270edo: Difference between revisions

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In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), and by extension 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 the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[~~nexus~~ comma]] (~~1771561~~/~~1769472~~), the [[~~quartisma~~]] (~~117440512~~/~~117406179~~), and the [[~~symbiotic comma~~]] (~~19712~~/~~19683~~). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)^4 and (20/11)^4 are a hair off, 50.4%).

In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)<sup>4</sup> and (20/11)<sup>4</sup> are a hair off, 50.4%).

Finally, in the [[13-limit]] it is not quite as accurate but still very accurate~~, achieving the lowest [[TE logflat badness]] in the 13-limit up until [[96478edo]]~~. 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. 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.

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The harmonics [[29/1|29]] and [[31/1|31]] are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency ([[29/26]] and [[31/26]] are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 [[35-odd-limit]]. Notably, 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 small enough as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error (assuming consistency) is only 2.{{overline|2}}{{c}}, yet having a step size that ''can'' be [[~~Just~~-noticeable difference|discernible]].

On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error (assuming consistency) is only 2.{{overline|2}}{{c}}, yet having a step size that ''can'' be [[just-noticeable difference|discernible]].

If, however, you want a edo for more rounded, consistent 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 with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.

@@ Line 9: / Line 9: @@
 In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), and by extension 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 the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[nexus comma]] (1771561/1769472), the [[quartisma]] (117440512/117406179), and the [[symbiotic comma]] (19712/19683). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)^4 and (20/11)^4 are a hair off, 50.4%).
+In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]),  the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)<sup>4</sup> and (20/11)<sup>4</sup> are a hair off, 50.4%).
-Finally, in the [[13-limit]] it is not quite as accurate but still very accurate, achieving the lowest [[TE logflat badness]] in the 13-limit up until [[96478edo]]. 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. 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.
@@ Line 19: / Line 19: @@
 The harmonics [[29/1|29]] and [[31/1|31]] are also more than 1/3-edostep sharp, but not as sharp as the 17 to incur inconsistency ([[29/26]] and [[31/26]] are critically sharp but still consistent). This makes 270edo consistent in the no-17/13 no-23 [[35-odd-limit]]. Notably, 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 small enough as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error (assuming consistency) is only 2.{{overline|2}}{{c}}, yet having a step size that ''can'' be [[Just-noticeable difference|discernible]].
+On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error (assuming consistency) is only 2.{{overline|2}}{{c}}, yet having a step size that ''can'' be [[just-noticeable difference|discernible]].
 If, however, you want a edo for more rounded, consistent 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 with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.