205edo: Difference between revisions

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== Theory ==

205 = 5 × 41, and 205edo shares its [[3/2|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.

Since {{nowrap|205 {{=}} 5 × 41}}, and 205edo shares its [[3/2|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.

In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]], [[640/637]], [[729/728]], [[847/845]], and [[1188/1183]] in the 13-limit.

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205edo can be notated with [[Ups and downs notation|ups and downs]] representing 5\205 = 1\41, and lifts and drops (written as / and \) representing 1\205. This has the advantage of building on a familiarity with 41edo, and is especially useful for [[Kite guitar|Kite guitarists]] who want to notate microbends more precisely.

{| class="wikitable"

|+

|-

! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! …

|-

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== Regular temperament properties ==

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

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

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

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

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

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

|-

~~! [[TE error|Absolute]] (¢)~~

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

|-

| 2.3.5

| 1600000/1594323, {{monzo| 38 -2 -15 }}

| {{mapping| 205 325 476 }}

| -0.106

| −0.106

| 0.141

| 2.41

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| 5632/5625, 14641/14580, 1600000/1594323

| {{mapping| 181 287 420 508 }}

| -0.002

| −0.002

| 0.218

| 3.72

|}

=== Rank-2 temperaments ===

{~~| class="wikitable center-all left-5"~~

~~|+Table of~~ rank-2 ~~temperaments by generator~~

~~! Periods per 8ve~~

! Generator*

! Cents*

! Associated Ratio*

~~! Temperaments~~

|-

| 1

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|-

| 5

| 63\205 (19\205)

| 63\205 (19\205)

| 368.780 (111.220)

| 368.780 (111.220)

| 1024/891 (16/15)

| 1024/891 (16/15)

| [[Quintosec]]

|-

| 41

| 66\205 (1\205)

| 66\205 (1\205)

| 386.341 (5.85)

| 386.341 (5.85)

| 5/4 (32805/32768)

| 5/4 (32805/32768)

| [[Countercomp]]

|}

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

== Scales ==

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* [http://tonalsoft.com/enc/m/mem.aspx mem, 205-edo] on [[Tonalsoft Encyclopedia]]

~~[[Category:Huntmic]]~~

[[Category:Amity]]

[[Category:Hemithirds]]

[[Category:Huntmic]]

[[Category:Laka]]

[[Category:Quanic]]

@@ Line 5: / Line 5: @@
 == Theory ==
-= 5 × 41, and 205edo shares its [[3/2|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.
+Since {{nowrap|205 {{=}} 5 &times; 41}}, and 205edo shares its [[3/2|fifth]] with [[41edo]]. It can serve as a tuning for various temperaments, such as [[amity]] or [[laka]], and supplies the [[optimal patent val]] for [[quanic]] in the 7-, 11-, 13-, 17- and 19-limit, and for 13-limit amity, as well as other temperaments [[tempering out]] the huntma, [[640/637]], the rank-5 temperament for which it also supplies the optimal patent val.
 In the 5-limit it tempers out 1600000/1594323, the [[amity comma]], and {{monzo| 38 -2 -15 }}, the [[hemithirds comma]], and is an excellent tuning for 5-limit amity. The [[patent val]] {{val| 205 325 476 576 709 759 }} tempers out [[4375/4374]], [[5120/5103]], [[6144/6125]] in the 7-limit; [[540/539]], 1331/1323, and 2420/2401 in the 11-limit; [[352/351]], [[640/637]], [[729/728]], [[847/845]], and [[1188/1183]] in the 13-limit.
@@ Line 28: / Line 28: @@
 edo can be notated with [[Ups and downs notation|ups and downs]] representing 5\205 = 1\41, and lifts and drops (written as / and \) representing 1\205. This has the advantage of building on a familiarity with 41edo, and is especially useful for [[Kite guitar|Kite guitarists]] who want to notate microbends more precisely.
 {| class="wikitable"
-|+
+|-
 ! 0 !! 1 !! 2 !! 3 !! 4 !! 5 !! 6 !! 7 !! 8 !! 9 !! 10 !! 11 !! 12 !! 13 !! 14 !! 15 !! 16 !! …
 |-
@@ Line 35: / Line 35: @@
 == Regular temperament properties ==
-{| class="wikitable center-4 center-5 center-6"
+{{comma basis begin}}
-! 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]] (%)
 |-
 | 2.3.5
 | 1600000/1594323, {{monzo| 38 -2 -15 }}
 | {{mapping| 205 325 476 }}
-| -0.106
+| &minus;0.106
 | 0.141
 | 2.41
@@ Line 56: / Line 47: @@
 | 5632/5625, 14641/14580, 1600000/1594323
 | {{mapping| 181 287 420 508 }}
-| -0.002
+| &minus;0.002
 | 0.218
 | 3.72
-|}
+{{comma basis end}}
 === Rank-2 temperaments ===
-{| class="wikitable center-all left-5"
+{{rank-2 begin}}
-|+Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
-! Generator*
-! Cents*
-! Associated<br>Ratio*
-! Temperaments
 |-
 | 1
@@ Line 95: / Line 80: @@
 |-
 | 5
-| 63\205<br>(19\205)
+| 63\205<br />(19\205)
-| 368.780<br>(111.220)
+| 368.780<br />(111.220)
-| 1024/891<br>(16/15)
+| 1024/891<br />(16/15)
 | [[Quintosec]]
 |-
 | 41
-| 66\205<br>(1\205)
+| 66\205<br />(1\205)
-| 386.341<br>(5.85)
+| 386.341<br />(5.85)
-| 5/4<br>(32805/32768)
+| 5/4<br />(32805/32768)
 | [[Countercomp]]
-|}
+{{rank-2 end}}
-<nowiki>*</nowiki> [[Normal lists|octave-reduced form]], reduced to the first half-octave, and [[Normal lists|minimal form]] in parentheses if it is distinct
+{{orf}}
 == Scales ==
@@ Line 166: / Line 151: @@
 * [http://tonalsoft.com/enc/m/mem.aspx mem, 205-edo] on [[Tonalsoft Encyclopedia]]
-[[Category:Huntmic]]
 [[Category:Amity]]
 [[Category:Hemithirds]]
+[[Category:Huntmic]]
 [[Category:Laka]]
 [[Category:Quanic]]