Gentle region (extended version)

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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"