Gentle region
IMPORTED REVISION FROM WIKISPACES
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- This revision was by author JosephRuhf and made on 2015-03-09 20:02:57 UTC.
- The original revision id was 543524956.
- The revision comment was:
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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. || 17\29 || || 703.448 || < 29 46 81| ||= 2\29 82.759 ||= 3\29 124.138 ||||= 6\29 248.276 || || || || 61\104 || 703.846 || < 104 165 292| ||= 20\312 76.923 ||= 33\208 132.692 ||= 20\104 230.769 ||= 33\104 265.385 || || || || 44\75 || 704 || < 75 119 211| ||= 14\225 74.667 ||= 17\150 136 ||= 14\75 224 ||= 17\75 272 || || || || 71\121 || 704.132 || < 121 192 340| ||= 23\363 76.033 ||= 27\242 133.884 ||= 23\121 228.099 ||= 27\121 267.769 || || || 27\46 || || 704.348 || < 46 73 129| ||= 3\46 78.261 ||= 5\46 130.435 ||= 9\46 234.783 ||= 10\46 260.87 || || || || || 704.426 || < 29 46 81|+ < 109 173 306| <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 || || || || 118\201 || 704.478 || < 201 319 564| ||= 13\201 77.612 ||= 22\201 131.343 ||= 39\201 232.836 ||= 44\201 262.687 || || || || 91\155 || 704.516 || < 155 246 435| ||= 10\155 77.419 ||= 17\155 131.613 ||= 30\155 232.258 ||= 34\155 263.226 || || || || 155\264 || 704.5455 || < 264 419 741| ||= 17\264 77.273 ||= 29\264 131.818 ||= 51\264 231.818 ||= 58\264 263.636 || || || 64\109 || || 704.587 || < 109 173 306| ||= 7\109 77.064 ||= 12\109 132.11 ||= 21\109 231.192 ||= 24\109 264.22 || Boundary of smaller "gentle region" || || || 165\281 || 704.626 || < 281 446 789| ||= 18\281 76.868 ||= 31\281 132.384 ||= 54\281 230.605 ||= 62\281 264.769 || || || || 101\172 || 704.651 || < 172 273 483| ||= 11\172 76.744 ||= 19\172 132.558 ||= 33\172 230.232 ||= 38\172 265.116 || || || || 138\235 || 704.681 || < 235 373 660| ||= 15\235 76.596 ||= 26\235 132.766 ||= 45\235 229.787 ||= 52\235 265.532 || || || || || 704.716 || < 80 127 225| + < 109 173 306| <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 || || || 37\63 || || 704.762 || < 63 100 177| ||= 4\63 76.1905 ||= 7\63 133.333 ||= 12\63 228.571 ||= 14\63 266.667 || || || || 121\206 || 704.854 || < 206 327 578| ||= 40\618 77.67 ||= 45\412 131.068 ||= 40\206 233.01 ||= 45\206 262.136 || || || || 84\143 || 704.895 || < 143 227 401| ||= 28\429 78.322 ||= 31\286 130.07 ||= 28\143 234.965 ||= 31\143 260.14 || || || || 131\223 || 704.933 || < 223 354 626| ||= 43\669 77.13 ||= 49\446 131.839 ||= 43\223 231.39 ||= 49\223 263.667 || || || 47\80 || || 705 || < 80 127 225| ||= 5\80 75 ||= 9\80 135 ||= 15\80 225 ||= 18\80 270 || 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>
<td>17\29<br />
</td>
<td><br />
</td>
<td>703.448<br />
</td>
<td>< 29 46 81|<br />
</td>
<td style="text-align: center;">2\29<br />
82.759<br />
</td>
<td style="text-align: center;">3\29<br />
124.138<br />
</td>
<td colspan="2" style="text-align: center;">6\29<br />
248.276<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>61\104<br />
</td>
<td>703.846<br />
</td>
<td>< 104 165 292|<br />
</td>
<td style="text-align: center;">20\312<br />
76.923<br />
</td>
<td style="text-align: center;">33\208<br />
132.692<br />
</td>
<td style="text-align: center;">20\104<br />
230.769<br />
</td>
<td style="text-align: center;">33\104<br />
265.385<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>44\75<br />
</td>
<td>704<br />
</td>
<td>< 75 119 211|<br />
</td>
<td style="text-align: center;">14\225<br />
74.667<br />
</td>
<td style="text-align: center;">17\150<br />
136<br />
</td>
<td style="text-align: center;">14\75<br />
224<br />
</td>
<td style="text-align: center;">17\75<br />
272<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>71\121<br />
</td>
<td>704.132<br />
</td>
<td>< 121 192 340|<br />
</td>
<td style="text-align: center;">23\363<br />
76.033<br />
</td>
<td style="text-align: center;">27\242<br />
133.884<br />
</td>
<td style="text-align: center;">23\121<br />
228.099<br />
</td>
<td style="text-align: center;">27\121<br />
267.769<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>27\46<br />
</td>
<td><br />
</td>
<td>704.348<br />
</td>
<td>< 46 73 129|<br />
</td>
<td style="text-align: center;">3\46<br />
78.261<br />
</td>
<td style="text-align: center;">5\46<br />
130.435<br />
</td>
<td style="text-align: center;">9\46<br />
234.783<br />
</td>
<td style="text-align: center;">10\46<br />
260.87<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>704.426<br />
</td>
<td>< 29 46 81|+<br />
< 109 173 306|<br />
<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<br />
</td>
<td style="text-align: center;">130.984<br />
</td>
<td style="text-align: center;">233.605<br />
</td>
<td style="text-align: center;">261.969<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>118\201<br />
</td>
<td>704.478<br />
</td>
<td>< 201 319 564|<br />
</td>
<td style="text-align: center;">13\201<br />
77.612<br />
</td>
<td style="text-align: center;">22\201<br />
131.343<br />
</td>
<td style="text-align: center;">39\201<br />
232.836<br />
</td>
<td style="text-align: center;">44\201<br />
262.687<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>91\155<br />
</td>
<td>704.516<br />
</td>
<td>< 155 246 435|<br />
</td>
<td style="text-align: center;">10\155<br />
77.419<br />
</td>
<td style="text-align: center;">17\155<br />
131.613<br />
</td>
<td style="text-align: center;">30\155<br />
232.258<br />
</td>
<td style="text-align: center;">34\155<br />
263.226<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>155\264<br />
</td>
<td>704.5455<br />
</td>
<td>< 264 419 741|<br />
</td>
<td style="text-align: center;">17\264<br />
77.273<br />
</td>
<td style="text-align: center;">29\264<br />
131.818<br />
</td>
<td style="text-align: center;">51\264<br />
231.818<br />
</td>
<td style="text-align: center;">58\264<br />
263.636<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>64\109<br />
</td>
<td><br />
</td>
<td>704.587<br />
</td>
<td>< 109 173 306|<br />
</td>
<td style="text-align: center;">7\109<br />
77.064<br />
</td>
<td style="text-align: center;">12\109<br />
132.11<br />
</td>
<td style="text-align: center;">21\109<br />
231.192<br />
</td>
<td style="text-align: center;">24\109<br />
264.22<br />
</td>
<td>Boundary of smaller "gentle region"<br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>165\281<br />
</td>
<td>704.626<br />
</td>
<td>< 281 446 789|<br />
</td>
<td style="text-align: center;">18\281<br />
76.868<br />
</td>
<td style="text-align: center;">31\281<br />
132.384<br />
</td>
<td style="text-align: center;">54\281<br />
230.605<br />
</td>
<td style="text-align: center;">62\281<br />
264.769<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>101\172<br />
</td>
<td>704.651<br />
</td>
<td>< 172 273 483|<br />
</td>
<td style="text-align: center;">11\172<br />
76.744<br />
</td>
<td style="text-align: center;">19\172<br />
132.558<br />
</td>
<td style="text-align: center;">33\172<br />
230.232<br />
</td>
<td style="text-align: center;">38\172<br />
265.116<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>138\235<br />
</td>
<td>704.681<br />
</td>
<td>< 235 373 660|<br />
</td>
<td style="text-align: center;">15\235<br />
76.596<br />
</td>
<td style="text-align: center;">26\235<br />
132.766<br />
</td>
<td style="text-align: center;">45\235<br />
229.787<br />
</td>
<td style="text-align: center;">52\235<br />
265.532<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td><br />
</td>
<td>704.716<br />
</td>
<td>< 80 127 225|<br />
+<br />
< 109 173 306|<br />
<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<br />
</td>
<td style="text-align: center;">133.012<br />
</td>
<td style="text-align: center;">229.26<br />
</td>
<td style="text-align: center;">266.024<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>37\63<br />
</td>
<td><br />
</td>
<td>704.762<br />
</td>
<td>< 63 100 177|<br />
</td>
<td style="text-align: center;">4\63<br />
76.1905<br />
</td>
<td style="text-align: center;">7\63<br />
133.333<br />
</td>
<td style="text-align: center;">12\63<br />
228.571<br />
</td>
<td style="text-align: center;">14\63<br />
266.667<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>121\206<br />
</td>
<td>704.854<br />
</td>
<td>< 206 327 578|<br />
</td>
<td style="text-align: center;">40\618<br />
77.67<br />
</td>
<td style="text-align: center;">45\412<br />
131.068<br />
</td>
<td style="text-align: center;">40\206<br />
233.01<br />
</td>
<td style="text-align: center;">45\206<br />
262.136<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>84\143<br />
</td>
<td>704.895<br />
</td>
<td>< 143 227 401|<br />
</td>
<td style="text-align: center;">28\429<br />
78.322<br />
</td>
<td style="text-align: center;">31\286<br />
130.07<br />
</td>
<td style="text-align: center;">28\143<br />
234.965<br />
</td>
<td style="text-align: center;">31\143<br />
260.14<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td><br />
</td>
<td>131\223<br />
</td>
<td>704.933<br />
</td>
<td>< 223 354 626|<br />
</td>
<td style="text-align: center;">43\669<br />
77.13<br />
</td>
<td style="text-align: center;">49\446<br />
131.839<br />
</td>
<td style="text-align: center;">43\223<br />
231.39<br />
</td>
<td style="text-align: center;">49\223<br />
263.667<br />
</td>
<td><br />
</td>
</tr>
<tr>
<td>47\80<br />
</td>
<td><br />
</td>
<td>705<br />
</td>
<td>< 80 127 225|<br />
</td>
<td style="text-align: center;">5\80<br />
75<br />
</td>
<td style="text-align: center;">9\80<br />
135<br />
</td>
<td style="text-align: center;">15\80<br />
225<br />
</td>
<td style="text-align: center;">18\80<br />
270<br />
</td>
<td>Boundary of larger "gentle region"<br />
</td>
</tr>
</table>
</body></html>