17edt: Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>

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>~~2012~~-05-~~26 21~~:57:16 UTC</tt>.<br>

: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-13 12:00:10 UTC</tt>.<br>

: The original revision id was <tt>~~339787378~~</tt>.<br>

: The original revision id was <tt>595261250</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>

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=Discussion=

17edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a ~~very~~ good approximation of 11/9, which is in fact the size of the large step.

17edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents),

leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to a 16/5 which is only .3 cents flat.

=Intervals=

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* Notes named so that C D E F G H J A B C = Lambda mode

It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar.

It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).

=Z function=

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<br />

<h1 id="toc1"><a name="Discussion"></a>Discussion</h1>

17edt is closely related to <a class="wiki_link" href="/13edt">13edt</a>, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a ~~very~~ good approximation of 11/9, which is in fact the size of the large step.<br />

17edt is closely related to <a class="wiki_link" href="/13edt">13edt</a>, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), <br />

leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to a 16/5 which is only .3 cents flat.<br />

<br />

<h1 id="toc2"><a name="Intervals"></a>Intervals</h1>

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<br />

<ul><li>Notes named so that C D E F G H J A B C = Lambda mode</li></ul>It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar.<br />

<ul><li>Notes named so that C D E F G H J A B C = Lambda mode</li></ul>It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).<br />

<br />

<h1 id="toc3"><a name="Z function"></a>Z function</h1>

@@ Line 1: / Line 1: @@
 <h2>IMPORTED REVISION FROM WIKISPACES</h2>
 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>2012-05-26 21:57:16 UTC</tt>.<br>
+: This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2016-10-13 12:00:10 UTC</tt>.<br>
-: The original revision id was <tt>339787378</tt>.<br>
+: The original revision id was <tt>595261250</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>
@@ Line 13: / Line 13: @@
 =Discussion=
-edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step.
+edt is closely related to [[13edt]], the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents),
+leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to a 16/5 which is only .3 cents flat.
 =Intervals=
@@ Line 54: / Line 55: @@
 * Notes named so that C D E F G H J A B C = Lambda mode
-It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar.
+It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).
 =Z function=
@@ Line 68: / Line 69: @@
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:2:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc1"&gt;&lt;a name="Discussion"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:2 --&gt;Discussion&lt;/h1&gt;
-edt is closely related to &lt;a class="wiki_link" href="/13edt"&gt;13edt&lt;/a&gt;, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly, in return for a very good approximation of 11/9, which is in fact the size of the large step.&lt;br /&gt;
+edt is closely related to &lt;a class="wiki_link" href="/13edt"&gt;13edt&lt;/a&gt;, the Bohlen-Pierce division, because they share the feature of tempering out 245/243. Both 13edt and 17edt have 4L+5s nonatonic modes, but whereas the ratio of large to small steps in 13edt is a calm 2:1, in 17edt it is a hard 3:1. Thus, the approximation of 5/3 and 7/3 suffers slightly in return for gaining a good approximation of 11/9 (given the context of the weak 5/3 and 7/3), which is in fact the size of the large step. However, by the coincidence of the 11-limit commas 17edt tempers out, 5/3 and 11/9 are off by practically the same amount in opposite directions (+10.7 cents and -11.8 cents), leading to an excellent approximation of 55/27 (only 1.1 cents flat), as are 11/9 and 9/7 (-11.8 cents and +12.4 cents), &lt;br /&gt;
+leading to an excellent approximation of 11/7 (only .6 cents flat) and these sum to a 16/5 which is only .3 cents flat.&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:4:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc2"&gt;&lt;a name="Intervals"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:4 --&gt;Intervals&lt;/h1&gt;
@@ Line 437: / Line 439: @@
 &lt;br /&gt;
-&lt;ul&gt;&lt;li&gt;Notes named so that C D E F G H J A B C = Lambda mode&lt;/li&gt;&lt;/ul&gt;It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar.&lt;br /&gt;
+&lt;ul&gt;&lt;li&gt;Notes named so that C D E F G H J A B C = Lambda mode&lt;/li&gt;&lt;/ul&gt;It's a weird coincidence how the schemes of 17edo and 17edt diatonicism are so similar and how their approximations of 9/7 are off by such similar amounts in opposite directions (17edo -11.6 cents and 17edt +12.4 cents).&lt;br /&gt;
 &lt;br /&gt;
 &lt;!-- ws:start:WikiTextHeadingRule:6:&amp;lt;h1&amp;gt; --&gt;&lt;h1 id="toc3"&gt;&lt;a name="Z function"&gt;&lt;/a&gt;&lt;!-- ws:end:WikiTextHeadingRule:6 --&gt;Z function&lt;/h1&gt;