270edo: Difference between revisions
Added back TE logflat badness mention (Come on there's nothing like it until the tens or hundreds of thousands!) |
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== Theory == | == Theory == | ||
270edo is an extremely strong [[13-limit]] | 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 [[ | |||
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 }}. | 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. | 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 | 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]]. | 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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== Notation == | == Notation == | ||
=== Ups and downs notation === | === Ups and downs notation === | ||
270edo can be notated using [[Kite's ups and downs notation|ups and downs]] with Stein-Zimmerman quarter-tone accidentals representing half- | 270edo can be notated using [[Kite's ups and downs notation|ups and downs]] with Stein-Zimmerman quarter-tone accidentals representing half-sharps and half-flats. These can be spoken as ''sha'' and ''fla''. For example, the note 12\270 above C is C downsha, and the note 39\270 above C is C shasharp. | ||
{{Ups and downs sharpness|270|true}} | {{Ups and downs sharpness|270|true}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses | <span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses symbols from the Promethean set. Since the apotome can be split in two, the Stein-Zimmermann half-sharp and half-flat may be used.</span> | ||
{| class="wikitable center-all" data-darkreader-inline-color="" | {| class="wikitable center-all" data-darkreader-inline-color="" | ||
! colspan="2" |+ edosteps | ! colspan="2" |+ edosteps | ||
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|- | |- | ||
! rowspan="3" |Symbol | ! rowspan="3" |Symbol | ||
!SZ | !Evo-SZ | ||
| rowspan="3" |<big>{{sagittal||(}}</big> | | rowspan="3" |<big>{{sagittal||(}}</big> | ||
| rowspan="3" |<big>{{sagittal|)|(}}</big> | | rowspan="3" |<big>{{sagittal|)|(}}</big> | ||
| rowspan="3" |<big>{{Sagittal|~|(}}</big> | | rowspan="3" |<big>{{Sagittal|~|(}}</big> | ||
| rowspan="3" |<big>{{Sagittal|~~|}}</big> | |||
| rowspan="3" |<big>{{Sagittal|/|}}</big> | | rowspan="3" |<big>{{Sagittal|/|}}</big> | ||
| rowspan="3" |<big>{{Sagittal||)}}</big> | | rowspan="3" |<big>{{Sagittal||)}}</big> | ||
| rowspan="3" |<big>{{sagittal||\}}</big> | | rowspan="3" |<big>{{sagittal||\}}</big> | ||
| rowspan="3" |<big>{{sagittal| | | rowspan="3" |<big>{{sagittal|~|)}}</big> | ||
| rowspan="3" |<big>{{sagittal|(|(}}</big> | | rowspan="3" |<big>{{sagittal|(|(}}</big> | ||
| rowspan="3" |<big>{{sagittal|//|}}</big> | | rowspan="3" |<big>{{sagittal|//|}}</big> | ||
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| rowspan="3" |<big>{{Sagittal|/|\}}</big> | | rowspan="3" |<big>{{Sagittal|/|\}}</big> | ||
|<big>{{Sagittal|t}}</big> | |<big>{{Sagittal|t}}</big> | ||
| | |{{Sagittal||(}}{{sagittal|t}} | ||
| | |{{Sagittal|)|(}}{{sagittal|t}} | ||
| | | rowspan="2" |{{sagittal|\\!}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|(!(}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|~!)}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|!/}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|!)}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|\!}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|~~!}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|~!(}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|)!(}}{{sagittal|#}} | ||
| | | rowspan="2" |{{sagittal|!(}}{{sagittal|#}} | ||
| rowspan="2" |<big>{{Sagittal|#}}</big> | | rowspan="2" |<big>{{Sagittal|#}}</big> | ||
|- | |- | ||
!Evo | !Evo | ||
| rowspan="2" |<big>{{sagittal|)/|\}}</big> | | rowspan="2" |<big>{{sagittal|)/|\}}</big> | ||
| | | rowspan="2" |<big>{{Sagittal|(|)}}</big> | ||
| rowspan="2" |<big>{{sagittal|(|\}}</big> | |||
|< | |||
| | |||
|- | |- | ||
!Revo | !Revo | ||
|<big>{{sagittal|)||(}}</big> | |<big>{{sagittal|)||(}}</big> | ||
|<big>{{sagittal|~||(}}</big> | |<big>{{sagittal|~||(}}</big> | ||
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|<big>{{Sagittal|||)}}</big> | |<big>{{Sagittal|||)}}</big> | ||
|<big>{{Sagittal|||\}}</big> | |<big>{{Sagittal|||\}}</big> | ||
|<big>{{sagittal|~||)}}</big> | |||
|<big>{{sagittal|(||(}}</big> | |<big>{{sagittal|(||(}}</big> | ||
|<big>{{sagittal|//||}}</big> | |<big>{{sagittal|//||}}</big> | ||
|<big>{{sagittal|/||)}}</big> | |<big>{{sagittal|/||)}}</big> | ||
|<big>{{Sagittal|/||\}}</big> | |<big>{{Sagittal|/||\}}</big> | ||
|} | |} | ||
Note that the Revo notation has matching flag sequences between the double-shaft symbols and a subsequence of the single-shaft symbols. | |||
<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span> | <span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span> | ||
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| 4.58 | | 4.58 | ||
|} | |} | ||
* 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, 13-limit and 19-limit up until [[342edo]], [[96478edo]] and [[3395edo]] respectively. | ||
* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better. | * 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]]. | * 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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| 315.{{overline|5}}<br>(48.{{overline|8}}) | | 315.{{overline|5}}<br>(48.{{overline|8}}) | ||
| 6/5<br>(36/35) | | 6/5<br>(36/35) | ||
| [[Ennealimmal]] / | | [[Ennealimmal]] / enneabiotic / ennealympic | ||
|- | |- | ||
| 10 | | 10 | ||