Gentle region (extended version): 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:

: This revision was by author [[User:~~xenwolf~~|~~xenwolf~~]] and made on <tt>~~2016~~-12-~~29 12~~:17:24 UTC</tt>.

: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2018-04-26 01:37:51 UTC</tt>.

: The original revision id was <tt>~~602895168~~</tt>.

: The original revision id was <tt>629084659</tt>.

: The revision comment was: <tt></tt>

The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.

<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">//This is an extended version of the **[[Gentle region]] article. **//

<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">//This is an extended version of the **[[Gentle region]] article.**//

----

[[@Margo Schulter]], in a [[@http://launch.groups.yahoo.com/group/tuning/message/38721|tuning list posting]], defined the "gentle region" of temperaments with a fifth as generator as that of fifths about 1.49 to 2.65 cents sharp; later [[@http://launch.groups.yahoo.com/group/tuning/message/105172|amending that]] to from 1.49 to 3.04 cents sharp. We can consider the first region to extend from fifths of size 17\29 to 64\109, and the extended region to reach 47\80. If we remove the restriction to tempering based on chains of fifths, we find that notable equal divisions in the smaller gentle region include multiples of [[@29edo]], [[@46edo]], [[@75edo]], [[@104edo]], [[@109edo]], [[@121edo]], [[@145edo|133edo]], [[@155edo]], [[@162edo]], [[@167edo]], [[@179edo]], [[@191edo]], [[@201edo]], [[@213edo]], [[@225edo]] and [[@237edo]], plus [[@63edo]] and [[@80edo]] in the extended region.

||||~ ~~Geneator~~ ||~ Cents ||~ 2-3-7(b)-11-13(b) ||||~ Half 8/7+ 1\3 7/61\3 8/7+ Half 7/6 ||~ 8/7+7/6 ||~ ||

||||~ Generator ||~ Cents ||~ 2-3-7(b)-11-13(b) ||~ Half 8/7+ 1\3 7/61\3 8/7+ Half 7/6 ||~ 8/7+7/6 ||~ ||

|| (7+10)\29 || ||= 289.655+413.793 || < 29 46 81 100 107| ||||= 2\29+3\29

82.759+124.138 ||= 6\29+6\29

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76.868+132.384 ||= 54\281+62\281

230.605+264.769 || ||

|| || ||= 286.101+418.533 || < 46 73 129 159 170|+< 63 100 177 218 233|φ ||||= 116.526+89.264

77.684+133.8965 ||= 233.052+267.793 || ||

|| || (41+60)\172 ||= 286.0465+418.605 || < 172 273 483 595 636| ||||= 33\344+38\516

Line 89:

114.286+88.88

4\63+7\63

~~~~76.1905+133.333~~~~ ||= 12\63+14\63

76.1905+133.333 ||= 12\63+14\63

228.571+266.667 || ||

|| || ||= 285.513+419.316 || < 46 73 129 159 170|+< 80 127 225 277 296|φ ||||= 113.7825+89.2075.855+133.80

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225+270 || Boundary of larger "gentle region" ||</pre></div>

<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>Gentle region (extended version)</title></head><body>This is an extended version of the <a class="wiki_link" href="/Gentle%20region">Gentle region</a> article.

<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>Gentle region (extended version)</title></head><body>This is an extended version of the <a class="wiki_link" href="/Gentle%20region">Gentle region</a> article.

<hr />

<a class="wiki_link" href="/Margo%20Schulter" target="_blank">Margo Schulter</a>, in a <a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/38721" rel="nofollow" target="_blank">tuning list posting</a>, defined the &quot;gentle region&quot; of temperaments with a fifth as generator as that of fifths about 1.49 to 2.65 cents sharp; later <a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/105172" rel="nofollow" target="_blank">amending that</a> to from 1.49 to 3.04 cents sharp. We can consider the first region to extend from fifths of size 17\29 to 64\109, and the extended region to reach 47\80. If we remove the restriction to tempering based on chains of fifths, we find that notable equal divisions in the smaller gentle region include multiples of <a class="wiki_link" href="/29edo" target="_blank">29edo</a>, <a class="wiki_link" href="/46edo" target="_blank">46edo</a>, <a class="wiki_link" href="/75edo" target="_blank">75edo</a>, <a class="wiki_link" href="/104edo" target="_blank">104edo</a>, <a class="wiki_link" href="/109edo" target="_blank">109edo</a>, <a class="wiki_link" href="/121edo" target="_blank">121edo</a>, <a class="wiki_link" href="/145edo" target="_blank">133edo</a>, <a class="wiki_link" href="/155edo" target="_blank">155edo</a>, <a class="wiki_link" href="/162edo" target="_blank">162edo</a>, <a class="wiki_link" href="/167edo" target="_blank">167edo</a>, <a class="wiki_link" href="/179edo" target="_blank">179edo</a>, <a class="wiki_link" href="/191edo" target="_blank">191edo</a>, <a class="wiki_link" href="/201edo" target="_blank">201edo</a>, <a class="wiki_link" href="/213edo" target="_blank">213edo</a>, <a class="wiki_link" href="/225edo" target="_blank">225edo</a> and <a class="wiki_link" href="/237edo" target="_blank">237edo</a>, plus <a class="wiki_link" href="/63edo" target="_blank">63edo</a> and <a class="wiki_link" href="/80edo" target="_blank">80edo</a> in the extended region.

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<tr>

<th colspan="2">~~Geneator~~

<th colspan="2">Generator

</th>

<th>Cents

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<th>2-3-7(b)-11-13(b)

</th>

<th ~~colspan="2"~~>Half 8/7+ 1\3 7/61\3 8/7+ Half 7/6

<th>Half 8/7+ 1\3 7/61\3 8/7+ Half 7/6

</th>

<th>8/7+7/6

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<td>&lt; 46 73 129 159 170|+&lt; 63 100 177 218 233|φ

</td>

<td colspan="2" style="text-align: center;">116.526+89.264

<td colspan="2" style="text-align: center;">116.526+89.264

77.684+133.8965

</td>

Line 509:

114.286+88.88

4\63+7\63

~~~~76.1905+133.333~~~~

76.1905+133.333

</td>

<td style="text-align: center;">12\63+14\63

@@ 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:xenwolf|xenwolf]] and made on <tt>2016-12-29 12:17:24 UTC</tt>.<br>
+: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2018-04-26 01:37:51 UTC</tt>.<br>
-: The original revision id was <tt>602895168</tt>.<br>
+: The original revision id was <tt>629084659</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">//This is an extended version of the **[[Gentle region]] article. **//
+<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">//This is an extended version of the **[[Gentle region]] article.**//
 ----
 [[@Margo Schulter]], in a [[@http://launch.groups.yahoo.com/group/tuning/message/38721|tuning list posting]], defined the "gentle region" of temperaments with a fifth as generator as that of fifths about 1.49 to 2.65 cents sharp; later [[@http://launch.groups.yahoo.com/group/tuning/message/105172|amending that]] to from 1.49 to 3.04 cents sharp. We can consider the first region to extend from fifths of size 17\29 to 64\109, and the extended region to reach 47\80. If we remove the restriction to tempering based on chains of fifths, we find that notable equal divisions in the smaller gentle region include multiples of [[@29edo]], [[@46edo]], [[@75edo]], [[@104edo]], [[@109edo]], [[@121edo]], [[@145edo|133edo]], [[@155edo]], [[@162edo]], [[@167edo]], [[@179edo]], [[@191edo]], [[@201edo]], [[@213edo]], [[@225edo]] and [[@237edo]], plus [[@63edo]] and [[@80edo]] in the extended region.
-||||~ Geneator ||~ Cents ||~ 2-3-7(b)-11-13(b) ||||~ &lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;Half 8/7+ 1\3 7/6&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;1\3 8/7+ Half 7/6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; ||~ 8/7+7/6 ||~   ||
+||||~ Generator ||~ Cents ||~ 2-3-7(b)-11-13(b) ||~ &lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;Half 8/7+ 1\3 7/6&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;1\3 8/7+ Half 7/6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt; ||~ 8/7+7/6 ||~   ||
 || (7+10)\29 ||   ||= 289.655+413.793 || &lt; 29 46 81 100 107| ||||= 2\29+&lt;span style="line-height: 1.5;"&gt;3\29&lt;/span&gt;
 .759+124.138 ||= 6\29+6\29
@@ Line 72: / Line 72: @@
 &lt;span style="background-color: rgba(255,255,255,0);"&gt;76.868+132.384&lt;/span&gt; ||= 54\281+62\281
 .605+264.769 ||   ||
 ||   ||   ||= 286.101+418.533 || &lt; 46 73 129 159 170|+&lt; 63 100 177 218 233|&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"&gt;φ&lt;/span&gt; ||||= 116.526+89.264
 &lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;77.684+133.8965&lt;/span&gt; ||= 233.052+267.793 ||   ||
 ||   || (41+60)\172 ||= 286.0465+418.605 || &lt; 172 273 483 595 636| ||||= 33\344+38\516
@@ Line 89: / Line 89: @@
 .286+88.88
 &lt;span style="background-color: rgba(255,255,255,0);"&gt;4\63+7\63&lt;/span&gt;
-&lt;span style="text-align: start;"&gt;&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.1905+133.333&lt;/span&gt;&lt;/span&gt; ||= 12\63+14\63
+&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;76.1905+133.333&lt;/span&gt; ||= 12\63+14\63
 .571+266.667 ||   ||
 ||   ||   ||= 285.513+419.316 || &lt; 46 73 129 159 170|+&lt; 80 127 225 277 296|&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"&gt;φ&lt;/span&gt; ||||= &lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;113.7825+89.20&lt;/span&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;&lt;span style="background-color: rgba(255,255,255,0);"&gt;75.855+133.80&lt;/span&gt;&lt;/span&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;
@@ Line 116: / Line 116: @@
 +270 || Boundary of larger "gentle region" ||</pre></div>
 <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;Gentle region (extended version)&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;This is an extended version of the &lt;strong&gt;&lt;a class="wiki_link" href="/Gentle%20region"&gt;Gentle region&lt;/a&gt; article. &lt;/strong&gt;&lt;/em&gt;&lt;br /&gt;
+<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;Gentle region (extended version)&lt;/title&gt;&lt;/head&gt;&lt;body&gt;&lt;em&gt;This is an extended version of the &lt;strong&gt;&lt;a class="wiki_link" href="/Gentle%20region"&gt;Gentle region&lt;/a&gt; article.&lt;/strong&gt;&lt;/em&gt; &lt;br /&gt;
 &lt;hr /&gt;
 &lt;a class="wiki_link" href="/Margo%20Schulter" target="_blank"&gt;Margo Schulter&lt;/a&gt;, in a &lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/38721" rel="nofollow" target="_blank"&gt;tuning list posting&lt;/a&gt;, defined the &amp;quot;gentle region&amp;quot; of temperaments with a fifth as generator as that of fifths about 1.49 to 2.65 cents sharp; later &lt;a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/105172" rel="nofollow" target="_blank"&gt;amending that&lt;/a&gt; to from 1.49 to 3.04 cents sharp. We can consider the first region to extend from fifths of size 17\29 to 64\109, and the extended region to reach 47\80. If we remove the restriction to tempering based on chains of fifths, we find that notable equal divisions in the smaller gentle region include multiples of &lt;a class="wiki_link" href="/29edo" target="_blank"&gt;29edo&lt;/a&gt;, &lt;a class="wiki_link" href="/46edo" target="_blank"&gt;46edo&lt;/a&gt;, &lt;a class="wiki_link" href="/75edo" target="_blank"&gt;75edo&lt;/a&gt;, &lt;a class="wiki_link" href="/104edo" target="_blank"&gt;104edo&lt;/a&gt;, &lt;a class="wiki_link" href="/109edo" target="_blank"&gt;109edo&lt;/a&gt;, &lt;a class="wiki_link" href="/121edo" target="_blank"&gt;121edo&lt;/a&gt;, &lt;a class="wiki_link" href="/145edo" target="_blank"&gt;133edo&lt;/a&gt;, &lt;a class="wiki_link" href="/155edo" target="_blank"&gt;155edo&lt;/a&gt;, &lt;a class="wiki_link" href="/162edo" target="_blank"&gt;162edo&lt;/a&gt;, &lt;a class="wiki_link" href="/167edo" target="_blank"&gt;167edo&lt;/a&gt;, &lt;a class="wiki_link" href="/179edo" target="_blank"&gt;179edo&lt;/a&gt;, &lt;a class="wiki_link" href="/191edo" target="_blank"&gt;191edo&lt;/a&gt;, &lt;a class="wiki_link" href="/201edo" target="_blank"&gt;201edo&lt;/a&gt;, &lt;a class="wiki_link" href="/213edo" target="_blank"&gt;213edo&lt;/a&gt;, &lt;a class="wiki_link" href="/225edo" target="_blank"&gt;225edo&lt;/a&gt; and &lt;a class="wiki_link" href="/237edo" target="_blank"&gt;237edo&lt;/a&gt;, plus &lt;a class="wiki_link" href="/63edo" target="_blank"&gt;63edo&lt;/a&gt; and &lt;a class="wiki_link" href="/80edo" target="_blank"&gt;80edo&lt;/a&gt; in the extended region.&lt;br /&gt;
@@ Line 124: / Line 124: @@
 &lt;table class="wiki_table"&gt;
      &lt;tr&gt;
-         &lt;th colspan="2"&gt;Geneator&lt;br /&gt;
+         &lt;th colspan="2"&gt;Generator&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;Cents&lt;br /&gt;
@@ Line 130: / Line 130: @@
          &lt;th&gt;2-3-7(b)-11-13(b)&lt;br /&gt;
 &lt;/th&gt;
-         &lt;th colspan="2"&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;Half 8/7+ 1\3 7/6&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;1\3 8/7+ Half 7/6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+         &lt;th&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;Half 8/7+ 1\3 7/6&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;1\3 8/7+ Half 7/6&lt;/span&gt;&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
 &lt;/th&gt;
          &lt;th&gt;8/7+7/6&lt;br /&gt;
@@ Line 432: / Line 432: @@
          &lt;td&gt;&amp;lt; 46 73 129 159 170|+&amp;lt; 63 100 177 218 233|&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"&gt;φ&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
-         &lt;td colspan="2" style="text-align: center;"&gt;116.526+89.264 &lt;br /&gt;
+         &lt;td colspan="2" style="text-align: center;"&gt;116.526+89.264&lt;br /&gt;
 &lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;77.684+133.8965&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
@@ Line 509: / Line 509: @@
 .286+88.88&lt;br /&gt;
 &lt;span style="background-color: rgba(255,255,255,0);"&gt;4\63+7\63&lt;/span&gt;&lt;br /&gt;
-&lt;span style="text-align: start;"&gt;&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.1905+133.333&lt;/span&gt;&lt;/span&gt;&lt;br /&gt;
+&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;76.1905+133.333&lt;/span&gt;&lt;br /&gt;
 &lt;/td&gt;
          &lt;td style="text-align: center;"&gt;12\63+14\63&lt;br /&gt;