Gentle region (extended version): Difference between revisions

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<h2>IMPORTED REVISION FROM WIKISPACES</h2>
''This is an extended version of the '''[[Gentle_region|Gentle region]] article.'''''
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
 
: This revision was by author [[User:mbattaglia1|mbattaglia1]] and made on <tt>2018-04-26 01:40:17 UTC</tt>.<br>
-----
: The original revision id was <tt>629084697</tt>.<br>
[[Margo_Schulter|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|29edo]], [[46edo|46edo]], [[75edo|75edo]], [[104edo|104edo]], [[109edo|109edo]], [[121edo|121edo]], [[145edo|133edo]], [[155edo|155edo]], [[162edo|162edo]], [[167edo|167edo]], [[179edo|179edo]], [[191edo|191edo]], [[201edo|201edo]], [[213edo|213edo]], [[225edo|225edo]] and [[237edo|237edo]], plus [[63edo|63edo]] and [[80edo|80edo]] in the extended region.
: 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>
{| class="wikitable"
<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.**//
! colspan="2" | Generator
----
! | Cents
[[@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.
! | 2-3-7(b)-11-13(b)
! colspan="2" | <span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"><span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">Half 8/7+ 1\3 7/6<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">1\3 8/7+ Half 7/6</span></span></span>
! | 8/7+7/6
! |
|-
| | (7+10)\29
| |
| style="text-align:center;" | 289.655+413.793
| | &lt; 29 46 81 100 107|
| colspan="2" style="text-align:center;" | 2\29+<span style="line-height: 1.5;">3\29</span>
 
82.759+124.138
| style="text-align:center;" | 6\29+6\29
 
248.276+248.276
| |
|-
| | (25+36)\104
| |
| style="text-align:center;" | 288.4615+415.385
| | &lt; 104 165 292 360 385|
| colspan="2" style="text-align:center;" | 10\104+23\312


||||~ 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;
82.759+124.138 ||= 6\29+6\29
248.276+248.276 ||  ||
|| (25+36)\104 ||  ||= 288.4615+415.385 || &lt; 104 165 292 360 385| ||||= 10\104+23\312
115.385+88.4615
115.385+88.4615
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;20\312+23\208&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;76.923+132.692&lt;/span&gt; ||= 20\104+23\104
<span style="background-color: rgba(255,255,255,0); text-align: start;">20\312+23\208</span>
230.769+265.385 ||   ||
 
|| (18+26)\75 ||   ||= 288+416 || &lt; 7&lt;span style="line-height: 1.5;"&gt;5 119 210~211 259 277|&lt;/span&gt; ||||= 7\75+17\225
<span style="background-color: rgba(255,255,255,0); text-align: start;">76.923+132.692</span>
| style="text-align:center;" | 20\104+23\104
 
230.769+265.385
| |  
|-
| | (18+26)\75
| |  
| style="text-align:center;" | 288+416
| | &lt; 7<span style="line-height: 1.5;">5 119 210~211 259 277|</span>
| colspan="2" style="text-align:center;" | 7\75+17\225
 
112+90.667
112+90.667
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;14\225+17\150&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;74.667+136&lt;/span&gt; ||= 14\75+17\75
<span style="background-color: rgba(255,255,255,0); text-align: start;">14\225+17\150</span>
224+272 ||   ||
 
||   || (47+68)\196 ||= 287.755+416.3265 || &lt;&lt;span style="line-height: 1.5;"&gt; 196 311 549-551 678 725|&lt;/span&gt; ||||= 37\392+44\588
<span style="background-color: rgba(255,255,255,0); text-align: start;">74.667+136</span>
| style="text-align:center;" | 14\75+17\75
 
224+272
| |  
|-
| |  
| | (47+68)\196
| style="text-align:center;" | 287.755+416.3265
| | &lt;<span style="line-height: 1.5;"> 196 311 549-551 678 725|</span>
| colspan="2" style="text-align:center;" | 37\392+44\588
 
113.265+89.796
113.265+89.796
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;37/588+22/196&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;75.51+134.694&lt;/span&gt; ||= 37/196+44/196
<span style="background-color: rgba(255,255,255,0); text-align: start;">37/588+22/196</span>
226.531+269.388 ||   ||
 
||   ||   ||= 287.713+416.382 || &lt; 29 46 81 100 107|+&lt; &lt;span style="line-height: 1.5;"&gt;46 73 129 159 170|&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small; line-height: 1.5;"&gt;φ&lt;/span&gt; ||||= 119.283+85.7795
<span style="background-color: rgba(255,255,255,0); text-align: start;">75.51+134.694</span>
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;79.522+128.769&lt;/span&gt; ||= 238.566+257.3385 ||   ||
| style="text-align:center;" | 37/196+44/196
||   || (29+42)\121 ||= 287.603+416.529 || &lt; 121 192 339~340 419 448| ||||= 23\242+9\121
 
226.531+269.388
| |  
|-
| |  
| |  
| style="text-align:center;" | 287.713+416.382
| | &lt; 29 46 81 100 107|+&lt; <span style="line-height: 1.5;">46 73 129 159 170|</span><span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small; line-height: 1.5;">φ</span>
| colspan="2" style="text-align:center;" | 119.283+85.7795
 
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">79.522+128.769</span>
| style="text-align:center;" | 238.566+257.3385
| |  
|-
| |  
| | (29+42)\121
| style="text-align:center;" | 287.603+416.529
| | &lt; 121 192 339~340 419 448|
| colspan="2" style="text-align:center;" | 23\242+9\121
 
114.05+89.256
114.05+89.256
&lt;span style="background-color: rgba(255,255,255,0);"&gt;23\363+27\242&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.033+133.884&lt;/span&gt; ||= 23\121+27\121
<span style="background-color: rgba(255,255,255,0);">23\363+27\242</span>
228.099+267.769 ||   ||
 
||   ||   ||= 287.267+416.978 || &lt; 29 46 81 100 107|+&lt; 109 173 306 377 403|&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"&gt;φ&lt;/span&gt; ||||= 116.8205+87.323
<span style="background-color: rgba(255,255,255,0);">76.033+133.884</span>
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;78.617+130.984&lt;/span&gt; ||= 235.85+261.969 ||   ||
| style="text-align:center;" | 23\121+27\121
|| (11+16)\46 ||   ||= 286.9565+417.391 || &lt; 46 73 129 159 170| ||||= 9\92+10\138
 
228.099+267.769
| |  
|-
| |  
| |  
| style="text-align:center;" | 287.267+416.978
| | &lt; 29 46 81 100 107|+&lt; 109 173 306 377 403|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>
| colspan="2" style="text-align:center;" | 116.8205+87.323
 
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">78.617+130.984</span>
| style="text-align:center;" | 235.85+261.969
| |  
|-
| | (11+16)\46
| |  
| style="text-align:center;" | 286.9565+417.391
| | &lt; 46 73 129 159 170|
| colspan="2" style="text-align:center;" | 9\92+10\138
 
117.391+86.9565
117.391+86.9565
&lt;span style="background-color: rgba(255,255,255,0);"&gt;3\46+5\46&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;78.261+130.435&lt;/span&gt; ||= 9\46+10\46
<span style="background-color: rgba(255,255,255,0);">3\46+5\46</span>
234.783+260.87 ||   ||
 
||   ||   ||= 286.587+417.884 || &lt; 29 46 81 100 107|+&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; ||||= 117.925+88.626
<span style="background-color: rgba(255,255,255,0);">78.261+130.435</span>
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;77.635+132.9395&lt;/span&gt; ||= 232.936+265.879 ||   ||
| style="text-align:center;" | 9\46+10\46
||   || (48+70)\201 ||= 286.567+417.91 || &lt; 201 319 564 695 703| ||||= 39\402+44\603
 
234.783+260.87
| |  
|-
| |  
| |  
| style="text-align:center;" | 286.587+417.884
| | &lt; 29 46 81 100 107|+&lt;63 100 177 218 233|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>
| colspan="2" style="text-align:center;" | 117.925+88.626
 
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.635+132.9395</span>
| style="text-align:center;" | 232.936+265.879
| |  
|-
| |  
| | (48+70)\201
| style="text-align:center;" | 286.567+417.91
| | &lt; 201 319 564 695 703|
| colspan="2" style="text-align:center;" | 39\402+44\603
 
116.418+87.56
116.418+87.56
&lt;span style="background-color: rgba(255,255,255,0);"&gt;13\201+22\201&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.612+131.343&lt;/span&gt; ||= 39\201+44\201
<span style="background-color: rgba(255,255,255,0);">13\201+22\201</span>
232.836+262.687 ||   ||
 
||   || (37+54)\155 ||= 286.452+418.0645 || &lt; 155 246 435 536 573| ||||= 15\155+34\465
<span style="background-color: rgba(255,255,255,0);">77.612+131.343</span>
| style="text-align:center;" | 39\201+44\201
 
232.836+262.687
| |  
|-
| |  
| | (37+54)\155
| style="text-align:center;" | 286.452+418.0645
| | &lt; 155 246 435 536 573|
| colspan="2" style="text-align:center;" | 15\155+34\465
 
116.129+87.742
116.129+87.742
&lt;span style="background-color: rgba(255,255,255,0);"&gt;10\155+17\15&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.419+131.613&lt;/span&gt; ||= 30\155+34\155
<span style="background-color: rgba(255,255,255,0);">10\155+17\15</span>
232.258+263.226 ||   ||
 
||   ||   ||= 286.387+418.151 || &lt; 46 73 129 159 170|+&lt; 109 173 306 377 403|&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;"&gt;φ&lt;/span&gt; ||||= 115.968+87.842
<span style="background-color: rgba(255,255,255,0);">77.419+131.613</span>
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;77.312+131.7365&lt;/span&gt; ||= 231.935+263.527 ||   ||
| style="text-align:center;" | 30\155+34\155
||   || (63+92)\264 ||= 286.364+418.182 || &lt; 264 419 741 913 976| ||||= 51\528+58\792
 
232.258+263.226
| |  
|-
| |  
| |  
| style="text-align:center;" | 286.387+418.151
| | &lt; 46 73 129 159 170|+&lt; 109 173 306 377 403|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>
| colspan="2" style="text-align:center;" | 115.968+87.842
 
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.312+131.7365</span>
| style="text-align:center;" | 231.935+263.527
| |  
|-
| |  
| | (63+92)\264
| style="text-align:center;" | 286.364+418.182
| | &lt; 264 419 741 913 976|
| colspan="2" style="text-align:center;" | 51\528+58\792
 
115.909+87.87
115.909+87.87
&lt;span style="background-color: rgba(255,255,255,0);"&gt;17\264+29\264&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.273+131.818&lt;/span&gt; ||= 51\264+58\264
<span style="background-color: rgba(255,255,255,0);">17\264+29\264</span>
231.818+263.636 ||   ||
 
|| (26+38)\109 ||   ||= 286.2385+418.349 || &lt; 109 173 306 377 403| ||||= 21\218+8\109
<span style="background-color: rgba(255,255,255,0);">77.273+131.818</span>
| style="text-align:center;" | 51\264+58\264
 
231.818+263.636
| |  
|-
| | (26+38)\109
| |  
| style="text-align:center;" | 286.2385+418.349
| | &lt; 109 173 306 377 403|
| colspan="2" style="text-align:center;" | 21\218+8\109
 
115.596+88.07
115.596+88.07
&lt;span style="background-color: rgba(255,255,255,0);"&gt;7\109+12\10&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.064+132.11&lt;/span&gt; ||= 21\109+24\109
<span style="background-color: rgba(255,255,255,0);">7\109+12\10</span>
231.192+264.22 || Boundary of smaller "gentle region" ||
 
||   || (67+98)\281 ||= 286.121+418.505 || &lt; 281 446 789 972 1039| ||||= 27\281+62\843
<span style="background-color: rgba(255,255,255,0);">77.064+132.11</span>
| style="text-align:center;" | 21\109+24\109
 
231.192+264.22
| | Boundary of smaller "gentle region"
|-
| |  
| | (67+98)\281
| style="text-align:center;" | 286.121+418.505
| | &lt; 281 446 789 972 1039|
| colspan="2" style="text-align:center;" | 27\281+62\843
 
115.3025+88.256
115.3025+88.256
&lt;span style="background-color: rgba(255,255,255,0);"&gt;18\281+31\28&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.868+132.384&lt;/span&gt; ||= 54\281+62\281
<span style="background-color: rgba(255,255,255,0);">18\281+31\28</span>
230.605+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
<span style="background-color: rgba(255,255,255,0);">76.868+132.384</span>
&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 ||   ||
| style="text-align:center;" | 54\281+62\281
||   || (41+60)\172 ||= 286.0465+418.605 || &lt; 172 273 483 595 636| ||||= 33\344+38\516
 
230.605+264.769
| |  
|-
| |  
| |  
| style="text-align:center;" | 286.101+418.533
| | &lt; 46 73 129 159 170|+&lt; 63 100 177 218 233|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>
| colspan="2" style="text-align:center;" | 116.526+89.264
 
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.684+133.8965</span>
| style="text-align:center;" | 233.052+267.793
| |  
|-
| |  
| | (41+60)\172
| style="text-align:center;" | 286.0465+418.605
| | &lt; 172 273 483 595 636|
| colspan="2" style="text-align:center;" | 33\344+38\516
 
115.116+88.372
115.116+88.372
&lt;span style="background-color: rgba(255,255,255,0);"&gt;11\172+19\17&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.744+132.558&lt;/span&gt; ||= 33\172+38\172
<span style="background-color: rgba(255,255,255,0);">11\172+19\17</span>
230.232+265.116 ||   ||
 
||   || (56+82)\235 ||= 285.957+418.723 || &lt; 235 373 660 813 869| ||||= 45\470+52\705
<span style="background-color: rgba(255,255,255,0);">76.744+132.558</span>
| style="text-align:center;" | 33\172+38\172
 
230.232+265.116
| |  
|-
| |  
| | (56+82)\235
| style="text-align:center;" | 285.957+418.723
| | &lt; 235 373 660 813 869|
| colspan="2" style="text-align:center;" | 45\470+52\705
 
114.894+81.511
114.894+81.511
&lt;span style="background-color: rgba(255,255,255,0);"&gt;15\235+26\23&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.596+132.766&lt;/span&gt; ||= 45\235+52\235
<span style="background-color: rgba(255,255,255,0);">15\235+26\23</span>
229.787+265.532 ||   ||
 
||   ||   ||= 285.852+418.864 || &lt; &lt;span style="line-height: 1.5;"&gt;109 173 306 377 403|+&lt;/span&gt;&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; ||||= 114.963+88.4675
<span style="background-color: rgba(255,255,255,0);">76.596+132.766</span>
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;76.642+132.701&lt;/span&gt; ||= 229.926+265.402 ||   ||
| style="text-align:center;" | 45\235+52\235
|| &lt;span style="display: block; text-align: center;"&gt;(15+22)\63&lt;/span&gt; ||   ||= &lt;span style="display: block; text-align: center;"&gt;285.714+419.048&lt;/span&gt; || &lt; 63 100 177 218 233| ||||= 6\63+14\189
 
229.787+265.532
| |  
|-
| |  
| |  
| style="text-align:center;" | 285.852+418.864
| | &lt; <span style="line-height: 1.5;">109 173 306 377 403|+</span>&lt; 63 100 177 218 233|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>
| colspan="2" style="text-align:center;" | 114.963+88.4675
 
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">76.642+132.701</span>
| style="text-align:center;" | 229.926+265.402
| |  
|-
| | <span style="display: block; text-align: center;">(15+22)\63</span>
| |  
| style="text-align:center;" | <span style="display: block; text-align: center;">285.714+419.048</span>
| | &lt; 63 100 177 218 233|
| colspan="2" style="text-align:center;" | 6\63+14\189
 
114.286+88.88
114.286+88.88
&lt;span style="background-color: rgba(255,255,255,0);"&gt;4\63+7\63&lt;/span&gt;
 
&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
<span style="background-color: rgba(255,255,255,0);">4\63+7\63</span>
228.571+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;
<span style="background-color: rgba(255,255,255,0); text-align: start;">76.1905+133.333</span>
&lt;/span&gt; ||= 227.565+267.606 ||   ||
| style="text-align:center;" | 12\63+14\63
||   || &lt;span style="display: block; text-align: center;"&gt;(49+72)\206&lt;/span&gt; ||= &lt;span style="display: block; text-align: center;"&gt;285.437+419.4175&lt;/span&gt; || &lt; 206 327 578~579 713 762| ||||= 20\206+15\206
 
228.571+266.667
| |  
|-
| |  
| |  
| style="text-align:center;" | 285.513+419.316
| | &lt; 46 73 129 159 170|+&lt; 80 127 225 277 296|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>
| colspan="2" style="text-align:center;" | <span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">113.7825+89.20</span><span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"><span style="background-color: rgba(255,255,255,0);">75.855+133.80</span></span><span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">
 
</span>
| style="text-align:center;" | 227.565+267.606
| |  
|-
| |  
| | <span style="display: block; text-align: center;">(49+72)\206</span>
| style="text-align:center;" | <span style="display: block; text-align: center;">285.437+419.4175</span>
| | &lt; 206 327 578~579 713 762|
| colspan="2" style="text-align:center;" | 20\206+15\206
 
116.505+87.37
116.505+87.37
&lt;span style="background-color: rgba(255,255,255,0);"&gt;40\618+45\41&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.67+131.068&lt;/span&gt; ||= 40\206+45\206
<span style="background-color: rgba(255,255,255,0);">40\618+45\41</span>
233.01+262.136 ||   ||
 
|| &lt;span style="display: block; text-align: center;"&gt;(34+50)\143&lt;/span&gt; ||   ||= &lt;span style="display: block; text-align: center;"&gt;285.315+419.58&lt;/span&gt; || &lt; 143 227 401~402 495 529| ||||= 14\143+31\429
<span style="background-color: rgba(255,255,255,0);">77.67+131.068</span>
| style="text-align:center;" | 40\206+45\206
 
233.01+262.136
| |  
|-
| | <span style="display: block; text-align: center;">(34+50)\143</span>
| |  
| style="text-align:center;" | <span style="display: block; text-align: center;">285.315+419.58</span>
| | &lt; 143 227 401~402 495 529|
| colspan="2" style="text-align:center;" | 14\143+31\429
 
117.4825+86.71
117.4825+86.71
&lt;span style="background-color: rgba(255,255,255,0);"&gt;28\429+31\28&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;78.322+130.07&lt;/span&gt; ||= 28\143+31\143
<span style="background-color: rgba(255,255,255,0);">28\429+31\28</span>
234.965+260.14 ||   ||
 
||   ||   ||= 285.234+419.688 || &lt; 63 100 177 218 233|+&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; ||||= 113.085+89.636
<span style="background-color: rgba(255,255,255,0);">78.322+130.07</span>
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;75.39+134.454&lt;/span&gt; ||= 226.169+268.909 ||   ||
| style="text-align:center;" | 28\143+31\143
|| &lt;span style="display: block; text-align: center;"&gt;(54+78)\223&lt;/span&gt; ||   ||= &lt;span style="display: block; text-align: center;"&gt;285.202+419.731&lt;/span&gt; || &lt; 223 354 626~627 771 825| ||||= 43\446+49\669
 
234.965+260.14
| |  
|-
| |  
| |  
| style="text-align:center;" | 285.234+419.688
| | &lt; 63 100 177 218 233|+&lt; 80 127 225 277 296|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>
| colspan="2" style="text-align:center;" | 113.085+89.636
 
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">75.39+134.454</span>
| style="text-align:center;" | 226.169+268.909
| |  
|-
| | <span style="display: block; text-align: center;">(54+78)\223</span>
| |  
| style="text-align:center;" | <span style="display: block; text-align: center;">285.202+419.731</span>
| | &lt; 223 354 626~627 771 825|
| colspan="2" style="text-align:center;" | 43\446+49\669
 
115.695+87.892
115.695+87.892
&lt;span style="background-color: rgba(255,255,255,0);"&gt;43\669+49\44&lt;/span&gt;
 
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.13+131.839&lt;/span&gt; ||= 43\223+49\223
<span style="background-color: rgba(255,255,255,0);">43\669+49\44</span>
231.39+263.677 ||   ||
 
|| &lt;span style="display: block; text-align: center;"&gt;(19+28)\80&lt;/span&gt; ||   ||= &lt;span style="display: block; text-align: center;"&gt;285+420&lt;/span&gt; || &lt; 80 127 225 277 296| ||||= 15\160+6\80
<span style="background-color: rgba(255,255,255,0);">77.13+131.839</span>
| style="text-align:center;" | 43\223+49\223
 
231.39+263.677
| |  
|-
| | <span style="display: block; text-align: center;">(19+28)\80</span>
| |  
| style="text-align:center;" | <span style="display: block; text-align: center;">285+420</span>
| | &lt; 80 127 225 277 296|
| colspan="2" style="text-align:center;" | 15\160+6\80
 
112.5+90
112.5+90
&lt;span style="background-color: rgba(255,255,255,0);"&gt;5\80+9\80&lt;/span&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;75+135&lt;/span&gt; ||= 15\80+18\80
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">&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;
&lt;br /&gt;


<span style="background-color: rgba(255,255,255,0);">5\80+9\80</span>


&lt;table class="wiki_table"&gt;
<span style="background-color: rgba(255,255,255,0);">75+135</span>
    &lt;tr&gt;
| style="text-align:center;" | 15\80+18\80
        &lt;th colspan="2"&gt;Generator&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;Cents&lt;br /&gt;
&lt;/th&gt;
        &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;th&gt;8/7+7/6&lt;br /&gt;
&lt;/th&gt;
        &lt;th&gt;&lt;br /&gt;
&lt;/th&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;(7+10)\29&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;289.655+413.793&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 29 46 81 100 107|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;2\29+&lt;span style="line-height: 1.5;"&gt;3\29&lt;/span&gt;&lt;br /&gt;
82.759+124.138&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;6\29+6\29&lt;br /&gt;
248.276+248.276&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;(25+36)\104&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;288.4615+415.385&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 104 165 292 360 385|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;10\104+23\312&lt;br /&gt;
115.385+88.4615&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;20\312+23\208&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;76.923+132.692&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;20\104+23\104&lt;br /&gt;
230.769+265.385&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;(18+26)\75&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;288+416&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 7&lt;span style="line-height: 1.5;"&gt;5 119 210~211 259 277|&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;7\75+17\225&lt;br /&gt;
112+90.667&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;14\225+17\150&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;74.667+136&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;14\75+17\75&lt;br /&gt;
224+272&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(47+68)\196&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;287.755+416.3265&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt;&lt;span style="line-height: 1.5;"&gt; 196 311 549-551 678 725|&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;37\392+44\588&lt;br /&gt;
113.265+89.796&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;37/588+22/196&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); text-align: start;"&gt;75.51+134.694&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;37/196+44/196&lt;br /&gt;
226.531+269.388&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;287.713+416.382&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 29 46 81 100 107|+&amp;lt; &lt;span style="line-height: 1.5;"&gt;46 73 129 159 170|&lt;/span&gt;&lt;span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small; line-height: 1.5;"&gt;φ&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;119.283+85.7795&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;79.522+128.769&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;238.566+257.3385&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(29+42)\121&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;287.603+416.529&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 121 192 339~340 419 448|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;23\242+9\121&lt;br /&gt;
114.05+89.256&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;23\363+27\242&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.033+133.884&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;23\121+27\121&lt;br /&gt;
228.099+267.769&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;287.267+416.978&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 29 46 81 100 107|+&amp;lt; 109 173 306 377 403|&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;116.8205+87.323&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;78.617+130.984&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;235.85+261.969&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;(11+16)\46&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.9565+417.391&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 46 73 129 159 170|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;9\92+10\138&lt;br /&gt;
117.391+86.9565&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;3\46+5\46&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;78.261+130.435&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;9\46+10\46&lt;br /&gt;
234.783+260.87&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.587+417.884&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 29 46 81 100 107|+&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;117.925+88.626&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;77.635+132.9395&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;232.936+265.879&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(48+70)\201&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.567+417.91&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 201 319 564 695 703|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;39\402+44\603&lt;br /&gt;
116.418+87.56&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;13\201+22\201&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.612+131.343&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;39\201+44\201&lt;br /&gt;
232.836+262.687&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(37+54)\155&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.452+418.0645&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 155 246 435 536 573|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;15\155+34\465&lt;br /&gt;
116.129+87.742&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;10\155+17\15&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.419+131.613&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;30\155+34\155&lt;br /&gt;
232.258+263.226&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.387+418.151&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 46 73 129 159 170|+&amp;lt; 109 173 306 377 403|&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;115.968+87.842&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;77.312+131.7365&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;231.935+263.527&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(63+92)\264&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.364+418.182&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 264 419 741 913 976|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;51\528+58\792&lt;br /&gt;
115.909+87.87&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;17\264+29\264&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.273+131.818&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;51\264+58\264&lt;br /&gt;
231.818+263.636&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;(26+38)\109&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.2385+418.349&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 109 173 306 377 403|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;21\218+8\109&lt;br /&gt;
115.596+88.07&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;7\109+12\10&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.064+132.11&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;21\109+24\109&lt;br /&gt;
231.192+264.22&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Boundary of smaller &amp;quot;gentle region&amp;quot;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(67+98)\281&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.121+418.505&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 281 446 789 972 1039|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;27\281+62\843&lt;br /&gt;
115.3025+88.256&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;18\281+31\28&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.868+132.384&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;54\281+62\281&lt;br /&gt;
230.605+264.769&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.101+418.533&lt;br /&gt;
&lt;/td&gt;
        &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;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;
        &lt;td style="text-align: center;"&gt;233.052+267.793&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(41+60)\172&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;286.0465+418.605&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 172 273 483 595 636|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;33\344+38\516&lt;br /&gt;
115.116+88.372&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;11\172+19\17&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.744+132.558&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;33\172+38\172&lt;br /&gt;
230.232+265.116&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;(56+82)\235&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;285.957+418.723&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 235 373 660 813 869|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;45\470+52\705&lt;br /&gt;
114.894+81.511&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;15\235+26\23&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;76.596+132.766&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;45\235+52\235&lt;br /&gt;
229.787+265.532&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;285.852+418.864&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; &lt;span style="line-height: 1.5;"&gt;109 173 306 377 403|+&lt;/span&gt;&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;114.963+88.4675&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;76.642+132.701&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;229.926+265.402&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;span style="display: block; text-align: center;"&gt;(15+22)\63&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;285.714+419.048&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 63 100 177 218 233|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;6\63+14\189&lt;br /&gt;
114.286+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="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;
228.571+266.667&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;285.513+419.316&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 46 73 129 159 170|+&amp;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;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&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;&lt;br /&gt;
&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;227.565+267.606&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;span style="display: block; text-align: center;"&gt;(49+72)\206&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;285.437+419.4175&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 206 327 578~579 713 762|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;20\206+15\206&lt;br /&gt;
116.505+87.37&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;40\618+45\41&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.67+131.068&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;40\206+45\206&lt;br /&gt;
233.01+262.136&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;span style="display: block; text-align: center;"&gt;(34+50)\143&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;285.315+419.58&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 143 227 401~402 495 529|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;14\143+31\429&lt;br /&gt;
117.4825+86.71&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;28\429+31\28&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;78.322+130.07&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;28\143+31\143&lt;br /&gt;
234.965+260.14&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;285.234+419.688&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 63 100 177 218 233|+&amp;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;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;113.085+89.636&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"&gt;75.39+134.454&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;226.169+268.909&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;span style="display: block; text-align: center;"&gt;(54+78)\223&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;285.202+419.731&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 223 354 626~627 771 825|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;43\446+49\669&lt;br /&gt;
115.695+87.892&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;43\669+49\44&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;77.13+131.839&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;43\223+49\223&lt;br /&gt;
231.39+263.677&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
    &lt;tr&gt;
        &lt;td&gt;&lt;span style="display: block; text-align: center;"&gt;(19+28)\80&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;&lt;span style="display: block; text-align: center;"&gt;285+420&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;&amp;lt; 80 127 225 277 296|&lt;br /&gt;
&lt;/td&gt;
        &lt;td colspan="2" style="text-align: center;"&gt;15\160+6\80&lt;br /&gt;
112.5+90&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;5\80+9\80&lt;/span&gt;&lt;br /&gt;
&lt;span style="background-color: rgba(255,255,255,0);"&gt;75+135&lt;/span&gt;&lt;br /&gt;
&lt;/td&gt;
        &lt;td style="text-align: center;"&gt;15\80+18\80&lt;br /&gt;
225+270&lt;br /&gt;
&lt;/td&gt;
        &lt;td&gt;Boundary of larger &amp;quot;gentle region&amp;quot;&lt;br /&gt;
&lt;/td&gt;
    &lt;/tr&gt;
&lt;/table&gt;


&lt;/body&gt;&lt;/html&gt;</pre></div>
225+270
| | Boundary of larger "gentle region"
|}

Revision as of 00:00, 17 July 2018

This is an extended version of the Gentle region article.

Margo Schulter, in a 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 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, 133edo, 155edo, 162edo, 167edo, 179edo, 191edo, 201edo, 213edo, 225edo and 237edo, plus 63edo and 80edo in the extended region.

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

248.276+248.276

(25+36)\104 288.4615+415.385 < 104 165 292 360 385| 10\104+23\312

115.385+88.4615

20\312+23\208

76.923+132.692

20\104+23\104

230.769+265.385

(18+26)\75 288+416 < 75 119 210~211 259 277| 7\75+17\225

112+90.667

14\225+17\150

74.667+136

14\75+17\75

224+272

(47+68)\196 287.755+416.3265 < 196 311 549-551 678 725| 37\392+44\588

113.265+89.796

37/588+22/196

75.51+134.694

37/196+44/196

226.531+269.388

287.713+416.382 < 29 46 81 100 107|+< 46 73 129 159 170|φ 119.283+85.7795

79.522+128.769

238.566+257.3385
(29+42)\121 287.603+416.529 < 121 192 339~340 419 448| 23\242+9\121

114.05+89.256

23\363+27\242

76.033+133.884

23\121+27\121

228.099+267.769

287.267+416.978 < 29 46 81 100 107|+< 109 173 306 377 403|φ 116.8205+87.323

78.617+130.984

235.85+261.969
(11+16)\46 286.9565+417.391 < 46 73 129 159 170| 9\92+10\138

117.391+86.9565

3\46+5\46

78.261+130.435

9\46+10\46

234.783+260.87

286.587+417.884 < 29 46 81 100 107|+<63 100 177 218 233|φ 117.925+88.626

77.635+132.9395

232.936+265.879
(48+70)\201 286.567+417.91 < 201 319 564 695 703| 39\402+44\603

116.418+87.56

13\201+22\201

77.612+131.343

39\201+44\201

232.836+262.687

(37+54)\155 286.452+418.0645 < 155 246 435 536 573| 15\155+34\465

116.129+87.742

10\155+17\15

77.419+131.613

30\155+34\155

232.258+263.226

286.387+418.151 < 46 73 129 159 170|+< 109 173 306 377 403|φ 115.968+87.842

77.312+131.7365

231.935+263.527
(63+92)\264 286.364+418.182 < 264 419 741 913 976| 51\528+58\792

115.909+87.87

17\264+29\264

77.273+131.818

51\264+58\264

231.818+263.636

(26+38)\109 286.2385+418.349 < 109 173 306 377 403| 21\218+8\109

115.596+88.07

7\109+12\10

77.064+132.11

21\109+24\109

231.192+264.22

Boundary of smaller "gentle region"
(67+98)\281 286.121+418.505 < 281 446 789 972 1039| 27\281+62\843

115.3025+88.256

18\281+31\28

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

115.116+88.372

11\172+19\17

76.744+132.558

33\172+38\172

230.232+265.116

(56+82)\235 285.957+418.723 < 235 373 660 813 869| 45\470+52\705

114.894+81.511

15\235+26\23

76.596+132.766

45\235+52\235

229.787+265.532

285.852+418.864 < 109 173 306 377 403|+< 63 100 177 218 233|φ 114.963+88.4675

76.642+132.701

229.926+265.402
(15+22)\63 285.714+419.048 < 63 100 177 218 233| 6\63+14\189

114.286+88.88

4\63+7\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

227.565+267.606
(49+72)\206 285.437+419.4175 < 206 327 578~579 713 762| 20\206+15\206

116.505+87.37

40\618+45\41

77.67+131.068

40\206+45\206

233.01+262.136

(34+50)\143 285.315+419.58 < 143 227 401~402 495 529| 14\143+31\429

117.4825+86.71

28\429+31\28

78.322+130.07

28\143+31\143

234.965+260.14

285.234+419.688 < 63 100 177 218 233|+< 80 127 225 277 296|φ 113.085+89.636

75.39+134.454

226.169+268.909
(54+78)\223 285.202+419.731 < 223 354 626~627 771 825| 43\446+49\669

115.695+87.892

43\669+49\44

77.13+131.839

43\223+49\223

231.39+263.677

(19+28)\80 285+420 < 80 127 225 277 296| 15\160+6\80

112.5+90

5\80+9\80

75+135

15\80+18\80

225+270

Boundary of larger "gentle region"
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