270edo: Difference between revisions

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== Theory ==

270edo is an extremely strong [[13-limit]] ~~and~~ 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]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit ''and'' 13-limit up until [[342edo]] and [[96478edo]] respectively.

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]].

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]].

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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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, the approximated [[17/1|harmonic 17]] is more than 1/3-edostep, but the [[19/1|harmonic 19]] is ~~extremely~~ accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. 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]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.

Beyond the 13-limit, the approximated [[17/1|harmonic 17]] is more than 1/3-edostep, but the [[19/1|harmonic 19]] is very accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. 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]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.

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]].

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* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo|72]] in the 11-limit, [[224edo|224]] in the 13-limit, and [[217edo|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo|342]], in the 13-limit [[494edo|494]], in the 23-limit [[282edo|282]]; and in the 19-limit, [[311edo|311]] for absolute error and [[581edo|581]] for relative error.

* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo|72]] in the 11-limit, [[224edo|224]] in the 13-limit, and [[217edo|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo|342]], in the 13-limit [[494edo|494]], in the 23-limit [[282edo|282]]; and in the 19-limit, [[311edo|311]] for absolute error and [[581edo|581]] for relative error. It is also a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit and 13-limit up until [[342edo]] and [[96478edo]] respectively.

* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.

* It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo|552]], and the least relative error until [[2190edo|2190]].

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 == Theory ==
-edo is an extremely strong [[13-limit]] and 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]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit ''and'' 13-limit up until [[342edo]] and [[96478edo]] respectively.
+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]].
-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]].
 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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 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, the approximated [[17/1|harmonic 17]] is more than 1/3-edostep, but the [[19/1|harmonic 19]] is extremely accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. 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]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.
+Beyond the 13-limit, the approximated [[17/1|harmonic 17]] is more than 1/3-edostep, but the [[19/1|harmonic 19]] is very accurately tuned. [[17/13]] and its [[octave complement]] [[26/17]] are the only inconsistently approximated [[21-odd-limit]] intervals, each barely missing the mark (50.4% relative error). The [[23/1|harmonic 23]] is more than 1/3-edostep flat, which incurs more inconsistencies in the next odd limits yet makes 270edo viable but tricky for the full [[23-limit]]. 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]]. If the full 23-limit is desired, then [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 are further tempered out.
 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]].
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-* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo|72]] in the 11-limit, [[224edo|224]] in the 13-limit, and [[217edo|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo|342]], in the 13-limit [[494edo|494]], in the 23-limit [[282edo|282]]; and in the 19-limit, [[311edo|311]] for absolute error and [[581edo|581]] for relative error.
+* 270et has lower [[Tenney-Euclidean temperament measures #TE simple badness|relative errors]] than any previous equal temperaments in the 11-, 13-, 19-, and 23-limit. It is the first to beat [[72edo|72]] in the 11-limit, [[224edo|224]] in the 13-limit, and [[217edo|217]] in the 19- and 23-limit. The next equal temperament that has lower absolute or relative error in the 11-limit is [[342edo|342]], in the 13-limit [[494edo|494]], in the 23-limit [[282edo|282]]; and in the 19-limit, [[311edo|311]] for absolute error and [[581edo|581]] for relative error. It is also a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit, and the edo with the lowest [[TE logflat badness]] in the 11-limit and 13-limit up until [[342edo]] and [[96478edo]] respectively.
 * 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.
 * It is best in the 2.3.5.7.11.13.19 subgroup, having the least absolute error until [[552edo|552]], and the least relative error until [[2190edo|2190]].