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27edo: Difference between revisions

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Move temperament measures to RTT properties section
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| +31
| +31
|-
|-
! colspan="2" | [[nearest edomapping]]
! colspan="2" | [[Nearest edomapping]]
| 27
| 27
| 16
| 16
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| 7
| 7
|-
|-
! colspan="2" | [[fifthspan]]
! colspan="2" | [[Fifthspan]]
| 0
| 0
| +1
| +1
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For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. See also the [[22edo]] page.
For a more complete list, see [[Ups and Downs Notation #Chords and Chord Progressions]]. See also the [[22edo]] page.


== Just approximation ==
== JI approximation ==


=== Selected just intervals by error ===
=== 15-odd-limit interval mappings ===
The following table shows how [[15-odd-limit intervals]] are represented in 27edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
The following table shows how [[15-odd-limit intervals]] are represented in 27edo. Prime harmonics are in '''bold'''; inconsistent intervals are in ''italic''.
{| class="wikitable center-all"
{| class="wikitable center-all"
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|}


=== Temperament measures ===
== Regular temperament properties ==
 
{| class="wikitable center-4 center-5 center-6"
The following table shows [[TE temperament measures]] (RMS normalized) of 27et.
! rowspan="2" | Subgroup
 
! rowspan="2" | [[Comma list]]
{| class="wikitable center-all"
! rowspan="2" | [[Mapping]]
! colspan="2" |
! rowspan="2" | Optimal<br>8ve stretch (¢)
! 3-limit
! colspan="2" | Tuning error
! 5-limit
|-
! 7-limit
! [[TE error|Absolute]] (¢)
! 2.3.5.7.13
! [[TE simple badness|Relative]] (%)
! 2.3.5.7.13.19
|-
|-
! colspan="2" |Octave stretch (¢)
| 2.3
| {{monzo| 43 -27 }}
| [{{val| 27 43 }}]
| -2.89
| -2.89
| 2.88
| 6.50
|-
| 2.3.5
| 128/125, 20000/19683
| [{{val| 27 43 63 }}]
| -3.88
| -3.88
| 2.74
| 6.19
|-
| 2.3.5.7
| 64/63, 126/125, 245/243
| [{{val| 27 43 63 76 }}]
| -3.70
| -3.70
| 2.39
| 5.40
|-
| 2.3.5.7.13
| 64/63, 91/90, 126/125, 169/168
| [{{val| 27 43 63 76 100 }}]
| -3.18
| -3.18
| 2.39
| 5.39
|-
| 2.3.5.7.13.19
| 64/63, 76/75, 91/90, 126/125, 169/168
| [{{val| 27 43 63 76 100 115 }}]
| -3.18
| -3.18
|-
! rowspan="2" |Error
! [[TE error|absolute]] (¢)
| 2.88
| 2.74
| 2.39
| 2.39
| 2.18
| 2.18
|-
! [[TE simple badness|relative]] (%)
| 6.50
| 6.19
| 5.40
| 5.39
| 4.92
| 4.92
|}
|}
* 27et (27eg val) is lower in relative error than any previous ETs in the 13-, 17-, and 19-limit. The next ETs in those subgroups are 31, 31, and 46, respectively.
* 27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next ET that does better in this subgroup is 53.


== Rank two temperaments ==
27et (27eg val) is lower in relative error than any previous equal temperaments in the 13-, 17-, and 19-limit. The next ETs in those subgroups are 31, 31, and 46, respectively.


27et is particularly strong in the 2.3.5.7.13.19 subgroup. The next ET that does better in this subgroup is 53.
=== Rank-2 temperaments ===
* [[List of 27edo rank two temperaments by badness]]
* [[List of 27edo rank two temperaments by badness]]
* [[List of edo-distinct 27e rank two temperaments]]
* [[List of edo-distinct 27e rank two temperaments]]
Important MOSes include:  
Important MOSes include:  
* [[superpyth]] diatonic 5L2s 5551551 (16\27, 1\1)
* [[superpyth]] diatonic 5L2s 5551551 (16\27, 1\1)
* [[superpyth]] chromatic 5L7s 4141414114141411 (16\27, 1\1)
* [[superpyth]] chromatic 5L7s 4141414114141411 (16\27, 1\1)
* [[superpyth]] hyperchromatic 5L12s 31131131131113113113111 (16\27, 1\1)
* [[superpyth]] hyperchromatic 5L12s 31131131131113113113111 (16\27, 1\1)
* Pathological [[superpyth]] [[A shruti list|shrutis]] 5L17s 211121112111211112111211121111 (16\27, 1\1)
* Pathological [[superpyth]] [[A shruti list|shrutis]] 5L17s 211121112111211112111211121111 (16\27, 1\1)
*[[augmented]] ([[augene]]) 3L6s 522522522 (2\27, 1\3)
* [[augmented]] ([[augene]]) 3L6s 522522522 (2\27, 1\3)
* [[augmented]] ([[augene]]) 3L9s 322232223222 (2\27, 1\3)
* [[augmented]] ([[augene]]) 3L9s 322232223222 (2\27, 1\3)
* [[augmented]] ([[augene]]) 3L12s 212222122221222 (2\27, 1\3)
* [[augmented]] ([[augene]]) 3L12s 212222122221222 (2\27, 1\3)
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|}
|}


== Commas ==
=== Commas ===
27 EDO [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val|27 43 63 76 93 100}}.)
27 EDO [[tempers out]] the following [[commas]]. (Note: This assumes the [[val]] {{val| 27 43 63 76 93 100 }}.)


{| class="commatable wikitable center-all left-3 right-4 left-6"
{| class="commatable wikitable center-all left-3 right-4 left-6"
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[[Category:27edo]]
[[Category:27edo]]
[[Category:augene]]
[[Category:Augene]]
[[Category:Equal divisions of the octave]]
[[Category:Equal divisions of the octave]]
[[Category:listen]]
[[Category:Listen]]
[[Category:theory]]
[[Category:Theory]]
[[Category:todo:unify_precision]]
[[Category:Twentuning]]
[[Category:twentuning]]
[[Category:Zeta]]
[[Category:zeta]]
Retrieved from "https://en.xen.wiki/w/27edo"