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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]]). This results in it being a record edo for [[Pepper ambiguity]] in the 11-, 13- and 15-odd-limit. 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]].
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 }}.  
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 [[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 [[breedsma]] (2401/2400), the [[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 [[septischisma]] (33554432/33480783).  


In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[nexus comma]] (1771561/1769472), the [[quartisma]] (117440512/117406179), and the [[symbiotic comma]] (19712/19683). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)^4 and (20/11)^4 are a hair off, 50.4%).
In the [[11-limit]], it tempers out the [[lehmerisma]] (3025/3024), the [[vishdelisma]] (5632/5625), the [[kalisma]] (9801/9800), the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). It is almost consistent to distance 3 in the [[11-odd-limit]], only barely missing [[11/9]] (off by 16.68%).  


Finally, in the [[13-limit]] it is not quite as accurate but still very accurate, achieving the lowest [[TE logflat badness]] in the 13-limit up until [[96478edo]]. 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]].  
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 [[Just-noticeable difference|discernible]].
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-apotomes:
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 alterations of the Promethian set. Since the apotome can be split in two, a SZ half-sharp and a half-flat may be used.</span>  
<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||\}}</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>
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| rowspan="3" |<big>{{Sagittal|/|\}}</big>
| rowspan="3" |<big>{{Sagittal|/|\}}</big>
|<big>{{Sagittal|t}}</big>
|<big>{{Sagittal|t}}</big>
|<small>{{Sagittal||(}}{{sagittal|t}}</small>
|{{Sagittal||(}}{{sagittal|t}}
|<small>{{Sagittal|)|(}}{{sagittal|t}}</small>
|{{Sagittal|)|(}}{{sagittal|t}}
|<small>{{Sagittal|)~|}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|\\!}}{{sagittal|#}}
|<small>{{Sagittal|~|(}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|(!(}}{{sagittal|#}}
|<small>{{Sagittal|/|}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|~!)}}{{sagittal|#}}
|<small>{{Sagittal||)}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|!/}}{{sagittal|#}}
|<small>{{Sagittal||\}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|!)}}{{sagittal|#}}
|<small>{{Sagittal|(|}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|\!}}{{sagittal|#}}
|<small>{{Sagittal|(|(}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|~~!}}{{sagittal|#}}
|<small>{{Sagittal|//|}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|~!(}}{{sagittal|#}}
|<small>{{Sagittal|/|)}}{{sagittal|t}}</small>
| rowspan="2" |{{sagittal|)!(}}{{sagittal|#}}
|<small>{{Sagittal|/|\}}{{sagittal|t}}</small>
| 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>
|<small>{{sagittal|\!/}}{{sagittal|#}}</small>
| rowspan="2" |<big>{{Sagittal|(|)}}</big>
|<small>{{sagittal|\!)}}{{sagittal|#}}</small>
| rowspan="2" |<big>{{sagittal|(|\}}</big>
|<small>{{sagittal|\\!}}{{sagittal|#}}</small>
|<small>{{sagittal|(!(}}{{sagittal|#}}</small>
|<small>{{sagittal|(!}}{{sagittal|#}}</small>
|<small>{{sagittal|!/}}{{sagittal|#}}</small>
|<small>{{sagittal|!)}}{{sagittal|#}}</small>
|<small>{{sagittal|\!}}{{sagittal|#}}</small>
|<small>{{sagittal|~!(}}{{sagittal|#}}</small>
|<small>{{sagittal|)~!}}{{sagittal|#}}</small>
|<small>{{sagittal|)!(}}{{sagittal|#}}</small>
|<small>{{sagittal|!(}}{{sagittal|#}}</small>
|-
|-
!Revo
!Revo
|<big>{{Sagittal|(|)}}</big>
|<big>{{sagittal|(|\}}</big>
|<big>{{sagittal|)||(}}</big>
|<big>{{sagittal|)||(}}</big>
|<big>{{sagittal|~||(}}</big>
|<big>{{sagittal|~||(}}</big>
Line 124: Line 112:
|<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>
|<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>  


Line 221: Line 211:
| 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]].
Line 244: Line 234:
|-
|-
| 5
| 5
| [[Vishnuzma|(20 digits)]]
| [[Vishnu comma|(20 digits)]]
| 3.34
| 3.34
| {{Monzo| 23 6 -14 }}
| {{Monzo| 23 6 -14 }}
| Sasepbigu
| Sasepbigu
| sg<sup>14</sup>
| sg<sup>14</sup>
| Vishnuzma
| Vishnu comma
|-
|-
| 7
| 7
Line 257: Line 247:
| Sasaru
| Sasaru
| ssr
| ssr
| Garischisma
| Septischisma
|-
| 7
| [[Quasiorwellisma|(16 digits)]]
| 3.73
| {{Monzo| 22 -1 -10 1 }}
| Sazoquinbigu
| szg<sup>10</sup>
| Quasiorwellisma
|-
|-
| 7
| 7
Line 274: Line 272:
| zy<sup>4</sup>
| zy<sup>4</sup>
| Ragisma
| Ragisma
|-
| 7
| [[Quasiorwellisma|(16 digits)]]
| 3.73
| {{Monzo| 22 -1 -10 1 }}
| Sazoquinbigu
| szg<sup>10</sup>
| Quasiorwellisma
|-
|-
| 11
| 11
Line 297: Line 287:
| Saloquagu
| Saloquagu
| s1og<sup>4</sup>
| s1og<sup>4</sup>
| Vishdel comma
| Vishdelisma
|-
|-
| 11
| 11
Line 355: Line 345:
| Minisma
| Minisma
|-
|-
|17
| 17
| [[12376/12375]]
| [[12376/12375]]
| 0.14
| 0.14
Line 371: Line 361:
| Password, Eratosthenes' comma
| Password, Eratosthenes' comma
|-
|-
|19
| 19
|[[11859211/11859210|(16 digits)]]
| [[11859211/11859210|(16 digits)]]
|0.00
| 0.00
|{{Monzo|-1 -4 -1 1 -4 1 0 4}}
| {{Monzo| -1 -4 -1 1 -4 1 0 4 }}
|<small>Quadno-athoquadlu-azogu</small>
| <small>Quadno-athoquadlu-azogu</small>
|<small>9o<sup>4</sup>3o1u<sup>4</sup>zg</small>
| <small>9o<sup>4</sup>3o1u<sup>4</sup>zg</small>
|Tredekisma
| Tredekisma
|-
|-
| 23
| 23
Line 456: Line 446:
| 25/19
| 25/19
| [[Vulture]]
| [[Vulture]]
|-
| 2
| 4\270
| 17.{{overline|7}}
| 99/98
| [[Quarvish]]
|-
|-
| 2
| 2
Line 467: Line 463:
| 71.{{overline|1}}
| 71.{{overline|1}}
| 25/24
| 25/24
| [[Vishnu]] / ananta / acyuta
| [[Vishnu]] / acyuta / ananta
|-
|-
| 2
| 2
| 112\270<br>(23\270)
| 23\270
| 497.{{overline|7}}<br>(102.{{overline|2}})
| 102.{{overline|2}}
| 4/3<br>(35/33)
| 35/33
| [[Gariwizmic]]
| [[Gariwizmic]]
|-
|-
Line 492: Line 488:
| 8/7
| 8/7
| [[Orga]]
| [[Orga]]
|-
| 2
| 131\270<br>(4\270)
| 582.{{overline|2}}<br>(17.{{overline|7}})
| 7/5<br>(99/98)
| [[Quarvish]]
|-
|-
| 3
| 3
Line 512: Line 502:
|-
|-
| 5
| 5
| 83\270<br>(25\270)
| 25\270
| 368.{{overline|8}}<br>(111.{{overline|1}})
| 111.{{overline|1}}
| 1024/891<br>(16/15)
| 16/15
| [[Quintosec]]
| [[Quintosec]]
|-
|-
| 6
| 6
| 112\270<br>(4\270)
| 16\270
| 497.{{overline|7}}<br>(97.{{overline|7}})
| 71.{{overline|1}}
| 4/3<br>(128/121)
| 25/24
| [[Trivish]]
|-
| 6
| 22\270
| 97.{{overline|7}}
| 128/121
| [[Sextile]]
| [[Sextile]]
|-
|-
| 9
| 9
| 71\270<br>(11\270)
| 11\270
| 315.{{overline|5}}<br>(48.{{overline|8}})
| 48.{{overline|8}}
| 6/5<br>(36/35)
| 36/35
| [[Ennealimmal]] / ennealimmia
| [[Ennealimmal]] / enneabiotic / ennealympic
|-
|-
| 10
| 10
| 16\270<br>(11\270)
| 2\270
| 71.{{overline|1}}<br>(48.{{overline|8}})
| 8.{{overline|8}}
| 25/24<br>(36/35)
| 196/195
| [[Decavish]]
| [[Decoid]]
|-
|-
| 10
| 10
| 56\270<br>(2\270)
| 10\270
| 248.{{overline|8}}<br>(8.{{overline|8}})
| 44.{{overline|4}}
| 15/13<br>(176/175)
| 40/39
| [[Decoid]]
| [[Deca]]
|-
|-
| 10
| 10
| 71\270<br>(10\270)
| 11\270
| 315.{{overline|5}}<br>(44.{{overline|4}})
| 48.{{overline|8}}
| 6/5<br>(40/39)
| 36/35
| [[Deca]]
| [[Decavish]]
|-
|-
| 18
| 18
| 71\270<br>(4\270)
| 2\270
| 248.{{overline|8}}<br>(17.{{overline|7}})
| 8.{{overline|8}}
| 15/13<br>(99/98)
| 1287/1280
| [[Hemiennealimmal]]
| [[Semihemiennealimmal]]
|-
|-
| 18
| 18
| 71\270<br>(2\270)
| 4\270
| 475.{{overline|5}}<br>(8.{{overline|8}})
| 17.{{overline|7}}
| 1053/800<br>(1287/1280)
| 99/98
| [[Semihemiennealimmal]]
| [[Hemiennealimmal]]
|-
|-
| 27
| 27
| 61\270<br>(1\270)
| 1\270
| 271.{{overline|1}}<br>(4.{{overline|4}})
| 4.{{overline|4}}
| 1375/1176<br>(385/384)
| 385/384
| [[Trinealimmal]]
| [[Trinealimmal]]
|-
|-
| 30
| 30
| 82\270<br>(1\270)
| 1\270
| 364.{{overline|4}}<br>(4.{{overline|4}})
| 4.{{overline|4}}
| 216/175<br>(385/384)
| 385/384
| [[Zinc]]
| [[Zinc]]
|-
| 30
| 2\270
| 8.{{overline|8}}
| 196/195
| [[Triacontoid]]
|-
|-
| 45
| 45
| 59\270<br>(1\270)
| 1\270
| 262.{{overline|2}}<br>(4.{{overline|4}})
| 4.{{overline|4}}
| 64/55<br>(385/384)
| 385/384
| [[Rhodium]]
| [[Rhodium]]
|}
|}
<nowiki/>* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]


== Scales ==
== 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
* [[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 ===
=== 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 ===
=== Other scales ===
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