270edo: Difference between revisions

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'''270edo''' divides the octave into 270 equal parts of 4.{{overline|4}} [[cent]]s each. It is an extremely strong [[13-limit]] system, distinct and [[consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being more in-tune than out-of-tune with 270edo 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 and the 23rd zeta peak edo]]. In the [[5-limit]] it tempers out the ennealimma, {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[Vishnuzmic family|vishnuzma]] or semisuper comma, {{monzo| 23 6 -14 }}. In the [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it supports [[ennealimmal]] temperament, the [[wizma]], 420175/419904 and the landscape comma, 250047/250000. In the [[11-limit]], it tempers out [[5632/5625]], [[3025/3024]] and [[9801/9800]], meaning it tempers the 4 smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374 and 9801/9800). 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. 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.

'''270edo''' divides the octave into 270 equal parts of 4.{{overline|4}} [[cent]]s each.

== Prime intervals ==

== 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 more in-tune than out-of-tune with 270edo 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 and the 23rd zeta peak edo]].

In the [[5-limit]] it tempers out the ennealimma, {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[Vishnuzmic family|vishnuzma]] or semisuper comma, {{monzo| 23 6 -14 }}.

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

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).

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.

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 intervals ===

== Divisors ==

270 is a very composite number, with divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135, and some of these form the periods of the period and generators for some of rank two temperaments 270 supports; these include [[~~Ragismic_microtemperaments~~#Ennealimmal|ennealimmal]], hemiennealimmal and [[~~The_Archipelago~~#Rank two temperaments|decitonic]]. This means that 270edo can be conceptualised as the superset/intersection of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right.

270 is a very composite number, with divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135, and some of these form the periods of the period and generators for some of rank two temperaments 270 supports; these include [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], hemiennealimmal and [[The Archipelago #Rank two temperaments|decitonic]]. This means that 270edo can be conceptualised as the superset/intersection of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right.

The prime factorization of 270 is:

Line 11:

Line 24:

Here may be found a [[~~Table_of_270edo_intervals|~~table of 270edo intervals]].

== Intervals ==

Here may be found a [[table of 270edo intervals]].

[[Category:270edo]]

[[Category:Equal divisions of the octave]]

[[Category:~~theory~~]]

[[Category:Theory]]

[[Category:~~zeta~~]]

[[Category:Zeta]]

@@ Line 1: / Line 1: @@
-'''270edo''' divides the octave into 270 equal parts of 4.{{overline|4}} [[cent]]s each. It is an extremely strong [[13-limit]] system, distinct and [[consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being more in-tune than out-of-tune with 270edo 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 and the 23rd zeta peak edo]]. In the [[5-limit]] it tempers out the ennealimma, {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[Vishnuzmic family|vishnuzma]] or semisuper comma, {{monzo| 23 6 -14 }}. In the [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it supports [[ennealimmal]] temperament, the [[wizma]], 420175/419904 and the landscape comma, 250047/250000. In the [[11-limit]], it tempers out [[5632/5625]], [[3025/3024]] and [[9801/9800]], meaning it tempers the 4 smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374 and 9801/9800). 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. 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.
+'''270edo''' divides the octave into 270 equal parts of 4.{{overline|4}} [[cent]]s each.
-== Prime intervals ==
+== 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 more in-tune than out-of-tune with 270edo 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 and the 23rd zeta peak edo]].
+In the [[5-limit]] it tempers out the ennealimma, {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[Vishnuzmic family|vishnuzma]] or semisuper comma, {{monzo| 23 6 -14 }}.
+In the [[7-limit]] it tempers out [[2401/2400]] and [[4375/4374]], so that it supports [[ennealimmal]] temperament; the [[wizma]], 420175/419904 and the [[landscape comma]], 250047/250000.
+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).
+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.
+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 intervals ===
 {{Primes in edo|270|prec=2}}
 == Divisors ==
-is a very composite number, with divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135, and some of these form the periods of the period and generators for some of rank two temperaments 270 supports; these include [[Ragismic_microtemperaments#Ennealimmal|ennealimmal]], hemiennealimmal and [[The_Archipelago#Rank two temperaments|decitonic]]. This means that 270edo can be conceptualised as the superset/intersection of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right.
+is a very composite number, with divisors 1, 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90 and 135, and some of these form the periods of the period and generators for some of rank two temperaments 270 supports; these include [[Ragismic microtemperaments #Ennealimmal|ennealimmal]], hemiennealimmal and [[The Archipelago #Rank two temperaments|decitonic]]. This means that 270edo can be conceptualised as the superset/intersection of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right.
 The prime factorization of 270 is:
@@ Line 11: / Line 24: @@
 <math>270 = 2 \cdot 3^{3} \cdot 5</math>
-Here may be found a [[Table_of_270edo_intervals|table of 270edo intervals]].
+== Intervals ==
+Here may be found a [[table of 270edo intervals]].
 [[Category:270edo]]
 [[Category:Equal divisions of the octave]]
-[[Category:theory]]
+[[Category:Theory]]
-[[Category:zeta]]
+[[Category:Zeta]]