5edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 238826189 - Original comment: ** |
Wikispaces>hstraub **Imported revision 239087229 - Original comment: ** |
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<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:hstraub|hstraub]] and made on <tt>2011-06-28 03:00:10 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>239087229</tt>.<br> | ||
: The revision comment was: <tt></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> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
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Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence. | Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain [[Bug family|bug temperament]], which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence. | ||
Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the 9-limit | Despite its lack of accuracy, 5EDO is the second [[The Riemann Zeta Function and Tuning#Zeta EDO lists|zeta integral edo]], after 2EDO. It also is the smallest equal division representing the [[9-limit]] [[consistent]]ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how [[4edo]] can be used, and which is discussed in that article, it can be used to represent [[7-limit]] intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the [[The Seven Limit Symmetrical Lattices|lattice]] of tetrads/pentads together with the number of scale steps in 5EDO. However, while [[2edo]] represents the [[3-limit]] consistently, [[3edo]] the [[5-limit]], [[4edo]] the [[7-limit]] and [[5edo]] the [[9-limit]], to represent the [[11-limit]] consistently with a [[patent val]] requires going all the way to [[22edo]]. | ||
==Cycles, Divisions== | ==Cycles, Divisions== | ||
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||~ Comma ||~ Value (cents) ||~ Name ||~ Second Name ||~ Third Name ||~ Val || | ||~ Comma ||~ Value (cents) ||~ Name ||~ Second Name ||~ Third Name ||~ Val || | ||
||= 256/243 ||> 90.225 || Limma || Pythagorean Minor 2nd || || | 8 -5 > || | ||= 256/243 ||> 90.225 || Limma || Pythagorean Minor 2nd || || | 8 -5 > || | ||
||= 81/80 ||> 21.506 || Syntonic Comma || Didymos Comma || Meantone Comma || | -4 4 -1 > || | ||= 81/80 ||> 21.506 || Syntonic Comma || Didymos Comma || Meantone Comma || | -4 4 -1 > || | ||
||= 2889416/2882415 ||> 4.200 || Vulture || || || | 24 -21 4 > || | ||= 2889416/2882415 ||> 4.200 || Vulture || || || | 24 -21 4 > || | ||
||= 36/35 ||> 48.770 || Septimal Quarter Tone || || || | 2 2 -1 -1 > || | ||= 36/35 ||> 48.770 || Septimal Quarter Tone || || || | 2 2 -1 -1 > || | ||
||= 49/48 ||> 35.697 || Slendro Diesis || || || | -4 -1 0 2 > || | ||= 49/48 ||> 35.697 || Slendro Diesis || || || | -4 -1 0 2 > || | ||
||= 64/63 ||> 27.264 || Septimal Comma || Archytas' Comma || Leipziger Komma || | 6 -2 0 -1 > || | ||= 64/63 ||> 27.264 || Septimal Comma || Archytas' Comma || Leipziger Komma || | 6 -2 0 -1 > || | ||
||= 245/243 ||> 14.191 || Sensamagic || || || | 0 -5 1 2 > || | ||= 245/243 ||> 14.191 || Sensamagic || || || | 0 -5 1 2 > || | ||
||= 1728/1715 ||> 13.074 || Orwellisma || Orwell Comma || || | 6 3 -1 -3 > || | ||= 1728/1715 ||> 13.074 || Orwellisma || Orwell Comma || || | 6 3 -1 -3 > || | ||
||= 1029/1024 ||> 8.433 || Gamelisma || || || | -10 1 0 3 > || | ||= 1029/1024 ||> 8.433 || Gamelisma || || || | -10 1 0 3 > || | ||
||= 19683/19600 ||> 7.316 || Cataharry || || || | -4 9 -2 -2 > || | ||= 19683/19600 ||> 7.316 || Cataharry || || || | -4 9 -2 -2 > || | ||
||= 5120/5103 ||> 5.758 || Hemifamity || || || | 10 -6 1 -1 > || | ||= 5120/5103 ||> 5.758 || Hemifamity || || || | 10 -6 1 -1 > || | ||
||= 1065875/1063543 ||> 3.792 || Wadisma || || || | -26 -1 1 9 > || | ||= 1065875/1063543 ||> 3.792 || Wadisma || || || | -26 -1 1 9 > || | ||
||= 420175/419904 ||> 1.117 || Wizma || || || | -6 -8 2 5 > || | ||= 420175/419904 ||> 1.117 || Wizma || || || | -6 -8 2 5 > || | ||
||= 99/98 ||> 17.576 || Mothwellsma || || || | -1 2 0 -2 1 > || | ||= 99/98 ||> 17.576 || Mothwellsma || || || | -1 2 0 -2 1 > || | ||
||= 896/891 ||> 9.688 || Pentacircle || || || | 7 -4 0 1 -1 > || | ||= 896/891 ||> 9.688 || Pentacircle || || || | 7 -4 0 1 -1 > || | ||
||= 385/384 ||> 4.503 || Keenanisma || || || | -7 -1 1 1 1 > || | ||= 385/384 ||> 4.503 || Keenanisma || || || | -7 -1 1 1 1 > || | ||
||= 441/440 ||> 3.930 || Werckisma || || || | -3 2 -1 2 -1 > || | ||= 441/440 ||> 3.930 || Werckisma || || || | -3 2 -1 2 -1 > || | ||
||= 3025/3024 ||> 0.572 || Lehmerisma || || || | -4 -3 2 -1 2 > || | ||= 3025/3024 ||> 0.572 || Lehmerisma || || || | -4 -3 2 -1 2 > || | ||
||= 91/90 ||> 19.130 || Superleap || || || | -1 -2 -1 1 0 1 > || | ||= 91/90 ||> 19.130 || Superleap || || || | -1 -2 -1 1 0 1 > || | ||
||= 676/675 ||> 2.563 || Parizeksma || || || | 2 -3 -2 0 0 2 > || || | ||= 676/675 ||> 2.563 || Parizeksma || || || | 2 -3 -2 0 0 2 > || ||</pre></div> | ||
</pre></div> | |||
<h4>Original HTML content:</h4> | <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>5edo</title></head><body><!-- ws:start:WikiTextTocRule:26:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | <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>5edo</title></head><body><!-- ws:start:WikiTextTocRule:26:&lt;img id=&quot;wikitext@@toc@@flat&quot; class=&quot;WikiMedia WikiMediaTocFlat&quot; title=&quot;Table of Contents&quot; src=&quot;/site/embedthumbnail/toc/flat?w=100&amp;h=16&quot;/&gt; --><!-- ws:end:WikiTextTocRule:26 --><!-- ws:start:WikiTextTocRule:27: --><a href="#x5 Equal Divisions of the Octave: Theory">5 Equal Divisions of the Octave: Theory</a><!-- ws:end:WikiTextTocRule:27 --><!-- ws:start:WikiTextTocRule:28: --><!-- ws:end:WikiTextTocRule:28 --><!-- ws:start:WikiTextTocRule:29: --><!-- ws:end:WikiTextTocRule:29 --><!-- ws:start:WikiTextTocRule:30: --><!-- ws:end:WikiTextTocRule:30 --><!-- ws:start:WikiTextTocRule:31: --><!-- ws:end:WikiTextTocRule:31 --><!-- ws:start:WikiTextTocRule:32: --><!-- ws:end:WikiTextTocRule:32 --><!-- ws:start:WikiTextTocRule:33: --> | <a href="#x5-edo in Musicmaking">5-edo in Musicmaking</a><!-- ws:end:WikiTextTocRule:33 --><!-- ws:start:WikiTextTocRule:34: --><!-- ws:end:WikiTextTocRule:34 --><!-- ws:start:WikiTextTocRule:35: --><!-- ws:end:WikiTextTocRule:35 --><!-- ws:start:WikiTextTocRule:36: --><!-- ws:end:WikiTextTocRule:36 --><!-- ws:start:WikiTextTocRule:37: --><!-- ws:end:WikiTextTocRule:37 --><!-- ws:start:WikiTextTocRule:38: --><!-- ws:end:WikiTextTocRule:38 --><!-- ws:start:WikiTextTocRule:39: --><!-- ws:end:WikiTextTocRule:39 --><!-- ws:start:WikiTextTocRule:40: --> | ||
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Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain <a class="wiki_link" href="/Bug%20family">bug temperament</a>, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.<br /> | Also tempered out is 27/25, if we temper this out in preference to 16/15 we obtain <a class="wiki_link" href="/Bug%20family">bug temperament</a>, which equates 10/9 with 6/5: it is a little more perverse even than father. Because these intervals are so large, this sort of analysis is less significant with 5 than it becomes with larger and more accurate divisions, but it still plays a role. For example, I-IV-V-I is the same as 1-III-V-I and involves triads with common intervals because of fourth-thirds equivalence.<br /> | ||
<br /> | <br /> | ||
Despite its lack of accuracy, 5EDO is the second <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">zeta integral edo</a>, after 2EDO. It also is the smallest equal division representing the 9-limit | Despite its lack of accuracy, 5EDO is the second <a class="wiki_link" href="/The%20Riemann%20Zeta%20Function%20and%20Tuning#Zeta EDO lists">zeta integral edo</a>, after 2EDO. It also is the smallest equal division representing the <a class="wiki_link" href="/9-limit">9-limit</a> <a class="wiki_link" href="/consistent">consistent</a>ly, giving a distinct value modulo five to 2, 3, 5, 7 and 9. Hence in a way similar to how <a class="wiki_link" href="/4edo">4edo</a> can be used, and which is discussed in that article, it can be used to represent <a class="wiki_link" href="/7-limit">7-limit</a> intervals in terms of their position in a pentad, by giving a triple of integers representing a pentad in the <a class="wiki_link" href="/The%20Seven%20Limit%20Symmetrical%20Lattices">lattice</a> of tetrads/pentads together with the number of scale steps in 5EDO. However, while <a class="wiki_link" href="/2edo">2edo</a> represents the <a class="wiki_link" href="/3-limit">3-limit</a> consistently, <a class="wiki_link" href="/3edo">3edo</a> the <a class="wiki_link" href="/5-limit">5-limit</a>, <a class="wiki_link" href="/4edo">4edo</a> the <a class="wiki_link" href="/7-limit">7-limit</a> and <a class="wiki_link" href="/5edo">5edo</a> the <a class="wiki_link" href="/9-limit">9-limit</a>, to represent the <a class="wiki_link" href="/11-limit">11-limit</a> consistently with a <a class="wiki_link" href="/patent%20val">patent val</a> requires going all the way to <a class="wiki_link" href="/22edo">22edo</a>.<br /> | ||
<br /> | <br /> | ||
<!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"></a><!-- ws:end:WikiTextHeadingRule:10 -->Cycles, Divisions</h2> | <!-- ws:start:WikiTextHeadingRule:10:&lt;h2&gt; --><h2 id="toc5"><a name="x5 Equal Divisions of the Octave: Theory-Cycles, Divisions"></a><!-- ws:end:WikiTextHeadingRule:10 -->Cycles, Divisions</h2> | ||