Gentle region
IMPORTED REVISION FROM WIKISPACES
This is an imported revision from Wikispaces. The revision metadata is included below for reference:
- This revision was by author JosephRuhf and made on 2015-11-17 17:11:39 UTC.
- The original revision id was 566835439.
- The revision comment was:
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.
Original Wikitext content:
[[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 8/7 ||~ 8/7+7/6 ||~ || || (7+10)\29 || ||= 289.655+413.793 || < 29 46 81 100 107| ||= 2\29+<span style="line-height: 1.5;">3\29</span> 82.759+124.138 ||= 6\29+6\29 248.276+248.276 || || || (25+36)\104 || ||= 288.4615+415.385 || < 104 165 292 360 385| ||= 20\312+<span style="line-height: 1.5;">33\208</span> 76.923+132.692 ||= 20\104+33\104 230.769+265.385 || || || (18+26)\75 || ||= 288+416 || < 7<span style="line-height: 1.5;">5 119 210~211 259 277|</span> ||= 14\225+<span style="line-height: 1.5;">17\150</span> 74.667+136 ||= 14\75+17\75 224+272 || || || || (47+68)\196 ||= 287.755+416.3265 || <<span style="line-height: 1.5;"> 196 311 551 678 725|</span> ||= 37/588+<span style="line-height: 1.5;">22/196</span> 75.51+134.694 ||= 37/196+44/196 226.531+269.388 || || || || ||= 287.713+416.382 || < 29 46 81 100 107|+< <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> ||= 79.522+128.769 ||= 238.566+257.3385 || || || || (29+42)\121 ||= 287.603+416.529 || < 121 192 340 419 448| ||= 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|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span> ||= 77.868+130.984 ||= 233.605+261.969 || || || (11+16)\46 || ||= 286.9565+417.391 || < 46 73 129 159 170| ||= 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|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span> ||= 77.635+131.297 ||= 232.936+262.593 || || || || (48+70)\201 ||= 286.567+417.91 || < 201 319 564 695 703| ||= 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| ||= 10\155+17\155 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|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span> ||= 77.312+131.8635 ||= 231.935+263.527 || || || || (63+92)\264 ||= 286.364+418.182 || < 264 419 741 913 976| ||= 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| ||= 7\109+12\109 77.06+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| ||= 18\281+31\281 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|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span> ||= 76.834+132.432 ||= 230.503+264.864 || || || || (41+60)\172 ||= 286.0465+418.605 || < 172 273 483 595 636| ||= 11\172+19\172 76.744+132.558 ||= 33\172+38\172 230.232+265.116 || || || || (56+82)\235 ||= 285.957+418.723 || < 235 373 660 813 869| ||= 15\235+26\235 76.596+132.766 ||= 45\235+52\235 229.787+265.532 || || || || ||= 285.852+418.864 || < <span style="line-height: 1.5;">109 173 306 377 403|+</span>< 63 100 177 218 233|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span> ||= 76.42+133.012 ||= 229.26+266.024 || || || <span style="display: block; text-align: center;">(15+22)\63</span> || ||= <span style="display: block; text-align: center;">285.714+419.048</span> || < 63 100 177 218 233| ||= 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|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span> ||= 75.855+133.803 ||= 227.565+267.606 || || || || <span style="display: block; text-align: center;">(49+72)\206</span> ||= <span style="display: block; text-align: center;">285.437+419.4175</span> || < 206 327 579 713 762| ||= 40\618+45\412 77.67+131.068 ||= 40\206+45\206 233.01+262.136 || || || <span style="display: block; text-align: center;">(34+50)\143</span> || ||= <span style="display: block; text-align: center;">285.315+419.58</span> || < 143 227 401~402 495 529| ||= 28\429+31\286 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|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span> ||= 75.056+134.454 ||= 226.169+268.909 || || || <span style="display: block; text-align: center;">(54+78)\223</span> || ||= <span style="display: block; text-align: center;">285.202+419.731</span> || < 223 354 626 771 825| ||= 43\669+49\446 77.13+131.839 ||= 43\223+49\223 231.39+263.667 || || || <span style="display: block; text-align: center;">(19+28)\80</span> || ||= <span style="display: block; text-align: center;">285+420</span> || < 80 127 225 277 296| ||= 5\80+9\80 75+135 ||= 15\80+18\80 225+270 || Boundary of larger "gentle region" ||
Original HTML content:
<html><head><title>Gentle region</title></head><body><a class="wiki_link" href="/Margo%20Schulter">Margo Schulter</a>, in a <a class="wiki_link_ext" href="http://launch.groups.yahoo.com/group/tuning/message/38721" rel="nofollow">tuning list posting</a>, defined the "gentle region" 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">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">29edo</a>, <a class="wiki_link" href="/46edo">46edo</a>, <a class="wiki_link" href="/75edo">75edo</a>, <a class="wiki_link" href="/104edo">104edo</a>, <a class="wiki_link" href="/109edo">109edo</a>, <a class="wiki_link" href="/121edo">121edo</a>, <a class="wiki_link" href="/145edo">133edo</a>, <a class="wiki_link" href="/155edo">155edo</a>, <a class="wiki_link" href="/162edo">162edo</a>, <a class="wiki_link" href="/167edo">167edo</a>, <a class="wiki_link" href="/179edo">179edo</a>, <a class="wiki_link" href="/191edo">191edo</a>, <a class="wiki_link" href="/201edo">201edo</a>, <a class="wiki_link" href="/213edo">213edo</a>, <a class="wiki_link" href="/225edo">225edo</a> and <a class="wiki_link" href="/237edo">237edo</a>, plus <a class="wiki_link" href="/63edo">63edo</a> and <a class="wiki_link" href="/80edo">80edo</a> in the extended region.<br />
<br />
<table class="wiki_table">
<tr>
<th colspan="2">Geneator<br />
</th>
<th>Cents<br />
</th>
<th>2-3-7(b)-11-13(b)<br />
</th>
<th>Half 8/7+1\3 8/7<br />
</th>
<th>8/7+7/6<br />
</th>
<th><br />
</th>
</tr>
<tr>
<td>(7+10)\29<br />
</td>
<td><br />
</td>
<td style="text-align: center;">289.655+413.793<br />
</td>
<td>< 29 46 81 100 107|<br />
</td>
<td style="text-align: center;">2\29+<span style="line-height: 1.5;">3\29</span><br />
82.759+124.138<br />
</td>
<td style="text-align: center;">6\29+6\29<br />
248.276+248.276<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>(25+36)\104<br />
</td>
<td><br />
</td>
<td style="text-align: center;">288.4615+415.385<br />
</td>
<td>< 104 165 292 360 385|<br />
</td>
<td style="text-align: center;">20\312+<span style="line-height: 1.5;">33\208</span><br />
76.923+132.692<br />
</td>
<td style="text-align: center;">20\104+33\104<br />
230.769+265.385<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>(18+26)\75<br />
</td>
<td><br />
</td>
<td style="text-align: center;">288+416<br />
</td>
<td>< 7<span style="line-height: 1.5;">5 119 210~211 259 277|</span><br />
</td>
<td style="text-align: center;">14\225+<span style="line-height: 1.5;">17\150</span><br />
74.667+136<br />
</td>
<td style="text-align: center;">14\75+17\75<br />
224+272<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(47+68)\196<br />
</td>
<td style="text-align: center;">287.755+416.3265<br />
</td>
<td><<span style="line-height: 1.5;"> 196 311 551 678 725|</span><br />
</td>
<td style="text-align: center;">37/588+<span style="line-height: 1.5;">22/196</span><br />
75.51+134.694<br />
</td>
<td style="text-align: center;">37/196+44/196<br />
226.531+269.388<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">287.713+416.382<br />
</td>
<td>< 29 46 81 100 107|+< <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><br />
</td>
<td style="text-align: center;">79.522+128.769<br />
</td>
<td style="text-align: center;">238.566+257.3385<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(29+42)\121<br />
</td>
<td style="text-align: center;">287.603+416.529<br />
</td>
<td>< 121 192 340 419 448|<br />
</td>
<td style="text-align: center;">23\363+27\242<br />
76.033+133.884<br />
</td>
<td style="text-align: center;">23\121+27\121<br />
228.099+267.769<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">287.267+416.978<br />
</td>
<td>< 29 46 81 100 107|+< 109 173 306 377 403|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><br />
</td>
<td style="text-align: center;">77.868+130.984<br />
</td>
<td style="text-align: center;">233.605+261.969<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>(11+16)\46<br />
</td>
<td><br />
</td>
<td style="text-align: center;">286.9565+417.391<br />
</td>
<td>< 46 73 129 159 170|<br />
</td>
<td style="text-align: center;">3\46+5\46<br />
78.261+130.435<br />
</td>
<td style="text-align: center;">9\46+10\46<br />
234.783+260.87<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">286.587+417.884<br />
</td>
<td>< 29 46 81 100 107|+<63 100 177 218 233|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><br />
</td>
<td style="text-align: center;">77.635+131.297<br />
</td>
<td style="text-align: center;">232.936+262.593<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(48+70)\201<br />
</td>
<td style="text-align: center;">286.567+417.91<br />
</td>
<td>< 201 319 564 695 703|<br />
</td>
<td style="text-align: center;">13\201+22\201<br />
77.612+131.343<br />
</td>
<td style="text-align: center;">39\201+44\201<br />
232.836+262.687<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(37+54)\155<br />
</td>
<td style="text-align: center;">286.452+418,0645<br />
</td>
<td>< 155 246 435 536 573|<br />
</td>
<td style="text-align: center;">10\155+17\155<br />
77.419+131.613<br />
</td>
<td style="text-align: center;">30\155+34\155<br />
232.258+263.226<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">286.387+418.151<br />
</td>
<td>< 46 73 129 159 170|+< 109 173 306 377 403|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><br />
</td>
<td style="text-align: center;">77.312+131.8635<br />
</td>
<td style="text-align: center;">231.935+263.527<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(63+92)\264<br />
</td>
<td style="text-align: center;">286.364+418.182<br />
</td>
<td>< 264 419 741 913 976|<br />
</td>
<td style="text-align: center;">17\264+29\264<br />
77.273+131.818<br />
</td>
<td style="text-align: center;">51\264+58\264<br />
231.818+263.636<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>(26+38)\109<br />
</td>
<td><br />
</td>
<td style="text-align: center;">286.2385+418.349<br />
</td>
<td>< 109 173 306 377 403|<br />
</td>
<td style="text-align: center;">7\109+12\109<br />
77.06+132.11<br />
</td>
<td style="text-align: center;">21\109+24\109<br />
231.192+264.22<br />
</td>
<td>Boundary of smaller "gentle region"<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(67+98)\281<br />
</td>
<td style="text-align: center;">286.121+418.505<br />
</td>
<td>< 281 446 789 972 1039|<br />
</td>
<td style="text-align: center;">18\281+31\281<br />
76.868+132.384<br />
</td>
<td style="text-align: center;">54\281+62\281<br />
230.605+264.769<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">286.101+418.533<br />
</td>
<td>< 46 73 129 159 170|+< 63 100 177 218 233|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><br />
</td>
<td style="text-align: center;">76.834+132.432<br />
</td>
<td style="text-align: center;">230.503+264.864<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(41+60)\172<br />
</td>
<td style="text-align: center;">286.0465+418.605<br />
</td>
<td>< 172 273 483 595 636|<br />
</td>
<td style="text-align: center;">11\172+19\172<br />
76.744+132.558<br />
</td>
<td style="text-align: center;">33\172+38\172<br />
230.232+265.116<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>(56+82)\235<br />
</td>
<td style="text-align: center;">285.957+418.723<br />
</td>
<td>< 235 373 660 813 869|<br />
</td>
<td style="text-align: center;">15\235+26\235<br />
76.596+132.766<br />
</td>
<td style="text-align: center;">45\235+52\235<br />
229.787+265.532<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">285.852+418.864<br />
</td>
<td>< <span style="line-height: 1.5;">109 173 306 377 403|+</span>< 63 100 177 218 233|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><br />
</td>
<td style="text-align: center;">76.42+133.012<br />
</td>
<td style="text-align: center;">229.26+266.024<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;">(15+22)\63</span><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">285.714+419.048</span><br />
</td>
<td>< 63 100 177 218 233|<br />
</td>
<td style="text-align: center;">4\63+7\63<br />
76.1905+133.333<br />
</td>
<td style="text-align: center;">12\63+14\63<br />
228.571+266.667<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">285.513+419.316<br />
</td>
<td>< 46 73 129 159 170|+< 80 127 225 277 296|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><br />
</td>
<td style="text-align: center;">75.855+133.803<br />
</td>
<td style="text-align: center;">227.565+267.606<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><span style="display: block; text-align: center;">(49+72)\206</span><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">285.437+419.4175</span><br />
</td>
<td>< 206 327 579 713 762|<br />
</td>
<td style="text-align: center;">40\618+45\412<br />
77.67+131.068<br />
</td>
<td style="text-align: center;">40\206+45\206<br />
233.01+262.136<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;">(34+50)\143</span><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">285.315+419.58</span><br />
</td>
<td>< 143 227 401~402 495 529|<br />
</td>
<td style="text-align: center;">28\429+31\286<br />
78.322+130.07<br />
</td>
<td style="text-align: center;">28\143+31\143<br />
234.965+260.14<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td style="text-align: center;">285.234+419.688<br />
</td>
<td>< 63 100 177 218 233|+< 80 127 225 277 296|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><br />
</td>
<td style="text-align: center;">75.056+134.454<br />
</td>
<td style="text-align: center;">226.169+268.909<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;">(54+78)\223</span><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">285.202+419.731</span><br />
</td>
<td>< 223 354 626 771 825|<br />
</td>
<td style="text-align: center;">43\669+49\446<br />
77.13+131.839<br />
</td>
<td style="text-align: center;">43\223+49\223<br />
231.39+263.667<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><span style="display: block; text-align: center;">(19+28)\80</span><br />
</td>
<td><br />
</td>
<td style="text-align: center;"><span style="display: block; text-align: center;">285+420</span><br />
</td>
<td>< 80 127 225 277 296|<br />
</td>
<td style="text-align: center;">5\80+9\80<br />
75+135<br />
</td>
<td style="text-align: center;">15\80+18\80<br />
225+270<br />
</td>
<td>Boundary of larger "gentle region"<br />
</td>
</tr>
</table>
</body></html>