167edo: Difference between revisions

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~~'''167edo''' is the [[~~EDO|~~equal division of the octave]] into~~ 167 ~~parts of 7.18562874251~~ [[~~cent]]s each. It [[tempering_out~~|tempers out]] the [[~~Würschmidt family|~~würschmidt comma]], 393216/390625 and ~~10737418240/10460353203~~ in the [[5-limit]]; [[2401/2400]], [[3136/3125]], and 179200/177147 in the [[7-limit]]; [[896/891]], 2200/2187, and 3388/3375 in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and 1716/1715 in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[Porwell temperaments|polypyth temperament]]; [[256/255]], 442/441, [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s 11-limit [[Breedsmic temperaments|unthirds temperament]].

== Theory ==

167et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, and the leapday comma, {{monzo| 31 -21 1 }}, in the [[5-limit]]; [[2401/2400]], [[3136/3125]], and 179200/177147 in the [[7-limit]]; [[896/891]], 2200/2187, and 3388/3375 in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and [[1716/1715]] in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[Porwell temperaments|polypyth temperament]]; [[256/255]], [[442/441]], [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s the 11-limit [[Breedsmic temperaments #Unthirds|unthirds temperament]].

167edo also has a very close approximation to the [[golden magic]] scale.

167edo ~~is the 39th [[prime EDO]].~~

=== Prime harmonics ===

{{Harmonics in equal|167|intervals=prime|columns=12|start=13|title=Approximation of prime harmonics in 167edo (continued)|collapsed=1}}

~~{{Harmonics in equal|167|intervals~~=~~prime|columns~~=~~13}}~~

=== Subsets and supersets ===

~~{{Harmonics in equal|167|intervals~~=~~prime|start~~=~~14|columns~~=~~12}}~~

167edo is the 39th [[prime edo]].

==Regular temperament properties==

== Regular temperament properties ==

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

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

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

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

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

|-

![[TE error|Absolute]] (¢)

! [[TE error|Absolute]] (¢)

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

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

|-

|2.3

| 2.3

|{{monzo|265 -167}}

| {{monzo| 265 -167 }}

|{{val|167 265}}

| {{val| 167 265 }}

| -0.7056

| 0.7052

| 9.81

|-

|2.3.5

| 2.3.5

|~~{{monzo|17 1 -8}}~~, {{monzo|14 -~~22 9~~}}

| 393216/390625, {{monzo| 31 -21 1 }}

|{{val|167 265 388}}

| {{val| 167 265 388 }}

| -0.7158

| 0.5759

| 8.01

|-

|2.3.5.7

| 2.3.5.7

|~~6144~~/~~6125~~, 3136/3125, 179200/177147

| 2401/2400, 3136/3125, 179200/177147

|{{val|167 265 388 469}}

| {{val| 167 265 388 469 }}

| -0.6467

| 0.5129

| 7.14

|-

|2.3.5.7.11

| 2.3.5.7.11

|896/891, 2200/2187, ~~6144~~/~~6125~~, ~~6250~~/~~6237~~

| 896/891, 2200/2187, 2401/2400, 3136/3125

|{{val|167 265 388 469 578}}

| {{val| 167 265 388 469 578 }}

| -0.6315

| 0.4598

| 6.40

|-

|2.3.5.7.11.13

| 2.3.5.7.11.13

|325/324, 352/351, ~~896~~/~~891~~, 1001/1000, ~~6656~~/~~6615~~

| 325/324, 352/351, 364/363, 1001/1000, 1716/1715

|{{val|167 265 388 469 578 618}}

| {{val| 167 265 388 469 578 618 }}

| -0.5349

| 0.4721

| 6.57

|-

|2.3.5.7.11.13.17

| 2.3.5.7.11.13.17

|325/324, 352/351, ~~896~~/~~891~~, ~~256~~/~~255~~, 1001/1000~~, 1225/1224~~

| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000

|{{val|167 265 388 469 578 618 683}}

| {{val| 167 265 388 469 578 618 683 }}

| -0.5573

| 0.4405

| 6.13

|}

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

~~[[Category:Prime EDO]]~~

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-'''167edo''' is the [[EDO|equal division of the octave]] into 167 parts of 7.18562874251 [[cent]]s each. It [[tempering_out|tempers out]] the [[Würschmidt family|würschmidt comma]], 393216/390625 and 10737418240/10460353203 in the [[5-limit]]; [[2401/2400]], [[3136/3125]], and 179200/177147 in the [[7-limit]]; [[896/891]], 2200/2187, and 3388/3375 in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and 1716/1715 in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[Porwell temperaments|polypyth temperament]]; [[256/255]], 442/441, [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s 11-limit [[Breedsmic temperaments|unthirds temperament]].
+{{EDO intro|167}}
+== Theory ==
+et [[tempering out|tempers out]] the [[würschmidt comma]], 393216/390625, and the leapday comma, {{monzo| 31 -21 1 }}, in the [[5-limit]]; [[2401/2400]], [[3136/3125]], and 179200/177147 in the [[7-limit]]; [[896/891]], 2200/2187, and 3388/3375 in the [[11-limit]]; [[325/324]], [[352/351]], [[364/363]], [[1001/1000]], and [[1716/1715]] in the [[13-limit]], providing the [[optimal patent val]] for 11- and 13-limit [[Porwell temperaments|polypyth temperament]]; [[256/255]], [[442/441]], [[595/594]], [[715/714]], and [[936/935]] in the [[17-limit]]. It also [[support]]s the 11-limit [[Breedsmic temperaments #Unthirds|unthirds temperament]].
 edo also has a very close approximation to the [[golden magic]] scale.
-edo is the 39th [[prime EDO]].
+=== Prime harmonics ===
+{{Harmonics in equal|167|intervals=prime|columns=12}}
+{{Harmonics in equal|167|intervals=prime|columns=12|start=13|title=Approximation of prime harmonics in 167edo (continued)|collapsed=1}}
-{{Harmonics in equal|167|intervals=prime|columns=13}}
+=== Subsets and supersets ===
-{{Harmonics in equal|167|intervals=prime|start=14|columns=12}}
+edo is the 39th [[prime edo]].
-==Regular temperament properties==
+== Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
-! rowspan="2" |[[Subgroup]]
+! rowspan="2" | [[Subgroup]]
-! rowspan="2" |[[Comma list|Comma List]]
+! rowspan="2" | [[Comma list|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
 |-
-![[TE error|Absolute]] (¢)
+! [[TE error|Absolute]] (¢)
-![[TE simple badness|Relative]] (%)
+! [[TE simple badness|Relative]] (%)
 |-
-|2.3
+| 2.3
-|{{monzo|265 -167}}
+| {{monzo| 265 -167 }}
-|{{val|167 265}}
+| {{val| 167 265 }}
 | -0.7056
 | 0.7052
 | 9.81
 |-
-|2.3.5
+| 2.3.5
-|{{monzo|17 1 -8}}, {{monzo|14 -22 9}}
+| 393216/390625, {{monzo| 31 -21 1 }}
-|{{val|167 265 388}}
+| {{val| 167 265 388 }}
 | -0.7158
 | 0.5759
 | 8.01
 |-
-|2.3.5.7
+| 2.3.5.7
-|6144/6125, 3136/3125, 179200/177147
+| 2401/2400, 3136/3125, 179200/177147
-|{{val|167 265 388 469}}
+| {{val| 167 265 388 469 }}
 | -0.6467
 | 0.5129
 | 7.14
 |-
-|2.3.5.7.11
+| 2.3.5.7.11
-|896/891, 2200/2187, 6144/6125, 6250/6237
+| 896/891, 2200/2187, 2401/2400, 3136/3125
-|{{val|167 265 388 469 578}}
+| {{val| 167 265 388 469 578 }}
 | -0.6315
 | 0.4598
 | 6.40
 |-
-|2.3.5.7.11.13
+| 2.3.5.7.11.13
-|325/324, 352/351, 896/891, 1001/1000, 6656/6615
+| 325/324, 352/351, 364/363, 1001/1000, 1716/1715
-|{{val|167 265 388 469 578 618}}
+| {{val| 167 265 388 469 578 618 }}
 | -0.5349
 | 0.4721
 | 6.57
 |-
-|2.3.5.7.11.13.17
+| 2.3.5.7.11.13.17
-|325/324, 352/351, 896/891, 256/255, 1001/1000, 1225/1224
+| 256/255, 325/324, 352/351, 364/363, 442/441, 1001/1000
-|{{val|167 265 388 469 578 618 683}}
+| {{val| 167 265 388 469 578 618 683 }}
 | -0.5573
 | 0.4405
 | 6.13
 |}
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Prime EDO]]