270edo: Difference between revisions
→Theory: wizma and landscape comma are implied commas of ennealimmal, so I'm grouping them together. +two 7-limit commas that aren't related to ennealimmal. |
→Theory: note the quality of 29 and 31. Extend the prime error table to include 47, the last prime of mode 24 |
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The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, including [[sinbadmic chords]] in the 13-odd-limit and [[island chords]] in the 15-odd-limit. | The excellent tuning accuracy does not bar it from the utility of [[essentially tempered chord]]s, 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]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, 1771/1768 in the 23-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]], [[1701/1700]], [[2025/2023]], [[2058/2057]], [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], [[1729/1728]] in the 19-limit; [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, 1771/1768 in the 23-limit. The [[29/1]] and [[31/1|31]] are also more than 1/3-edostep sharp, but not as sharp as 17 to incur inconsistency with the lower primes. In fact, 270edo is consistent in the no-17 no-23 [[35-odd-limit]]. We may note it tempers out [[784/783]], [[900/899]], and [[1024/1023]]. | ||
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. | ||
=== Prime harmonics === | === Prime harmonics === | ||
{{Harmonics in equal|270|prec=3|columns= | {{Harmonics in equal|270|prec=3|columns=15}} | ||
=== Subsets and supersets === | === Subsets and supersets === | ||