Xenharmonic Wiki

360edo: Difference between revisions

← Older edit
VisualWikitext
Wikispaces>genewardsmith
**Imported revision 339578486 - 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"
 
(49 intermediate revisions by 11 users not shown)
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>2012-05-25 14:41:19 UTC</tt>.<br>
 
: The original revision id was <tt>339578486</tt>.<br>
== Theory ==
: The revision comment was: <tt></tt><br>
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.  
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>
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.
<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, but extending beyond the 5-limit does not support any higher limit misty temperament. Two seven limit temperaments it supports and 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 -118||; 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.</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}. 
<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;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;
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.
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, but extending beyond the 5-limit does not support any higher limit misty temperament. Two seven limit temperaments it supports and 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 -118||; 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.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
=== 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"