135edo: Difference between revisions

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'''135edo''' is the [[EDO|equal division of the octave]] into 135 parts of 8.8888 cents each. It tempers out 32805/32768 (schisma) and 30517578125/29386561536 (quintriyo comma) in the 5-limit; [[225/224]], 3125/3087, and 28824005/28697814 in the 7-limit. It is consistent to the 7-limit, but there is a large relative delta for 5th harmonic. Using the patent val, it tempers out [[385/384]], 540/539, 2200/2187, 12005/11979 and [[Quartisma|117440512/117406179]] in the 11-limit; 275/273, [[325/324]], 352/351, and 729/728 in the 13-limit. Using the 135c val, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; 126/125, 10976/10935, and 589824/588245 in the 7-limit; [[176/175]], 441/440, 14641/14580 and 16384/16335 in the 11-limit; 196/195, [[351/350]], [[352/351]], 676/675, and 6656/6655 in the 13-limit.

[[~~Category:Equal divisions~~ of the ~~octave~~]]

== Theory ==

[[~~Category:Quartismic~~]]

135edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo|270et]], 135et makes most sense to use as a [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]] [[regular temperament|temperament]], where it is characterized by [[tempering out]] the [[garischisma]], the [[septiennealimma]], the [[symbiotic comma]], the [[argyria]], the [[chrysia]], and the [[olympia]]. On top of this, it also has fairly good approximations to primes [[17/1|17]], [[29/1|29]], and [[31/1|31]].

If we consider the full 13-limit, the flat-tending {{val| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.

Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (pentadecal comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], [[2200/2187]], [[12005/11979]] and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.

Using the 135c val, it tempers out 1594323/1562500 ([[unicorn comma]]) and 50331648/48828125 ([[magus comma]]) in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.

=== Prime harmonics ===

{{Harmonics in equal|135|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 135edo (continued)}}

=== Subsets and supersets ===

Since 135 factors into primes as {{nowrap| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, notably provides extremely good corrections for the approximation to harmonics 5, 13, and 19.

== Regular temperament properties ==

{| class="wikitable center-4 center-5 center-6"

|-

! rowspan="2" | [[Subgroup]]

! rowspan="2" | [[Comma list]]

! rowspan="2" | [[Mapping]]

! rowspan="2" | Optimal<br>8ve stretch (¢)

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

! [[TE simple badness|Relative]] (%)

|-

| 2.3

| {{Monzo| 214 -135 }}

| {{Mapping| 135 214 }}

| −0.0843

| 0.0843

| 0.95

|-

| 2.3.7

| 33554432/33480783, 40353607/40310784

| {{Mapping| 135 214 379 }}

| −0.0637

| 0.0747

| 0.84

|-

| 2.3.7.11

| 19712/19683, 41503/41472, 43923/43904

| {{Mapping| 135 214 379 467 }}

| −0.0328

| 0.0840

| 0.94

|-

| 2.3.7.11.17

| 1089/1088, 2058/2057, 5832/5831, 19712/19683

| {{Mapping| 135 214 379 467 552 }}

| −0.1100

| 0.1716

| 1.93

|}

== Instruments ==

* [[Lumatone mapping for 135edo]]

@@ Line 1: / Line 1: @@
-'''135edo''' is the [[EDO|equal division of the octave]] into 135 parts of 8.8888 cents each. It tempers out 32805/32768 (schisma) and 30517578125/29386561536 (quintriyo comma) in the 5-limit; [[225/224]], 3125/3087, and 28824005/28697814 in the 7-limit. It is consistent to the 7-limit, but there is a large relative delta for 5th harmonic. Using the patent val, it tempers out [[385/384]], 540/539, 2200/2187, 12005/11979 and [[Quartisma|117440512/117406179]] in the 11-limit; 275/273, [[325/324]], 352/351, and 729/728 in the 13-limit. Using the 135c val, it tempers out 1594323/1562500 and 50331648/48828125 in the 5-limit; 126/125, 10976/10935, and 589824/588245 in the 7-limit; [[176/175]], 441/440, 14641/14580 and 16384/16335 in the 11-limit; 196/195, [[351/350]], [[352/351]], 676/675, and 6656/6655 in the 13-limit.
+{{Infobox ET}}
+{{ED intro}}
-[[Category:Equal divisions of the octave]]
+== Theory ==
-[[Category:Quartismic]]
+edo is [[consistent]] to the [[7-odd-limit]], but with large relative error for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s. As every other step of the full [[13-limit]] monster – [[270edo|270et]], 135et makes most sense to use as a [[2.3.7.11 subgroup|2.3.7.11-]][[subgroup]] [[regular temperament|temperament]], where it is characterized by [[tempering out]] the [[garischisma]], the [[septiennealimma]], the [[symbiotic comma]], the [[argyria]], the [[chrysia]], and the [[olympia]]. On top of this, it also has fairly good approximations to primes [[17/1|17]], [[29/1|29]], and [[31/1|31]].
+If we consider the full 13-limit, the flat-tending {{val| 135 214 313 379 467 '''499''' }} (135f) and the sharp-tending {{val| 135 214 '''314''' 379 467 500 }} (135c) are reasonable choices.
+Using the 135f val, it tempers out 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (pentadecal comma) in the 5-limit; [[225/224]], [[3125/3087]], and 28824005/28697814 in the 7-limit, [[385/384]], [[540/539]], [[2200/2187]], [[12005/11979]] and the [[quartisma]] in the 11-limit; [[169/168]] and [[364/363]] in the 13-limit.
+Using the 135c val, it tempers out 1594323/1562500 ([[unicorn comma]]) and 50331648/48828125 ([[magus comma]]) in the 5-limit; [[126/125]], [[10976/10935]], and [[589824/588245]] in the 7-limit; [[176/175]], [[441/440]], [[14641/14580]] and [[16384/16335]] in the 11-limit; [[196/195]], [[351/350]], [[352/351]], [[676/675]], and [[6656/6655]] in the 13-limit.
+=== Prime harmonics ===
+{{Harmonics in equal|135|columns=11}}
+{{Harmonics in equal|135|columns=11|start=12|collapsed=true|title=Approximation of prime harmonics in 135edo (continued)}}
+=== Subsets and supersets ===
+Since 135 factors into primes as {{nowrap| 3<sup>3</sup> × 5 }}, 135edo has subset edos {{EDOs| 3, 5, 9, 15, 27, and 45 }}. [[270edo]], which doubles it, notably provides extremely good corrections for the approximation to harmonics 5, 13, and 19.
+== Regular temperament properties ==
+{| class="wikitable center-4 center-5 center-6"
+|-
+! rowspan="2" | [[Subgroup]]
+! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Mapping]]
+! rowspan="2" | Optimal<br>8ve stretch (¢)
+! colspan="2" | Tuning error
+|-
+! [[TE error|Absolute]] (¢)
+! [[TE simple badness|Relative]] (%)
+|-
+| 2.3
+| {{Monzo| 214 -135 }}
+| {{Mapping| 135 214 }}
+| −0.0843
+| 0.0843
+| 0.95
+|-
+| 2.3.7
+| 33554432/33480783, 40353607/40310784
+| {{Mapping| 135 214 379 }}
+| −0.0637
+| 0.0747
+| 0.84
+|-
+| 2.3.7.11
+| 19712/19683, 41503/41472, 43923/43904
+| {{Mapping| 135 214 379 467 }}
+| −0.0328
+| 0.0840
+| 0.94
+|-
+| 2.3.7.11.17
+| 1089/1088, 2058/2057, 5832/5831, 19712/19683
+| {{Mapping| 135 214 379 467 552 }}
+| −0.1100
+| 0.1716
+| 1.93
+|}
+== Instruments ==
+* [[Lumatone mapping for 135edo]]