Ed5/3: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
The '''equal division of 5/3''' ('''ed5/3''') is a [[tuning]] obtained by dividing the [[5/3|just major sixth (5/3)]] into a number of [[equal]] steps.
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-11-30 16:35:55 UTC</tt>.<br>
: The original revision id was <tt>601057772</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">&lt;span style="font-size: 19.5px;"&gt;Division of a sixth (e. g. 5/3 or 11/7) into n equal parts&lt;/span&gt;


== Properties ==
Division of 5/3 into equal parts does not necessarily imply directly using this interval as an [[equivalence]]. Many, though not all, ed5/3 scales have a perceptually important [[Pseudo-octave|false octave]], with various degrees of accuracy.


Division of e. g. the 5:3 or the 11:7 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of [[equivalence]] has not even been posed yet. The utility of 5:3 or 11:7 or another sixth as a base though, is apparent by being named directly in the standard definition of such as the octave based [[Sensi|sensi]] temperament. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy, but which context(s), if any, it is very perceptually important in is as yet an open question.
5/3 is the most consonant interval in between 1/1 and 2/1, so this suggests it could be useful either as an equivalence, or as just an important structural feature.


Incidentally, one way to treat 5/3 or 11/7 as an equivalence is the use of the 6:7:8:(10) or 7:8:9:(11) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224) or four 9/7 to get to 9/8 (tempering out the comma 5929/5832). So, doing this yields 7, 9, and 16 note MOS, the 16 note MOS of the two temperaments being mirror images of each other (7L 9s for ed(5/3)s vs 9L 7s for ed(11/7)s). While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it (even better than for edfs as the generator it uses is an excellent fit for heptatonic MOS) if it hasn't been named yet.</pre></div>
[[Joseph Ruhf]] suggested the use of the 6:7:8:(10) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone as a way to evoke 5/3-equivalence. Though it could also be used just as a useful sonority, even without equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224). So, doing this yields 7-, 9-, and 16-note [[mos]] either way, the 16-note mos being 7L 9s. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it (even better than for [[edf]]s as the generator it uses is an excellent fit for heptatonic mos) though it is, technically speaking, micro-[[7L 2s|armotonic]].
<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;edVI&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;span style="font-size: 19.5px;"&gt;Division of a sixth (e. g. 5/3 or 11/7) into n equal parts&lt;/span&gt;&lt;br /&gt;
If we instead opt to continue using 4:5:6 as the fundamental sonority, then it will take three 3/2 to get to 5/4, resulting in [[Blackcomb]] temperament that tempers out the comma 250/243. This yields mos scales of 4, 5, 6, 11, 16, and 21 notes. Although, it should be noted that doing this will often create a pseudo-octave unlike the 6:7:8 approach.
&lt;br /&gt;
 
&lt;br /&gt;
ED5/3 tuning systems that accurately represent the intervals 5/4 and 4/3 include: [[7ed5/3]] (7.30 cent error), [[9ed5/3]] (6.73 cent error), and [[16ed5/3]] (0.59 cent error).
Division of e. g. the 5:3 or the 11:7 into equal parts can be conceived of as to directly use this interval as an equivalence, or not. The question of &lt;a class="wiki_link" href="/equivalence"&gt;equivalence&lt;/a&gt; has not even been posed yet. The utility of 5:3 or 11:7 or another sixth as a base though, is apparent by being named directly in the standard definition of such as the octave based &lt;a class="wiki_link" href="/Sensi"&gt;sensi&lt;/a&gt; temperament. Many, though not all, of these scales have a pseudo (false) octave, with various degrees of accuracy, but which context(s), if any, it is very perceptually important in is as yet an open question.&lt;br /&gt;
 
&lt;br /&gt;
[[7ed5/3]], [[9ed5/3]], and [[16ed5/3]] are to the [[Ed5/3|division of 5/3]] what [[5edo]], [[7edo]], and [[12edo]] are to the [[EDO|division of 2/1]].
Incidentally, one way to treat 5/3 or 11/7 as an equivalence is the use of the 6:7:8:(10) or 7:8:9:(11) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224) or four 9/7 to get to 9/8 (tempering out the comma 5929/5832). So, doing this yields 7, 9, and 16 note MOS, the 16 note MOS of the two temperaments being mirror images of each other (7L 9s for ed(5/3)s vs 9L 7s for ed(11/7)s). While the notes are rather closer together, the scheme is uncannily similar to meantone. &amp;quot;Microdiatonic&amp;quot; might be a good term for it (even better than for edfs as the generator it uses is an excellent fit for heptatonic MOS) if it hasn't been named yet.&lt;/body&gt;&lt;/html&gt;</pre></div>
 
== Individual pages for ed5/3's ==
{| class="wikitable center-all"
|+ style=white-space:nowrap | 0…49
| [[0ed5/3|0]]
| [[1ed5/3|1]]
| [[2ed5/3|2]]
| [[3ed5/3|3]]
| [[4ed5/3|4]]
| [[5ed5/3|5]]
| [[6ed5/3|6]]
| [[7ed5/3|7]]
| [[8ed5/3|8]]
| [[9ed5/3|9]]
|-
| [[10ed5/3|10]]
| [[11ed5/3|11]]
| [[12ed5/3|12]]
| [[13ed5/3|13]]
| [[14ed5/3|14]]
| [[15ed5/3|15]]
| [[16ed5/3|16]]
| [[17ed5/3|17]]
| [[18ed5/3|18]]
| [[19ed5/3|19]]
|-
| [[20ed5/3|20]]
| [[21ed5/3|21]]
| [[22ed5/3|22]]
| [[23ed5/3|23]]
| [[24ed5/3|24]]
| [[25ed5/3|25]]
| [[26ed5/3|26]]
| [[27ed5/3|27]]
| [[28ed5/3|28]]
| [[29ed5/3|29]]
|-
| [[30ed5/3|30]]
| [[31ed5/3|31]]
| [[32ed5/3|32]]
| [[33ed5/3|33]]
| [[34ed5/3|34]]
| [[35ed5/3|35]]
| [[36ed5/3|36]]
| [[37ed5/3|37]]
| [[38ed5/3|38]]
| [[39ed5/3|39]]
|-
| [[40ed5/3|40]]
| [[41ed5/3|41]]
| [[42ed5/3|42]]
| [[43ed5/3|43]]
| [[44ed5/3|44]]
| [[45ed5/3|45]]
| [[46ed5/3|46]]
| [[47ed5/3|47]]
| [[48ed5/3|48]]
| [[49ed5/3|49]]
|}
 
[[Category:Ed5/3's| ]]
<!-- main article -->
[[Category:Edonoi]]
[[Category:Lists of scales]]
 
 
{{todo|inline=1|cleanup|explain edonoi|text=Most people do not think 5/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property ''other than'' equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.}}

Latest revision as of 18:40, 1 August 2025

The equal division of 5/3 (ed5/3) is a tuning obtained by dividing the just major sixth (5/3) into a number of equal steps.

Properties

Division of 5/3 into equal parts does not necessarily imply directly using this interval as an equivalence. Many, though not all, ed5/3 scales have a perceptually important false octave, with various degrees of accuracy.

5/3 is the most consonant interval in between 1/1 and 2/1, so this suggests it could be useful either as an equivalence, or as just an important structural feature.

Joseph Ruhf suggested the use of the 6:7:8:(10) chord as the fundamental complete sonority in a very similar way to the 4:5:6:(8) chord in meantone as a way to evoke 5/3-equivalence. Though it could also be used just as a useful sonority, even without equivalence. Whereas in meantone it takes four 3/2 to get to 5/1, here it takes four 4/3 to get to 8/7 (tempering out the comma 225/224). So, doing this yields 7-, 9-, and 16-note mos either way, the 16-note mos being 7L 9s. While the notes are rather closer together, the scheme is uncannily similar to meantone. "Microdiatonic" might be a good term for it (even better than for edfs as the generator it uses is an excellent fit for heptatonic mos) though it is, technically speaking, micro-armotonic.

If we instead opt to continue using 4:5:6 as the fundamental sonority, then it will take three 3/2 to get to 5/4, resulting in Blackcomb temperament that tempers out the comma 250/243. This yields mos scales of 4, 5, 6, 11, 16, and 21 notes. Although, it should be noted that doing this will often create a pseudo-octave unlike the 6:7:8 approach.

ED5/3 tuning systems that accurately represent the intervals 5/4 and 4/3 include: 7ed5/3 (7.30 cent error), 9ed5/3 (6.73 cent error), and 16ed5/3 (0.59 cent error).

7ed5/3, 9ed5/3, and 16ed5/3 are to the division of 5/3 what 5edo, 7edo, and 12edo are to the division of 2/1.

Individual pages for ed5/3's

0…49
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49


Todo: cleanup , explain edonoi

Most people do not think 5/3 sounds like an equivalence, so there must be some other reason why people are dividing it — some property other than equivalence that makes people want to divide it. Please add to this page an explanation of what that reason is... The page also needs a general overall cleanup.

Retrieved from "https://en.xen.wiki/index.php?title=Ed5/3&oldid=205841"