270edo: Difference between revisions

Line 9:

== Theory ==

270edo is an extremely strong [[13-limit]] system, ~~distinct and~~ [[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-limits. 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]].

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 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-limits. 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 [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s [[ennealimmal]] temperament; ~~the~~ [[wizma]] ~~(420175/419904~~) and ~~the~~ [[landscape comma]] ~~(250047/250000~~).

In the [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s [[ennealimmal]] temperament; 420175/419904 ([[wizma]]) and 250047/250000 ([[landscape comma]]).

In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers the 4 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 [[symbiosma]] (19712/19683) is tempered out as well.

In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers 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 [[symbiosma]] (19712/19683) is tempered out as well.

Finally, in the [[13-limit]] it ~~isn't~~ 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.

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.

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 EDO as opposed to a highly 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 ===

=== Divisors ===

@@ Line 9: / Line 9: @@
 == Theory ==
-edo is an extremely strong [[13-limit]] system, distinct and [[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-limits. 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]].
+edo 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 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-limits. 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 [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s [[ennealimmal]] temperament; the [[wizma]] (420175/419904) and the [[landscape comma]] (250047/250000).
+In the [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it [[support]]s [[ennealimmal]] temperament; 420175/419904 ([[wizma]]) and 250047/250000 ([[landscape comma]]).
-In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers the 4 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 [[symbiosma]] (19712/19683) is tempered out as well.
+In the [[11-limit]], it tempers out [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers 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 [[symbiosma]] (19712/19683) is tempered out as well.
-Finally, in the [[13-limit]] it isn't 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.
+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.
+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 EDO as opposed to a highly 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 ===
-{{Primes in edo|270|prec=3}}
+{{Harmonics in equal|270|prec=3}}
 === Divisors ===