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<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:hstraub|hstraub]] and made on <tt>2010-08-13 07:23:05 UTC</tt>.<br>
 
: The original revision id was <tt>156403257</tt>.<br>
205edo's step size is called a '''mem''' when used as an [[interval size unit]].
: 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>
== Theory ==
<h4>Original Wikitext content:</h4>
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.
<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 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.</pre></div>
 
<h4>Original HTML content:</h4>
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.
<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 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
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, 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}}
{{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 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 [[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 ===
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<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
|}
 
=== 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
 
== Scales ==
=== 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
 
=== Amity (58\205) mos ===
; 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
 
=== Hemithirds (33\205) mos ===
; 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
 
=== 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
 
=== Myna (53\205) mos ===
; 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
 
=== 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
 
== 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]]
Retrieved from "https://en.xen.wiki/w/205edo"