270edo: Difference between revisions

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== Theory ==

270edo 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~~ ([[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]] ~~(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 the lehmerisma ([[3025/3024]]), the ~~vishdel comma (~~[[5632/5625]]), ~~and~~ the kalisma ([[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~~. 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 ~~approxiamted~~ [[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.

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 so small 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}}. 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.

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 ===

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== 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.

=== Sagittal notation ===

The [[Sagittal notation]] for ~~311edo~~ uses ~~alterations of~~ the ~~Promethian~~ set. Since the apotome can be split in two, a half-sharp and a half-flat may be used.

The [[Sagittal notation]] 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.

{| class="wikitable center-all" data-darkreader-inline-color=""

! colspan="2" |+ edosteps

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|26

|-

! rowspan="3" |~~Sym.~~

! 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>

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| rowspan="3" |<big>{{Sagittal|/|\}}</big>

|<big>{{Sagittal|t}}</big>

|~~~~{{Sagittal||(}}{{sagittal|t}}~~~~

|{{Sagittal||(}}{{sagittal|t}}

|~~~~{{Sagittal|)|(}}{{sagittal|t}}~~~~

|{{Sagittal|)|(}}{{sagittal|t}}

|~~~~{{~~Sagittal|)~~~|}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|\\!}}{{sagittal|#}}

|~~~~{{~~Sagittal|~~~|(}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|(!(}}{{sagittal|#}}

|~~~~{{~~Sagittal|/~~|}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|~!)}}{{sagittal|#}}

|~~~~{{~~Sagittal~~||)}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|!/}}{{sagittal|#}}

|~~~~{{~~Sagittal~~||\}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|!)}}{{sagittal|#}}

|~~~~{{~~Sagittal|(~~|}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|\!}}{{sagittal|#}}

|~~~~{{~~Sagittal~~|~~(|(~~}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|~~!}}{{sagittal|#}}

|~~~~{{~~Sagittal|//~~|}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|~!(}}{{sagittal|#}}

|~~~~{{~~Sagittal|/~~|)}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|)!(}}{{sagittal|#}}

|~~~~{{~~Sagittal~~|~~/|\~~}}{{sagittal|t}}~~~~

| rowspan="2" |{{sagittal|!(}}{{sagittal|#}}

| rowspan="2" |<big>{{Sagittal|#}}</big>

|-

!Evo

| rowspan="2" |<big>{{sagittal|)/|\}}</big>

|~~{{sagittal|\!/}}{{sagittal|#}}~~

| rowspan="2" |<big>{{Sagittal|(|)}}</big>

~~|{{sagittal|\!)}}{{sagittal|#}}~~

| rowspan="2" |<big>{{sagittal|(|\}}</big>

~~|{{sagittal|\\!}}{{sagittal|#}}~~

~~|{{sagittal|(!(}}{{sagittal|#}}~~

~~|{{sagittal|(!}}{{sagittal|#}}~~

~~|{{sagittal|!/}}{{sagittal|#}}~~

~~|{{sagittal|!)}}{{sagittal|#}}~~

|<~~small~~>{{~~sagittal~~|~~\!}}{{sagittal|#}}~~

~~|{{sagittal|~!~~(~~}}{{sagittal|#}}~~

~~|{{sagittal~~|)~~~!}}{{sagittal|#~~}}</~~small~~>

|~~{{sagittal|)!(}}{{sagittal~~|~~#}}~~<~~/small>~~

~~|<small~~>{{sagittal|!(~~}}{{sagittal~~|#}}</~~small~~>

|-

!Revo

~~|<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>

|}

Note that the Revo notation has matching flag sequences between the double-shaft symbols and a subsequence of the single-shaft symbols.

Alternate spellings in the Promethean set (comma tempered out):

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=== 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-odd-limit, using the 270i val with the sharp mapping of 23.

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 ==

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| 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.

* It is ~~even better~~ 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]].

* It is also notable in the 17-limit, with lower absolute errors than smaller equal temperaments despite inconsistency in the corresponding odd limit.

* 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 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

| ssy4

| Vulture comma

|-

| 5

| [[Vishnu comma|(20 digits)]]

| 3.34

| {{Monzo| 23 6 -14 }}

| Sasepbigu

| sg14

| 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

| szg10

| Quasiorwellisma

|-

| 7

| [[2401/2400]]

| 0.72

| {{Monzo| -5 -1 -2 4 }}

| Bizozogu

| z4gg

| Breedsma

|-

| 7

| [[4375/4374]]

| 0.40

| {{Monzo| -1 -7 4 1 }}

| Zoquadyo

| zy4

| Ragisma

|-

| 11

| <abbr title="94489280512/94143178827">[[Pythrabian comma|(22 digits)]]</abbr>

| 6.35

| {{Monzo| 33 -23 0 0 1 }}

| Trisalo

| s1o3

| Pythrabian comma

|-

| 11

| [[5632/5625]]

| 2.15

| {{Monzo| 9 -2 -4 0 1 }}

| Saloquagu

| s1og4

| Vishdelisma

|-

| 11

| [[Nexus comma|(12 digits)]]

| 2.04

| {{Monzo| -16 -3 0 0 6 }}

| Tribilo

| 1o3

| 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)2

| 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

| 3o1ozg3

| 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

| 7o3o1uzg3

| 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 }}

| Quadno-athoquadlu-azogu

| 9o43o1u4zg

| 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

|}

=== Rank-2 temperaments ===

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| 1

| 1\270

| 4.44

| 4.{{overline|4}}

| 385/384

| [[Keenanose]]

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| 1

| 29\270

| 128.89

| 128.{{overline|8}}

| 14/13

| [[Tertiathirds]]

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| 1

| 61\270

| 271.11

| 271.{{overline|1}}

| 90/77

| [[Quasiorwell]]

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| 1

| 71\270

| 315.56

| 315.{{overline|5}}

| 6/5

| [[Acrokleismic]] / counteracro

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| 1

| 79\270

| 351.11

| 351.{{overline|1}}

| 49/40

| [[Newt]]

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| 1

| 97\270

| 431.11

| 431.{{overline|1}}

| 77/60

| [[Lockerbie]]

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| 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]]

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| 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]]

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| 2

| 47\270

| 208.89

| 208.{{overline|8}}

| 44/39

| [[Abigail]]

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| 2

| 52\270

| 231.11

| 231.{{overline|1}}

| 8/7

| [[Orga]]

|-

~~| 2~~

~~| 131\270 (4\270)~~

~~| 582.22 (17.78)~~

~~| 7/5 (99/98)~~

~~| [[Quarvish]]~~

|-

| 3

| 17\270

| 75.56

| 75.{{overline|5}}

| 24/23

| [[Terture]]

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| 3

| 31\270

| 137.78

| 137.{{overline|7}}

| 13/12

| [[Avicenna]]

|-

| 5

| ~~83\270 (~~25\270)

| 25\270

| ~~368.89 (~~111.~~11)~~

| 111.{{overline|1}}

| ~~1024/891 (~~16/15)

| 16/15

| [[Quintosec]]

|-

| 6

| ~~112~~\270~~ (4~~\270)

| 16\270

| ~~497.78 (~~97.~~78)~~

| 71.{{overline|1}}

| ~~4/3 (~~128/121)

| 25/24

| [[Trivish]]

|-

| 6

| 22\270

| 97.{{overline|7}}

| 128/121

| [[Sextile]]

|-

| 9

| ~~71\270 (~~11\270)

| 11\270

| ~~315.56 (~~48.~~89)~~

| 48.{{overline|8}}

| ~~6/5 (~~36/35)

| 36/35

| [[Ennealimmal]] / ~~ennealimmia~~

| [[Ennealimmal]] / enneabiotic / ennealympic

|-

| 10

| 16\270~~ (11\270)~~

| 2\270

| 71.~~11 (48.89)~~

| 8.{{overline|8}}

| ~~25/24 (36~~/~~35)~~

| 196/195

| [[~~Decavish~~]]

| [[Decoid]]

|-

| 10

| 56\270~~ (2\270)~~

| 10\270

| ~~248~~.~~89 (8.89)~~

| 44.{{overline|4}}

| ~~15/13 (176~~/~~175)~~

| 40/39

| [[~~Decoid~~]]

| [[Deca]]

|-

| 10

| 71\270~~ (10\270)~~

| 11\270

| ~~315~~.~~56 (44.44)~~

| 48.{{overline|8}}

| ~~6/5 (40~~/~~39)~~

| 36/35

| [[~~Deca~~]]

| [[Decavish]]

|-

| 18

| 71\270~~ (4\270)~~

| 2\270

| ~~248~~.~~89 (17.78)~~

| 8.{{overline|8}}

| ~~15/13 (99~~/~~98)~~

| 1287/1280

| [[~~Hemiennealimmal~~]]

| [[Semihemiennealimmal]]

|-

| 18

| 71\270~~ (2\270)~~

| 4\270

| ~~475~~.~~56 (8.89)~~

| 17.{{overline|7}}

| ~~1053/800 (1287~~/~~1280)~~

| 99/98

| [[~~Semihemiennealimmal~~]]

| [[Hemiennealimmal]]

|-

| 27

| ~~61\270 (~~1\270)

| 1\270

| ~~271~~.~~11 (~~4~~.44)~~

| 4.{{overline|4}}

| ~~1375/1176 (~~385/384)

| 385/384

| [[Trinealimmal]]

|-

| 30

| ~~82\270 (~~1\270)

| 1\270

| ~~364~~.~~44 (~~4~~.44)~~

| 4.{{overline|4}}

| ~~216/175 (~~385/384)

| 385/384

| [[Zinc]]

|-

| 30

| 2\270

| 8.{{overline|8}}

| 196/195

| [[Triacontoid]]

|-

| 45

| ~~59\270 (~~1\270)

| 1\270

| ~~262~~.~~22 (~~4~~.44)~~

| 4.{{overline|4}}

| ~~64/55 (~~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

=== Harmonic scales ===

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

=== [[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 ===

* [[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 ==

@@ 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 ([[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.
+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 the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), and the kalisma ([[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. 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 approxiamted [[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.
+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 so small 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}}. 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.
+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 35: / Line 37: @@
 == Notation ==
 === Ups and downs notation ===
-edo can be notated using [[Kite's ups and downs notation|ups and downs]] with Stein-Zimmerman quarter-tone accidentals representing half-apotomes:
+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 311edo uses alterations of the Promethian set. Since the apotome can be split in two, a 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=""
 ! colspan="2" |+ edosteps
@@ Line 69: / Line 71: @@
 |26
 |-
-! rowspan="3" |Sym.
+! 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>
@@ Line 84: / Line 86: @@
 | rowspan="3" |<big>{{Sagittal|/|\}}</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>
 |-
 !Evo
 | 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
-|<big>{{Sagittal|(|)}}</big>
-|<big>{{sagittal|(|\}}</big>
 |<big>{{sagittal|)||(}}</big>
 |<big>{{sagittal|~||(}}</big>
@@ Line 122: / 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>
 |}
+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>
@@ Line 147: / Line 139: @@
 === Higher-limit JI ===
-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-odd-limit, using the 270i val with the sharp mapping of 23.
+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.
+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 ==
@@ Line 217: / Line 211: @@
 | 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.
-* It is even better 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]].
-* It is also notable in the 17-limit, with lower absolute errors than smaller equal temperaments despite inconsistency in the corresponding odd limit.
+* 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 234: / Line 407: @@
 | 1
 | 1\270
-| 4.44
+| 4.{{overline|4}}
 | 385/384
 | [[Keenanose]]
@@ Line 240: / Line 413: @@
 | 1
 | 29\270
-| 128.89
+| 128.{{overline|8}}
 | 14/13
 | [[Tertiathirds]]
@@ Line 246: / Line 419: @@
 | 1
 | 61\270
-| 271.11
+| 271.{{overline|1}}
 | 90/77
 | [[Quasiorwell]]
@@ Line 252: / Line 425: @@
 | 1
 | 71\270
-| 315.56
+| 315.{{overline|5}}
 | 6/5
 | [[Acrokleismic]] / counteracro
@@ Line 258: / Line 431: @@
 | 1
 | 79\270
-| 351.11
+| 351.{{overline|1}}
 | 49/40
 | [[Newt]]
@@ Line 264: / Line 437: @@
 | 1
 | 97\270
-| 431.11
+| 431.{{overline|1}}
 | 77/60
 | [[Lockerbie]]
@@ Line 270: / 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 282: / 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 294: / Line 479: @@
 | 2
 | 47\270
-| 208.89
+| 208.{{overline|8}}
 | 44/39
 | [[Abigail]]
@@ Line 300: / 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 318: / 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
+=== 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
+=== [[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 ===
+* [[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 ==

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