270edo: Difference between revisions

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== Theory ==

270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency #Consistency to distance d|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]]). It is 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]].

270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency #Consistency to distance d|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]], [[CWE tuning|CWE]] yields only 0.4% error from 270edo for both intervals). It is 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]].

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

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== Scales ==

=== ~~Mos~~ scales ===

=== [[MOS]] scales ===

* [[Ennealimmal]][45]: ~~2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2~~

* [[Abigail]][86]: 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5

* [[Avicenna]][96]: 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3

* [[Eagle]][84]: 4 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1

* [[Ennealimmal]][27]: 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8

* Ennealimmal[45]: 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8

* [[Quasiorwell]][84]: 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1

* [[Quintosec]][25]: 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4

* Quintosec[55]: 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4

* Quintosec[65]: 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4

* [[Vishnu]][34]: 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7

=== Harmonic scales ===

270edo very accurately approximates ~~the~~ mode 16 of [[harmonic series]]. ~~The scale in adjacent steps is 24, 22, 21, 20, 19, 18, 17, 17, 16, 15, 15, 14, 14, 13, 13, 12. Four~~ interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28, and 30/29~31/30.

270edo very accurately approximates mode 16 of the [[harmonic series]]. Only four interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28 and 30/29~31/30. The scale in adjacent steps is:

* 24 22 21 20 19 18 17 17 16 15 15 14 14 13 13 12

It further does decently in the mode 24. The scale in adjacent steps is 16, 15, 15, 14, 14, 13, 13, 12, 12, 12, 11, 11, 11, 10, 10, 10, 10, 9, 9, 9, 9, 9, 8, 8.

It further does decently in the mode 24. The scale in adjacent steps is:

* 16 15 15 14 14 13 13 12 12 12 11 11 11 10 10 10 10 9 9 9 9 9 8 8

=== [[Historical temperaments|Historical temperament]] approximations ===

* [[Kirnberger III]]: 20 23 23 21 25 21 24 21 22 24 21 25

* [[Septenarius]]: 20 24 23 22 23 22 23 21 23 24 22 23

* [[Vallotti]]: 21 23 23 21 25 20 24 22 22 24 20 25

* [[Werckmeister III]]: 20 23 23 22 24 20 25 21 22 24 22 24

=== Other scales ===

@@ Line 3: / Line 3: @@
 == Theory ==
-edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency #Consistency to distance d|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]]). It is 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]].
+edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] and almost [[Consistency #Consistency to distance d|consistent to distance 2]] in it, missing [[15/13]] and [[26/15]] as they have 25.8% error ([[tempering out]] [[676/675]], [[CWE tuning|CWE]] yields only 0.4% error from 270edo for both intervals). It is 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]].
 In the [[5-limit]] it tempers out the [[ennealimma]], {{monzo| 1 -27 18 }}, the [[vulture comma]], {{monzo| 24 -21 4 }}, and the [[vishnuzma]], {{monzo| 23 6 -14 }}.
@@ Line 582: / Line 582: @@
 == Scales ==
-=== Mos scales ===
+=== [[MOS]] scales ===
-* [[Ennealimmal]][45]: 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2 2 12 2 12 2
+* [[Abigail]][86]: 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5
+* [[Avicenna]][96]: 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3
+* [[Eagle]][84]: 4 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1
+* [[Ennealimmal]][27]: 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8
+* Ennealimmal[45]: 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8
+* [[Quasiorwell]][84]: 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1
+* [[Quintosec]][25]: 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4
+* Quintosec[55]: 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4
+* Quintosec[65]: 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4
 * [[Vishnu]][34]: 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7
 === Harmonic scales ===
-edo very accurately approximates the mode 16 of [[harmonic series]]. The scale in adjacent steps is 24, 22, 21, 20, 19, 18, 17, 17, 16, 15, 15, 14, 14, 13, 13, 12. Four interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28, and 30/29~31/30.
+edo very accurately approximates mode 16 of the [[harmonic series]]. Only four interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28 and 30/29~31/30. The scale in adjacent steps is:
+* 24 22 21 20 19 18 17 17 16 15 15 14 14 13 13 12
-It further does decently in the mode 24. The scale in adjacent steps is 16, 15, 15, 14, 14, 13, 13, 12, 12, 12, 11, 11, 11, 10, 10, 10, 10, 9, 9, 9, 9, 9, 8, 8.
+It further does decently in the mode 24. The scale in adjacent steps is:
+* 16 15 15 14 14 13 13 12 12 12 11 11 11 10 10 10 10 9 9 9 9 9 8 8
+=== [[Historical temperaments|Historical temperament]] approximations ===
+* [[Kirnberger III]]: 20 23 23 21 25 21 24 21 22 24 21 25
+* [[Septenarius]]: 20 24 23 22 23 22 23 21 23 24 22 23
+* [[Vallotti]]: 21 23 23 21 25 20 24 22 22 24 20 25
+* [[Werckmeister III]]: 20 23 23 22 24 20 25 21 22 24 22 24
 === Other scales ===

270edo: Difference between revisions

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