | 150.943 | 12/11 | Altinex |
| 3 | 22\159 | 166.038 | 11/10 | Tritricot |
| 3 | 42\159 (11\159) |
316.981 (83.019) |
6/5 (21/20) |
Tritikleismic |
| 3 | 66\159 (13\159) |
498.113 (98.113) |
4/3 (35/33) |
Term / terminal |
| 53 | 31\159 (1\159) |
233.962 (7.547) |
8/7 (225/224) |
Schismerc / cartography |
Music
The songs below are written in approximations of 159edo that differ from the actual 159edo by only fractions of a cent.
Articles
- 79-Tone Tuning & Theory for Turkish Maqam Music – Ozan Yarman's dissertation
- Search For A Theoretical Model Conforming To Turkish Maqam Music Practice: A Selection Of Fixed-Pitch Settings From 34-tone Equal Temperament To The 79-tone Tuning – also by Ozan Yarman, gives a summary.
- Letter to Ozan Yarman by Margo Schulter (permalink)