Gentle region (extended version)
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.
Gentle-tempered tone systems are thus "mild" (or, as the name says, "gentle") versions of Superpyth temperament. They allow harmony in the style of medieval Pythagorean harmony, usable for "Neo-gothic" harmony systems; besides, they are possible temperament frameworks for middle-eastern (Arabic, Turkish, Persian) tuning systems, with the special property of delivering a common framework for both Arabic and Turkish music, differing in the degree of tempering. When the tempering of the fifth is "very gentle"/near-just, the interval notated as C-Fb in standard sheet notation (8 fifths down) will be close to a 5/4 major third, as used in Turkish music; while sharper tempering 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 larger Pythagorean - or super-Pythagorean - major third.)
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| | 5\52+23\312
115.385+88.4615 5\78+23\208 76.923+132.692 |
5\52+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+11/98 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+5\69
117.391+86.9565 3\46+5\46 78.261+130.435 |
9\46+5\23
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| | 13\134+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| | 3\31+34\465
116.129+87.742 2\31+17\155 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+29\396
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\109 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\281 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+19\258
115.116+88.372 11\172+19\172 76.744+132.558 |
33\172+38\172
230.232+265.116 |
||
| (56+82)\235 | 285.957+418.723 | < 235 373 660 813 869| | 9\94+52\705
114.894+81.511 3\47+26\235 76.596+132.766 |
9\47+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| | 2\21+2\27
114.286+88.889 4\63+1\9 76.1905+133.333 |
4\21+2\9
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| | 10\103+15\206
116.505+87.37 20\309+45\412 77.67+131.068 |
20\103+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\286 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\446 77.13+131.839 |
43\223+49\223
231.39+263.677 |
||
| (19+28)\80 | 285+420 | < 80 127 225 277 296| | 3\32+3\40
112.5+90 1\16+9\80 75+135 |
3\16+9\40
225+270 |
Boundary of larger "gentle region" | |
| (4+6)\17 | 282.353+423.529 | < 17 27 48 60 63| | 1\17+4\51
70.588+93.1765 3\34+1\17 105.882+70.588 |
3\17+4\17
211.765+282.353 |
||