Gentle region (extended version): Difference between revisions

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 [[Arabic, Turkish, Persian| 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 {{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.

! Cents

! 2-3-7(b)-11-13(b)

! 8/7+7/6

! Notes

| (7+10)\29

|  

| style="text-align: center;" | 289.655+413.793

| {{val| 29 46 81 100 107 }}

| style="text-align: center;" | 2\29+3\29

| style="text-align: center;" | 6\29+6\29

|  

| (25+36)\104

|  

| style="text-align: center;" | 288.4615+415.385

| {{val| 104 165 292 360 385 }}

| style="text-align: center;" | 5\52+''23\312''

| style="text-align: center;" | 5\52+23\104

|  

| (18+26)\75

|  

| style="text-align: center;" | 288+416

| < 75 119 210~211 259 277|  

| style="text-align: center;" | 7\75+''17\225''

| style="text-align: center;" | 14\75+17\75

|  

|  

| (47+68)\196

| style="text-align: center;" | 287.755+416.3265

| < 196 311 549-551 678 725|  

| style="text-align: center;" | ''37\392+44\588''

| style="text-align: center;" | 37/196+44/196

|  

|  

|  

| style="text-align: center;" | 287.713+416.382

| {{val| 29 46 81 100 107 }} + {{val| 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>

| style="text-align: center;" | 119.283+85.7795

| style="text-align: center;" | 238.566+257.3385

|  

|  

| (29+42)\121

| style="text-align: center;" | 287.603+416.529

| {{val| 121 192 339~340 419 448 }}

| style="text-align: center;" | ''23\242''+9\121

| style="text-align: center;" | 23\121+27\121

|  

|  

|  

| style="text-align: center;" | 287.267+416.978

| {{val| 29 46 81 100 107 }} + {{val| 109 173 306 377 403 }}<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>

| style="text-align: center;" | 116.8205+87.323

| style="text-align: center;" | 235.85+261.969

|  

| (11+16)\46

|  

| style="text-align: center;" | 286.9565+417.391

| {{val| 46 73 129 159 170 }}

| style="text-align: center;" | ''9\92+5\69''

| style="text-align: center;" | 9\46+5\23

|  

|  

|  

| style="text-align: center;" | 286.587+417.884

| {{val| 29 46 81 100 107 }} + {{val| 63 100 177 218 233 }}<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>

| style="text-align: center;" | 117.925+88.626

| style="text-align: center;" | 232.936+265.879

|  

|  

| (48+70)\201

| style="text-align: center;" | 286.567+417.91

| {{val| 201 319 564 695 703 }}

| style="text-align: center;" | ''13\134+44\603''

| style="text-align: center;" | 39\201+44\201

|  

|  

| (37+54)\155

| style="text-align: center;" | 286.452+418.0645

| {{val| 155 246 435 536 573 }}

| style="text-align: center;" | 3\31+''34\465''

| style="text-align: center;" | 30\155+34\155

|  

|  

|  

| style="text-align: center;" | 286.387+418.151

| {{val| 46 73 129 159 170 }} + {{val| 109 173 306 377 403 }}<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>

| style="text-align: center;" | 115.968+87.842

| style="text-align: center;" | 231.935+263.527

|  

|  

| (63+92)\264

| style="text-align: center;" | 286.364+418.182

| {{val| 264 419 741 913 976 }}

| style="text-align: center;" | ''51\528+29\396''

| style="text-align: center;" | 51\264+58\264

|  

| (26+38)\109

|  

| style="text-align: center;" | 286.2385+418.349

| {{val| 109 173 306 377 403 }}

| style="text-align: center;" | ''21\218''+8\109

| style="text-align: center;" | 21\109+24\109

| Boundary of smaller "gentle region"

|  

| (67+98)\281

| style="text-align: center;" | 286.121+418.505

| {{val| 281 446 789 972 1039 }}

| style="text-align: center;" | 27\281+''62\843''

| style="text-align: center;" | 54\281+62\281

|  

|  

|  

| style="text-align: center;" | 286.101+418.533

| {{val| 46 73 129 159 170 }} + {{val| 63 100 177 218 233 }}<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>

| style="text-align: center;" | 116.526+89.264

| style="text-align: center;" | 233.052+267.793

|  

|  

| (41+60)\172

| style="text-align: center;" | 286.0465+418.605

| {{val| 172 273 483 595 636 }}

| style="text-align: center;" | ''33\344+19\258''

| style="text-align: center;" | 33\172+38\172

|  

|  

| (56+82)\235

| style="text-align: center;" | 285.957+418.723

| {{val| 235 373 660 813 869 }}

| style="text-align: center;" | ''9\94+52\705''

| style="text-align: center;" | 9\47+52\235

|  

|  

|  

| style="text-align: center;" | 285.852+418.864

| {{val| 109 173 306 377 403 }} + {{val| 63 100 177 218 233 }}<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>

| style="text-align: center;" | 114.963+88.4675

| style="text-align: center;" | 229.926+265.402

|  

| <span style="display: block; text-align: center;">(15+22)\63</span>

|  

| style="text-align: center;" | <span style="display: block; text-align: center;">285.714+419.048</span>

| {{val| 63 100 177 218 233 }}

| style="text-align: center;" | 2\21+''2\27''

| style="text-align: center;" | 4\21+2\9

|  

|  

|  

| style="text-align: center;" | 285.513+419.316

| {{val| 46 73 129 159 170 }} + {{val| 80 127 225 277 296 }}<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>

| 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>

| style="text-align: center;" | 227.565+267.606

|  

|  

| <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>

| {{val| 206 327 578~579 713 762 }}

| style="text-align: center;" | 10\103+15\206

| style="text-align: center;" | 20\103+45\206

|  

| <span style="display: block; text-align: center;">(34+50)\143</span>

|  

| style="text-align: center;" | <span style="display: block; text-align: center;">285.315+419.58</span>

| {{val| 143 227 401~402 495 529 }}

| style="text-align: center;" | ''14\143+31\429''

| style="text-align: center;" | 28\143+31\143

|  

|  

|  

| style="text-align: center;" | 285.234+419.688

| {{val| 63 100 177 218 233 }} + {{val| 80 127 225 277 296 }}<span style="background-color: #ffffff; color: #222222; font-family: arial,sans-serif; font-size: small;">φ</span>

| style="text-align: center;" | 113.085+89.636

| style="text-align: center;" | 226.169+268.909

|  

| <span style="display: block; text-align: center;">(54+78)\223</span>

|  

| style="text-align: center;" |  

| {{val| 223 354 626~627 771 825 }}

| style="text-align: center;" | ''43\446+49\669''

| style="text-align: center;" | 43\223+49\223

|  

| <span style="display: block; text-align: center;">(19+28)\80</span>

|  

| style="text-align: center;" | <span style="display: block; text-align: center;">285+420</span>

| {{val| 80 127 225 277 296 }}

| style="text-align: center;" | ''3\32+3\40''

| style="text-align: center;" | 3\16+9\40

| Boundary of larger "gentle region"

| style="text-align: center;" | (4+6)\17

| style="text-align: center;" | 282.353+423.529

| style="text-align: center;" | 1\17+''4\51''

| style="text-align: center;" | 3\17+4\17