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~~<h2>IMPORTED REVISION FROM WIKISPACES</h2>~~

~~This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>~~

~~: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-09-28 08:50:21 UTC</tt>.<br>~~

~~: The original revision id was <tt>454888364</tt>.<br>~~

~~: The revision comment was: <tt></tt><br>~~

~~The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>~~

~~<h4>Original Wikitext content:</h4>~~

<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">205EDO divides the octave into 205 parts of 5.854 cents each. It can serve as a tuning for various temperaments, such as [[Ragismic microtemperaments|amity]] or [[Hemifamity family|laka]], and supplies the optimal patent val for [[Hemifamity temperaments#Quanic|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 five temperament for which it also supplies the optimal patent val. In the 5-limit it tempers out 1600000/1594323, the amity comma, and |38 -2 -15>, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val tempers out 4375/4374, 5120/5103, 6144/6125 and 540/539. Using its alternative mapping <205 325 476 525| it can also be used for [[Gamelismic clan|hemithirds temperament]]; the 13-limit version of this, <205 325 476 575 709 759|, is especially noteworthy. It factors into primes as 5 * 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 [[mina|minas]].

~~205et tempers out 540/539, so that it allows [[swetismic chords]]; 640/637, so that it allows [[huntmic chords]]; 352/351, so that it allows~~ [[~~minthmic chords~~]]~~; 1188/1183 and 540/539, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. This makes it a tuning of exceptional fludity for its degree of accuracy~~.

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

=~~Temperament generators and Tonal Plexus~~=

== Theory ==

205edo ~~is the default tuning for the~~ [[~~http://www.h-pi.com~~/~~TPX28features.html~~|~~Tonal Plexus midi controller~~]]~~. See the~~ [[~~http://www.h-pi.com/theory/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~~.

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.

~~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~~ [[~~http:~~//~~en.wikipedia.org/wiki~~/~~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~~.

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.

~~=205edo MOS=~~

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.

~~==Quanic~~ (~~24\~~205) ~~MOS==~~

~~===17 note===~~

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.

~~11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13~~

===~~26 note~~===

=== Odd harmonics ===

11 11 ~~2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2~~

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

==~~Amity (58\205) MOS~~==

=== 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.

~~===~~11 ~~note===~~

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.

~~27 27 4 27 27 4 27 27 4 27 4~~

===~~18 note~~===

=== Temperament generators and Tonal Plexus ===

~~23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4~~

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.

~~===25 note===~~

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.

19 4 4 ~~19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4~~ 4 4

==~~Hemithirds (33\205) MOS~~==

=== Subsets and supersets ===

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.

==~~=13 note=~~==

== Intervals ==

~~26 7 26 7 26 7 26 7 26 7 26 7 7~~

===~~19 note~~===

== Notation ==

~~19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7~~

=== Ups and downs ===

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.

=~~==25 note===~~

{| class="wikitable"

~~12 7 7~~ 7 12 ~~7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7~~

|-

! 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

|

|}

===~~31 notes~~===

== Regular temperament properties ==

5 ~~7 7 7 7~~ 5 ~~7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7~~

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

|-

! 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

| 0.141

| 2.41

|-

| 2.3.5.11

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

| {{mapping| 181 287 420 508 }}

| −0.002

| 0.218

| 3.72

|}

==~~Meantone~~ (~~119~~\205) ~~MOS==~~

=== Rank-2 temperaments ===

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

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

|-

! Periods<br />per 8ve

! Generator*

! Cents*

! Associated<br />ratio*

! Temperament

|-

| 1

| 6\205

| 35.122

| 45/44

| [[Gammic]] (205e)

|-

| 1

| 24\205

| 140.488

| 13/12

| [[Quanic]] (205)

|-

| 1

| 33\205

| 193.171

| 28/25

| [[Luna]] / [[lunatic]] (205) / [[hemithirds]] (205d)

|-

| 1

| 58\205

| 339.512

| 128/105

| [[Amity]] (205)

|-

| 5

| 63\205<br />(19\205)

| 368.780<br />(111.220)

| 1024/891<br />(16/15)

| [[Quintosec]]

|-

| 41

| 66\205<br />(1\205)

| 386.341<br />(5.85)

| 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

===~~12 note~~===

== Scales ==

13 20 13 20 13 ~~20 20~~ 13 20 13 ~~20 20~~

=== Quanic (24\205) mos ===

; 17-note

: 11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13

; 26-note

: 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2

===~~19 note~~===

=== Amity (58\205) mos ===

~~13 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7~~

; 11-note

: 27 27 4 27 27 4 27 27 4 27 4

; 18-note

: 23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4

; 25-note

: 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4

===~~31 notes~~===

=== Hemithirds (33\205) mos ===

6 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7

; 13-note

: 26 7 26 7 26 7 26 7 26 7 26 7 7

; 19-note

: 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7

; 25-note

: 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7

; 31-notes

: 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7

==~~Myna~~ (53\205) ~~MOS~~==

=== Meantone (119\205) mos ===

; 12-note

: 13 20 13 20 13 20 20 13 20 13 20 20

; 19-note

: 13 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7

; 31-note

: 6 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7

===~~11 note~~===

=== Myna (53\205) mos ===

7 7 39 7 7 39 7 7 39 7 39

; 11-note

: 7 7 39 7 7 39 7 7 39 7 39

; 15-note

: 7 7 7 32 7 7 7 32 7 7 7 32 7 7 32

; 19-note

: 7 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25

; 23-note

: 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18

; 27-note

: 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11

; 31-note

: 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4

===~~15 note~~===

=== Porcupine (28\205) mos ===

~~7 7 7 32 7 7 7 32 7 7 7 32 7 7 32~~

; 15-note

: 19 9 19 9 19 9 19 9 19 9 19 9 19 9 9

; 22-note

: 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9

; 29-note

: 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

==~~=19 note=~~==

== External links ==

~~7 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25~~

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

~~===23 note===~~

[[Category:Amity]]

~~7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18~~

[[Category:Hemithirds]]

[[Category:Huntmic]]

~~===27 note===~~

[[Category:Laka]]

~~7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11~~

[[Category:Quanic]]

~~===31 notes===~~

~~7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4~~

~~==Porcupine (28\205) MOS==~~

~~===15 note===~~

~~19 9 19 9 19 9 19 9 19 9 19 9 19 9 9~~

~~===22 note===~~

~~10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9~~

~~===29 note===~~

~~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</pre></div>~~

~~<h4>Original HTML content:</h4>~~

~~<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border~~: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>205edo</title></head><body>205EDO divides the octave into 205 parts of 5.854 cents each. It can serve as a tuning for various temperaments, such as <a class="wiki_link" href="/Ragismic%20microtemperaments">amity</a> or <a class="wiki_link" href="/Hemifamity%20family">laka</a>, and supplies the optimal patent val for <a class="wiki_link" href="/Hemifamity%20temperaments#Quanic">quanic</a> 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 five temperament for which it also supplies the optimal patent val. In the 5-limit it tempers out 1600000/1594323, the amity comma, and |38 -2 -15&gt;, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val tempers out 4375/4374, 5120/5103, 6144/6125 and 540/539. Using its alternative mapping &lt;205 325 476 525| it can also be used for <a class="wiki_link" href="/Gamelismic%20clan">hemithirds temperament</a>; the 13-limit version of this, &lt;205 325 476 575 709 759|, is especially noteworthy. It factors into primes as 5 * 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 <a class="wiki_link" href="/mina">minas</a>.<br />

~~<br />~~

205et tempers out 540/539, so that it allows <a class="wiki_link" href="/swetismic%20chords">swetismic chords</a>; 640/637, so that it allows <a class="wiki_link" href="/huntmic%20chords">huntmic chords</a>; 352/351, so that it allows <a class="wiki_link" href="/minthmic%20chords">minthmic chords</a>; 1188/1183 and 540/539, so that it allows <a class="wiki_link" href="/kestrel%20chords">kestrel chords</a>; and 847/845, so that it allows the <a class="wiki_link" href="/cuthbert%20triad">cuthbert triad</a>. This makes it a tuning of exceptional fludity for its degree of accuracy.<br />

~~<br />~~

<h1 id="toc0"><a name="Temperament generators and Tonal Plexus"></a>Temperament generators and Tonal Plexus</h1>

205edo is the default tuning for the <a class="wiki_link_ext" href="http://www.h-pi.com/TPX28features.html" rel="nofollow">Tonal Plexus midi controller</a>. See the <a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem1.html" rel="nofollow">theory part</a> 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 <a class="wiki_link" href="/meantone">meantone</a> fifth in 119\205, an excellent <a class="wiki_link" href="/myna">myna</a> generator in 53\205, and a very good <a class="wiki_link" href="/porcupine">porcupine</a> 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.<br />

~~<br />~~

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 <a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Nicola_Vicentino" rel="nofollow">Nicola Vicentino</a>'s <a class="wiki_link_ext" href="http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx" rel="nofollow">adaptive-JI scheme of 1555</a>. 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.<br />

~~<br />~~

~~<h1 id="toc1"><a name="x205edo MOS"></a>205edo MOS</h1>~~

<h2 id="toc2"><a name="x205edo MOS-Quanic (24\205) MOS"></a>Quanic (24\205) MOS</h2>

~~<br />~~

<h3 id="toc3"><a name="x205edo MOS-Quanic (24\205) MOS-17 note"></a>17 note</h3>

~~11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13<br />~~

~~<br />~~

<h3 id="toc4"><a name="x205edo MOS-Quanic (24\205) MOS-26 note"></a>26 note</h3>

~~11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2<br />~~

~~<br />~~

<h2 id="toc5"><a name="x205edo MOS-Amity (58\205) MOS"></a>Amity (58\205) MOS</h2>

~~<br />~~

<h3 id="toc6"><a name="x205edo MOS-Amity (58\205) MOS-11 note"></a>11 note</h3>

~~27 27 4 27 27 4 27 27 4 27 4<br />~~

~~<br />~~

<h3 id="toc7"><a name="x205edo MOS-Amity (58\205) MOS-18 note"></a>18 note</h3>

~~23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4<br />~~

~~<br />~~

~~<h3 id="toc8"><a name="x205edo MOS-~~Amity ~~(58\205) MOS-25 note"></a>25 note</h3>~~

~~19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4<br />~~

~~<br />~~

~~<h2 id="toc9"><a name="x205edo MOS-Hemithirds (33\205) MOS"></a>~~Hemithirds ~~(33\205) MOS</h2>~~

~~<br />~~

<h3 id="toc10"><a name="x205edo MOS-Hemithirds (33\205) MOS-13 note"></a>13 note</h3>

~~26 7 26 7 26 7 26 7 26 7 26 7 7<br />~~

~~<br />~~

~~<h3 id="toc11"><a name="x205edo MOS-Hemithirds (33\205) MOS-19 note"></a>19 note</h3>~~

~~19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7<br />~~

~~<br />~~

<h3 id="toc12"><a name="x205edo MOS-Hemithirds (33\205) MOS-25 note"></a>25 note</h3>

~~12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7<br />~~

~~<br />~~

<h3 id="toc13"><a name="x205edo MOS-Hemithirds (33\205) MOS-31 notes"></a>31 notes</h3>

~~5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7<br />~~

~~<br />~~

<h2 id="toc14"><a name="x205edo MOS-Meantone (119\205) MOS"></a>Meantone (119\205) MOS</h2>

~~<br />~~

<h3 id="toc15"><a name="x205edo MOS-Meantone (119\205) MOS-12 note"></a>12 note</h3>

~~13 20 13 20 13 20 20 13 20 13 20 20<br />~~

~~<br />~~

<h3 id="toc16"><a name="x205edo MOS-Meantone (119\205) MOS-19 note"></a>19 note</h3>

~~13 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7<br />~~

~~<br />~~

<h3 id="toc17"><a name="x205edo MOS-Meantone (119\205) MOS-31 notes"></a>31 notes</h3>

~~6 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7<br />~~

~~<br />~~

<h2 id="toc18"><a name="x205edo MOS-Myna (53\205) MOS"></a>Myna (53\205) MOS</h2>

~~<br />~~

<h3 id="toc19"><a name="x205edo MOS-Myna (53\205) MOS-11 note"></a>11 note</h3>

~~7 7 39 7 7 39 7 7 39 7 39<br />~~

~~<br />~~

<h3 id="toc20"><a name="x205edo MOS-Myna (53\205) MOS-15 note"></a>15 note</h3>

~~7 7 7 32 7 7 7 32 7 7 7 32 7 7 32<br />~~

~~<br />~~

<h3 id="toc21"><a name="x205edo MOS-Myna (53\205) MOS-19 note"></a>19 note</h3>

~~7 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25<br />~~

~~<br />~~

<h3 id="toc22"><a name="x205edo MOS-Myna (53\205) MOS-23 note"></a>23 note</h3>

~~7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18<br />~~

~~<br />~~

<h3 id="toc23"><a name="x205edo MOS-Myna (53\205) MOS-27 note"></a>27 note</h3>

~~7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11<br />~~

~~<br />~~

<h3 id="toc24"><a name="x205edo MOS-Myna (53\205) MOS-31 notes"></a>31 notes</h3>

~~7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4<br />~~

~~<br />~~

<h2 id="toc25"><a name="x205edo MOS-Porcupine (28\205) MOS"></a>Porcupine (28\205) MOS</h2>

~~<br />~~

<h3 id="toc26"><a name="x205edo MOS-Porcupine (28\205) MOS-15 note"></a>15 note</h3>

~~19 9 19 9 19 9 19 9 19 9 19 9 19 9 9<br />~~

~~<br />~~

<h3 id="toc27"><a name="x205edo MOS-Porcupine (28\205) MOS-22 note"></a>22 note</h3>

~~10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9<br />~~

~~<br />~~

<h3 id="toc28"><a name="x205edo MOS-Porcupine (28\205) MOS-29 note"></a>29 note</h3>

~~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</body></html></pre></div>~~

@@ Line 1: / Line 1: @@
-<h2>IMPORTED REVISION FROM WIKISPACES</h2>
+{{Infobox ET}}
-This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
+{{ED intro}}
-: This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2013-09-28 08:50:21 UTC</tt>.<br>
-: The original revision id was <tt>454888364</tt>.<br>
-: The revision comment was: <tt></tt><br>
-The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
-<h4>Original Wikitext content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;white-space: pre-wrap ! important" class="old-revision-html">205EDO divides the octave into 205 parts of 5.854 cents each. It can serve as a tuning for various temperaments, such as [[Ragismic microtemperaments|amity]] or [[Hemifamity family|laka]], and supplies the optimal patent val for [[Hemifamity temperaments#Quanic|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 five temperament for which it also supplies the optimal patent val. In the 5-limit it tempers out 1600000/1594323, the amity comma, and |38 -2 -15&gt;, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val tempers out 4375/4374, 5120/5103, 6144/6125 and 540/539. Using its alternative mapping &lt;205 325 476 525| it can also be used for [[Gamelismic clan|hemithirds temperament]]; the 13-limit version of this, &lt;205 325 476 575 709 759|, is especially noteworthy. It factors into primes as 5 * 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 [[mina|minas]].
-et tempers out 540/539, so that it allows [[swetismic chords]]; 640/637, so that it allows [[huntmic chords]]; 352/351, so that it allows [[minthmic chords]]; 1188/1183 and 540/539, so that it allows [[kestrel chords]]; and 847/845, so that it allows the [[cuthbert triad]]. This makes it a tuning of exceptional fludity for its degree of accuracy.
+edo's step size is called a '''mem''' when used as an [[interval size unit]].
-=Temperament generators and Tonal Plexus=
+== Theory ==
-edo is the default tuning for the [[http://www.h-pi.com/TPX28features.html|Tonal Plexus midi controller]]. See the [[http://www.h-pi.com/theory/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.
+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.
-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 [[http://en.wikipedia.org/wiki/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.
+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.
-=205edo MOS=
+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.
-==Quanic (24\205) MOS==
-===17 note===
+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.
-13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13
-===26 note===
+=== Odd harmonics ===
-11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2
+{{Harmonics in equal|205|columns=11}}
+{{Harmonics in equal|205|columns=11|start=12|collapsed=1|title = Approximation of odd harmonics in 205edo (continued)}}
-==Amity (58\205) MOS==
+=== 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.
-===11 note===
+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.
-27 4 27 27 4 27 27 4 27 4
-===18 note===
+=== Temperament generators and Tonal Plexus ===
-4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4
+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.
-===25 note===
+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.
-4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4
-==Hemithirds (33\205) MOS==
+=== Subsets and supersets ===
+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.
-===13 note===
+== Intervals ==
-7 26 7 26 7 26 7 26 7 26 7 7
+{{Interval table}}
-===19 note===
+== Notation ==
-7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7
+=== Ups and downs ===
+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.
-===25 note===
+{| class="wikitable"
-7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7
+|-
+! 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
+|
+|}
-===31 notes===
+== Regular temperament properties ==
-7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7
+{| class="wikitable center-4 center-5 center-6"
+|-
+! 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
+| 0.141
+| 2.41
+|-
+| 2.3.5.11
+| 5632/5625, 14641/14580, 1600000/1594323
+| {{mapping| 181 287 420 508 }}
+| −0.002
+| 0.218
+| 3.72
+|}
-==Meantone (119\205) MOS==
+=== Rank-2 temperaments ===
+{| class="wikitable center-all left-5"
+|+ style="font-size: 105%;" | Table of rank-2 temperaments by generator
+|-
+! Periods<br />per 8ve
+! Generator*
+! Cents*
+! Associated<br />ratio*
+! Temperament
+|-
+| 1
+| 6\205
+| 35.122
+| 45/44
+| [[Gammic]] (205e)
+|-
+| 1
+| 24\205
+| 140.488
+| 13/12
+| [[Quanic]] (205)
+|-
+| 1
+| 33\205
+| 193.171
+| 28/25
+| [[Luna]] / [[lunatic]] (205) / [[hemithirds]] (205d)
+|-
+| 1
+| 58\205
+| 339.512
+| 128/105
+| [[Amity]] (205)
+|-
+| 5
+| 63\205<br />(19\205)
+| 368.780<br />(111.220)
+| 1024/891<br />(16/15)
+| [[Quintosec]]
+|-
+| 41
+| 66\205<br />(1\205)
+| 386.341<br />(5.85)
+| 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
-===12 note===
+== Scales ==
-20 13 20 13 20 20 13 20 13 20 20
+=== Quanic (24\205) mos ===
+; 17-note
+: 11 13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13
+; 26-note
+: 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2
-===19 note===
+=== Amity (58\205) mos ===
-13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7
+; 11-note
+: 27 27 4 27 27 4 27 27 4 27 4
+; 18-note
+: 23 4 23 4 4 23 4 23 4 4 23 4 23 4 4 23 4 4
+; 25-note
+: 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 19 4 4 4 19 4 4 4
-===31 notes===
+=== Hemithirds (33\205) mos ===
-7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7
+; 13-note
+: 26 7 26 7 26 7 26 7 26 7 26 7 7
+; 19-note
+: 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 19 7 7 7
+; 25-note
+: 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 12 7 7 7 7
+; 31-notes
+: 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 5 7 7 7 7 7
-==Myna (53\205) MOS==
+=== Meantone (119\205) mos ===
+; 12-note
+: 13 20 13 20 13 20 20 13 20 13 20 20
+; 19-note
+: 13 13 7 13 13 7 13 13 7 13 7 13 13 7 13 13 7 13 7
+; 31-note
+: 6 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7 6 7 6 7 7 6 7 6 7 7 6 7 7
-===11 note===
+=== Myna (53\205) mos ===
-7 39 7 7 39 7 7 39 7 39
+; 11-note
+: 7 7 39 7 7 39 7 7 39 7 39
+; 15-note
+: 7 7 7 32 7 7 7 32 7 7 7 32 7 7 32
+; 19-note
+: 7 7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25
+; 23-note
+: 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18
+; 27-note
+: 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11
+; 31-note
+: 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4
-===15 note===
+=== Porcupine (28\205) mos ===
-7 7 32 7 7 7 32 7 7 7 32 7 7 32
+; 15-note
+: 19 9 19 9 19 9 19 9 19 9 19 9 19 9 9
+; 22-note
+: 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9
+; 29-note
+: 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
-===19 note===
+== External links ==
-7 7 7 25 7 7 7 7 25 7 7 7 7 25 7 7 7 25
+* [http://tonalsoft.com/enc/m/mem.aspx mem, 205-edo] on [[Tonalsoft Encyclopedia]]
-===23 note===
+[[Category:Amity]]
-7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 7 18 7 7 7 7 18
+[[Category:Hemithirds]]
+[[Category:Huntmic]]
-===27 note===
+[[Category:Laka]]
-7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 7 11 7 7 7 7 7 11
+[[Category:Quanic]]
-===31 notes===
-7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 7 4 7 7 7 7 7 7 4
-==Porcupine (28\205) MOS==
-===15 note===
-9 19 9 19 9 19 9 19 9 19 9 19 9 9
-===22 note===
-9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 10 9 9 9
-===29 note===
-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</pre></div>
-<h4>Original HTML content:</h4>
-<div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html">&lt;html&gt;&lt;head&gt;&lt;title&gt;205edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;205EDO divides the octave into 205 parts of 5.854 cents each. It can serve as a tuning for various temperaments, such as &lt;a class="wiki_link" href="/Ragismic%20microtemperaments"&gt;amity&lt;/a&gt; or &lt;a class="wiki_link" href="/Hemifamity%20family"&gt;laka&lt;/a&gt;, and supplies the optimal patent val for &lt;a class="wiki_link" href="/Hemifamity%20temperaments#Quanic"&gt;quanic&lt;/a&gt; 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 five temperament for which it also supplies the optimal patent val. In the 5-limit it tempers out 1600000/1594323, the amity comma, and |38 -2 -15&amp;gt;, the hemithirds comma, and is an excellent tuning for 5-limit amity. The patent val tempers out 4375/4374, 5120/5103, 6144/6125 and 540/539. Using its alternative mapping &amp;lt;205 325 476 525| it can also be used for &lt;a class="wiki_link" href="/Gamelismic%20clan"&gt;hemithirds temperament&lt;/a&gt;; the 13-limit version of this, &amp;lt;205 325 476 575 709 759|, is especially noteworthy. It factors into primes as 5 * 41, a fact some advocates of the division make use of; it is also 2460/12, so that a single step is precisely 12 &lt;a class="wiki_link" href="/mina"&gt;minas&lt;/a&gt;.&lt;br /&gt;
-&lt;br /&gt;
-et tempers out 540/539, so that it allows &lt;a class="wiki_link" href="/swetismic%20chords"&gt;swetismic chords&lt;/a&gt;; 640/637, so that it allows &lt;a class="wiki_link" href="/huntmic%20chords"&gt;huntmic chords&lt;/a&gt;; 352/351, so that it allows &lt;a class="wiki_link" href="/minthmic%20chords"&gt;minthmic chords&lt;/a&gt;; 1188/1183 and 540/539, so that it allows &lt;a class="wiki_link" href="/kestrel%20chords"&gt;kestrel chords&lt;/a&gt;; and 847/845, so that it allows the &lt;a class="wiki_link" href="/cuthbert%20triad"&gt;cuthbert triad&lt;/a&gt;. This makes it a tuning of exceptional fludity for its degree of accuracy.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:0:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc0"&gt;&lt;a name="Temperament generators and Tonal Plexus"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:0 --&gt;Temperament generators and Tonal Plexus&lt;/h1&gt;
-edo is the default tuning for the &lt;a class="wiki_link_ext" href="http://www.h-pi.com/TPX28features.html" rel="nofollow"&gt;Tonal Plexus midi controller&lt;/a&gt;. See the &lt;a class="wiki_link_ext" href="http://www.h-pi.com/theory/huntsystem1.html" rel="nofollow"&gt;theory part&lt;/a&gt; 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 &lt;a class="wiki_link" href="/meantone"&gt;meantone&lt;/a&gt; fifth in 119\205, an excellent &lt;a class="wiki_link" href="/myna"&gt;myna&lt;/a&gt; generator in 53\205, and a very good &lt;a class="wiki_link" href="/porcupine"&gt;porcupine&lt;/a&gt; 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.&lt;br /&gt;
-&lt;br /&gt;
-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 &lt;a class="wiki_link_ext" href="http://en.wikipedia.org/wiki/Nicola_Vicentino" rel="nofollow"&gt;Nicola Vicentino&lt;/a&gt;'s &lt;a class="wiki_link_ext" href="http://www.tonalsoft.com/monzo/vicentino/vicentino.aspx" rel="nofollow"&gt;adaptive-JI scheme of 1555&lt;/a&gt;. 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.&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="x205edo MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;205edo MOS&lt;/h1&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc2"&gt;&lt;a name="x205edo MOS-Quanic (24\205) MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Quanic (24\205) MOS&lt;/h2&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc3"&gt;&lt;a name="x205edo MOS-Quanic (24\205) MOS-17 note"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;17 note&lt;/h3&gt;
-13 11 13 11 13 11 13 11 13 11 13 11 13 11 13 13&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:8:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc4"&gt;&lt;a name="x205edo MOS-Quanic (24\205) MOS-26 note"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:8 --&gt;26 note&lt;/h3&gt;
-11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 11 2 11 2&lt;br /&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:10:&amp;lt;h2&amp;gt; --&gt;&lt;h2 id="toc5"&gt;&lt;a name="x205edo MOS-Amity (58\205) MOS"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:10 --&gt;Amity (58\205) MOS&lt;/h2&gt;
-&lt;br /&gt;
-&lt;!-- ws:start:WikiTextHeadingRule:12:&amp;lt;h3&amp;gt; --&gt;&lt;h3 id="toc6"&gt;&lt;a name="x205edo MOS-Amity (58\205) MOS-11 note"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:12 --&gt;11 note&lt;/h3&gt;
-27 4 27 27 4 27 27 4 27 4&lt;br /&gt;
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