270edo: Difference between revisions
m →Theory: - typo |
→Rank-2 temperaments: + trivish. Also convert the table to minimal form since we're already reducing it to the first semi-octave |
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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]]). 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 [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000) | In the [[7-limit]] it tempers out the [[2401/2400|breedsma]] (2401/2400), the [[4375/4374|ragisma]] (4375/4374), and by extension the [[wizma]] (420175/419904), and the [[landscape comma]] (250047/250000) so that it [[support]]s [[ennealimmal]] temperament. It also tempers out the [[quasiorwellisma]] (29360128/29296875) and the [[garischisma]] (33554432/33480783). | ||
In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), | In the [[11-limit]], it tempers out the lehmerisma ([[3025/3024]]), the vishdel comma ([[5632/5625]]), the kalisma ([[9801/9800]]), the [[symbiotic comma]] (19712/19683), the [[nexus comma]] (1771561/1769472), and the [[quartisma]] (117440512/117406179). Notably, it is consistent to distance 3 in the [[11-odd-limit]], and almost to distance 4 ((11/10)<sup>4</sup> and (20/11)<sup>4</sup> are a hair off, 50.4%). | ||
Finally, in the [[13-limit]] it is | 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. | ||
| Line 19: | Line 19: | ||
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 === | |||
{| class="commatable wikitable center-1 center-2 right-3 center-6" | |||
! [[Harmonic limit|Prime<br>limit]] | |||
! [[Ratio]]<ref group="note">{{rd}}</ref> | |||
! [[Cent]]s | |||
! [[Monzo]] | |||
! colspan="2" | [[Kite's color notation|Color name]] | |||
! Name(s) | |||
|- | |||
| 5 | |||
| <abbr title="10485760000/10460353203">[[Vulture comma|(22 digits)]]</abbr> | |||
| 4.20 | |||
| {{Monzo| 24 -21 4 }} | |||
| Sasaquadyo | |||
| ssy<sup>4</sup> | |||
| Vulture comma | |||
|- | |||
| 5 | |||
| [[Vishnuzma|(20 digits)]] | |||
| 3.34 | |||
| {{Monzo| 23 6 -14 }} | |||
| Sasepbigu | |||
| sg<sup>14</sup> | |||
| Vishnuzma | |||
|- | |||
| 7 | |||
| [[33554432/33480783|(16 digits)]] | |||
| 3.80 | |||
| {{Monzo| 25 -14 0 -1 }} | |||
| Sasaru | |||
| ssr | |||
| Garischisma | |||
|- | |||
| 7 | |||
| [[2401/2400]] | |||
| 0.72 | |||
| {{Monzo| -5 -1 -2 4 }} | |||
| Bizozogu | |||
| z<sup>4</sup>gg | |||
| Breedsma | |||
|- | |||
| 7 | |||
| [[4375/4374]] | |||
| 0.40 | |||
| {{Monzo| -1 -7 4 1 }} | |||
| Zoquadyo | |||
| zy<sup>4</sup> | |||
| Ragisma | |||
|- | |||
| 7 | |||
| [[Quasiorwellisma|(16 digits)]] | |||
| 3.73 | |||
| {{Monzo| 22 -1 -10 1 }} | |||
| Sazoquinbigu | |||
| szg<sup>10</sup> | |||
| Quasiorwellisma | |||
|- | |||
| 11 | |||
| <abbr title="94489280512/94143178827">[[Pythrabian comma|(22 digits)]]</abbr> | |||
| 6.35 | |||
| {{Monzo| 33 -23 0 0 1 }} | |||
| Trisalo | |||
| s1o<sup>3</sup> | |||
| Pythrabian comma | |||
|- | |||
| 11 | |||
| [[5632/5625]] | |||
| 2.15 | |||
| {{Monzo| 9 -2 -4 0 1 }} | |||
| Saloquagu | |||
| s1og<sup>4</sup> | |||
| Vishdel comma | |||
|- | |||
| 11 | |||
| [[Nexus comma|(12 digits)]] | |||
| 2.04 | |||
| {{Monzo| -16 -3 0 0 6 }} | |||
| Tribilo | |||
| 1o<sup>3</sup> | |||
| Nexus comma | |||
|- | |||
| 11 | |||
| [[3025/3024]] | |||
| 0.57 | |||
| {{Monzo| -4 -3 2 -1 2 }} | |||
| Loloruyoyo | |||
| 1ooryy | |||
| Lehmerisma | |||
|- | |||
| 11 | |||
| [[9801/9800]] | |||
| 0.18 | |||
| {{Monzo| -3 4 -2 -2 2 }} | |||
| Bilorugu | |||
| (1org)<sup>2</sup> | |||
| Kalisma | |||
|- | |||
| 13 | |||
| [[676/675]] | |||
| 2.56 | |||
| {{Monzo| 2 -3 -2 0 0 2 }} | |||
| Bithogu | |||
| 3oogg | |||
| Island comma, parizeksma | |||
|- | |||
| 13 | |||
| [[1001/1000]] | |||
| 1.73 | |||
| {{Monzo| -3 0 -3 1 1 1 }} | |||
| Tholozotrigu | |||
| 3o1ozg<sup>3</sup> | |||
| Fairytale comma, sinbadma | |||
|- | |||
| 13 | |||
| [[2080/2079]] | |||
| 0.83 | |||
| {{Monzo| 5 -3 1 -1 -1 1 }} | |||
| Tholuruyo | |||
| 3o1ury | |||
| Ibnsinma, sinaisma | |||
|- | |||
| 13 | |||
| [[4096/4095]] | |||
| 0.42 | |||
| {{Monzo| 12 -2 -1 -1 0 -1 }} | |||
| Sathurugu | |||
| s3urg | |||
| Minisma | |||
|- | |||
|17 | |||
| [[12376/12375]] | |||
| 0.14 | |||
| {{Monzo| 3 -2 -3 1 -1 1 1 }} | |||
| Sotholuzotrigu | |||
| 7o3o1uzg<sup>3</sup> | |||
| Flashma | |||
|- | |||
| 19 | |||
| [[1216/1215]] | |||
| 1.42 | |||
| 2.3.5.19 {{Monzo| 6 -5 -1 1 }} | |||
| Sanogu | |||
| s9og | |||
| Password, Eratosthenes' comma | |||
|- | |||
|19 | |||
|[[11859211/11859210|(16 digits)]] | |||
|0.00 | |||
|{{Monzo|-1 -4 -1 1 -4 1 0 4}} | |||
|<small>Quadno-athoquadlu-azogu</small> | |||
|<small>9o<sup>4</sup>3o1u<sup>4</sup>zg</small> | |||
|Tredekisma | |||
|- | |||
| 23 | |||
| [[529/528]] | |||
| 3.24 | |||
| 2.3.11.23 {{monzo| -4 -1 -1 2 }} | |||
| Bitwetho-alu | |||
| 23oo1u | |||
| Preziosisma | |||
|- | |||
| 29 | |||
| [[784/783]] | |||
| 2.20 | |||
| 2.3.7.29 {{monzo| 4 -3 2 -1 }} | |||
| Twenuzozo | |||
| 23uzz | |||
| Biminorisma | |||
|- | |||
| 31 | |||
| [[621/620]] | |||
| 2.79 | |||
| 2.3.5.23.31 {{monzo| -2 3 -1 1 -1 }} | |||
| Thiwutwethogu | |||
| 31u23og | |||
| Owowhatsthisma | |||
|} | |||
<references group="note"/> | |||
=== Rank-2 temperaments === | === Rank-2 temperaments === | ||
| Line 236: | Line 407: | ||
| 1 | | 1 | ||
| 1\270 | | 1\270 | ||
| 4. | | 4.{{overline|4}} | ||
| 385/384 | | 385/384 | ||
| [[Keenanose]] | | [[Keenanose]] | ||
| Line 242: | Line 413: | ||
| 1 | | 1 | ||
| 29\270 | | 29\270 | ||
| 128. | | 128.{{overline|8}} | ||
| 14/13 | | 14/13 | ||
| [[Tertiathirds]] | | [[Tertiathirds]] | ||
| Line 248: | Line 419: | ||
| 1 | | 1 | ||
| 61\270 | | 61\270 | ||
| 271. | | 271.{{overline|1}} | ||
| 90/77 | | 90/77 | ||
| [[Quasiorwell]] | | [[Quasiorwell]] | ||
| Line 254: | Line 425: | ||
| 1 | | 1 | ||
| 71\270 | | 71\270 | ||
| 315. | | 315.{{overline|5}} | ||
| 6/5 | | 6/5 | ||
| [[Acrokleismic]] / counteracro | | [[Acrokleismic]] / counteracro | ||
| Line 260: | Line 431: | ||
| 1 | | 1 | ||
| 79\270 | | 79\270 | ||
| 351. | | 351.{{overline|1}} | ||
| 49/40 | | 49/40 | ||
| [[Newt]] | | [[Newt]] | ||
| Line 266: | Line 437: | ||
| 1 | | 1 | ||
| 97\270 | | 97\270 | ||
| 431. | | 431.{{overline|1}} | ||
| 77/60 | | 77/60 | ||
| [[Lockerbie]] | | [[Lockerbie]] | ||
| Line 272: | 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 284: | Line 461: | ||
| 2 | | 2 | ||
| 16\270 | | 16\270 | ||
| 71. | | 71.{{overline|1}} | ||
| 25/24 | | 25/24 | ||
| [[Vishnu]] / ananta / | | [[Vishnu]] / acyuta / ananta | ||
|- | |||
| 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 296: | Line 479: | ||
| 2 | | 2 | ||
| 47\270 | | 47\270 | ||
| 208. | | 208.{{overline|8}} | ||
| 44/39 | | 44/39 | ||
| [[Abigail]] | | [[Abigail]] | ||
| Line 302: | 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 320: | 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}} | ||
| | | 176/175 | ||
| [[ | | [[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]] | ||
|- | |- | ||
| 45 | | 45 | ||
| | | 1\270 | ||
| | | 4.{{overline|4}} | ||
| | | 385/384 | ||
| [[Rhodium]] | | [[Rhodium]] | ||
|} | |} | ||
<nowiki/>* [[ | <nowiki/>* In [[normal forms #Minimal-generator form|minimal-generator form]] | ||
== Scales == | == Scales == | ||
| Line 398: | Line 581: | ||
=== Harmonic scales === | === Harmonic scales === | ||
270edo accurately approximates the mode 16 of [[harmonic series]]. The scale in adjacent steps is 24, 22, 21, 20, 19, 18, 17, 17, 16, 15, 15, 14, 14, 13, 13, 12. Four interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28, and 30/29~31/30. | 270edo very accurately approximates the mode 16 of [[harmonic series]]. The scale in adjacent steps is 24, 22, 21, 20, 19, 18, 17, 17, 16, 15, 15, 14, 14, 13, 13, 12. Four interval pairs are conflated: 23/22~24/23, 26/25~27/26, 28/27~29/28, and 30/29~31/30. | ||
It further does decently in the mode 24. The scale in adjacent steps is 16, 15, 15, 14, 14, 13, 13, 12, 12, 12, 11, 11, 11, 10, 10, 10, 10, 9, 9, 9, 9, 9, 8, 8. | 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. | ||