Gentle region (extended version): Difference between revisions
Jump to navigation
Jump to search
Wikispaces>FREEZE No edit summary |
CritDeathX (talk | contribs) Added links to the Yahoo groups archive |
||
| Line 2: | Line 2: | ||
----- | ----- | ||
[[Margo_Schulter|Margo Schulter]], in a [ | [[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" | {| class="wikitable" | ||
| Line 16: | Line 16: | ||
| | | | | | ||
| style="text-align:center;" | 289.655+413.793 | | style="text-align:center;" | 289.655+413.793 | ||
| | < 29 46 81 100 107| | | | <<nowiki> 29 46 81 100 107|</nowiki> | ||
| colspan="2" style="text-align:center;" | 2\29+<span style="line-height: 1.5;">3\29</span> | | 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 | | style="text-align:center;" | 288.4615+415.385 | ||
| | < 104 165 292 360 385| | | | <<nowiki> 104 165 292 360 385|</nowiki> | ||
| colspan="2" style="text-align:center;" | 10\104+23\312 | | colspan="2" style="text-align:center;" | 10\104+23\312 | ||
| Line 44: | Line 44: | ||
| | | | | | ||
| style="text-align:center;" | 288+416 | | style="text-align:center;" | 288+416 | ||
| | < 7<span style="line-height: 1.5;">5 119 210~211 259 277|</span> | | | < 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 | | colspan="2" style="text-align:center;" | 7\75+17\225 | ||
| Line 60: | Line 60: | ||
| | (47+68)\196 | | | (47+68)\196 | ||
| style="text-align:center;" | 287.755+416.3265 | | style="text-align:center;" | 287.755+416.3265 | ||
| | <<span style="line-height: 1.5;"> 196 311 549-551 678 725|</span> | | | <<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 | | colspan="2" style="text-align:center;" | 37\392+44\588 | ||
| Line 76: | Line 76: | ||
| | | | | | ||
| style="text-align:center;" | 287.713+416.382 | | style="text-align:center;" | 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> | | | <<nowiki> 29 46 81 100 107|+</nowiki>< <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 | | colspan="2" style="text-align:center;" | 119.283+85.7795 | ||
| Line 86: | Line 86: | ||
| | (29+42)\121 | | | (29+42)\121 | ||
| style="text-align:center;" | 287.603+416.529 | | style="text-align:center;" | 287.603+416.529 | ||
| | < 121 192 339~340 419 448| | | | <<nowiki> 121 192 339~340 419 448|</nowiki> | ||
| colspan="2" style="text-align:center;" | 23\242+9\121 | | colspan="2" style="text-align:center;" | 23\242+9\121 | ||
| Line 102: | Line 102: | ||
| | | | | | ||
| style="text-align:center;" | 287.267+416.978 | | style="text-align:center;" | 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> | | | <<nowiki> 29 46 81 100 107|+</nowiki><<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 | | colspan="2" style="text-align:center;" | 116.8205+87.323 | ||
| Line 112: | Line 112: | ||
| | | | | | ||
| style="text-align:center;" | 286.9565+417.391 | | style="text-align:center;" | 286.9565+417.391 | ||
| | < 46 73 129 159 170| | | | <<nowiki> 46 73 129 159 170|</nowiki> | ||
| colspan="2" style="text-align:center;" | 9\92+10\138 | | colspan="2" style="text-align:center;" | 9\92+10\138 | ||
| Line 128: | Line 128: | ||
| | | | | | ||
| style="text-align:center;" | 286.587+417.884 | | style="text-align:center;" | 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> | | | <<nowiki> 29 46 81 100 107|+</nowiki><<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 | | colspan="2" style="text-align:center;" | 117.925+88.626 | ||
| Line 138: | Line 138: | ||
| | (48+70)\201 | | | (48+70)\201 | ||
| style="text-align:center;" | 286.567+417.91 | | style="text-align:center;" | 286.567+417.91 | ||
| | < 201 319 564 695 703| | | | <<nowiki> 201 319 564 695 703|</nowiki> | ||
| colspan="2" style="text-align:center;" | 39\402+44\603 | | colspan="2" style="text-align:center;" | 39\402+44\603 | ||
| Line 154: | Line 154: | ||
| | (37+54)\155 | | | (37+54)\155 | ||
| style="text-align:center;" | 286.452+418.0645 | | style="text-align:center;" | 286.452+418.0645 | ||
| | < 155 246 435 536 573| | | | <<nowiki> 155 246 435 536 573|</nowiki> | ||
| colspan="2" style="text-align:center;" | 15\155+34\465 | | colspan="2" style="text-align:center;" | 15\155+34\465 | ||
| Line 170: | Line 170: | ||
| | | | | | ||
| style="text-align:center;" | 286.387+418.151 | | style="text-align:center;" | 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> | | | <<nowiki> 46 73 129 159 170|+</nowiki><<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 | | colspan="2" style="text-align:center;" | 115.968+87.842 | ||
| Line 180: | Line 180: | ||
| | (63+92)\264 | | | (63+92)\264 | ||
| style="text-align:center;" | 286.364+418.182 | | style="text-align:center;" | 286.364+418.182 | ||
| | < 264 419 741 913 976| | | | <<nowiki> 264 419 741 913 976|</nowiki> | ||
| colspan="2" style="text-align:center;" | 51\528+58\792 | | colspan="2" style="text-align:center;" | 51\528+58\792 | ||
| Line 196: | Line 196: | ||
| | | | | | ||
| style="text-align:center;" | 286.2385+418.349 | | style="text-align:center;" | 286.2385+418.349 | ||
| | < 109 173 306 377 403| | | | <<nowiki> 109 173 306 377 403|</nowiki> | ||
| colspan="2" style="text-align:center;" | 21\218+8\109 | | colspan="2" style="text-align:center;" | 21\218+8\109 | ||
| Line 212: | Line 212: | ||
| | (67+98)\281 | | | (67+98)\281 | ||
| style="text-align:center;" | 286.121+418.505 | | style="text-align:center;" | 286.121+418.505 | ||
| | < 281 446 789 972 1039| | | | <<nowiki> 281 446 789 972 1039|</nowiki> | ||
| colspan="2" style="text-align:center;" | 27\281+62\843 | | colspan="2" style="text-align:center;" | 27\281+62\843 | ||
| Line 228: | Line 228: | ||
| | | | | | ||
| style="text-align:center;" | 286.101+418.533 | | style="text-align:center;" | 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> | | | <<nowiki> 46 73 129 159 170|+</nowiki><<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 | | colspan="2" style="text-align:center;" | 116.526+89.264 | ||
| Line 238: | Line 238: | ||
| | (41+60)\172 | | | (41+60)\172 | ||
| style="text-align:center;" | 286.0465+418.605 | | style="text-align:center;" | 286.0465+418.605 | ||
| | < 172 273 483 595 636| | | | <<nowiki> 172 273 483 595 636|</nowiki> | ||
| colspan="2" style="text-align:center;" | 33\344+38\516 | | colspan="2" style="text-align:center;" | 33\344+38\516 | ||
| Line 254: | Line 254: | ||
| | (56+82)\235 | | | (56+82)\235 | ||
| style="text-align:center;" | 285.957+418.723 | | style="text-align:center;" | 285.957+418.723 | ||
| | < 235 373 660 813 869| | | | <<nowiki> 235 373 660 813 869|</nowiki> | ||
| colspan="2" style="text-align:center;" | 45\470+52\705 | | colspan="2" style="text-align:center;" | 45\470+52\705 | ||
| Line 270: | Line 270: | ||
| | | | | | ||
| style="text-align:center;" | 285.852+418.864 | | style="text-align:center;" | 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> | | | < <span style="line-height: 1.5;"><nowiki>109 173 306 377 403|+</nowiki></span><<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 | | 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> | | style="text-align:center;" | <span style="display: block; text-align: center;">285.714+419.048</span> | ||
| | < 63 100 177 218 233| | | | <<nowiki> 63 100 177 218 233|</nowiki> | ||
| colspan="2" style="text-align:center;" | 6\63+14\189 | | colspan="2" style="text-align:center;" | 6\63+14\189 | ||
| Line 296: | Line 296: | ||
| | | | | | ||
| style="text-align:center;" | 285.513+419.316 | | style="text-align:center;" | 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> | | | <<nowiki> 46 73 129 159 170|+</nowiki><<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 | | style="text-align:center;" | 227.565+267.606 | ||
| | | | | | ||
| Line 306: | Line 304: | ||
| | <span style="display: block; text-align: center;">(49+72)\206</span> | | | <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> | | style="text-align:center;" | <span style="display: block; text-align: center;">285.437+419.4175</span> | ||
| | < 206 327 578~579 713 762| | | | <<nowiki> 206 327 578~579 713 762|</nowiki> | ||
| colspan="2" style="text-align:center;" | 20\206+15\206 | | 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> | | style="text-align:center;" | <span style="display: block; text-align: center;">285.315+419.58</span> | ||
| | < 143 227 401~402 495 529| | | | <<nowiki> 143 227 401~402 495 529|</nowiki> | ||
| colspan="2" style="text-align:center;" | 14\143+31\429 | | colspan="2" style="text-align:center;" | 14\143+31\429 | ||
| Line 338: | Line 336: | ||
| | | | | | ||
| style="text-align:center;" | 285.234+419.688 | | style="text-align:center;" | 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> | | | <<nowiki> 63 100 177 218 233|+</nowiki><<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 | | 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> | | style="text-align:center;" | <span style="display: block; text-align: center;">285.202+419.731</span> | ||
| | < 223 354 626~627 771 825| | | | <<nowiki> 223 354 626~627 771 825|</nowiki> | ||
| colspan="2" style="text-align:center;" | 43\446+49\669 | | 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> | | style="text-align:center;" | <span style="display: block; text-align: center;">285+420</span> | ||
| | < 80 127 225 277 296| | | | <<nowiki> 80 127 225 277 296|</nowiki> | ||
| colspan="2" style="text-align:center;" | 15\160+6\80 | | colspan="2" style="text-align:center;" | 15\160+6\80 | ||
Revision as of 22:15, 28 May 2020
This is an extended version of the Gentle region article.
Margo Schulter, in a 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 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, 133edo, 155edo, 162edo, 167edo, 179edo, 191edo, 201edo, 213edo, 225edo and 237edo, plus 63edo and 80edo in the extended region.
| Generator | Cents | 2-3-7(b)-11-13(b) | Half 8/7+ 1\3 7/61\3 8/7+ Half 7/6 | 8/7+7/6 | |||
|---|---|---|---|---|---|---|---|
| (7+10)\29 | 289.655+413.793 | < 29 46 81 100 107| | 2\29+3\29
82.759+124.138 |
6\29+6\29
248.276+248.276 |
|||
| (25+36)\104 | 288.4615+415.385 | < 104 165 292 360 385| | 10\104+23\312
115.385+88.4615 20\312+23\208 76.923+132.692 |
20\104+23\104
230.769+265.385 |
|||
| (18+26)\75 | 288+416 | < 75 119 210~211 259 277| | 7\75+17\225
112+90.667 14\225+17\150 74.667+136 |
14\75+17\75
224+272 |
|||
| (47+68)\196 | 287.755+416.3265 | < 196 311 549-551 678 725| | 37\392+44\588
113.265+89.796 37/588+22/196 75.51+134.694 |
37/196+44/196
226.531+269.388 |
|||
| 287.713+416.382 | < 29 46 81 100 107|+< 46 73 129 159 170|φ | 119.283+85.7795
79.522+128.769 |
238.566+257.3385 | ||||
| (29+42)\121 | 287.603+416.529 | < 121 192 339~340 419 448| | 23\242+9\121
114.05+89.256 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|φ | 116.8205+87.323
78.617+130.984 |
235.85+261.969 | ||||
| (11+16)\46 | 286.9565+417.391 | < 46 73 129 159 170| | 9\92+10\138
117.391+86.9565 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|φ | 117.925+88.626
77.635+132.9395 |
232.936+265.879 | ||||
| (48+70)\201 | 286.567+417.91 | < 201 319 564 695 703| | 39\402+44\603
116.418+87.56 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| | 15\155+34\465
116.129+87.742 10\155+17\15 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|φ | 115.968+87.842
77.312+131.7365 |
231.935+263.527 | ||||
| (63+92)\264 | 286.364+418.182 | < 264 419 741 913 976| | 51\528+58\792
115.909+87.87 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| | 21\218+8\109
115.596+88.07 7\109+12\10 77.064+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| | 27\281+62\843
115.3025+88.256 18\281+31\28 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|φ | 116.526+89.264
77.684+133.8965 |
233.052+267.793 | ||||
| (41+60)\172 | 286.0465+418.605 | < 172 273 483 595 636| | 33\344+38\516
115.116+88.372 11\172+19\17 76.744+132.558 |
33\172+38\172
230.232+265.116 |
|||
| (56+82)\235 | 285.957+418.723 | < 235 373 660 813 869| | 45\470+52\705
114.894+81.511 15\235+26\23 76.596+132.766 |
45\235+52\235
229.787+265.532 |
|||
| 285.852+418.864 | < 109 173 306 377 403|+< 63 100 177 218 233|φ | 114.963+88.4675
76.642+132.701 |
229.926+265.402 | ||||
| (15+22)\63 | 285.714+419.048 | < 63 100 177 218 233| | 6\63+14\189
114.286+88.88 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|φ | 113.7825+89.2075.855+133.80 | 227.565+267.606 | ||||
| (49+72)\206 | 285.437+419.4175 | < 206 327 578~579 713 762| | 20\206+15\206
116.505+87.37 40\618+45\41 77.67+131.068 |
40\206+45\206
233.01+262.136 |
|||
| (34+50)\143 | 285.315+419.58 | < 143 227 401~402 495 529| | 14\143+31\429
117.4825+86.71 28\429+31\28 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|φ | 113.085+89.636
75.39+134.454 |
226.169+268.909 | ||||
| (54+78)\223 | 285.202+419.731 | < 223 354 626~627 771 825| | 43\446+49\669
115.695+87.892 43\669+49\44 77.13+131.839 |
43\223+49\223
231.39+263.677 |
|||
| (19+28)\80 | 285+420 | < 80 127 225 277 296| | 15\160+6\80
112.5+90 5\80+9\80 75+135 |
15\80+18\80
225+270 |
Boundary of larger "gentle region" | ||