270edo: Difference between revisions
→Theory: rework to better reflect its 19-limit capability |
Added and reworded to make more succint |
||
| Line 3: | Line 3: | ||
== Theory == | == Theory == | ||
270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error except [[15/13]] and [[26/15]] which barely miss ( | 270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error except [[15/13]] and [[26/15]] which barely miss ([[tempering out]] [[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]], making it a strict zeta edo. It also has the lowest 13-limit logflat badness of any edo until [[96478edo|96478]]. | ||
In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[vishnuzma]] (a.k.a. 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]] (a.k.a. semisuper comma), {{monzo| 23 6 -14 }}. | ||
| Line 9: | Line 9: | ||
In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000), so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783). | In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000), so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783). | ||
In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]] | In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), and the kalisma ([[9801/9800]]). In addition to these, it also tempers out both the [[nexus comma]] (1771561/1769472) and the [[quartisma]] (117440512/117406179), which, in turn means that the [[symbiotic comma]] (19712/19683) is tempered out as well. | ||
Finally, in the [[13-limit]] it is not quite as accurate but still very accurate | Finally, in the [[13-limit]] it is not quite as accurate but still very accurate. It tempers out [[676/675]], [[1001/1000]], [[1716/1715]], and [[2080/2079]], making it an [[The Archipelago|archipelago]] tuning, and the [[optimal patent val]] for some of the archipelago temperaments such as [[hemiennealimmal]], [[vulture]], [[eagle]], and [[avicenna (temperament)|avicenna]]. | ||
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, | 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. | ||
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 | 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. | |||
=== Prime harmonics === | === Prime harmonics === | ||
| Line 111: | Line 113: | ||
| 4.58 | | 4.58 | ||
|} | |} | ||
* 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 | * 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 even | * It is even better in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo|552]], and the least relative error until [[2190edo|2190]]. | ||
* It is also | * It is also great in the 17-limit, with lower absolute errors than smaller ETs despite inconsistency in the corresponding odd limit. | ||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||