270edo: Difference between revisions

Beyond the 13-limit, [[17/1|harmonic 17]] is almost 4/5-edostep sharp, but the [[19/1|harmonic 19]] is very accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. It tempers out [[715/714]], [[936/935]], [[1089/1088]], [[1225/1224]], [[1701/1700]], [[2025/2023]], [[2058/2057]], and [[2431/2430]] in the [[17-limit]]; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the [[19-limit]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.  

The harmonics [[29/1|29]] and [[31/1|31]] are also sharp, but not as sharp as the 17, so 29/13 and 31/13 are critically sharp. This makes 270edo consistent in the no-17 no-23 [[35-odd-limit]]. It tempers out [[784/783]], [[900/899]], and [[1024/1023]], while inflating [[841/840]] and [[961/960]].  

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}}{{c}}. 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 (and 19) with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.

| {{monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }}

| {{mapping| 270 428 627 }}

| {{mapping| 270 428 627 758 }}

| {{mapping| 270 428 627 758 934 }}

| {{mapping| 270 428 627 758 934 999 }}

| {{mapping| 270 428 627 758 934 999 1147 }}

| {{mapping| 270 428 627 758 934 999 1104 }}

| {{mapping| 270 428 627 758 934 999 1104 1147 }}

| {{mapping| 270 428 627 758 934 999 1104 1147 1221 }}

* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo|72]] in the 11-limit, [[224edo|224]] in the 13-limit, and [[217edo|217]] in the 19- and 23-limit. The next equal temperament that has lower error (any error) in the 11-limit is [[342edo|342]], in the 13-limit [[494edo|494]], in the 23-limit [[282edo|282]]; and in the 19-limit, [[311edo|311]] for absolute error and [[581edo|581]] for relative error.

* It is also great in the 17-limit, with lower absolute errors than smaller ETs despite inconsistency in the corresponding odd limit.