270edo: Difference between revisions
Improve wording to avoid an implication of causality; misc. wording improvements |
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Despite the excellent tuning accuracy, however, [[essentially tempered chord]]s exist, including [[sinbadmic chords]] in the 13-odd-limit and [[island chords]] in the 15-odd-limit. | Despite the excellent tuning accuracy, however, [[essentially tempered chord]]s exist, including [[sinbadmic chords]] in the 13-odd-limit and [[island chords]] in the 15-odd-limit. | ||
Beyond the 13-limit, the [[17/1|17]] is more than 1/3-edostep sharp of just, and while [[19/1|19]] is accurately tuned, the [[23/1|23]] is more than 1/3-edostep flat of just. [[17/13]], [[23/15]], and [[23/17]] are all the inconsistently approximated 23-odd-limit intervals, making 270edo a somewhat viable but tricky full 23-limit system. It tempers out [[715/714]], [[936/935]], [[1089/1088]], [[1225/1224]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; [[460/459]], [[897/896]], [[1288/1287]], 1311/1309, 1771/1768 in the 23-limit. | |||
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 very 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 very 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. | ||
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270 is a very composite number. The prime factorization is 270 = 2 × 3<sup>3</sup> × 5, with divisors {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135 }}. This means that 270edo can be conceptualised as the superset of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right. | 270 is a very composite number. The prime factorization is 270 = 2 × 3<sup>3</sup> × 5, with divisors {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135 }}. This means that 270edo can be conceptualised as the superset of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right. | ||
[[540edo]], which divides the edostep in two, provides good correction for | [[540edo]], which divides the edostep in two, provides good correction for harmonics 17, 23, and beyond. | ||
== Intervals == | == Intervals == | ||