159edo: Difference between revisions
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'''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 [[cent]]s each. | ||
== Theory == | == Theory == | ||
As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 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. | As the step size of 159edo is simultaneously above the average peak [http://musictheory.zentral.zone/huntsystem2.html#2 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 === | |||
{{Primes in edo|159|prec=2|columns=11}} | |||
=== Mappings === | === Mappings === | ||
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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. | 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 [[ | Notably, 159edo provides the [[optimal patent val]] for 11-limit guiron and 13-limit tritikleismic, as well as the 13-limit rank three temperament [[Gamelismic family #Portending|portending]]. In addition to this, it also supports [[Turkish maqam music temperaments|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|arabic/turkish/persian]] music. | ||
=== MOSes and | === MOSes and other scales === | ||
No less than five possible generators for [[5L 2s|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 diatonic scale that is slightly harder 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 related non-MOS scales like the Ptolemaic-Auric Diatonic Scale. | No less than five possible generators for [[5L 2s|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 diatonic scale that is slightly harder 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 related non-MOS scales like the Ptolemaic-Auric Diatonic Scale. | ||
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== Intervals == | == Intervals == | ||
{{main|Table of 159edo Intervals}} | {{main|Table of 159edo Intervals}} | ||
== Notation == | == Notation == | ||