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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]]). 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 }}.  
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 [[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%).
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 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]].   
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]].   
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=== 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
Line 72: Line 72:
|-
|-
! 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>
Line 86: Line 86:
| 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>
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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>
|<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 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 ===
Retrieved from "https://en.xen.wiki/w/270edo"