179edo: Difference between revisions

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== Theory ==

179edo ~~doesn't~~ approximate well any odd harmonic up to 23, best being [[21/16]] with 22% error. Nonetheless, it is consistent in the 7-limit and there are a number of temperaments to be considered.

179edo does not approximate well any odd [[harmonic]] up to 23, best being [[21/16]] with 22% error. Nonetheless, it is [[consistent]] in the [[7-odd-limit]] and there are a number of temperaments to be considered.

~~179edo~~ tempers out the parakleisma, ~~1224440064/1220703125~~ in the 5-limit, and [[support]]s parakleismic and its ~~extensions~~, providing the optimal patent val for 11- and 13-limit [[~~Ragismic_microtemperaments~~|parkleismic ~~temperament~~]]. In the 7-limit it tempers out 3136/3125, 4375/4374 and 10976/10935, in the 11-limit 176/175 and 1375/1372 and in the 13-limit 169/168, 325/324, 351/350 and 352/351, providing the optimal patent val for 11- and 13-limit [[~~Valinorismic_temperaments#Ulmo|~~ulmo ~~temperament~~]].

The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }} in the 5-limit, and [[support]]s [[parakleismic]] and its [[extension]]s, providing the [[optimal patent val]] for 11- and 13-limit [[Ragismic microtemperaments #parkleismic|parkleismic]] temperament. In the 7-limit it tempers out [[3136/3125]], [[4375/4374]] and [[10976/10935]], in the 11-limit [[176/175]] and 1375/1372 and in the 13-limit [[169/168]], [[325/324]], [[351/350]] and [[352/351]], providing the optimal patent val for 11- and 13-limit [[ulmo]] temperament.

=== Odd harmonics ===

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179edo is the 41st [[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 8ve Stretch (¢)

! rowspan="2" | Optimal 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|284 -179}}

| {{monzo| 284 -179 }}

|{{~~val~~|179 284}}

| {{mapping| 179 284 }}

| -0.6169

| 0.6166

| 9.20

|-

|2.3.5

| 2.3.5

|{{monzo|20 -17 3}}, {{monzo|28 -3 -10}}

| {{monzo| 20 -17 3 }}, {{monzo| 28 -3 -10 }}

|{{~~val~~|179 284 416}}

| {{mapping| 179 284 416 }}

| -0.7718

| 0.5490

| 8.19

|-

|2.3.5.7

| 2.3.5.7

|3136/3125, 4375/4374, 10976/10935

| 3136/3125, 4375/4374, 10976/10935

|{{~~val~~|179 284 416 503}}

| {{mapping| 179 284 416 503 }}

| -0.8673

| 0.5034

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|+Table of rank-2 temperaments by generator

! Periods per 8ve

! Generator~~ (reduced)~~

! Generator*

! Cents~~ (reduced)~~

! Cents*

! Associated ~~ratio~~

! Associated Ratio*

! Temperaments

|-

|1

| 1

|35\179

| 35\179

|234.64

| 234.64

|8/7

| 8/7

|[[Rodan]]

| [[Rodan]] (179d)

|-

|1

| 1

|47\179

| 47\179

|315.08

| 315.08

|6/5

| 6/5

|[[Parakleismic]]

| [[Parakleismic]]

|-

|1

| 1

|79\~~167~~

| 79\179

|529.61

| 529.61

|512/375

| 512/375

|[[Mabila]]

| [[Mabila]]

|}

<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct

== Music ==

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* "in a hopeless situation" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/787NGqkI5FHjQcfEOu4y4w Spotify] | [https://francium223.bandcamp.com/track/in-a-hopeless-situation Bandcamp] | [https://www.youtube.com/watch?v=kGdWmP81jwc YouTube]

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

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

[[Category:Listen]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
 {{EDO intro|179}}
 == Theory ==
-edo doesn't approximate well any odd harmonic up to 23, best being [[21/16]] with 22% error. Nonetheless, it is consistent in the 7-limit and there are a number of temperaments to be considered.
+edo does not approximate well any odd [[harmonic]] up to 23, best being [[21/16]] with 22% error. Nonetheless, it is [[consistent]] in the [[7-odd-limit]] and there are a number of temperaments to be considered.
-edo tempers out the parakleisma, 1224440064/1220703125 in the 5-limit, and [[support]]s parakleismic and its extensions, providing the optimal patent val for 11- and 13-limit [[Ragismic_microtemperaments|parkleismic temperament]]. In the 7-limit it tempers out 3136/3125, 4375/4374 and 10976/10935, in the 11-limit 176/175 and 1375/1372 and in the 13-limit 169/168, 325/324, 351/350 and 352/351, providing the optimal patent val for 11- and 13-limit [[Valinorismic_temperaments#Ulmo|ulmo temperament]].
+The equal temperament [[tempering out|tempers out]] the [[parakleisma]], {{monzo| 8 14 -13 }} in the 5-limit, and [[support]]s [[parakleismic]] and its [[extension]]s, providing the [[optimal patent val]] for 11- and 13-limit [[Ragismic microtemperaments #parkleismic|parkleismic]] temperament. In the 7-limit it tempers out [[3136/3125]], [[4375/4374]] and [[10976/10935]], in the 11-limit [[176/175]] and 1375/1372 and in the 13-limit [[169/168]], [[325/324]], [[351/350]] and [[352/351]], providing the optimal patent val for 11- and 13-limit [[ulmo]] temperament.
 === Odd harmonics ===
@@ Line 12: / Line 13: @@
 edo is the 41st [[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|284 -179}}
+| {{monzo| 284 -179 }}
-|{{val|179 284}}
+| {{mapping| 179 284 }}
 | -0.6169
 | 0.6166
 | 9.20
 |-
-|2.3.5
+| 2.3.5
-|{{monzo|20 -17 3}}, {{monzo|28 -3 -10}}
+| {{monzo| 20 -17 3 }}, {{monzo| 28 -3 -10 }}
-|{{val|179 284 416}}
+| {{mapping| 179 284 416 }}
 | -0.7718
 | 0.5490
 | 8.19
 |-
-|2.3.5.7
+| 2.3.5.7
-|3136/3125, 4375/4374, 10976/10935
+| 3136/3125, 4375/4374, 10976/10935
-|{{val|179 284 416 503}}
+| {{mapping| 179 284 416 503 }}
 | -0.8673
 | 0.5034
@@ Line 49: / Line 50: @@
 |+Table of rank-2 temperaments by generator
 ! Periods<br>per 8ve
-! Generator<br>(reduced)
+! Generator*
-! Cents<br>(reduced)
+! Cents*
-! Associated<br>ratio
+! Associated<br>Ratio*
 ! Temperaments
 |-
-|1
+| 1
-|35\179
+| 35\179
-|234.64
+| 234.64
-|8/7
+| 8/7
-|[[Rodan]]
+| [[Rodan]] (179d)
 |-
-|1
+| 1
-|47\179
+| 47\179
-|315.08
+| 315.08
-|6/5
+| 6/5
-|[[Parakleismic]]
+| [[Parakleismic]]
 |-
-|1
+| 1
-|79\167
+| 79\179
-|529.61
+| 529.61
-|512/375
+| 512/375
-|[[Mabila]]
+| [[Mabila]]
 |}
+<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
 == Music ==
@@ Line 77: / Line 79: @@
 * "in a hopeless situation" from ''hope in dark times'' (2024) – [https://open.spotify.com/track/787NGqkI5FHjQcfEOu4y4w Spotify] | [https://francium223.bandcamp.com/track/in-a-hopeless-situation Bandcamp] | [https://www.youtube.com/watch?v=kGdWmP81jwc YouTube]
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
-[[Category:Prime EDO]]
 [[Category:Listen]]