27edo: Difference between revisions
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27edo, with its 400 cent major third, tempers out the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4. | 27edo, with its 400 cent major third, tempers out the lesser diesis, [[128/125]], and the septimal comma, [[64/63]], and hence [[126/125]] as well. These it shares with 12edo, making some relationships familiar, and they both support the [[augene]] temperament. It shares with [[22edo]] tempering out the allegedly Bohlen-Pierce comma [[245/243]] as well as 64/63, so that they both support the [[superpyth]] temperament, with four quite sharp "superpythagorean" fifths giving a sharp [[9/7]] in place of meantone's 5/4. | ||
Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly | Though 27edo's [[7-limit]] tuning is not highly accurate, it nonetheless is the smallest equal division to represent the 7-odd-limit both [[consistent]]ly and distinctly—that is, everything in the [[7-odd-limit]] diamond is uniquely represented by a certain number of steps of 27edo. It also represents the 13th harmonic very well, and performs quite decently as a 2.3.5.7.13.19 (no-11s, no-17s 19-limit) temperament. It also approximates [[19/10]], [[19/12]], and [[19/14]], so {{dash|0, 7, 13, 25|med}} does quite well as a 10:12:14:19 chord, with the major seventh 25\27 being less than one cent off from 19/10. Octave-inverted, these also form a quite convincing approximation of the main Bohlen–Pierce triads, 3:5:7 and 5:7:9, making 27 the smallest edo that can simulate tritave harmony, although it rapidly becomes rough if extended to the 11 and above, unlike a true tritave based system. | ||
Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant. | Its step, as well as the octave-inverted and octave-equivalent versions of it, has some of the highest [[harmonic entropy]] possible and thus is, in theory, one of the most dissonant intervals possible, assuming the relatively common values of {{nowrap|''a'' {{=}} 2}} and {{nowrap|''s'' {{=}} 1%}}. This property is shared with all edos between around 24 and 30. Intervals smaller than this tend to be perceived as unison and are more consonant as a result; intervals larger than this have less "tension" and thus are also more consonant. | ||
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|- | |- | ||
| fourthward wa | | fourthward wa | ||
| {a, b}, b < | | {a, b}, b < −1 | ||
| 32/27, 16/9 | | 32/27, 16/9 | ||
|- | |- | ||
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=== 6L 1s (archeotonic) notation === | === 6L 1s (archeotonic) notation === | ||
The notation of Tetracot[7]. Notes are denoted as LLLLLLs = CDEFGABC, and raising and lowering by a chroma (L | The notation of Tetracot[7]. Notes are denoted as {{nowrap|LLLLLLs {{=}} CDEFGABC}}, and raising and lowering by a chroma {{nowrap|(L − s)}}, 1 step in this instance, is denoted by ♯ and ♭. | ||
{| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed" | {| class="wikitable center-1 right-2 center-3 mw-collapsible mw-collapsed" | ||
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| 2/1 | | 2/1 | ||
|} | |} | ||
{{clear}} | {{clear}} | ||
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== Regular temperament properties == | == Regular temperament properties == | ||
{| class="wikitable center-4 center-5 center-6" | {| class="wikitable center-4 center-5 center-6" | ||
|- | |||
! rowspan="2" | [[Subgroup]] | ! rowspan="2" | [[Subgroup]] | ||
! rowspan="2" | [[Comma list]] | ! rowspan="2" | [[Comma list]] | ||
! rowspan="2" | [[Mapping]] | ! rowspan="2" | [[Mapping]] | ||
! rowspan="2" | Optimal<br>8ve stretch (¢) | ! rowspan="2" | Optimal<br />8ve stretch (¢) | ||
! colspan="2" | Tuning error | ! colspan="2" | Tuning error | ||
|- | |- | ||
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{| class="wikitable center-all left-3 left-4" | {| class="wikitable center-all left-3 left-4" | ||
|- | |- | ||
! Periods<br>per 8ve | ! Periods<br />per 8ve | ||
! Generator | ! Generator | ||
! Temperaments | ! Temperaments | ||
! | ! MOS scales | ||
|- | |- | ||
| 1 | | 1 | ||
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| 1 | | 1 | ||
| 5\27 | | 5\27 | ||
| [[Machine]] (27)<br>[[Kumonga]] (27e) | | [[Machine]] (27)<br />[[Kumonga]] (27e) | ||
| [[1L 4s]], [[5L 1s]], [[5L 6s]], [[11L 5s]] | | [[1L 4s]], [[5L 1s]], [[5L 6s]], [[11L 5s]] | ||
|- | |- | ||
| 1 | | 1 | ||
| 7\27 | | 7\27 | ||
| [[Myna]] (27e) / coleto (27e) / myno (27)<br>[[Oolong]] (27e) | | [[Myna]] (27e) / coleto (27e) / myno (27)<br />[[Oolong]] (27e) | ||
| [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]] | | [[4L 3s]], [[4L 7s]], [[4L 11s]], [[4L 15s]], [[4L 19s]] | ||
|- | |- | ||
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| 3 | | 3 | ||
| 4\27 | | 4\27 | ||
| [[Oodako]] (27e)<br>[[Terrain]] | | [[Oodako]] (27e)<br />[[Terrain]] | ||
| [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15s]] | | [[3L 3s]], [[6L 3s]], [[6L 9s]], [[6L 15s]] | ||
|- | |- | ||
| 9 | | 9 | ||
| 1\27 | | 1\27 | ||
| [[Niner]] (27e)<br>[[Ennealimmal]] (out of tune) | | [[Niner]] (27e)<br />[[Ennealimmal]] (out of tune) | ||
| [[9L 9s]] | | [[9L 9s]] | ||
|} | |} | ||
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{| class="commatable wikitable center-all left-3 right-4 left-6" | {| class="commatable wikitable center-all left-3 right-4 left-6" | ||
|- | |- | ||
! [[Harmonic limit|Prime<br>limit]] | ! [[Harmonic limit|Prime<br />limit]] | ||
! [[Ratio]]<ref group="note">{{rd}}</ref> | ! [[Ratio]]<ref group="note">{{rd}}</ref> | ||
! [[Monzo]] | ! [[Monzo]] | ||
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=== MOS scales === | === MOS scales === | ||
{{Main|List of MOS scales in 27edo}} | {{Main|List of MOS scales in 27edo}} | ||
* Superpyth pentatonic | * Superpyth pentatonic – Superpyth[5] [[2L 3s]] (gen = 11\27): 5 5 6 5 6 | ||
* Superpyh diatonic | * Superpyh diatonic – Superpyth[7] [[5L 2s]] (gen = 11\27): 5 5 1 5 5 5 1 | ||
* Superpyth chromatic | * Superpyth chromatic – Superpyth[12] [[5L 7s]] (gen = 11\27): 4 1 1 4 1 4 1 4 1 1 4 1 | ||
* Superpyth hyperchromatic | * Superpyth hyperchromatic – Superpyth[17] [[5L 12s]] (gen = 11\27): 1 3 1 1 3 1 1 1 3 1 1 3 1 1 3 1 1 | ||
* Augene[6] [[3L 3s]] (period = 9\27, gen = 2\27): 7 2 7 2 7 2 | * Augene[6] [[3L 3s]] (period = 9\27, gen = 2\27): 7 2 7 2 7 2 | ||
* Augene[9] [[3L 6s]] (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2 | * Augene[9] [[3L 6s]] (period = 9\27, gen = 2\27): 5 2 2 5 2 2 5 2 2 | ||
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* enharmonic trichord octave species: 9 2 5 9 2, 2 9 5 2 9 | * enharmonic trichord octave species: 9 2 5 9 2, 2 9 5 2 9 | ||
* 5-limit / pental double harmonic hexatonic (Augmented[6] [[4M]]): 2 7 2 7 7 2, 7 7 2 2 7 2 | * 5-limit / pental double harmonic hexatonic (Augmented[6] [[4M]]): 2 7 2 7 7 2, 7 7 2 2 7 2 | ||
* Superpyth melodic minor | * Superpyth melodic minor – Superpyth 2|4 #6 #7 or 5|1 b3: 5 1 5 5 5 5 1 | ||
* Superpyth harmonic minor | * Superpyth harmonic minor – Superpyth 2|4 #7: 5 1 5 5 1 9 1 | ||
* Superpyth harmonic major | * Superpyth harmonic major – Superpyth 5|1 b6: 5 5 1 5 1 9 1 | ||
* Superpyth double harmonic major | * Superpyth double harmonic major – Superpyth 5|1 b2 b6: 1 9 1 5 1 9 1 | ||
* [[Zarlino]] / Ptolemy diatonic, "just" major: 5 4 2 5 4 5 2 | * [[Zarlino]] / Ptolemy diatonic, "just" major: 5 4 2 5 4 5 2 | ||
* "Just" minor (inverse of "just" major): 5 2 4 5 2 5 4 | * "Just" minor (inverse of "just" major): 5 2 4 5 2 5 4 | ||