270edo: Difference between revisions
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== Theory == | == Theory == | ||
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 ( | 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 (corresponding to the fact of 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, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]]. | ||
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 }}. | ||
In the [[7-limit]] it tempers out 2401/2400 | 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]], meaning it tempers out the four smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374 and 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. | In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers out the four smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374, and 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, as 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]]. | Finally, in the [[13-limit]] it is not quite as accurate but still very accurate, as 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, harmonic [[17/1|17]] is more than 1/3-edostep sharp of just, and while harmonic [[19/1|19]] is accurately tuned, [[23/1|harmonic 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]], and [[2431/2430]] in the 17-limit; [[1216/1215]], [[1331/1330]], [[1521/1520]], [[1540/1539]], and [[1729/1728]] in the 19-limit; and [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 in the 23-limit. Harmonics [[29/1|29]] 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 that 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. | ||
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=== Subsets and supersets === | === Subsets and supersets === | ||
270 is a very composite number. The prime factorization is {{nowrap|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 {{nowrap|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, and [[810edo]], which divides the edostep in three, provide good correction for harmonics 17, 23, and beyond. | [[540edo]], which divides the edostep in two, and [[810edo]], which divides the edostep in three, provide good correction for harmonics 17, 23, and beyond. | ||