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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]].   
Line 234: 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 247: 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 264: 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
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| Saloquagu
| Saloquagu
| s1og<sup>4</sup>
| s1og<sup>4</sup>
| Vishdel comma
| Vishdelisma
|-
|-
| 11
| 11
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| Minisma
| Minisma
|-
|-
|17
| 17
| [[12376/12375]]
| [[12376/12375]]
| 0.14
| 0.14
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| 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
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| 25/19
| 25/19
| [[Vulture]]
| [[Vulture]]
|-
| 2
| 4\270
| 17.{{overline|7}}
| 99/98
| [[Quarvish]]
|-
|-
| 2
| 2
Line 457: 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]]
|-
|-
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| 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
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|-
|-
| 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]] / enneabiotic / ennealympic
| [[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"