118edo: Difference between revisions

Line 1:

~~The '''~~118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.

== Theory ==

Line 17:

Line 16:

=== Prime harmonics ===

=== Subsets and supersets ===

118edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11.

== Intervals ==

Line 1,002:

Line 1,004:

== 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 8ve ~~stretch~~ (¢)

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

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

! colspan="2" | Tuning Error

|-

! [[TE error|Absolute]] (¢)

Line 1,039:

Line 1,041:

| 3.89

|-

| 2.3.5.7.11.13

| style="border-top: double;" | 2.3.5.7.11.13

| 196/195, 352/351, 384/384, 625/624, 729/728

| style="border-top: double;" | 196/195, 352/351, 384/384, 625/624, 729/728

| [{{val| 118 187 274 331 408 437 }}] (118)

| style="border-top: double;" | [{{val| 118 187 274 331 408 437 }}] (118)

| +0.125

| style="border-top: double;" | +0.125

| 0.604

| style="border-top: double;" | 0.604

| 5.93

| style="border-top: double;" | 5.93

|-

| 2.3.5.7.11.13

| style="border-top: double;" | 2.3.5.7.11.13

| 169/168, 325/324, 364/363, 385/384, 3136/3125

| style="border-top: double;" | 169/168, 325/324, 364/363, 385/384, 3136/3125

| [{{val| 118 187 274 331 408 436 }}] (118f)

| style="border-top: double;" | [{{val| 118 187 274 331 408 436 }}] (118f)

| +0.583

| style="border-top: double;" | +0.583

| 0.650

| style="border-top: double;" | 0.650

| 6.39

| style="border-top: double;" | 6.39

|-

| 2.3.5.7.11.17

| style="border-top: double;" | 2.3.5.7.11.17

| 289/288, 385/384, 441/440, 561/560, 3136/3125

| style="border-top: double;" | 289/288, 385/384, 441/440, 561/560, 3136/3125

| [{{val| 118 187 274 331 408 482 }}]

| style="border-top: double;" | [{{val| 118 187 274 331 408 482 }}]

| +0.417

| style="border-top: double;" | +0.417

| 0.399

| style="border-top: double;" | 0.399

| 3.92

| style="border-top: double;" | 3.92

|-

| 2.3.5.7.11.17.19

Line 1,067:

Line 1,069:

| 3.69

|}

* 118et is lower in relative error than any previous ~~ETs~~ in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.

=== Rank-2 temperaments ===

Line 1,073:

Line 1,075:

|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator (~~reduced~~)

! Generator (Reduced)

! Cents (~~reduced~~)

! Cents (Reduced)

! Associated ~~ratio~~

! Associated Ratio

! Temperaments

|-

Line 1,172:

Line 1,174:

* [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]

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

[[Category:Gamelismic]]

[[Category:Guiron]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
+{{EDO intro|118}}
-The '''118 equal divisions of the octave''' ('''118edo'''), or the '''118(-tone) equal temperament''' ('''118tet''', '''118et''') when viewed from a [[regular temperament]] perspective, is the [[equal division of the octave]] into 118 parts of about 10.2 [[cent]]s each.
 == Theory ==
@@ Line 17: / Line 16: @@
 === Prime harmonics ===
-{{harmonics in equal|118}}
+{{Harmonics in equal|118}}
+=== Subsets and supersets ===
+edo contains [[2edo]] and [[59edo]] as subsets. Its multiples, [[236edo]], [[354edo]] and [[472edo]] are all of various interests, each providing distinct interpretations of harmonics 7 and 11.
 == Intervals ==
@@ Line 1,002: / Line 1,004: @@
 == 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 1,039: / Line 1,041: @@
 | 3.89
 |-
-| 2.3.5.7.11.13
+| style="border-top: double;" | 2.3.5.7.11.13
-| 196/195, 352/351, 384/384, 625/624, 729/728
+| style="border-top: double;" | 196/195, 352/351, 384/384, 625/624, 729/728
-| [{{val| 118 187 274 331 408 437 }}] (118)
+| style="border-top: double;" | [{{val| 118 187 274 331 408 437 }}] (118)
-| +0.125
+| style="border-top: double;" | +0.125
-| 0.604
+| style="border-top: double;" | 0.604
-| 5.93
+| style="border-top: double;" | 5.93
 |-
-| 2.3.5.7.11.13
+| style="border-top: double;" | 2.3.5.7.11.13
-| 169/168, 325/324, 364/363, 385/384, 3136/3125
+| style="border-top: double;" | 169/168, 325/324, 364/363, 385/384, 3136/3125
-| [{{val| 118 187 274 331 408 436 }}] (118f)
+| style="border-top: double;" | [{{val| 118 187 274 331 408 436 }}] (118f)
-| +0.583
+| style="border-top: double;" | +0.583
-| 0.650
+| style="border-top: double;" | 0.650
-| 6.39
+| style="border-top: double;" | 6.39
 |-
-| 2.3.5.7.11.17
+| style="border-top: double;" | 2.3.5.7.11.17
-| 289/288, 385/384, 441/440, 561/560, 3136/3125
+| style="border-top: double;" | 289/288, 385/384, 441/440, 561/560, 3136/3125
-| [{{val| 118 187 274 331 408 482 }}]
+| style="border-top: double;" | [{{val| 118 187 274 331 408 482 }}]
-| +0.417
+| style="border-top: double;" | +0.417
-| 0.399
+| style="border-top: double;" | 0.399
-| 3.92
+| style="border-top: double;" | 3.92
 |-
 | 2.3.5.7.11.17.19
@@ Line 1,067: / Line 1,069: @@
 | 3.69
 |}
-* 118et is lower in relative error than any previous ETs in the 5-limit. Not until [[171edo|171]] do we find a better ET in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.
+* 118et is lower in relative error than any previous equal temperaments in the 5-limit. Not until [[171edo|171]] do we find a better one in terms of absolute error, and not until [[441edo|441]] do we find one in terms of relative error.
 === Rank-2 temperaments ===
@@ Line 1,073: / Line 1,075: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator<br>(Reduced)
-! Cents<br>(reduced)
+! Cents<br>(Reduced)
-! Associated<br>ratio
+! Associated<br>Ratio
 ! Temperaments
 |-
@@ Line 1,172: / Line 1,174: @@
 * [https://www.youtube.com/watch?v=eYnSsOnRZIs Pops] by [[Mercury Amalgam]]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
 [[Category:Gamelismic]]
 [[Category:Guiron]]