@@ Line 3: / Line 3: @@
 == Theory ==
-edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] with all intervals in the 15-odd-limit being approximated with less than 25% relative error except [[15/13]] and [[26/15]] which barely miss (corresponding to the [[tempering out]] of [[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, and is the first [[Trivial temperament|non-trivial]] edo to be consistent in the 16-[[Odd prime sum limit|odd-prime-sum-limit]].
+edo 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]] (a.k.a. semisuper comma), {{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), 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 [[3025/3024]], [[5632/5625]], and [[9801/9800]], meaning it tempers out the four smallest [[superparticular]] commas in the 11-limit (2401/2400, 3025/3024, 4375/4374, and 9801/9800). In addition to these, it also tempers out both the [[nexus comma]] (1771561/1769472) and the [[quartisma]] (117440512/117406179), which, in turn means that the [[symbiotic comma]] (19712/19683) is tempered out as well.
+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, as 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.
-Beyond the 13-limit, the approximated [[17/1|harmonic 17]] is more than 1/3-edostep sharp of just, and while the [[19/1|harmonic 19]] is accurately tuned, the [[23/1|harmonic 23]] is more than 1/3-edostep flat of just. [[17/13]], [[23/15]], and [[23/17]] are all the inconsistently approximated 23-odd-limit intervals, making 270edo a somewhat viable but tricky full 23-limit system. 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; and [[460/459]], [[529/528]], [[736/735]], [[897/896]], [[1288/1287]], 1311/1309, and 1771/1768 in the 23-limit. The harmonics [[29/1|29]] and [[31/1|31]] are also more than 1/3-edostep sharp, but not as sharp as 17 to incur inconsistency with the lower primes. In fact, 270edo is consistent in the no-17 no-23 [[35-odd-limit]]. We may note that it tempers out [[784/783]], [[900/899]], and [[1024/1023]].
+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.
-On top of this, its step size is so small as to arguably give a good enough approximation for any relatively simple JI consonance, as the maximum error is only 2.{{overline|2}}¢. If, however, you want an edo for 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 very accurately, 311edo approximates the first 42 but not as accurately – strongly favouring the approximation of as many harmonics as possible.
+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 ===
@@ Line 29: / Line 33: @@
 == Intervals ==
-As 270edo is a large edo, its intervals can be found on a separate page: [[table of 270edo 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 ===
-edo can be notated using [[Kite's ups and downs notation|ups and downs]] with quarter-tone accidentals:
+edo 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 ==
-=== 15-odd-limit interval mappings ===
+=== 23-odd-limit interval mappings ===
-{{15-odd-limit|270|15}}
+{{15-odd-limit|270|23}}
-=== Zeta peak index ===
+=== Higher-limit JI ===
-{{ZPI
+edo'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.
-| zpi = 1936
-| steps = 270.017794631965
+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.
-| step size = 4.44415154799558
-| tempered height = 13.370691
-| pure height = 12.895981
-| integral = 1.998595
-| gap = 22.286976
-| octave = 1199.92091795881
-| consistent = 22
-| distinct = 22
-}}
 == Regular temperament properties ==
@@ Line 66: / Line 156: @@
 |-
 | 2.3.5
-| {{monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }}
+| {{Monzo| 23 6 -14 }}, {{monzo| 24 -21 4 }}
-| {{mapping| 270 428 627 }}
+| {{Mapping| 270 428 627 }}
 | −0.1069
 | 0.0759
@@ Line 74: / Line 164: @@
 | 2.3.5.7
 | 2401/2400, 4375/4374, 29360128/29296875
-| {{mapping| 270 428 627 758 }}
+| {{Mapping| 270 428 627 758 }}
 | −0.0858
 | 0.0752
@@ Line 81: / Line 171: @@
 | 2.3.5.7.11
 | 2401/2400, 3025/3024, 4375/4374, 5632/5625
-| {{mapping| 270 428 627 758 934 }}
+| {{Mapping| 270 428 627 758 934 }}
 | −0.0567
 | 0.0889
@@ Line 88: / Line 178: @@
 | 2.3.5.7.11.13
 | 676/675, 1001/1000, 1716/1715, 3025/3024, 4096/4095
-| {{mapping| 270 428 627 758 934 999 }}
+| {{Mapping| 270 428 627 758 934 999 }}
 | −0.0235
 | 0.1100
@@ Line 95: / Line 185: @@
 | 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 }}
+| {{Mapping| 270 428 627 758 934 999 1147 }}
 | −0.0290
 | 0.1028
@@ Line 102: / Line 192: @@
 | 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 }}
+| {{Mapping| 270 428 627 758 934 999 1104 }}
 | −0.0799
 | 0.1718
@@ Line 109: / Line 199: @@
 | 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 }}
+| {{Mapping| 270 428 627 758 934 999 1104 1147 }}
 | −0.0777
 | 0.1608
@@ Line 116: / Line 206: @@
 | 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 }}
+| {{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 does better in terms of either 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.
-* It is even more prominent in the 2.3.5.7.11.13.19 subgroup. Not until [[552edo|552]] do we reach a better equal temperament in terms of absolute error, and not until [[2190edo|2190]] do we reach one in terms of relative error.
+* 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better.
-* It is also prominent in the 17-limit, with lower absolute errors than any previous equal temperaments, despite inconsistency in the corresponding odd limit.
+* 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
+| [[Vishnu comma|(20 digits)]]
+| 3.34
+| {{Monzo| 23 6 -14 }}
+| Sasepbigu
+| sg<sup>14</sup>
+| Vishnu comma
+|-
+| 7
+| [[33554432/33480783|(16 digits)]]
+| 3.80
+| {{Monzo| 25 -14 0 -1 }}
+| Sasaru
+| ssr
+| Septischisma
+|-
+| 7
+| [[Quasiorwellisma|(16 digits)]]
+| 3.73
+| {{Monzo| 22 -1 -10 1 }}
+| Sazoquinbigu
+| szg<sup>10</sup>
+| Quasiorwellisma
+|-
+| 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
+|-
+| 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>
+| Vishdelisma
+|-
+| 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 ===
@@ Line 137: / Line 407: @@
 | 1
 | 1\270
-| 4.44
+| 4.{{overline|4}}
 | 385/384
 | [[Keenanose]]
@@ Line 143: / Line 413: @@
 | 1
 | 29\270
-| 128.89
+| 128.{{overline|8}}
 | 14/13
 | [[Tertiathirds]]
@@ Line 149: / Line 419: @@
 | 1
 | 61\270
-| 271.11
+| 271.{{overline|1}}
 | 90/77
 | [[Quasiorwell]]
@@ Line 155: / Line 425: @@
 | 1
 | 71\270
-| 315.56
+| 315.{{overline|5}}
 | 6/5
 | [[Acrokleismic]] / counteracro
@@ Line 161: / Line 431: @@
 | 1
 | 79\270
-| 351.11
+| 351.{{overline|1}}
 | 49/40
 | [[Newt]]
@@ Line 167: / Line 437: @@
 | 1
 | 97\270
-| 431.11
+| 431.{{overline|1}}
 | 77/60
 | [[Lockerbie]]
@@ Line 173: / Line 443: @@
 | 1
 | 107\270
-| 475.56
+| 475.{{overline|5}}
 | 25/19
 | [[Vulture]]
+|-
+| 2
+| 4\270
+| 17.{{overline|7}}
+| 99/98
+| [[Quarvish]]
 |-
 | 2
 | 14\270
-| 62.22
+| 62.{{overline|2}}
 | 28/27
 | [[Eagle]]
@@ Line 185: / Line 461: @@
 | 2
 | 16\270
-| 71.11
+| 71.{{overline|1}}
 | 25/24
-| [[Vishnu]] / ananta / acyuta
+| [[Vishnu]] / acyuta / ananta
+|-
+| 2
+| 23\270
+| 102.{{overline|2}}
+| 35/33
+| [[Gariwizmic]]
 |-
 | 2
 | 28\270
-| 124.44
+| 124.{{overline|4}}
 | 275/256
 | [[Semivulture]]
@@ Line 197: / Line 479: @@
 | 2
 | 47\270
-| 208.89
+| 208.{{overline|8}}
 | 44/39
 | [[Abigail]]
@@ Line 203: / Line 485: @@
 | 2
 | 52\270
-| 231.11
+| 231.{{overline|1}}
 | 8/7
 | [[Orga]]
-|-
-| 2
-| 131\270<br>(4\270)
-| 582.22<br>(17.78)
-| 7/5<br>(99/98)
-| [[Quarvish]]
 |-
 | 3
 | 17\270
-| 75.56
+| 75.{{overline|5}}
 | 24/23
 | [[Terture]]
@@ Line 221: / Line 497: @@
 | 3
 | 31\270
-| 137.78
+| 137.{{overline|7}}
 | 13/12
 | [[Avicenna]]
 |-
 | 5
-| 83\270<br>(25\270)
+| 25\270
-| 368.89<br>(111.11)
+| 111.{{overline|1}}
-| 1024/891<br>(16/15)
+| 16/15
 | [[Quintosec]]
 |-
 | 6
-| 112\270<br>(4\270)
+| 16\270
-| 497.78<br>(97.78)
+| 71.{{overline|1}}
-| 4/3<br>(128/121)
+| 25/24
+| [[Trivish]]
+|-
+| 6
+| 22\270
+| 97.{{overline|7}}
+| 128/121
 | [[Sextile]]
 |-
 | 9
-| 71\270<br>(11\270)
+| 11\270
-| 315.56<br>(48.89)
+| 48.{{overline|8}}
-| 6/5<br>(36/35)
+| 36/35
-| [[Ennealimmal]] / ennealimmia
+| [[Ennealimmal]] / enneabiotic / ennealympic
 |-
 | 10
-| 16\270<br>(11\270)
+| 2\270
-| 71.11<br>(48.89)
+| 8.{{overline|8}}
-| 25/24<br>(36/35)
+| 196/195
-| [[Decavish]]
+| [[Decoid]]
 |-
 | 10
-| 56\270<br>(2\270)
+| 10\270
-| 248.89<br>(8.89)
+| 44.{{overline|4}}
-| 15/13<br>(176/175)
+| 40/39
-| [[Decoid]]
+| [[Deca]]
 |-
 | 10
-| 71\270<br>(10\270)
+| 11\270
-| 315.56<br>(44.44)
+| 48.{{overline|8}}
-| 6/5<br>(40/39)
+| 36/35
-| [[Deca]]
+| [[Decavish]]
 |-
 | 18
-| 71\270<br>(4\270)
+| 2\270
-| 248.89<br>(17.78)
+| 8.{{overline|8}}
-| 15/13<br>(99/98)
+| 1287/1280
-| [[Hemiennealimmal]]
+| [[Semihemiennealimmal]]
 |-
 | 18
-| 71\270<br>(2\270)
+| 4\270
-| 475.56<br>(8.89)
+| 17.{{overline|7}}
-| 1053/800<br>(1287/1280)
+| 99/98
-| [[Semihemiennealimmal]]
+| [[Hemiennealimmal]]
 |-
 | 27
-| 61\270<br>(1\270)
+| 1\270
-| 271.11<br>(4.44)
+| 4.{{overline|4}}
-| 1375/1176<br>(385/384)
+| 385/384
 | [[Trinealimmal]]
 |-
 | 30
-| 82\270<br>(1\270)
+| 1\270
-| 364.44<br>(4.44)
+| 4.{{overline|4}}
-| 216/175<br>(385/384)
+| 385/384
 | [[Zinc]]
+|-
+| 30
+| 2\270
+| 8.{{overline|8}}
+| 196/195
+| [[Triacontoid]]
 |-
 | 45
-| 59\270<br>(1\270)
+| 1\270
-| 262.22<br>(4.44)
+| 4.{{overline|4}}
-| 64/55<br>(385/384)
+| 385/384
 | [[Rhodium]]
 |}
-<nowiki/>* [[Normal lists|Octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if distinct
+<nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]]
 == 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
+=== [[MOS]] scales ===
-* [[Maeve Gutierrez#Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] (''octave reduced: 37 23 93 65 52'')
+* [[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
-* [[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
+* [[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
-== Commas of the 2.3.5.7.11.13 subgroup tempered out in 270edo patent val ==
+=== Harmonic scales ===
+edo 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
-Commas with numerator <= 2^20:
+=== [[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
-{| class="wikitable mw-collapsible mw-collapsed sortable"
+=== Other scales ===
-! [[Ratio]]
+* [[Maeve Gutierrez #Gutierrez-Lambeth quasi-subharmonic pentatonic|Gutierrez-Lambeth quasi-subharmonic pentatonic]] (''octave reduced: 37 23 93 65 52'')
-! [[Monzo]]
+* [[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
-! [[Cent]]s
-|-
-| [[676/675]]
-| [2 -3 -2 0 0 2⟩
-| 2.563
-|-
-| [[1001/1000]]
-| [-3 0 -3 1 1 1⟩
-| 1.730
-|-
-| [[1716/1715]]
-| [2 1 -1 -3 1 1⟩
-| 1.009
-|-
-| [[2080/2079]]
-| [5 -3 1 -1 -1 1⟩
-| 0.833
-|-
-| [[2401/2400]]
-| [-5 -1 -2 4⟩
-| 0.721
-|-
-| [[3025/3024]]
-| [-4 -3 2 -1 2⟩
-| 0.572
-|-
-| [[4096/4095]]
-| [12 -2 -1 -1 0 -1⟩
-| 0.423
-|-
-| [[4225/4224]]
-| [-7 -1 2 0 -1 2⟩
-| 0.410
-|-
-| [[4375/4374]]
-| [-1 -7 4 1⟩
-| 0.396
-|-
-| [[4459/4455]]
-| [0 -4 -1 3 -1 1⟩
-| 1.554
-|-
-| [[5632/5625]]
-| [9 -2 -4 0 1⟩
-| 2.153
-|-
-| [[6656/6655]]
-| [9 0 -1 0 -3 1⟩
-| 0.260
-|-
-| [[9801/9800]]
-| [-3 4 -2 -2 2⟩
-| 0.177
-|-
-| [[10648/10647]]
-| [3 -2 0 -1 3 -2⟩
-| 0.163
-|-
-| [[10985/10976]]
-| [-5 0 1 -3 0 3⟩
-| 1.419
-|-
-| [[14641/14625]]
-| [0 -2 -3 0 4 -1⟩
-| 1.893
-|-
-| [[15379/15360]]
-| [-10 -1 -1 1 0 3⟩
-| 2.140
-|-
-| [[17303/17280]]
-| [-7 -3 -1 0 3 1⟩
-| 2.303
-|-
-| [[19712/19683]]
-| [8 -9 0 1 1⟩
-| 2.549
-|-
-| [[20449/20412]]
-| [-2 -6 0 -1 2 2⟩
-| 3.135
-|-
-| [[21632/21609]]
-| [7 -2 0 -4 0 2⟩
-| 1.842
-|-
-| [[28561/28512]]
-| [-5 -4 0 0 -1 4⟩
-| 2.973
-|-
-| [[34398/34375]]
-| [1 3 -5 2 -1 1⟩
-| 1.158
-|-
-| [[35035/34992]]
-| [-4 -7 1 2 1 1⟩
-| 2.126
-|-
-| [[35750/35721]]
-| [1 -6 3 -2 1 1⟩
-| 1.405
-|-
-| [[40656/40625]]
-| [4 1 -5 1 2 -1⟩
-| 1.321
-|-
-| [[41503/41472]]
-| [-9 -4 0 3 2⟩
-| 1.294
-|-
-| [[42592/42525]]
-| [5 -5 -2 -1 3⟩
-| 2.725
-|-
-| [[43904/43875]]
-| [7 -3 -3 3 0 -1⟩
-| 1.144
-|-
-| [[43923/43904]]
-| [-7 1 0 -3 4⟩
-| 0.749
-|-
-| [[43940/43923]]
-| [2 -1 1 0 -4 3⟩
-| 0.670
-|-
-| [[47432/47385]]
-| [3 -6 -1 2 2 -1⟩
-| 1.716
-|-
-| [[50193/50176]]
-| [-10 3 0 -2 1 2⟩
-| 0.586
-|-
-| [[59150/59049]]
-| [1 -10 2 1 0 2⟩
-| 2.959
-|-
-| [[59319/59290]]
-| [-1 3 -1 -2 -2 3⟩
-| 0.847
-|-
-| [[61347/61250]]
-| [-1 1 -4 -2 2 2⟩
-| 2.740
-|-
-| [[67392/67375]]
-| [6 4 -3 -2 -1 1⟩
-| 0.437
-|-
-| [[75712/75625]]
-| [6 0 -4 1 -2 2⟩
-| 1.990
-|-
-| [[86515/86436]]
-| [-2 -2 1 -4 3 1⟩
-| 1.582
-|-
-| [[102487/102400]]
-| [-12 0 -2 1 4⟩
-| 1.470
-|-
-| [[105644/105625]]
-| [2 0 -4 4 1 -2⟩
-| 0.311
-|-
-| [[109512/109375]]
-| [3 4 -6 -1 0 2⟩
-| 2.167
-|-
-| [[117649/117612]]
-| [-2 -5 0 6 -2⟩
-| 0.545
-|-
-| [[123201/123200]]
-| [-6 6 -2 -1 -1 2⟩
-| 0.014
-|-
-| [[131072/130977]]
-| [17 -5 0 -2 -1⟩
-| 1.255
-|-
-| [[151263/151250]]
-| [-1 2 -4 5 -2⟩
-| 0.149
-|-
-| [[154880/154791]]
-| [8 -5 1 -2 2 -1⟩
-| 0.995
-|-
-| [[160083/160000]]
-| [-8 3 -4 2 2⟩
-| 0.898
-|-
-| [[169169/168750]]
-| [-1 -3 -5 1 1 3⟩
-| 4.293
-|-
-| [[172032/171875]]
-| [13 1 -6 1 -1⟩
-| 1.581
-|-
-| [[180224/180075]]
-| [14 -1 -2 -4 1⟩
-| 1.432
-|-
-| [[196625/196608]]
-| [-16 -1 3 0 2 1⟩
-| 0.150
-|-
-| [[199927/199650]]
-| [-1 -1 -2 1 -3 4⟩
-| 2.400
-|-
-| [[200704/200475]]
-| [12 -6 -2 2 -1⟩
-| 1.976
-|-
-| [[212992/212625]]
-| [14 -5 -3 -1 0 1⟩
-| 2.986
-|-
-| [[250047/250000]]
-| [-4 6 -6 3⟩
-| 0.325
-|-
-| [[256000/255879]]
-| [11 -9 3 0 0 -1⟩
-| 0.818
-|-
-| [[264992/264627]]
-| [5 -7 0 2 -2 2⟩
-| 2.386
-|-
-| [[274625/274428]]
-| [-2 -4 3 -1 -2 3⟩
-| 1.242
-|-
-| [[281216/280665]]
-| [7 -6 -1 -1 -1 3⟩
-| 3.395
-|-
-| [[312741/312500]]
-| [-2 7 -7 0 1 1⟩
-| 1.335
-|-
-| [[314171/313600]]
-| [-8 0 -2 -2 1 4⟩
-| 3.149
-|-
-| [[322102/321489]]
-| [1 -8 0 -2 5⟩
-| 3.298
-|-
-| [[369754/369603]]
-| [1 -7 0 5 1 -2⟩
-| 0.707
-|-
-| [[371293/370440]]
-| [-3 -3 -1 -3 0 5⟩
-| 3.982
-|-
-| [[386672/385875]]
-| [4 -2 -3 -3 1 3⟩
-| 3.572
-|-
-| [[391314/390625]]
-| [1 1 -8 2 3⟩
-| 3.051
-|-
-| [[405769/405000]]
-| [-3 -4 -4 4 0 2⟩
-| 3.284
-|-
-| [[405769/405504]]
-| [-12 -2 0 4 -1 2⟩
-| 1.131
-|-
-| [[420175/419904]]
-| [-6 -8 2 5⟩
-| 1.117
-|-
-| [[422576/421875]]
-| [4 -3 -6 4 1⟩
-| 2.874
-|-
-| [[456533/455625]]
-| [0 -6 -4 3 3⟩
-| 3.447
-|-
-| [[456533/456300]]
-| [-2 -3 -2 3 3 -2⟩
-| 0.884
-|-
-| [[456976/455625]]
-| [4 -6 -4 0 0 4⟩
-| 5.126
-|-
-| [[456976/456533]]
-| [4 0 0 -3 -3 4⟩
-| 1.679
-|-
-| [[468512/468195]]
-| [5 -1 -1 -4 4 -1⟩
-| 1.172
-|-
-| [[499408/499125]]
-| [4 -1 -3 4 -3 1⟩
-| 0.981
-|-
-| [[512435/511758]]
-| [-1 -9 1 1 4 -1⟩
-| 2.289
-|-
-| [[532400/531441]]
-| [4 -12 2 0 3⟩
-| 3.121
-|-
-| [[532480/531441]]
-| [13 -12 1 0 0 1⟩
-| 3.381
-|-
-| [[557568/557375]]
-| [9 2 -3 -3 2 -1⟩
-| 0.599
-|-
-| [[644204/643125]]
-| [2 -1 -4 -3 5⟩
-| 2.902
-|-
-| [[650000/649539]]
-| [4 -10 5 0 -1 1⟩
-| 1.228
-|-
-| [[655473/655360]]
-| [-17 1 -1 5 0 1⟩
-| 0.298
-|-
-| [[704704/703125]]
-| [6 -2 -7 1 2 1⟩
-| 3.883
-|-
-| [[714025/713097]]
-| [0 -3 2 -4 -1 4⟩
-| 2.252
-|-
-| [[742586/741125]]
-| [1 0 -3 -2 -2 5⟩
-| 3.409
-|-
-| [[761332/759375]]
-| [2 -5 -5 0 4 1⟩
-| 4.456
-|-
-| [[766656/765625]]
-| [6 2 -6 -2 3⟩
-| 2.330
-|-
-| [[768320/767637]]
-| [6 -10 1 4 0 -1⟩
-| 1.540
-|-
-| [[771147/770000]]
-| [-4 3 -4 -1 -1 4⟩
-| 2.577
-|-
-| [[819819/819200]]
-| [-15 2 -2 2 1 2⟩
-| 1.308
-|-
-| [[823680/823543]]
-| [7 2 1 -7 1 1⟩
-| 0.288
-|-
-| [[823875/823543]]
-| [0 1 3 -7 0 3⟩
-| 0.698
-|-
-| [[861224/859375]]
-| [3 0 -7 2 -1 3⟩
-| 3.721
-|-
-| [[942513/941192]]
-| [-3 1 0 -6 1 4⟩
-| 2.428
-|-
-| [[966680/964467]]
-| [3 -9 1 -2 1 3⟩
-| 3.968
-|-
-| [[968877/968000]]
-| [-6 2 -3 2 -2 3⟩
-| 1.568
-|-
-| [[1002001/1000000]]
-| [-6 0 -6 2 2 2⟩
-| 3.461
-|-
-| [[1016064/1015625]]
-| [8 4 -7 2 0 -1⟩
-| 0.748
-|-
-| [[1038336/1037575]]
-| [11 1 -2 -3 -2 2⟩
-| 1.269
-|}
 == External links ==

270edo: Difference between revisions

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