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| <h2>IMPORTED REVISION FROM WIKISPACES</h2>
| | {{interwiki |
| This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br>
| | | de = Gentle |
| : This revision was by author [[User:JosephRuhf|JosephRuhf]] and made on <tt>2015-10-31 12:14:39 UTC</tt>.<br>
| | | en = Gentle region |
| : The original revision id was <tt>564564145</tt>.<br>
| | | es = |
| : The revision comment was: <tt></tt><br>
| | | ja = |
| The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br>
| | }} |
| <h4>Original Wikitext content:</h4>
| | ''For an alternative version of the page, see: [[Gentle region (extended version)]].'' |
| <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">[[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.
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|
| || 17\29 || || || 703.448 || < 29 46 81| ||= 2\29
| | The '''gentle region''' refers to the set of tuning systems generated by fifths in the region between the fifths of [[29edo]] (~703.4c) and [[17edo]] (~705.9c), which generate [[neogothic]] (specifically, neomajor and neominor) thirds. The region was defined by [[Margo Schulter]] in a [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_105200.html#105202 tuning list posting], originally defined as the region between 1.49 to 2.65 cents sharp of a just fifth (~703.4 to ~704.6 cents), before being [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106239.html#106239 revised] to 1.49 to 3.04 cents sharp (~703.4 to 705 cents). The tuning range shown on this page shows tunings as sharp as 17edo. |
| 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 || < 7<span style="line-height: 1.5;">5 119 211|</span> ||= 14\225
| |
| 74.667 ||= 17\150
| |
| 136 ||= 14\75
| |
| 224 ||= 17\75
| |
| 272 || ||
| |
| || || || || 704.0585 || < 17 27 48|*pi+<span style="background-color: #ffffff;">< 24 38 67|</span> ||= 69.372 ||= 143.912 ||= 208.117 ||= 287.824 || ||
| |
| || || || 115\196 || 704.082 || <<span style="line-height: 1.5;"> 196 311 551|</span> ||= 37/588
| |
| 75.51 ||= 22/196 (11/98)
| |
| 134.694 ||= 37/196
| |
| 226.531 ||= 44/196 (11/49)
| |
| 269.388 || ||
| |
| || || || || 704.095 || < 17 27 48|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><span style="line-height: 1.5;">+</span>
| |
| <span style="background-color: #ffffff;">< 12 19 34|</span> ||= 69.397 ||= 143.857 ||= 208.191 ||= 287.731 || ||
| |
| || || 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 || ||
| |
| || || || || || < 17 27 48|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>+
| |
| <span style="background-color: #ffffff;">< 29 46 81|</span> ||= 76.834 ||= 132.432 ||= <span style="text-align: center;">230.503</span> ||= <span style="text-align: center;">264.864</span> || ||
| |
| || || 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" ||</pre></div>
| |
| <h4>Original HTML content:</h4>
| |
| <div style="width:100%; max-height:400pt; overflow:auto; background-color:#f8f9fa; border: 1px solid #eaecf0; padding:0em"><pre style="margin:0px;border:none;background:none;word-wrap:break-word;width:200%;white-space: pre-wrap ! important" class="old-revision-html"><html><head><title>Gentle region</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 />
| |
|
| |
|
| | Gentle tuning systems are thus "mild" (or, as the name says, "gentle") versions of tuning systems like [[Superpyth]] temperament. They allow harmony in the style of medieval Pythagorean harmony, usable for neogothic harmony systems; besides, they are possible temperament frameworks for [[Arabic music|Arabic]] and [[Turkish music|Turkish]] tuning systems, with the special property of delivering a common framework for both, differing in the degree of tempering. |
|
| |
|
| <table class="wiki_table">
| | When the tempering of the fifth is "very gentle"/near-just as in 29edo, the interval notated as C-Fb in standard sheet notation (8 fifths down) will be close to a 5/4 major third (implying [[schismic]] temperament), as used in Turkish music; sharper tempering as in 17edo will give this interval the character of a neutral third, as important in Arabic music. (The interval notated as C-E will have the character of a pythagorean or neomajor third.) |
| <tr>
| |
| <td>17\29<br />
| |
| </td>
| |
| <td><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><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><br />
| |
| </td>
| |
| <td>704<br />
| |
| </td>
| |
| <td>&lt; 7<span style="line-height: 1.5;">5 119 211|</span><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 />
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| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
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| <td><br />
| |
| </td>
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| <td>704.0585<br />
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| </td>
| |
| <td>&lt; 17 27 48|*pi+<span style="background-color: #ffffff;">&lt; 24 38 67|</span><br />
| |
| </td>
| |
| <td style="text-align: center;">69.372<br />
| |
| </td>
| |
| <td style="text-align: center;">143.912<br />
| |
| </td>
| |
| <td style="text-align: center;">208.117<br />
| |
| </td>
| |
| <td style="text-align: center;">287.824<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
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| <td>115\196<br />
| |
| </td>
| |
| <td>704.082<br />
| |
| </td>
| |
| <td>&lt;<span style="line-height: 1.5;"> 196 311 551|</span><br />
| |
| </td>
| |
| <td style="text-align: center;">37/588<br />
| |
| 75.51<br />
| |
| </td>
| |
| <td style="text-align: center;">22/196 (11/98)<br />
| |
| 134.694<br />
| |
| </td>
| |
| <td style="text-align: center;">37/196<br />
| |
| 226.531<br />
| |
| </td>
| |
| <td style="text-align: center;">44/196 (11/49)<br />
| |
| 269.388<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
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| <td><br />
| |
| </td>
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| <td>704.095<br />
| |
| </td>
| |
| <td>&lt; 17 27 48|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span><span style="line-height: 1.5;">+</span><br />
| |
| <span style="background-color: #ffffff;">&lt; 12 19 34|</span><br />
| |
| </td>
| |
| <td style="text-align: center;">69.397<br />
| |
| </td>
| |
| <td style="text-align: center;">143.857<br />
| |
| </td>
| |
| <td style="text-align: center;">208.191<br />
| |
| </td>
| |
| <td style="text-align: center;">287.731<br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
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| </td>
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| <td>71\121<br />
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| </td>
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| <td><br />
| |
| </td>
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| <td>704.132<br />
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| </td>
| |
| <td>&lt; 121 192 340|<br />
| |
| </td>
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| <td style="text-align: center;">23\363<br />
| |
| 76.033<br />
| |
| </td>
| |
| <td style="text-align: center;">27\242<br />
| |
| 133.884<br />
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| </td>
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| <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>
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| </tr>
| |
| <tr>
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| <td>27\46<br />
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| </td>
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| <td><br />
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| </td>
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| <td><br />
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| </td>
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| <td>704.348<br />
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| </td>
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| <td>&lt; 46 73 129|<br />
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| </td>
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| <td style="text-align: center;">3\46<br />
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| 78.261<br />
| |
| </td>
| |
| <td style="text-align: center;">5\46<br />
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| 130.435<br />
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| </td>
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| <td style="text-align: center;">9\46<br />
| |
| 234.783<br />
| |
| </td>
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| <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><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><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><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><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><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><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><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| <td>&lt; 17 27 48|<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>+<br />
| |
| <span style="background-color: #ffffff;">&lt; 29 46 81|</span><br />
| |
| </td>
| |
| <td style="text-align: center;">76.834<br />
| |
| </td>
| |
| <td style="text-align: center;">132.432<br />
| |
| </td>
| |
| <td style="text-align: center;"><span style="text-align: center;">230.503</span><br />
| |
| </td>
| |
| <td style="text-align: center;"><span style="text-align: center;">264.864</span><br />
| |
| </td>
| |
| <td><br />
| |
| </td>
| |
| </tr>
| |
| <tr>
| |
| <td><br />
| |
| </td>
| |
| <td>101\172<br />
| |
| </td>
| |
| <td><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><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><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><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><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><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><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><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>
| |
|
| |
|
| </body></html></pre></div>
| | We can consider the originally-defined gentle 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 {{EDOs| 29, 46, 75, 104, 109, 121, 145, 155, 162, 167, 179, 191, 201, 213, 225 and 237, plus 63 and 80 }} in the extended region. |
| | |
| | The extended gentle region is further divided into two subregions: |
| | * "Lower gentle": 703.4{{c}} (near 17\29) to 704.3{{c}} (near 27\46) |
| | * "Upper gentle": 704.3{{c}} to 705.0{{c}} (near 10\17, or more accurately 47\80). |
| | [[46edo]] is effectively the boundary between lower and upper neogothic. |
| | |
| | {| class="wikitable" |
| | |- |
| | ! colspan="4" | EDO Generator |
| | ! Fifth |
| | ! Dim 4th |
| | ! Comments |
| | |- |
| | | 17\29 |
| | | |
| | | |
| | | |
| | | 703.448 |
| | | 372.416 |
| | | |
| | |- |
| | | |
| | | |
| | | |
| | | |
| | |703.711 |
| | |370.312 |
| | |Margo Schulter's MET-24 fifth |
| | |- |
| | | |
| | | |
| | | |
| | | 61\104 |
| | | 703.846 |
| | | 369.231 |
| | | Neo-gothic theory of harmony |
| | |- |
| | | |
| | | |
| | | 44\75 |
| | | |
| | | 704.000 |
| | | 368.000 |
| | | |
| | |- |
| | | |
| | | |
| | | |
| | | 71\121 |
| | | 704.132 |
| | | 366.942 |
| | | |
| | |- |
| | | |
| | | 27\46 |
| | | |
| | | |
| | | 704.348 |
| | | 365.217 |
| | | |
| | |- |
| | | |
| | | |
| | | |
| | | 64\109 |
| | | 704.587 |
| | | 363.303 |
| | | Boundary of smaller "gentle region" |
| | |- |
| | | |
| | | |
| | | 37\63 |
| | | |
| | | 704.762 |
| | | 361.905 |
| | | |
| | |- |
| | | |
| | | |
| | | |
| | | 47\80 |
| | | 705.000 |
| | | 360.000 |
| | | Boundary of larger "gentle region" |
| | |- |
| | | 10\17 |
| | | |
| | | |
| | | |
| | | 705.882 |
| | | 352.941 |
| | | |
| | |} |
| | |
| | == See also == |
| | * [[Interseptimal]] |
| | |
| | [[Category:EDO theory pages]] |
| | [[Category:Fifth]] |
| | [[Category:Gentle]] |