270edo: Difference between revisions
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{{Infobox ET | |||
| Prime factorization = 2 × 3<sup>3</sup> × 5 | |||
| Step size = 4.44444¢ | |||
| Fifth = 158\270 (702.22¢) (→ [[135edo|79\135]]) | |||
| Semitones = 26:20 (115.56¢ : 88.89¢) | |||
| Consistency = 15 | |||
}} | |||
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. | 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. | ||
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On top of this, its step size is so small as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error is only 2.{{overline|2}}¢. If, however, you want an edo for very high-limit use, the obvious alternative choice is [[311edo]], which is in many ways dual to 270edo as it emphasizes consistency and accuracy in very high-prime-limit and high-odd-limit situations at the expense of lower ones, and is a prime EDO as opposed to a highly composite one. While 270edo approximates the first 16 harmonics very accurately, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible. | On top of this, its step size is so small as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error is only 2.{{overline|2}}¢. If, however, you want an edo for very high-limit use, the obvious alternative choice is [[311edo]], which is in many ways dual to 270edo as it emphasizes consistency and accuracy in very high-prime-limit and high-odd-limit situations at the expense of lower ones, and is a prime EDO as opposed to a highly composite one. While 270edo approximates the first 16 harmonics very accurately, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible. | ||
=== Prime | === Prime harmonics === | ||
{{Primes in edo|270|prec=3}} | {{Primes in edo|270|prec=3}} | ||
== Divisors == | === Divisors === | ||
270 is a very composite number, with divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135, and some of these form the periods of | 270 is a very composite number. The prime factorization is: 270 = 2 × 3<sup>3</sup> × 5, with divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135, and some of these form the periods of some rank two temperaments 270 supports; these include [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], hemiennealimmal and [[The Archipelago #Rank two temperaments|decitonic]]. This means that 270edo can be conceptualised as the superset/intersection of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right. | ||
== Intervals == | == Intervals == | ||