270edo: Difference between revisions
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'''270edo''' divides the octave into 270 equal parts of 4.{{overline|4}} [[cent]]s each. | The '''270 equal divisions of the octave''' ('''270edo'''), or the '''270(-tone) equal temperament''' ('''270tet''', '''270et''') when viewed from a [[regular temperament]] perspective, divides the [[octave]] into 270 [[equal]] parts of 4.{{overline|4}} [[cent]]s each. | ||
== Theory == | == Theory == | ||
270edo is an extremely strong [[13-limit]] system, distinct and [[consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (and which can be interpreted as the result of tempering [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limits. It is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo and the 18th zeta peak integer edo]]. | 270edo is an extremely strong [[13-limit]] system, distinct and [[consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error with only the exception of [[15/13]] which barely misses (and which can be interpreted as the result of tempering [[676/675]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limits. It is [[The Riemann Zeta Function and Tuning #Zeta EDO lists|the 11th zeta gap edo, the 13th zeta integral edo, the 23rd zeta peak edo and the 18th zeta peak integer edo]]. | ||
In the [[5-limit]] it tempers out the ennealimma, {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[ | In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[vishnuzma]] (aka semisuper comma), {{monzo| 23 6 -14 }}. | ||
In the [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it supports [[ennealimmal]] temperament; the [[wizma]] (420175/419904) and the [[landscape comma]] (250047/250000). | In the [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it supports [[ennealimmal]] temperament; the [[wizma]] (420175/419904) and the [[landscape comma]] (250047/250000). | ||
In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers the 4 smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374 and 9801/9800). | In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers the 4 smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374 and 9801/9800). In addition to these, it also tempers out both the [[nexus comma]] (1771561/1769472) and the [[quartisma]] (117440512/117406179), which, in turn means that the [[symbiosma]] (19712/19683) is tempered out as well. | ||
Finally, in the [[13-limit]] it isn't quite as accurate but still very accurate, as it tempers out [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]], making it an [[The Archipelago|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments. | Finally, in the [[13-limit]] it isn't quite as accurate but still very accurate, as it tempers out [[676/675]], [[1001/1000]], [[1716/1715]] and [[2080/2079]], making it an [[The Archipelago|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments. | ||