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360edo: Difference between revisions

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**Imported revision 586018267 - Original comment: **
m Text replacement - "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct" to "Octave-reduced form, reduced to the first half-octave, and minimal form in parentheses if distinct"
 
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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:MasonGreen1|MasonGreen1]] and made on <tt>2016-06-23 11:11:08 UTC</tt>.<br>
: The original revision id was <tt>586018267</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">360edo divides the octave into steps of 3 1/3 cents. Its 23-limit patent val is &lt;360 571 836 1011 1245 1332 1471 1529 1628|.
This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421. Its 5-limit patent val supports misty temperament, and the 7-limit patent val supports the 63&amp;99 temperament with wedgie &lt;&lt;9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&amp;360 = &lt;&lt;11 76 51 95 50 -95|| and 12&amp;360 = &lt;&lt;12 -48 -108 -104 -205 -116||; neither is very good though 41&amp;360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. Much better is [[Hemimage temperaments#Degrees|degrees temperament]], the 80&amp;140 temperament, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit. 360 has many proper divisors: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180.


360edo is used in the [[https://en.wikipedia.org/wiki/Eyeborg|eyeborg]], which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.</pre></div>
== Theory ==
<h4>Original HTML content:</h4>
360edo is [[consistent]] to the [[7-odd-limit]], but [[harmonic]] [[3/1|3]] is about halfway between its steps. It can also be used with 2.5.9.13 subgroup.  
<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;360edo&lt;/title&gt;&lt;/head&gt;&lt;body&gt;360edo divides the octave into steps of 3 1/3 cents. Its 23-limit patent val is &amp;lt;360 571 836 1011 1245 1332 1471 1529 1628|.&lt;br /&gt;
 
This val tempers out the kalisma, the triaphonisma, the septendecimal bridge comma, the misty comma, hemimage, dimicomp, 2*(14/15)^10, 289/288, 352/351, 589824/588245 and 2560000000/2542277421. Its 5-limit patent val supports misty temperament, and the 7-limit patent val supports the 63&amp;amp;99 temperament with wedgie &amp;lt;&amp;lt;9 -36 9 -78 -11 122|| which tempers out misty but has a period of 1/9 rather than 1/3 octave. Two other seven limit temperaments it supports and also provides the optimal patent val for are 41&amp;amp;360 = &amp;lt;&amp;lt;11 76 51 95 50 -95|| and 12&amp;amp;360 = &amp;lt;&amp;lt;12 -48 -108 -104 -205 -116||; neither is very good though 41&amp;amp;360 has a TE badness lower than any alternative 7-limit temperament for which 360 gives the optimal patent val. Much better is &lt;a class="wiki_link" href="/Hemimage%20temperaments#Degrees"&gt;degrees temperament&lt;/a&gt;, the 80&amp;amp;140 temperament, for which 360 supplies the optimal patent val in the 11-limit and which it supports and provides an excellent tuning for in the 13-limit. 360 has many proper divisors: 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180.&lt;br /&gt;
In the 5-limit, the [[patent val]] [[support]]s the [[misty]] temperament, and in the 7-limit 360edo supports the [[trimisty]] (name proposed by Eliora) 63 & 99 temperament with the comma basis {[[10976/10935]], 2097152/2083725}, which is similar to the misty temperament but has a period of 1/9- rather than 1/3-octave.
&lt;br /&gt;
 
360edo is used in the &lt;a class="wiki_link_ext" href="https://en.wikipedia.org/wiki/Eyeborg" rel="nofollow"&gt;eyeborg&lt;/a&gt;, which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.&lt;/body&gt;&lt;/html&gt;</pre></div>
360edo provides the optimal patent val in the 11-limit, and otherwise a good tuning in the 13-limit for [[degrees]], the {{nowrap|140 &amp; 220}} temperament with period 1\20. Aside from that, it provides the optimal patent val for the {{nowrap|41 &amp; 360}} temperament with comma basis {10976/10935, 16384000000/16209796869}, on which it has lower badness than any other 7-limit temperament for which 360edo gives the optimal patent val. It also supports {{nowrap|12 &amp; 360}} with the comma basis {[[390625/388962]], 67108864/66430125}. 
 
Aside from the patent val, there is a number of mappings to be considered. The 360d val, {{val|360 571 836 '''1010'''}}, tempers out 3136/3125, 5120/5103, and extends the misty temperament in to the 7-limit. It is also a tuning for the 12th-octave [[magnesium]] temperament.  
 
=== Odd harmonics ===
{{Harmonics in equal|360}}
 
=== Subsets and supersets ===
360 is the 13th [[highly composite edo]], with many proper divisors: {{EDOs| 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 }}. One step of 360edo is known as '''the Dröbisch angle''', an [[interval size measure]] first proposed by Moritz Dröbisch in the 19th century at first merely by the name "angle".
 
== Table of intervals ==
[[Eliora]] proposes notating 360edo with calendar dates, Jan 1 being the tonic, Jan 2 being the next step, etc, and each month having even 30 days. The notation is convenient because 1 month in this scenario is equal to 1 semitone, and corresponds to [[12edo]].
 
Any other notation system involving the number 360 can also be used.
 
See: [[Table of 360edo intervals]]
 
== Regular temperament properties ==
=== 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*
! Temperaments
|-
| 1
| 119\360
| 396.67
| 44/35
| [[Squarschmidt]]
|-
| 2
| 53\360
| 176.67
| 448/405
| Quatracot
|-
| 3
| 149\360<br />(29\360)
| 703.33<br />(303.33)
| 4/3<br />(135/128)
| [[Misty]]
|-
| 4
| 23\360
| 76.67
| 4302592/4100625
| [[Reenactment]]
|-
| 9
| 149\360<br />(29\360)
| 703.33<br />(36.67)
| 4/3<br />(135/128)
| [[Trimisty]]
|-
| 12
| 73\360<br />(13\360)
| 243.333<br />(43.333)
| 3145728/2734375<br />(?)
| [[Magnesium]] (360d)
|-
| 20
| 149\360<br />(5\360)
| 703.33<br />(43.33)
| 4/3<br />(126/125)
| [[Degrees]]
|}
<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
 
== Music ==
; [[User:Eliora|Eliora]]
* [https://www.youtube.com/watch?v=VSKqwJkWu_U ''Idyllic Tribe''] (2022)
 
== Application as a logarithmic scale outside of music ==
360edo is used in the {{w|eyeborg}}, which maps its scale degrees onto color hues, thus converting color into sound waves. The device was originally intended to help colorblind individuals.
 
[[Category:Sonifications]]
[[Category:Listen]]
{{Todo| cleanup |comment=move trimisty away}}
Retrieved from "https://en.xen.wiki/w/360edo"