135edo: Difference between revisions

Line 1:

~~The '''135 equal divisions of the octave''' ('''135edo'''), or the '''135(-tone) equal temperament''' ('''135tet''', '''135et''') when viewed from a [[regular temperament]] perspective, is the [[~~EDO~~|equal division of the octave]] into 135 parts of about 8.89 [[cent]]s each.~~

== Theory ==

135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the 5th and ~~the~~ 13th ~~harmonics~~.

135edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s.

Using the 135f [[val]] {{val| 135 214 313 379 467 '''499''' }}, which tends flat, 135et 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, [[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 135f [[val]] {{val| 135 214 313 379 467 '''499''' }}, which tends flat, 135et [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (quintriyo 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 {{val| 135 214 '''314''' 379 467 500 }}, which tends sharp, 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.

Using the 135c val {{val| 135 214 '''314''' 379 467 500 }}, which tends sharp, 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.

As every other step of the full 13-limit monster – [[270edo|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it tempers out the [[garischisma]] and the [[symbiotic comma]].

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== Regular temperament properties ==

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

! rowspan="2" | Subgroup

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

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

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

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

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

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

! colspan="2" | Tuning ~~error~~

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

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| 2.3

| {{monzo| 214 -135 }}

| [{{~~val~~| 135 214 }}]

| {{mapping| 135 214 }}

| -0.0843

| 0.0843

Line 34:

| 2.3.7

| 33554432/33480783, 40353607/40310784

| [{{~~val~~| 135 214 379 }}]

| {{mapping| 135 214 379 }}

| -0.0637

| 0.0747

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| 2.3.7.11

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

| [{{~~val~~| 135 214 379 467 }}]

| {{mapping| 135 214 379 467 }}

| -0.0328

| 0.0840

| 0.94

|}

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

~~[[Category:Quartismic]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-The '''135 equal divisions of the octave''' ('''135edo'''), or the '''135(-tone) equal temperament''' ('''135tet''', '''135et''') when viewed from a [[regular temperament]] perspective, is the [[EDO|equal division of the octave]] into 135 parts of about 8.89 [[cent]]s each.
+{{EDO intro}}
 == Theory ==
-edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the 5th and the 13th harmonics.
+edo is [[consistent]] to the [[7-odd-limit]], but there is a large relative delta for the [[5/1|5th]] and [[13/1|13th]] [[harmonic]]s.
-Using the 135f [[val]] {{val| 135 214 313 379 467 '''499''' }}, which tends flat, 135et 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, [[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 135f [[val]] {{val| 135 214 313 379 467 '''499''' }}, which tends flat, 135et [[tempering out|tempers out]] 32805/32768 ([[schisma]]) and {{monzo| -11 -15 15 }} (quintriyo 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 {{val| 135 214 '''314''' 379 467 500 }}, which tends sharp, 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.
+Using the 135c val {{val| 135 214 '''314''' 379 467 500 }}, which tends sharp, 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.
 As every other step of the full 13-limit monster – [[270edo|270et]], 135et probably makes more sense as a 2.3.7.11 [[subgroup]] temperament, where it tempers out the [[garischisma]] and the [[symbiotic comma]].
@@ Line 16: / Line 16: @@
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" | Subgroup
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" | [[Comma list]]
+! rowspan="2" | [[Comma list|Comma List]]
 ! rowspan="2" | [[Mapping]]
-! rowspan="2" | Optimal<br>8ve stretch (¢)
+! rowspan="2" | Optimal<br>8ve Stretch (¢)
-! colspan="2" | Tuning error
+! colspan="2" | Tuning Error
 |-
 ! [[TE error|Absolute]] (¢)
@@ Line 27: / Line 27: @@
 | 2.3
 | {{monzo| 214 -135 }}
-| [{{val| 135 214 }}]
+| {{mapping| 135 214 }}
 | -0.0843
 | 0.0843
@@ Line 34: / Line 34: @@
 | 2.3.7
 | 33554432/33480783, 40353607/40310784
-| [{{val| 135 214 379 }}]
+| {{mapping| 135 214 379 }}
 | -0.0637
 | 0.0747
@@ Line 41: / Line 41: @@
 | 2.3.7.11
 | 19712/19683, 41503/41472, 43923/43904
-| [{{val| 135 214 379 467 }}]
+| {{mapping| 135 214 379 467 }}
 | -0.0328
 | 0.0840
 | 0.94
 |}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Quartismic]]