270edo: Difference between revisions
Commas and more, possibly not the most reasonable, feel free to add/remove |
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== Theory == | == Theory == | ||
270edo is an extremely strong [[13-limit]] system, [[distinctly consistent]] through the [[15-odd-limit]] | 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]] | 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 [[ | 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 ( | 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 | 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. | 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. | ||
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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]]. | 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 | 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 === | === Prime harmonics === | ||
| Line 35: | Line 37: | ||
== Notation == | == Notation == | ||
=== Ups and downs 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- | 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}} | {{Ups and downs sharpness|270|true}} | ||
=== Sagittal notation === | === Sagittal notation === | ||
<span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses | <span data-darkreader-inline-color="">The</span> [[Sagittal notation]] <span data-darkreader-inline-color="">for 270edo uses symbols from the Promethean set. Since the apotome can be split in two, the Stein-Zimmermann half-sharp and half-flat may be used.</span> | ||
{| class="wikitable center-all" data-darkreader-inline-color="" | {| class="wikitable center-all" data-darkreader-inline-color="" | ||
! colspan="2" |+ edosteps | ! colspan="2" |+ edosteps | ||
| Line 70: | Line 72: | ||
|- | |- | ||
! rowspan="3" |Symbol | ! rowspan="3" |Symbol | ||
!SZ | !Evo-SZ | ||
| rowspan="3" |<big>{{sagittal||(}}</big> | | rowspan="3" |<big>{{sagittal||(}}</big> | ||
| rowspan="3" |<big>{{sagittal|)|(}}</big> | | rowspan="3" |<big>{{sagittal|)|(}}</big> | ||
| rowspan="3" |<big>{{Sagittal|~|(}}</big> | | rowspan="3" |<big>{{Sagittal|~|(}}</big> | ||
| rowspan="3" |<big>{{Sagittal|~~|}}</big> | |||
| rowspan="3" |<big>{{Sagittal|/|}}</big> | | rowspan="3" |<big>{{Sagittal|/|}}</big> | ||
| rowspan="3" |<big>{{Sagittal||)}}</big> | | rowspan="3" |<big>{{Sagittal||)}}</big> | ||
| rowspan="3" |<big>{{sagittal||\}}</big> | | rowspan="3" |<big>{{sagittal||\}}</big> | ||
| rowspan="3" |<big>{{sagittal| | | 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> | | rowspan="3" |<big>{{Sagittal|/|\}}</big> | ||
|<big>{{Sagittal|t}}</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> | | rowspan="2" |<big>{{Sagittal|#}}</big> | ||
|- | |- | ||
!Evo | !Evo | ||
| rowspan="2" |<big>{{sagittal|)/|\}}</big> | | rowspan="2" |<big>{{sagittal|)/|\}}</big> | ||
| | | rowspan="2" |<big>{{Sagittal|(|)}}</big> | ||
| rowspan="2" |<big>{{sagittal|(|\}}</big> | |||
|< | |||
| | |||
|- | |- | ||
!Revo | !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> | |<big>{{sagittal|//||}}</big> | ||
|<big>{{sagittal|/||)}}</big> | |<big>{{sagittal|/||)}}</big> | ||
|<big>{{Sagittal|/||\}}</big> | |<big>{{Sagittal|/||\}}</big> | ||
|} | |} | ||
Note that the Revo notation has matching flag sequences between the double-shaft symbols and a subsequence of the single-shaft symbols. | |||
<span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span> | <span data-darkreader-inline-color="">Alternate spellings in the Promethean set (comma tempered out):</span> | ||
| Line 147: | Line 139: | ||
=== Higher-limit JI === | === 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. 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]]. | 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 == | == Regular temperament properties == | ||
| Line 219: | Line 211: | ||
| 4.58 | | 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. | * 23-limit is not the subgroup it does best, with the no-23 29- and 31-limit approximated even better. | ||
* It is | * 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 === | === Commas === | ||
{| class="commatable wikitable center-1 center-2 right- | {| class="commatable wikitable center-1 center-2 right-3 center-6" | ||
![[Harmonic limit|Prime | ! [[Harmonic limit|Prime<br>limit]] | ||
! [[Ratio]]<ref group="note">{{rd}}</ref> | |||
![[Ratio]]<ref group="note">{{rd}}</ref> | ! [[Cent]]s | ||
![[Cent | ! [[Monzo]] | ||
![[Monzo]] | ! colspan="2" | [[Kite's color notation|Color name]] | ||
! colspan="2" |[[Color name]] | ! Name(s) | ||
!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 | | 5 | ||
| | | [[Vishnu comma|(20 digits)]] | ||
| | | 3.34 | ||
|{{Monzo| | | {{Monzo| 23 6 -14 }} | ||
| | | Sasepbigu | ||
| | | sg<sup>14</sup> | ||
| | | Vishnu comma | ||
|- | |- | ||
| | | 7 | ||
|[[ | | [[33554432/33480783|(16 digits)]] | ||
|3. | | 3.80 | ||
| | | {{Monzo| 25 -14 0 -1 }} | ||
| | | Sasaru | ||
| | | ssr | ||
| | | Septischisma | ||
|- | |- | ||
|7 | | 7 | ||
|[[ | | [[Quasiorwellisma|(16 digits)]] | ||
|3. | | 3.73 | ||
|{{ | | {{Monzo| 22 -1 -10 1 }} | ||
| | | Sazoquinbigu | ||
| | | szg<sup>10</sup> | ||
| | | Quasiorwellisma | ||
|- | |- | ||
|7 | | 7 | ||
|[[2401/2400]] | | [[2401/2400]] | ||
|0.72 | | 0.72 | ||
|{{ | | {{Monzo| -5 -1 -2 4 }} | ||
|Bizozogu | | Bizozogu | ||
|z<sup>4</sup>gg | | z<sup>4</sup>gg | ||
|Breedsma | | Breedsma | ||
|- | |- | ||
|7 | | 7 | ||
|[[4375/4374]] | | [[4375/4374]] | ||
|0.40 | | 0.40 | ||
|{{Monzo|-1 -7 4 1}} | | {{Monzo| -1 -7 4 1 }} | ||
|Zoquadyo | | Zoquadyo | ||
|zy<sup>4</sup> | | zy<sup>4</sup> | ||
|Ragisma | | 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 | | 11 | ||
| | | [[5632/5625]] | ||
| | | 2.15 | ||
|{{Monzo| | | {{Monzo| 9 -2 -4 0 1 }} | ||
| | | Saloquagu | ||
| | | s1og<sup>4</sup> | ||
| | | Vishdelisma | ||
|- | |- | ||
|11 | | 11 | ||
|[[ | | [[Nexus comma|(12 digits)]] | ||
|2. | | 2.04 | ||
| | | {{Monzo| -16 -3 0 0 6 }} | ||
| | | Tribilo | ||
| | | 1o<sup>3</sup> | ||
| | | Nexus comma | ||
|- | |- | ||
|11 | | 11 | ||
|[[ | | [[3025/3024]] | ||
| | | 0.57 | ||
| | | {{Monzo| -4 -3 2 -1 2 }} | ||
| | | Loloruyoyo | ||
| | | 1ooryy | ||
| | | Lehmerisma | ||
|- | |- | ||
|11 | | 11 | ||
|[[ | | [[9801/9800]] | ||
|0. | | 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 | | 13 | ||
|[[ | | [[1001/1000]] | ||
| | | 1.73 | ||
|{{Monzo| | | {{Monzo| -3 0 -3 1 1 1 }} | ||
| | | Tholozotrigu | ||
| | | 3o1ozg<sup>3</sup> | ||
| | | Fairytale comma, sinbadma | ||
|- | |- | ||
|13 | | 13 | ||
|[[ | | [[2080/2079]] | ||
| | | 0.83 | ||
|{{Monzo|-3 | | {{Monzo| 5 -3 1 -1 -1 1 }} | ||
| | | Tholuruyo | ||
| | | 3o1ury | ||
| | | Ibnsinma, sinaisma | ||
|- | |- | ||
|13 | | 13 | ||
|[[ | | [[4096/4095]] | ||
|0. | | 0.42 | ||
|{{Monzo| | | {{Monzo| 12 -2 -1 -1 0 -1 }} | ||
| | | Sathurugu | ||
| | | s3urg | ||
| | | Minisma | ||
|- | |- | ||
| | | 17 | ||
|[[ | | [[12376/12375]] | ||
|0. | | 0.14 | ||
|{{Monzo| | | {{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 | | 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 | | 23 | ||
|[[529/528]] | | [[529/528]] | ||
|3.24 | | 3.24 | ||
|2.3.11.23 | | 2.3.11.23 {{monzo| -4 -1 -1 2 }} | ||
|Bitwetho-alu | | Bitwetho-alu | ||
|23oo1u | | 23oo1u | ||
|Preziosisma | | Preziosisma | ||
|- | |- | ||
|29 | | 29 | ||
|[[784/783]] | | [[784/783]] | ||
|2.20 | | 2.20 | ||
|2.3.7.29 | | 2.3.7.29 {{monzo| 4 -3 2 -1 }} | ||
|Twenuzozo | | Twenuzozo | ||
|23uzz | | 23uzz | ||
|Biminorisma | | Biminorisma | ||
|- | |- | ||
|31 | | 31 | ||
|[[621/620]] | | [[621/620]] | ||
|2.79 | | 2.79 | ||
|2.3.5.23.31 | | 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }} | ||
|Thiwutwethogu | | Thiwutwethogu | ||
|31u23og | | 31u23og | ||
|Owowhatsthisma | | Owowhatsthisma | ||
|} | |} | ||
<references group="note"/> | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
| Line 407: | Line 407: | ||
| 1 | | 1 | ||
| 1\270 | | 1\270 | ||
| 4. | | 4.{{overline|4}} | ||
| 385/384 | | 385/384 | ||
| [[Keenanose]] | | [[Keenanose]] | ||
| Line 413: | Line 413: | ||
| 1 | | 1 | ||
| 29\270 | | 29\270 | ||
| 128. | | 128.{{overline|8}} | ||
| 14/13 | | 14/13 | ||
| [[Tertiathirds]] | | [[Tertiathirds]] | ||
| Line 419: | Line 419: | ||
| 1 | | 1 | ||
| 61\270 | | 61\270 | ||
| 271. | | 271.{{overline|1}} | ||
| 90/77 | | 90/77 | ||
| [[Quasiorwell]] | | [[Quasiorwell]] | ||
| Line 425: | Line 425: | ||
| 1 | | 1 | ||
| 71\270 | | 71\270 | ||
| 315. | | 315.{{overline|5}} | ||
| 6/5 | | 6/5 | ||
| [[Acrokleismic]] / counteracro | | [[Acrokleismic]] / counteracro | ||
| Line 431: | Line 431: | ||
| 1 | | 1 | ||
| 79\270 | | 79\270 | ||
| 351. | | 351.{{overline|1}} | ||
| 49/40 | | 49/40 | ||
| [[Newt]] | | [[Newt]] | ||
| Line 437: | Line 437: | ||
| 1 | | 1 | ||
| 97\270 | | 97\270 | ||
| 431. | | 431.{{overline|1}} | ||
| 77/60 | | 77/60 | ||
| [[Lockerbie]] | | [[Lockerbie]] | ||
| Line 443: | Line 443: | ||
| 1 | | 1 | ||
| 107\270 | | 107\270 | ||
| 475. | | 475.{{overline|5}} | ||
| 25/19 | | 25/19 | ||
| [[Vulture]] | | [[Vulture]] | ||
|- | |||
| 2 | |||
| 4\270 | |||
| 17.{{overline|7}} | |||
| 99/98 | |||
| [[Quarvish]] | |||
|- | |- | ||
| 2 | | 2 | ||
| 14\270 | | 14\270 | ||
| 62. | | 62.{{overline|2}} | ||
| 28/27 | | 28/27 | ||
| [[Eagle]] | | [[Eagle]] | ||
| Line 455: | Line 461: | ||
| 2 | | 2 | ||
| 16\270 | | 16\270 | ||
| 71. | | 71.{{overline|1}} | ||
| 25/24 | | 25/24 | ||
| [[Vishnu]] / ananta | | [[Vishnu]] / acyuta / ananta | ||
|- | |- | ||
|2 | | 2 | ||
| | | 23\270 | ||
| 102.{{overline|2}} | |||
| | | 35/33 | ||
| [[Gariwizmic]] | |||
| | |||
| | |||
|- | |- | ||
| 2 | | 2 | ||
| 28\270 | | 28\270 | ||
| 124. | | 124.{{overline|4}} | ||
| 275/256 | | 275/256 | ||
| [[Semivulture]] | | [[Semivulture]] | ||
| Line 475: | Line 479: | ||
| 2 | | 2 | ||
| 47\270 | | 47\270 | ||
| 208. | | 208.{{overline|8}} | ||
| 44/39 | | 44/39 | ||
| [[Abigail]] | | [[Abigail]] | ||
| Line 481: | Line 485: | ||
| 2 | | 2 | ||
| 52\270 | | 52\270 | ||
| 231. | | 231.{{overline|1}} | ||
| 8/7 | | 8/7 | ||
| [[Orga]] | | [[Orga]] | ||
|- | |- | ||
| 3 | | 3 | ||
| 17\270 | | 17\270 | ||
| 75. | | 75.{{overline|5}} | ||
| 24/23 | | 24/23 | ||
| [[Terture]] | | [[Terture]] | ||
| Line 499: | Line 497: | ||
| 3 | | 3 | ||
| 31\270 | | 31\270 | ||
| 137. | | 137.{{overline|7}} | ||
| 13/12 | | 13/12 | ||
| [[Avicenna]] | | [[Avicenna]] | ||
|- | |- | ||
| 5 | | 5 | ||
| | | 25\270 | ||
| | | 111.{{overline|1}} | ||
| | | 16/15 | ||
| [[Quintosec]] | | [[Quintosec]] | ||
|- | |- | ||
| 6 | | 6 | ||
| | | 16\270 | ||
| | | 71.{{overline|1}} | ||
| | | 25/24 | ||
| [[Trivish]] | |||
|- | |||
| 6 | |||
| 22\270 | |||
| 97.{{overline|7}} | |||
| 128/121 | |||
| [[Sextile]] | | [[Sextile]] | ||
|- | |- | ||
| 9 | | 9 | ||
| | | 11\270 | ||
| | | 48.{{overline|8}} | ||
| | | 36/35 | ||
| [[Ennealimmal]] / | | [[Ennealimmal]] / enneabiotic / ennealympic | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 2\270 | ||
| | | 8.{{overline|8}} | ||
| | | 196/195 | ||
| [[ | | [[Decoid]] | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 10\270 | ||
| | | 44.{{overline|4}} | ||
| | | 40/39 | ||
| [[ | | [[Deca]] | ||
|- | |- | ||
| 10 | | 10 | ||
| | | 11\270 | ||
| | | 48.{{overline|8}} | ||
| | | 36/35 | ||
| [[ | | [[Decavish]] | ||
|- | |- | ||
| 18 | | 18 | ||
| | | 2\270 | ||
| | | 8.{{overline|8}} | ||
| | | 1287/1280 | ||
| [[ | | [[Semihemiennealimmal]] | ||
|- | |- | ||
| 18 | | 18 | ||
| | | 4\270 | ||
| | | 17.{{overline|7}} | ||
| | | 99/98 | ||
| [[ | | [[Hemiennealimmal]] | ||
|- | |- | ||
| 27 | | 27 | ||
| | | 1\270 | ||
| | | 4.{{overline|4}} | ||
| | | 385/384 | ||
| [[Trinealimmal]] | | [[Trinealimmal]] | ||
|- | |- | ||
| 30 | | 30 | ||
| | | 1\270 | ||
| | | 4.{{overline|4}} | ||
| | | 385/384 | ||
| [[Zinc]] | | [[Zinc]] | ||
|- | |||
| 30 | |||
| 2\270 | |||
| 8.{{overline|8}} | |||
| 196/195 | |||
| [[Triacontoid]] | |||
|- | |- | ||
| 45 | | 45 | ||
| | | 1\270 | ||
| | | 4.{{overline|4}} | ||
| | | 385/384 | ||
| [[Rhodium]] | | [[Rhodium]] | ||
|} | |} | ||
<nowiki/>* [[ | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
== Scales == | == Scales == | ||
=== | === [[MOS]] scales === | ||
* [[Ennealimmal]][45]: | * [[Abigail]][86]: 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 5 1 5 1 5 1 5 1 5 1 5 1 5 1 5 | ||
* [[Avicenna]][96]: 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 3 3 3 3 3 1 3 3 3 3 3 | |||
* [[Eagle]][84]: 4 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 4 4 1 4 1 | |||
* [[Ennealimmal]][27]: 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 11 11 8 | |||
* Ennealimmal[45]: 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 8 3 8 3 8 | |||
* [[Quasiorwell]][84]: 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 1 7 1 7 1 1 7 1 1 7 1 7 1 1 7 1 | |||
* [[Quintosec]][25]: 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4 4 21 4 21 4 | |||
* Quintosec[55]: 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 4 4 9 4 4 4 4 4 9 4 4 | |||
* Quintosec[65]: 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 4 4 4 5 4 4 4 4 4 5 4 4 4 | |||
* [[Vishnu]][34]: 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 | * [[Vishnu]][34]: 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 9 7 | ||
=== Harmonic scales === | === Harmonic scales === | ||
270edo accurately approximates | 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 === | === Other scales === | ||