Gentle region (extended version): Difference between revisions

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-[[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.
+[[Margo_Schulter|Margo Schulter]], in a [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_105200.html#105202 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 [https://yahootuninggroupsultimatebackup.github.io/tuning/topicId_106239.html#106239 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.
 {| class="wikitable"
@@ Line 16: / Line 16: @@
 | |
 | style="text-align:center;" | 289.655+413.793
-| | &lt; 29 46 81 100 107|
+| | &lt;<nowiki> 29 46 81 100 107|</nowiki>
 | colspan="2" style="text-align:center;" | 2\29+<span style="line-height: 1.5;">3\29</span>
@@ Line 28: / Line 28: @@
 | |
 | style="text-align:center;" | 288.4615+415.385
-| | &lt; 104 165 292 360 385|
+| | &lt;<nowiki> 104 165 292 360 385|</nowiki>
 | colspan="2" style="text-align:center;" | 10\104+23\312
@@ Line 44: / Line 44: @@
 | |
 | style="text-align:center;" | 288+416
-| | &lt; 7<span style="line-height: 1.5;">5 119 210~211 259 277|</span>
+| | &lt; 7<span style="line-height: 1.5;"><nowiki>5 119 210~211 259 277|</nowiki></span>
 | colspan="2" style="text-align:center;" | 7\75+17\225
@@ Line 60: / Line 60: @@
 | | (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>
+| | &lt;<span style="line-height: 1.5;"><nowiki> 196 311 549-551 678 725|</nowiki></span>
 | colspan="2" style="text-align:center;" | 37\392+44\588
@@ Line 76: / Line 76: @@
 | |
 | 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>
+| | &lt;<nowiki> 29 46 81 100 107|+</nowiki>&lt; <span style="line-height: 1.5;"><nowiki>46 73 129 159 170|</nowiki></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
@@ Line 86: / Line 86: @@
 | | (29+42)\121
 | style="text-align:center;" | 287.603+416.529
-| | &lt; 121 192 339~340 419 448|
+| | &lt;<nowiki> 121 192 339~340 419 448|</nowiki>
 | colspan="2" style="text-align:center;" | 23\242+9\121
@@ Line 102: / Line 102: @@
 | |
 | 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>
+| | &lt;<nowiki> 29 46 81 100 107|+</nowiki>&lt;<nowiki> 109 173 306 377 403|</nowiki><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
@@ Line 112: / Line 112: @@
 | |
 | style="text-align:center;" | 286.9565+417.391
-| | &lt; 46 73 129 159 170|
+| | &lt;<nowiki> 46 73 129 159 170|</nowiki>
 | colspan="2" style="text-align:center;" | 9\92+10\138
@@ Line 128: / Line 128: @@
 | |
 | 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>
+| | &lt;<nowiki> 29 46 81 100 107|+</nowiki>&lt;<nowiki>63 100 177 218 233|</nowiki><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
@@ Line 138: / Line 138: @@
 | | (48+70)\201
 | style="text-align:center;" | 286.567+417.91
-| | &lt; 201 319 564 695 703|
+| | &lt;<nowiki> 201 319 564 695 703|</nowiki>
 | colspan="2" style="text-align:center;" | 39\402+44\603
@@ Line 154: / Line 154: @@
 | | (37+54)\155
 | style="text-align:center;" | 286.452+418.0645
-| | &lt; 155 246 435 536 573|
+| | &lt;<nowiki> 155 246 435 536 573|</nowiki>
 | colspan="2" style="text-align:center;" | 15\155+34\465
@@ Line 170: / Line 170: @@
 | |
 | 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>
+| | &lt;<nowiki> 46 73 129 159 170|+</nowiki>&lt;<nowiki> 109 173 306 377 403|</nowiki><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
@@ Line 180: / Line 180: @@
 | | (63+92)\264
 | style="text-align:center;" | 286.364+418.182
-| | &lt; 264 419 741 913 976|
+| | &lt;<nowiki> 264 419 741 913 976|</nowiki>
 | colspan="2" style="text-align:center;" | 51\528+58\792
@@ Line 196: / Line 196: @@
 | |
 | style="text-align:center;" | 286.2385+418.349
-| | &lt; 109 173 306 377 403|
+| | &lt;<nowiki> 109 173 306 377 403|</nowiki>
 | colspan="2" style="text-align:center;" | 21\218+8\109
@@ Line 212: / Line 212: @@
 | | (67+98)\281
 | style="text-align:center;" | 286.121+418.505
-| | &lt; 281 446 789 972 1039|
+| | &lt;<nowiki> 281 446 789 972 1039|</nowiki>
 | colspan="2" style="text-align:center;" | 27\281+62\843
@@ Line 228: / Line 228: @@
 | |
 | 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>
+| | &lt;<nowiki> 46 73 129 159 170|+</nowiki>&lt;<nowiki> 63 100 177 218 233|</nowiki><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
@@ Line 238: / Line 238: @@
 | | (41+60)\172
 | style="text-align:center;" | 286.0465+418.605
-| | &lt; 172 273 483 595 636|
+| | &lt;<nowiki> 172 273 483 595 636|</nowiki>
 | colspan="2" style="text-align:center;" | 33\344+38\516
@@ Line 254: / Line 254: @@
 | | (56+82)\235
 | style="text-align:center;" | 285.957+418.723
-| | &lt; 235 373 660 813 869|
+| | &lt;<nowiki> 235 373 660 813 869|</nowiki>
 | colspan="2" style="text-align:center;" | 45\470+52\705
@@ Line 270: / Line 270: @@
 | |
 | 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>
+| | &lt; <span style="line-height: 1.5;"><nowiki>109 173 306 377 403|+</nowiki></span>&lt;<nowiki> 63 100 177 218 233|</nowiki><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
@@ Line 280: / Line 280: @@
 | |
 | style="text-align:center;" | <span style="display: block; text-align: center;">285.714+419.048</span>
-| | &lt; 63 100 177 218 233|
+| | &lt;<nowiki> 63 100 177 218 233|</nowiki>
 | colspan="2" style="text-align:center;" | 6\63+14\189
@@ Line 296: / Line 296: @@
 | |
 | 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>
+| | &lt;<nowiki> 46 73 129 159 170|+</nowiki>&lt;<nowiki> 80 127 225 277 296|</nowiki><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;">
+| 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>
-</span>
 | style="text-align:center;" | 227.565+267.606
 | |
@@ Line 306: / Line 304: @@
 | | <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|
+| | &lt;<nowiki> 206 327 578~579 713 762|</nowiki>
 | colspan="2" style="text-align:center;" | 20\206+15\206
@@ Line 322: / Line 320: @@
 | |
 | style="text-align:center;" | <span style="display: block; text-align: center;">285.315+419.58</span>
-| | &lt; 143 227 401~402 495 529|
+| | &lt;<nowiki> 143 227 401~402 495 529|</nowiki>
 | colspan="2" style="text-align:center;" | 14\143+31\429
@@ Line 338: / Line 336: @@
 | |
 | 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>
+| | &lt;<nowiki> 63 100 177 218 233|+</nowiki>&lt;<nowiki> 80 127 225 277 296|</nowiki><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
@@ Line 348: / Line 346: @@
 | |
 | style="text-align:center;" | <span style="display: block; text-align: center;">285.202+419.731</span>
-| | &lt; 223 354 626~627 771 825|
+| | &lt;<nowiki> 223 354 626~627 771 825|</nowiki>
 | colspan="2" style="text-align:center;" | 43\446+49\669
@@ Line 364: / Line 362: @@
 | |
 | style="text-align:center;" | <span style="display: block; text-align: center;">285+420</span>
-| | &lt; 80 127 225 277 296|
+| | &lt;<nowiki> 80 127 225 277 296|</nowiki>
 | colspan="2" style="text-align:center;" | 15\160+6\80