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270edo: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
{{Infobox ET}}
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
{{ED intro}} 270edo's step size is called a '''tredek''' when used as an [[interval size unit]].
: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-27 02:16:52 UTC</tt>.<br>
 
: The original revision id was <tt>238900915</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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]].  
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
 
<h4>Original Wikitext content:</h4>
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 }}.
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">The 270 equal division divides the octave into 270 equal parts of 4.4444 cents each. It is a very strong [[13-limit]] system, distinct and consistent through the 15 odd limit, and is the thirteenth [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]]. In the 5-limit it tempers out the ennealimma, |1 -27 18&gt;, the vulture comma, |24 -21 4&gt;, and the vishnuzma or semisuper comma, |23 6 -14&gt;. In the 7-limit it tempers out 2401/2400 and 4375/4374, so that it supports ennealimmal temperament, the wizma, 420175/419904 and the landscape comma, 250047/250000. In the 11-limit, it tempers out 5632/5625, 3025/3024 and 9801/9800. Finally in the 13-limit 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.</pre></div>
 
<h4>Original HTML content:</h4>
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).  
<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;270edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;The 270 equal division divides the octave into 270 equal parts of 4.4444 cents each. It is a very strong &lt;a class="wiki_link" href="/13-limit"&gt;13-limit&lt;/a&gt; system, distinct and consistent through the 15 odd limit, and is the thirteenth &lt;a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists"&gt;zeta integral edo&lt;/a&gt;. In the 5-limit it tempers out the ennealimma, |1 -27 18&amp;gt;, the vulture comma, |24 -21 4&amp;gt;, and the vishnuzma or semisuper comma, |23 6 -14&amp;gt;. In the 7-limit it tempers out 2401/2400 and 4375/4374, so that it supports ennealimmal temperament, the wizma, 420175/419904 and the landscape comma, 250047/250000. In the 11-limit, it tempers out 5632/5625, 3025/3024 and 9801/9800. Finally in the 13-limit it tempers out 676/675, 1001/1000, 1716/1715 and 2080/2079, making it an &lt;a class="wiki_link" href="/The%20Archipelago"&gt;archipelago&lt;/a&gt; tuning, and the optimal patent val for some of the archipelago temperaments.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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 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.
 
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]].
 
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.
 
=== Prime harmonics ===
{{Harmonics in equal|270|prec=3|intervals=prime|columns=11}}
{{Harmonics in equal|270|prec=3|intervals=prime|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 270edo (continued)}}
 
=== Subsets and supersets ===
270 is a very composite number. The prime factorization is {{nowrap|270 {{=}} 2 &times; 3<sup>3</sup> &times; 5}}, with divisors {{EDOs| 2, 3, 5, 6, 9, 10, 15, 18, 27, 30, 45, 54, 90, and 135 }}. This means that 270edo can be conceptualised as the superset of, for example, [[10edo]] and [[27edo]], which are both interesting and somewhat peculiar in their own right.
 
[[540edo]], which divides the edostep in two, and [[810edo]], which divides the edostep in three, provide good correction for harmonics 17, 23, and beyond.
 
== Intervals ==
As 270edo is a large edo, its intervals can be found on a separate page: [[Table of 270edo intervals]].
 
== 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-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}}
 
=== Sagittal notation ===
<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=""
! colspan="2" |+ edosteps
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|13
|14
|15
|16
|17
|18
|19
|20
|21
|22
|23
|24
|25
|26
|-
! rowspan="3" |Symbol
!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>
|<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>
|-
!Evo
| rowspan="2" |<big>{{sagittal|)/|\}}</big>
| rowspan="2" |<big>{{Sagittal|(|)}}</big>
| rowspan="2" |<big>{{sagittal|(|\}}</big>
|-
!Revo
|<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="">{{sagittal|)|}}</span> = <span data-darkreader-inline-color="">{{sagittal||(}}</span> (2621440/2617839)
* {{sagittal|)|(}} = {{sagittal|~|}} (1949696/1948617)
* {{Sagittal|/|}} = <span data-darkreader-inline-color="">{{sagittal|)|~}}</span> ([[1216/1215]])
* {{Sagittal|~|(}} = {{Sagittal|~~|}} (22528/22491)
* {{sagittal||\}} = {{sagittal|)|)}} ([[1540/1539]])
* {{sagittal|(|}} = {{sagittal|~|)}} ([[19712/19683]])
* {{sagittal|(|(}} = {{sagittal|~|\}} (20493/20480)
* {{Sagittal|/|)}} = {{sagittal|)//|}} = {{sagittal|(|~}} ([[729/728]]) (1540/1539)
* {{Sagittal|/|\}} = {{sagittal|(/|}} ([[131072/130977]]) ''([[3969/3968]])''
 
See [[Sagittal notation#Revo|apotome complements]] for equivalent accidental pairs.
 
== Approximation to JI ==
=== 23-odd-limit interval mappings ===
{{15-odd-limit|270|23}}
 
=== Higher-limit JI ===
270edo's approximation to higher harmonics, starting from 29, demonstrates a somewhat monotonous sharp tendency. This allows it to be considered as a temperament of very high limits – specifically the [[53-limit]]. In fact, 270edo is the first edo to be [[diamond monotone|monotonic]] in the 47- through 51-odd-limit, using the 270i val with the sharp mapping of 23.
 
For primes 37 and 41, this means the pairs [[37/36]] and [[38/37]], and the pairs [[41/40]] and [[42/41]], are distinct, observing [[1369/1368]] ({{S|37}}) and [[1681/1680]] ({{S|41}}). In fact 38/37, [[39/38]], [[40/39]], and 41/40 are tempered together. The sharp mapping for prime 23 is required here so that [[37/33]] (198.071{{C}} just) is not tuned wider [[46/41]] (199.212{{C}} just). Prime 43 then fits naturally with 42/41, [[43/42]], [[44/43]], and [[45/44]] all tempered together, while 47 may be added such that [[48/47]] is tempered together with [[49/48]], [[50/49]], and [[51/50]]. Again the sharp mapping for prime 23 is required so that [[46/45]] is tempered together with 45/44 and that [[47/46]] is tempered together with 48/47. Prime 53, if desired, is tuned with [[51/50]]~[[53/52]] and [[52/51]]~[[54/53]], so monotonicity is unavoidably lost in the 53-odd-limit.
 
== Regular temperament properties ==
{| class="wikitable center-4 center-5 center-6"
|-
! rowspan="2" | [[Subgroup]]
! rowspan="2" | [[Comma list]]
! rowspan="2" | [[Mapping]]
! rowspan="2" | Optimal<br>8ve stretch (¢)
! colspan="2" | Tuning error
|-
! [[TE error|Absolute]] (¢)
! [[TE simple badness|Relative]] (%)
|-
| 2.3.5
| {{Monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }}
| {{Mapping| 270 428 627 }}
| −0.1069
| 0.0759
| 1.71
|-
| 2.3.5.7
| 2401/2400, 4375/4374, 29360128/29296875
| {{Mapping| 270 428 627 758 }}
| −0.0858
| 0.0752
| 1.69
|-
| 2.3.5.7.11
| 2401/2400, 3025/3024, 4375/4374, 5632/5625
| {{Mapping| 270 428 627 758 934 }}
| −0.0567
| 0.0889
| 2.00
|-
| 2.3.5.7.11.13
| 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095
| {{Mapping| 270 428 627 758 934 999 }}
| −0.0235
| 0.1100
| 2.48
|-
| 2.3.5.7.11.13.19
| 676/675, 1001/1000, 1216/1215, 1331/1330, 1540/1539, 1729/1728
| {{Mapping| 270 428 627 758 934 999 1147 }}
| −0.0290
| 0.1028
| 2.31
|- style="border-top: double;"
| 2.3.5.7.11.13.17
| 676/675, 715/714, 936/935, 1001/1000, 1225/1224, 4096/4095
| {{Mapping| 270 428 627 758 934 999 1104 }}
| −0.0799
| 0.1718
| 3.86
|-
| 2.3.5.7.11.13.17.19
| 676/675, 715/714, 936/935, 1001/1000, 1216/1215, 1225/1224, 1331/1330
| {{Mapping| 270 428 627 758 934 999 1104 1147 }}
| −0.0777
| 0.1608
| 3.62
|-
| 2.3.5.7.11.13.17.19.23
| 460/459, 529/528, 676/675, 715/714, 736/735, 936/935, 1001/1000, 1216/1215
| {{Mapping| 270 428 627 758 934 999 1104 1147 1221 }}
| −0.0296
| 0.2037
| 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. 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.
* 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 also notable in the 17-limit, with lower absolute errors than smaller equal temperaments despite inconsistency in the [[17-odd-limit|corresponding odd limit]].
 
=== Commas ===
{| class="commatable wikitable center-1 center-2 right-3 center-6"
! [[Harmonic limit|Prime<br>limit]]
! [[Ratio]]<ref group="note">{{rd}}</ref>
! [[Cent]]s
! [[Monzo]]
! colspan="2" | [[Kite's color notation|Color name]]
! Name(s)
|-
| 5
| <abbr title="10485760000/10460353203">[[Vulture comma|(22 digits)]]</abbr>
| 4.20
| {{Monzo| 24 -21 4 }}
| Sasaquadyo
| ssy<sup>4</sup>
| Vulture comma
|-
| 5
| [[Vishnuzma|(20 digits)]]
| 3.34
| {{Monzo| 23 6 -14 }}
| Sasepbigu
| sg<sup>14</sup>
| Vishnuzma
|-
| 7
| [[33554432/33480783|(16 digits)]]
| 3.80
| {{Monzo| 25 -14 0 -1 }}
| Sasaru
| ssr
| Garischisma
|-
| 7
| [[2401/2400]]
| 0.72
| {{Monzo| -5 -1 -2 4 }}
| Bizozogu
| z<sup>4</sup>gg
| Breedsma
|-
| 7
| [[4375/4374]]
| 0.40
| {{Monzo| -1 -7 4 1 }}
| Zoquadyo
| zy<sup>4</sup>
| Ragisma
|-
| 7
| [[Quasiorwellisma|(16 digits)]]
| 3.73
| {{Monzo| 22 -1 -10 1 }}
| Sazoquinbigu
| szg<sup>10</sup>
| Quasiorwellisma
|-
| 11
| <abbr title="94489280512/94143178827">[[Pythrabian comma|(22 digits)]]</abbr>
| 6.35
| {{Monzo| 33 -23 0 0 1 }}
| Trisalo
| s1o<sup>3</sup>
| Pythrabian comma
|-
| 11
| [[5632/5625]]
| 2.15
| {{Monzo| 9 -2 -4 0 1 }}
| Saloquagu
| s1og<sup>4</sup>
| Vishdel comma
|-
| 11
| [[Nexus comma|(12 digits)]]
| 2.04
| {{Monzo| -16 -3 0 0 6 }}
| Tribilo
| 1o<sup>3</sup>
| Nexus comma
|-
| 11
| [[3025/3024]]
| 0.57
| {{Monzo| -4 -3 2 -1 2 }}
| Loloruyoyo
| 1ooryy
| Lehmerisma
|-
| 11
| [[9801/9800]]
| 0.18
| {{Monzo| -3 4 -2 -2 2 }}
| Bilorugu
| (1org)<sup>2</sup>
| Kalisma
|-
| 13
| [[676/675]]
| 2.56
| {{Monzo| 2 -3 -2 0 0 2 }}
| Bithogu
| 3oogg
| Island comma, parizeksma
|-
| 13
| [[1001/1000]]
| 1.73
| {{Monzo| -3 0 -3 1 1 1 }}
| Tholozotrigu
| 3o1ozg<sup>3</sup>
| Fairytale comma, sinbadma
|-
| 13
| [[2080/2079]]
| 0.83
| {{Monzo| 5 -3 1 -1 -1 1 }}
| Tholuruyo
| 3o1ury
| Ibnsinma, sinaisma
|-
| 13
| [[4096/4095]]
| 0.42
| {{Monzo| 12 -2 -1 -1 0 -1 }}
| Sathurugu
| s3urg
| Minisma
|-
|17
| [[12376/12375]]
| 0.14
| {{Monzo| 3 -2 -3 1 -1 1 1 }}
| Sotholuzotrigu
| 7o3o1uzg<sup>3</sup>
| Flashma
|-
| 19
| [[1216/1215]]
| 1.42
| 2.3.5.19 {{Monzo| 6 -5 -1 1 }}
| Sanogu
| s9og
| Password, Eratosthenes' comma
|-
|19
|[[11859211/11859210|(16 digits)]]
|0.00
|{{Monzo|-1 -4 -1 1 -4 1 0 4}}
|<small>Quadno-athoquadlu-azogu</small>
|<small>9o<sup>4</sup>3o1u<sup>4</sup>zg</small>
|Tredekisma
|-
| 23
| [[529/528]]
| 3.24
| 2.3.11.23 {{monzo| -4 -1 -1 2 }}
| Bitwetho-alu
| 23oo1u
| Preziosisma
|-
| 29
| [[784/783]]
| 2.20
| 2.3.7.29 {{monzo| 4 -3 2 -1 }}
| Twenuzozo
| 23uzz
| Biminorisma
|-
| 31
| [[621/620]]
| 2.79
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }}
| Thiwutwethogu
| 31u23og
| Owowhatsthisma
|}
<references group="note"/>
 
=== Rank-2 temperaments ===
{| class="wikitable center-all left-5"
|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
|-
! Periods<br>per 8ve
! Generator*
! Cents*
! Associated<br>ratio*
! Temperament
|-
| 1
| 1\270
| 4.{{overline|4}}
| 385/384
| [[Keenanose]]
|-
| 1
| 29\270
| 128.{{overline|8}}
| 14/13
| [[Tertiathirds]]
|-
| 1
| 61\270
| 271.{{overline|1}}
| 90/77
| [[Quasiorwell]]
|-
| 1
| 71\270
| 315.{{overline|5}}
| 6/5
| [[Acrokleismic]] / counteracro
|-
| 1
| 79\270
| 351.{{overline|1}}
| 49/40
| [[Newt]]
|-
| 1
| 97\270
| 431.{{overline|1}}
| 77/60
| [[Lockerbie]]
|-
| 1
| 107\270
| 475.{{overline|5}}
| 25/19
| [[Vulture]]
|-
| 2
| 14\270
| 62.{{overline|2}}
| 28/27
| [[Eagle]]
|-
| 2
| 16\270
| 71.{{overline|1}}
| 25/24
| [[Vishnu]] / ananta / acyuta
|-
| 2
| 112\270<br>(23\270)
| 497.{{overline|7}}<br>(102.{{overline|2}})
| 4/3<br>(35/33)
| [[Gariwizmic]]
|-
| 2
| 28\270
| 124.{{overline|4}}
| 275/256
| [[Semivulture]]
|-
| 2
| 47\270
| 208.{{overline|8}}
| 44/39
| [[Abigail]]
|-
| 2
| 52\270
| 231.{{overline|1}}
| 8/7
| [[Orga]]
|-
| 2
| 131\270<br>(4\270)
| 582.{{overline|2}}<br>(17.{{overline|7}})
| 7/5<br>(99/98)
| [[Quarvish]]
|-
| 3
| 17\270
| 75.{{overline|5}}
| 24/23
| [[Terture]]
|-
| 3
| 31\270
| 137.{{overline|7}}
| 13/12
| [[Avicenna]]
|-
| 5
| 83\270<br>(25\270)
| 368.{{overline|8}}<br>(111.{{overline|1}})
| 1024/891<br>(16/15)
| [[Quintosec]]
|-
| 6
| 112\270<br>(4\270)
| 497.{{overline|7}}<br>(97.{{overline|7}})
| 4/3<br>(128/121)
| [[Sextile]]
|-
| 9
| 71\270<br>(11\270)
| 315.{{overline|5}}<br>(48.{{overline|8}})
| 6/5<br>(36/35)
| [[Ennealimmal]] / enneabiotic / ennealympic
|-
| 10
| 16\270<br>(11\270)
| 71.{{overline|1}}<br>(48.{{overline|8}})
| 25/24<br>(36/35)
| [[Decavish]]
|-
| 10
| 56\270<br>(2\270)
| 248.{{overline|8}}<br>(8.{{overline|8}})
| 15/13<br>(176/175)
| [[Decoid]]
|-
| 10
| 71\270<br>(10\270)
| 315.{{overline|5}}<br>(44.{{overline|4}})
| 6/5<br>(40/39)
| [[Deca]]
|-
| 18
| 71\270<br>(4\270)
| 248.{{overline|8}}<br>(17.{{overline|7}})
| 15/13<br>(99/98)
| [[Hemiennealimmal]]
|-
| 18
| 71\270<br>(2\270)
| 475.{{overline|5}}<br>(8.{{overline|8}})
| 1053/800<br>(1287/1280)
| [[Semihemiennealimmal]]
|-
| 27
| 61\270<br>(1\270)
| 271.{{overline|1}}<br>(4.{{overline|4}})
| 1375/1176<br>(385/384)
| [[Trinealimmal]]
|-
| 30
| 82\270<br>(1\270)
| 364.{{overline|4}}<br>(4.{{overline|4}})
| 216/175<br>(385/384)
| [[Zinc]]
|-
| 45
| 59\270<br>(1\270)
| 262.{{overline|2}}<br>(4.{{overline|4}})
| 64/55<br>(385/384)
| [[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
 
== 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
* [[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 ===
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.
 
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.
 
=== Other scales ===
* [[Maeve Gutierrez #Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] (''octave reduced: 37 23 93 65 52'')
* [[Maeve Gutierrez #Moonglade scale|Gutierrez Moonglade scale]] (24 tones): 3 17 22 3 20 5 17 25 5 14 22 5 3 16 18 4 19 5 5 12 2 6 17 5
 
== External links ==
* [http://tonalsoft.com/enc/t/tredek.aspx tredek, 270-edo] on [[Tonalsoft Encyclopedia]]
 
[[Category:Avicenna (temperament)]]
[[Category:Eagle]]
[[Category:Hemiennealimmal]]
[[Category:Vulture]]
Retrieved from "https://en.xen.wiki/w/270edo"