Gentle region: Difference between revisions

Revision as of 13:54, 30 October 2015

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[[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 (5\26)
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 (5\23)
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 (13\67)
232.836 ||= 44\201
262.687 ||   ||
||   || 91\155 || 704.516 || < 155 246 435| ||= 10\155 (2\31)
77.419 ||= 17\155
131.613 ||= 30\155 (6\31)
232.258 ||= 34\155
263.226 ||   ||
||   || 155\264 || 704.5455 || < 264 419 741| ||= 17\264
77.273 ||= 29\264
131.818 ||= 51\264 (17\88)
231.818 ||= 58\264 (29\132)
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 (19\86)
265.116 ||   ||
||   || 138\235 || 704.681 || < 235 373 660| ||= 15\235 (3\47)
76.596 ||= 26\235
132.766 ||= 45\235 (9/45)
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 (1\9)
133.333 ||= 12\63 (4\21)
228.571 ||= 14\63 (2\9)
266.667 ||   ||
||   || 121\206 || 704.854 || < 206 327 578| ||= 40\618 (20\309)
77.67 ||= 45\412
131.068 ||= 40\206 (20\103)
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 (1\16)
75 ||= 9\80
135 ||= 15\80 (3\16)
225 ||= 18\80 (9\40)
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 &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">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>&lt; 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>&lt; 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 (5\26)<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>&lt; 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>&lt; 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>&lt; 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 (5\23)<br />
260.87<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td><br />
</td>
        <td>704.426<br />
</td>
        <td>&lt; 29 46 81|+<br />
&lt; 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>&lt; 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 (13\67)<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>&lt; 155 246 435|<br />
</td>
        <td style="text-align: center;">10\155 (2\31)<br />
77.419<br />
</td>
        <td style="text-align: center;">17\155<br />
131.613<br />
</td>
        <td style="text-align: center;">30\155 (6\31)<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>&lt; 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 (17\88)<br />
231.818<br />
</td>
        <td style="text-align: center;">58\264 (29\132)<br />
263.636<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td>64\109<br />
</td>
        <td><br />
</td>
        <td>704.587<br />
</td>
        <td>&lt; 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 &quot;gentle region&quot;<br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>165\281<br />
</td>
        <td>704.626<br />
</td>
        <td>&lt; 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>&lt; 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 (19\86)<br />
265.116<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>138\235<br />
</td>
        <td>704.681<br />
</td>
        <td>&lt; 235 373 660|<br />
</td>
        <td style="text-align: center;">15\235 (3\47)<br />
76.596<br />
</td>
        <td style="text-align: center;">26\235<br />
132.766<br />
</td>
        <td style="text-align: center;">45\235 (9/45)<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>&lt; 80 127 225|<br />
+<br />
&lt; 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>&lt; 63 100 177|<br />
</td>
        <td style="text-align: center;">4\63<br />
76.1905<br />
</td>
        <td style="text-align: center;">7\63 (1\9)<br />
133.333<br />
</td>
        <td style="text-align: center;">12\63 (4\21)<br />
228.571<br />
</td>
        <td style="text-align: center;">14\63 (2\9)<br />
266.667<br />
</td>
        <td><br />
</td>
    </tr>
    <tr>
        <td><br />
</td>
        <td>121\206<br />
</td>
        <td>704.854<br />
</td>
        <td>&lt; 206 327 578|<br />
</td>
        <td style="text-align: center;">40\618 (20\309)<br />
77.67<br />
</td>
        <td style="text-align: center;">45\412<br />
131.068<br />
</td>
        <td style="text-align: center;">40\206 (20\103)<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>&lt; 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>&lt; 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>&lt; 80 127 225|<br />
</td>
        <td style="text-align: center;">5\80 (1\16)<br />
75<br />
</td>
        <td style="text-align: center;">9\80<br />
135<br />
</td>
        <td style="text-align: center;">15\80 (3\16)<br />
225<br />
</td>
        <td style="text-align: center;">18\80 (9\40)<br />
270<br />
</td>
        <td>Boundary of larger &quot;gentle region&quot;<br />
</td>
    </tr>
</table>

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Gentle region: Difference between revisions

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IMPORTED REVISION FROM WIKISPACES

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