205edo: Difference between revisions

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205edo's step size is called a '''mem''' when used as an [[interval size unit]].

== Theory ==

205edo 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-~~limits~~, 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}}, 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.

Using its alternative mapping {{val| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]], and [[364/363]] to the comma list.

Using its alternative mapping {{val| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]] and [[364/363]] to the comma list.

Anyway, ~~assume~~ the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.

Anyway, assuming the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.

=== Odd harmonics ===

{{Harmonics in equal|205|columns=11|start=12|collapsed=1|title = Approximation of odd harmonics in 205edo (continued)}}

=== Structural properties ===

205edo contains a very accurate approximation of the [[2.3.5.11 subgroup]], inheriting the perfect fifth from 41edo. The patent val mappings of primes 7 and 13 can then be found by mapping [[7/5]] to the Pythagorean diminished fifth, and [[13/11]] at the Pythagorean minor third, thus tempering out [[5120/5103]] and [[352/351]], as well as [[847/845]] and [[2080/2079]]. In fact, it is the last edo tempering out 5120/5103 to map both [[7/5]] and [[1024/729]] consistently. It also supports the [[counterpyth]] mapping of prime 19.

Its step size represents several important intervals, such as the septimal kleisma [[225/224]], and the keenanisma [[385/384]]. Notably, the mappings of primes 5, 7, 11, 13, and 19 all differ from their nearest 41edo step by 1 step of 205edo, so 205edo can be considered as 41edo with fine-tuning, similarly to how [[217edo]] can be considered as 31edo with fine-tuning. The intervals [[11/10]], [[12/11]], [[13/12]], [[14/13]], and [[15/14]] are mapped equidistant, corresponding to [[121/120]], [[144/143]], [[169/168]], and [[196/195]] all being mapped to 2 steps. The mappings of 17 and 19 are accurate, with 15/14, [[16/15]], [[17/16]], [[18/17]], [[19/18]], and [[20/19]] all spaced apart from each other by one step. Overall, despite the sharpness of its 7 and 13, 205edo does fairly well in a range of prime limits.

=== Temperament generators and Tonal Plexus ===

205edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.

The 119\205 meantone fifth is extremely close to the 1/4-comma fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.

The 119\205 meantone fifth is extremely close to the [[1/4-comma meantone]] fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.

=== Subsets and supersets ===

205 factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.

205 factors into primes as [[5edo|5]] × [[41edo|41]], a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.

== Intervals ==

== Notation ==

=== Ups and downs ===

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.

205edo can be notated with [[Ups and downs notation|ups and downs]] representing {{nowrap|5\205 {{=}} 1\41}}, and lifts and drops (written as / and \) representing 1\205. Alternatively, ups and downs represent 1\205 and the quintuple-arrow symbols quip and quid (> and <) represent {{nowrap|5\205 {{=}} 1\41}}. Both notations have the advantage of building on a familiarity with 41edo The first notation is especially useful for [[Kite guitar]]ists 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 !! …

|-

| P1 || /1 || //1 || ^\\1 || ^\1 || ^1 || ^/1 || ^//1 || v\\m2 || v\m2 || vm2 || v/m2 || v//m2 || \\m2 || \m2 || m2 || /m2 || …

|-

| P1

| ^1

| ^^1

| ^^^1

| v>1

| >1

| ^>1

| ^^>1

| vv<m2

| v<m2

| <m2

| ^<m2

| vvvm2

| vvm2

| vm2

| m2

| ^m2

|

|}

== 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~~ (¢)

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

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

! colspan="2" | Tuning error

|-

! [[TE error|Absolute]] (¢)

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

|-

| 2.3.5

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

| [{{~~val~~| 205 325 476 }}]

| {{mapping| 205 325 476 }}

| -0.106

| −0.106

| 0.141

| 2.41

Line 53:

Line 84:

| 2.3.5.11

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

| [{{~~val~~| 181 287 420 508 }}]

| {{mapping| 181 287 420 508 }}

| -0.002

| −0.002

| 0.218

| 3.72

Line 61:

Line 92:

=== Rank-2 temperaments ===

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

|+Table of rank-2 temperaments by generator

|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator

! Periods per 8ve

|-

! Generator~~ (Reduced)~~

! Periods per 8ve

! Cents~~ (Reduced)~~

! Generator*

! Associated ~~Ratio~~

! Cents*

! ~~Temperaments~~

! Associated ratio*

! Temperament

|-

| 1

Line 93:

Line 125:

|-

| 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 />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct

== Scales ==

Line 160:

Line 193:

: 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 9

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

== External links ==

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

[[Category:Amity]]

[[Category:Hemithirds]]

[[Category:Huntmic]]

[[Category:Laka]]

[[Category:Quanic]]

@@ Line 1: / Line 1: @@
 {{Infobox ET}}
-{{EDO intro|205}}
+{{ED intro}}
+edo's step size is called a '''mem''' when used as an [[interval size unit]].
 == Theory ==
-edo 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-limits, 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}}, 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.
-Using its alternative mapping {{val| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]], and [[364/363]] to the comma list.
+Using its alternative mapping {{val| 205 325 476 '''575''' }} (205d) it can also be used for [[hemithirds]] temperament. This extension tempers out [[385/384]], [[441/440]], and 3388/3375 in the 11-limit. The 13-limit version of this, {{val| 205 325 476 '''575''' 709 759 }} (205d), is especially noteworthy, where it tempers out [[196/195]] and [[1001/1000]]. Another 13-limit extension is {{val| 205 325 476 '''575''' 709 '''758''' }} (205df), where it adds [[325/324]] and [[364/363]] to the comma list.
-Anyway, assume the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.
+Anyway, assuming the patent val, 205et tempers out 540/539, so that it allows [[swetismic chords]]; 729/728, so that it allows [[squbemic chords]]; [[640/637]], so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. In the alternative 205df val, it allows [[marveltwin chords]], [[keenanismic chords]], [[gentle chords]], and [[werckismic chords]]. This makes it a tuning of exceptional fludity for its degree of accuracy.
 === Odd harmonics ===
-{{Harmonics in equal|205}}
+{{Harmonics in equal|205|columns=11}}
+{{Harmonics in equal|205|columns=11|start=12|collapsed=1|title = Approximation of odd harmonics in 205edo (continued)}}
+=== Structural properties ===
+edo contains a very accurate approximation of the [[2.3.5.11 subgroup]], inheriting the perfect fifth from 41edo. The patent val mappings of primes 7 and 13 can then be found by mapping [[7/5]] to the Pythagorean diminished fifth, and [[13/11]] at the Pythagorean minor third, thus tempering out [[5120/5103]] and [[352/351]], as well as [[847/845]] and [[2080/2079]]. In fact, it is the last edo tempering out 5120/5103 to map both [[7/5]] and [[1024/729]] consistently. It also supports the [[counterpyth]] mapping of prime 19.
+Its step size represents several important intervals, such as the septimal kleisma [[225/224]], and the keenanisma [[385/384]]. Notably, the mappings of primes 5, 7, 11, 13, and 19 all differ from their nearest 41edo step by 1 step of 205edo, so 205edo can be considered as 41edo with fine-tuning, similarly to how [[217edo]] can be considered as 31edo with fine-tuning. The intervals [[11/10]], [[12/11]], [[13/12]], [[14/13]], and [[15/14]] are mapped equidistant, corresponding to [[121/120]], [[144/143]], [[169/168]], and [[196/195]] all being mapped to 2 steps. The mappings of 17 and 19 are accurate, with 15/14, [[16/15]], [[17/16]], [[18/17]], [[19/18]], and [[20/19]] all spaced apart from each other by one step. Overall, despite the sharpness of its 7 and 13, 205edo does fairly well in a range of prime limits.
 === Temperament generators and Tonal Plexus ===
 edo is the default tuning for the [http://hpi.zentral.zone/tonalplexus Tonal Plexus midi controller]. See the [http://musictheory.zentral.zone/huntsystem1.html theory part] on the same website. Aside from the 24\205 generator of quanic, the 58\205 generator of amity, and the 33\205 generator of hemithirds, 205edo supplies an excellent [[meantone]] fifth in 119\205, an excellent [[myna]] generator in 53\205, and a very good [[porcupine]] generator with 28\205, which is also an excellent generator for the higher-limit extension porky, and when sliced in half to 14\205, can even be used for nautilus. These facts are all potentially of significance to anyone using a 205edo based system such as the Tonal Plexus.
-The 119\205 meantone fifth is extremely close to the 1/4-comma fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.
+The 119\205 meantone fifth is extremely close to the [[1/4-comma meantone]] fifth, being only 0.007 cents sharp of it. Moreover the steps are half a cent flat of 1/4 of a syntonic comma. This makes the Tonal Plexus keyboard potentially of use in implementing [[Wikipedia: Nicola Vicentino|Nicola Vicentino]]'s [http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx adaptive-JI scheme of 1555]. It also means that authentic 1/4-comma meantone tuning is, for practical purposes, available in 205 and allows for historically authentic performances of 1/4-comma music on the historically newfangled Tonal Plexus.
 === Subsets and supersets ===
-factors into primes as 5 × 41, a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.
+factors into primes as [[5edo|5]] × [[41edo|41]], a fact some advocates of the division make use of; it is also [[2460edo|2460/12]], so that a single step is precisely 12 [[mina]]s.
+== Intervals ==
+{{Interval table}}
 == Notation ==
 === Ups and downs ===
-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.
+edo can be notated with [[Ups and downs notation|ups and downs]] representing {{nowrap|5\205 {{=}} 1\41}}, and lifts and drops (written as / and \) representing 1\205. Alternatively, ups and downs represent 1\205 and the quintuple-arrow symbols quip and quid (> and <) represent {{nowrap|5\205 {{=}} 1\41}}. Both notations have the advantage of building on a familiarity with 41edo The first notation is especially useful for [[Kite guitar]]ists 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 !! …
 |-
 | P1 || /1 || //1 || ^\\1 || ^\1 || ^1 || ^/1 || ^//1 || v\\m2 || v\m2 || vm2 || v/m2 || v//m2 || \\m2 || \m2 || m2 || /m2 || …
+|-
+| P1
+| ^1
+| ^^1
+| ^^^1
+| v>1
+| >1
+| ^>1
+| ^^>1
+| vv<m2
+| v<m2
+| <m2
+| ^<m2
+| vvvm2
+| vvm2
+| vm2
+| m2
+| ^m2
+|
 |}
 == Regular temperament properties ==
 {| class="wikitable center-4 center-5 center-6"
+|-
 ! rowspan="2" | [[Subgroup]]
 ! rowspan="2" | [[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]] (¢)
 ! [[TE simple badness|Relative]] (%)
 |-
 | 2.3.5
 | 1600000/1594323, {{monzo| 38 -2 -15 }}
-| [{{val| 205 325 476 }}]
+| {{mapping| 205 325 476 }}
-| -0.106
+| −0.106
 | 0.141
 | 2.41
@@ Line 53: / Line 84: @@
 | 2.3.5.11
 | 5632/5625, 14641/14580, 1600000/1594323
-| [{{val| 181 287 420 508 }}]
+| {{mapping| 181 287 420 508 }}
-| -0.002
+| −0.002
 | 0.218
 | 3.72
@@ Line 61: / Line 92: @@
 === Rank-2 temperaments ===
 {| class="wikitable center-all left-5"
-|+Table of rank-2 temperaments by generator
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
-! Periods<br>per 8ve
+|-
-! Generator<br>(Reduced)
+! Periods<br />per 8ve
-! Cents<br>(Reduced)
+! Generator*
-! Associated<br>Ratio
+! Cents*
-! Temperaments
+! Associated<br />ratio*
+! Temperament
 |-
 | 1
@@ Line 93: / Line 125: @@
 |-
 | 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]]
 |}
+<nowiki />* [[Normal forms #Equave-reduced-generator form|Octave-reduced form]], reduced to the first half-octave, and [[normal forms #Minimal-generator form|minimal form]] in parentheses if distinct
 == Scales ==
@@ Line 160: / Line 193: @@
 : 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 1 9 9 9 9
-[[Category:Equal divisions of the octave|###]] <!-- 3-digit number -->
+== External links ==
+* [http://tonalsoft.com/enc/m/mem.aspx mem, 205-edo] on [[Tonalsoft Encyclopedia]]
 [[Category:Amity]]
 [[Category:Hemithirds]]
+[[Category:Huntmic]]
 [[Category:Laka]]
 [[Category:Quanic]]