270edo: Difference between revisions
Formatted and added extra things |
→Rank-2 temperaments: + trivish. Also convert the table to minimal form since we're already reducing it to the first semi-octave |
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== Theory == | == 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]]) | 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]]. | ||
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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In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), and by extension the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000) so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783). | In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), and by extension the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000) so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783). | ||
In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[ | In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)<sup>4</sup> and (20/11)<sup>4</sup> are a hair off, 50.4%). | ||
Finally, in the [[13-limit]] it is | Finally, in the [[13-limit]] it is slightly worse but still excellent. 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 such as [[hemiennealimmal]], [[vulture]], [[eagle]], and [[avicenna (temperament)|avicenna]]. | ||
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. | ||
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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]]. | ||
On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error (assuming consistency) is only 2.{{overline|2}}{{c}}, yet having a step size that ''can'' be [[ | On top of this, its step size is small enough as to arguably give a good enough approximation for any relatively simple JI consonance (beyond the 13-limit on which it is spot on), as the maximum error (assuming consistency) is only 2.{{overline|2}}{{c}}, yet having a step size that ''can'' be [[just-noticeable difference|discernible]]. | ||
If, however, you want a edo for more rounded, consistent 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 very composite one. While 270edo approximates the first 16 harmonics with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible. | If, however, you want a edo for more rounded, consistent 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 very composite one. While 270edo approximates the first 16 harmonics with astounding accuracy, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible. | ||
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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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| 25/19 | | 25/19 | ||
| [[Vulture]] | | [[Vulture]] | ||
|- | |||
| 2 | |||
| 4\270 | |||
| 17.{{overline|7}} | |||
| 99/98 | |||
| [[Quarvish]] | |||
|- | |- | ||
| 2 | | 2 | ||
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| 71.{{overline|1}} | | 71.{{overline|1}} | ||
| 25/24 | | 25/24 | ||
| [[Vishnu]] / ananta | | [[Vishnu]] / acyuta / ananta | ||
|- | |- | ||
| 2 | | 2 | ||
| | | 23\270 | ||
| | | 102.{{overline|2}} | ||
| | | 35/33 | ||
| [[Gariwizmic]] | | [[Gariwizmic]] | ||
|- | |- | ||
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| 8/7 | | 8/7 | ||
| [[Orga]] | | [[Orga]] | ||
|- | |- | ||
| 3 | | 3 | ||
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|- | |- | ||
| 5 | | 5 | ||
| | | 25\270 | ||
| | | 111.{{overline|1}} | ||
| | | 16/15 | ||
| [[Quintosec]] | | [[Quintosec]] | ||
|- | |- | ||
| 6 | | 6 | ||
| | | 16\270 | ||
| | | 71.{{overline|1}} | ||
| | | 25/24 | ||
| [[Trivish]] | |||
|- | |||
| 6 | |||
| 22\270 | |||
| 97.{{overline|7}} | |||
| 128/121 | |||
| [[Sextile]] | | [[Sextile]] | ||
|- | |- | ||
| 9 | | 9 | ||
| | | 11\270 | ||
| | | 48.{{overline|8}} | ||
| | | 36/35 | ||
| [[Ennealimmal]] / | | [[Ennealimmal]] / enneabiotic / ennealympic | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 2\270 | ||
| | | 8.{{overline|8}} | ||
| | | 176/175 | ||
| [[ | | [[Decoid]] | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 10\270 | ||
| | | 44.{{overline|4}} | ||
| | | 40/39 | ||
| [[ | | [[Deca]] | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 11\270 | ||
| | | 48.{{overline|8}} | ||
| | | 36/35 | ||
| [[ | | [[Decavish]] | ||
|- | |- | ||
| 18 | | 18 | ||
| | | 2\270 | ||
| | | 8.{{overline|8}} | ||
| | | 1287/1280 | ||
| [[ | | [[Semihemiennealimmal]] | ||
|- | |- | ||
| 18 | | 18 | ||
| | | 4\270 | ||
| | | 17.{{overline|7}} | ||
| | | 99/98 | ||
| [[ | | [[Hemiennealimmal]] | ||
|- | |- | ||
| 27 | | 27 | ||
| | | 1\270 | ||
| | | 4.{{overline|4}} | ||
| | | 385/384 | ||
| [[Trinealimmal]] | | [[Trinealimmal]] | ||
|- | |- | ||
| 30 | | 30 | ||
| | | 1\270 | ||
| 4.{{overline|4}} | |||
| | | 385/384 | ||
| [[Zinc]] | | [[Zinc]] | ||
|- | |- | ||
| 45 | | 45 | ||
| | | 1\270 | ||
| | | 4.{{overline|4}} | ||
| | | 385/384 | ||
| [[Rhodium]] | | [[Rhodium]] | ||
|} | |} | ||
<nowiki/>* | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
== Scales == | == Scales == | ||