5edo: Difference between revisions
Wikispaces>genewardsmith **Imported revision 238825589 - Original comment: ** |
Wikispaces>genewardsmith **Imported revision 238826189 - Original comment: ** |
||
| Line 1: | Line 1: | ||
<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:genewardsmith|genewardsmith]] and made on <tt>2011-06-26 15: | : This revision was by author [[User:genewardsmith|genewardsmith]] and made on <tt>2011-06-26 15:17:45 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>238826189</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> | ||
| Line 51: | Line 51: | ||
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 | 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 consistently, 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== | ||
| Line 229: | Line 229: | ||
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 | 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 consistently, 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 7-limit 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 3-limit consistently, <a class="wiki_link" href="/3edo">3edo</a> the 5-limit, <a class="wiki_link" href="/4edo">4edo</a> the 7-limit and <a class="wiki_link" href="/5edo">5edo</a> the 9-limit, to represent the 11-limit 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> | ||