270edo: Difference between revisions
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== Theory == | == Theory == | ||
270edo is an extremely strong [[13-limit]] system, | 270edo is an extremely strong [[13-limit]] system, [[consistency|distinctly 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-limit. 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 [[vishnuzma]] (aka semisuper comma), {{monzo| 23 6 -14 }}. | 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 }}. | ||
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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. | ||
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 | 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 === | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
270 is a very composite number. The prime factorization is | 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 the 17th and 23rd harmonics. | [[540edo]], which divides the edostep in two, provides good correction for the 17th and 23rd harmonics. | ||
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| 2.3.5 | | 2.3.5 | ||
| {{monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }} | | {{monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }} | ||
| | | {{mapping| 270 428 627 }} | ||
| -0.1069 | | -0.1069 | ||
| 0.0759 | | 0.0759 | ||
| Line 48: | Line 48: | ||
| 2.3.5.7 | | 2.3.5.7 | ||
| 2401/2400, 4375/4374, 29360128/29296875 | | 2401/2400, 4375/4374, 29360128/29296875 | ||
| | | {{mapping| 270 428 627 758 }} | ||
| -0.0858 | | -0.0858 | ||
| 0.0752 | | 0.0752 | ||
| Line 55: | Line 55: | ||
| 2.3.5.7.11 | | 2.3.5.7.11 | ||
| 2401/2400, 3025/3024, 4375/4374, 5632/5625 | | 2401/2400, 3025/3024, 4375/4374, 5632/5625 | ||
| | | {{mapping| 270 428 627 758 934 }} | ||
| -0.0567 | | -0.0567 | ||
| 0.0889 | | 0.0889 | ||
| Line 62: | Line 62: | ||
| 2.3.5.7.11.13 | | 2.3.5.7.11.13 | ||
| 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095 | | 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095 | ||
| | | {{mapping| 270 428 627 758 934 999 }} | ||
| -0.0235 | | -0.0235 | ||
| 0.1100 | | 0.1100 | ||
| Line 69: | Line 69: | ||
| 2.3.5.7.11.13.19 | | 2.3.5.7.11.13.19 | ||
| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728 | | 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728 | ||
| | | {{mapping| 270 428 627 758 934 999 1147 }} | ||
| -0.0290 | | -0.0290 | ||
| 0.1028 | | 0.1028 | ||
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|+Table of rank-2 temperaments by generator | |+Table of rank-2 temperaments by generator | ||
! Periods<br>per 8ve | ! Periods<br>per 8ve | ||
! Generator | ! Generator* | ||
! Cents | ! Cents* | ||
! Associated<br>Ratio | ! Associated<br>Ratio* | ||
! Temperaments | ! Temperaments | ||
|- | |- | ||
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| [[Rhodium]] | | [[Rhodium]] | ||
|} | |} | ||
<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct | |||