Gentle region (extended version): Difference between revisions

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[[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.
[[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.


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.)
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 [[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.
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.


{| class="wikitable"
{| class="wikitable"
|-
|-
! colspan="2" | Generator
! colspan="2" | Generator
! | Cents
! Cents
! | 2-3-7(b)-11-13(b)
! 2-3-7(b)-11-13(b)
! <span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"><span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">Half 8/7+ 1\3 7/6<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">1\3 8/7+ Half 7/6</span></span></span>
! <span style="background-color: rgba(255,255,255,0); display: block; text-align: center;"><span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">Half 8/7+ 1\3 7/6<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">1\3 8/7+ Half 7/6</span></span></span>
! |8/7+7/6
! 8/7+7/6
! |
! Notes
|-
|-
| |(7+10)\29
| (7+10)\29
| |
|  
| style="text-align:center;" |289.655+413.793
| style="text-align: center;" | 289.655+413.793
| |&lt;<nowiki> 29 46 81 100 107|</nowiki>
| {{val| 29 46 81 100 107 }}
| style="text-align:center;" |2\29+<span style="line-height: 1.5;">3\29</span>
| style="text-align: center;" | 2\29+3\29


82.759+124.138
82.759+124.138
| style="text-align:center;" |6\29+6\29
| style="text-align: center;" | 6\29+6\29


248.276+248.276
248.276+248.276
| |
|  
|-
|-
| |(25+36)\104
| (25+36)\104
| |
|  
| style="text-align:center;" |288.4615+415.385
| style="text-align: center;" | 288.4615+415.385
| |&lt;<nowiki> 104 165 292 360 385|</nowiki>
| {{val| 104 165 292 360 385 }}
| style="text-align:center;" |5\52+''23\312''
| style="text-align: center;" | 5\52+''23\312''


115.385+''88.4615''
115.385+''88.4615''
Line 40: Line 40:


<span style="background-color: rgba(255,255,255,0); text-align: start;">''76.923+132.692''</span>
<span style="background-color: rgba(255,255,255,0); text-align: start;">''76.923+132.692''</span>
| style="text-align:center;" |5\52+23\104
| style="text-align: center;" | 5\52+23\104


230.769+265.385
230.769+265.385
| |
|  
|-
|-
| |(18+26)\75
| (18+26)\75
| |
|  
| style="text-align:center;" |288+416
| style="text-align: center;" | 288+416
| |&lt; 7<span style="line-height: 1.5;"><nowiki>5 119 210~211 259 277|</nowiki></span>
| &lt; 75 119 210~211 259 277|  
| style="text-align:center;" |7\75+''17\225''
| style="text-align: center;" | 7\75+''17\225''


112+''90.667''
112+''90.667''
Line 56: Line 56:


<span style="background-color: rgba(255,255,255,0); text-align: start;">''74.667+136''</span>
<span style="background-color: rgba(255,255,255,0); text-align: start;">''74.667+136''</span>
| style="text-align:center;" |14\75+17\75
| style="text-align: center;" | 14\75+17\75


224+272
224+272
| |
|  
|-
|-
| |
|  
| |(47+68)\196
| (47+68)\196
| style="text-align:center;" |287.755+416.3265
| style="text-align: center;" | 287.755+416.3265
| |&lt;<span style="line-height: 1.5;"><nowiki> 196 311 549-551 678 725|</nowiki></span>
| &lt; 196 311 549-551 678 725|  
| style="text-align:center;" |''37\392+44\588''
| style="text-align: center;" | ''37\392+44\588''


''113.265+89.796''
''113.265+89.796''
Line 72: Line 72:


<span style="background-color: rgba(255,255,255,0); text-align: start;">''75.51''+134.694</span>
<span style="background-color: rgba(255,255,255,0); text-align: start;">''75.51''+134.694</span>
| style="text-align:center;" |37/196+44/196
| style="text-align: center;" | 37/196+44/196


226.531+269.388
226.531+269.388
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" |287.713+416.382
| style="text-align: center;" | 287.713+416.382
| |&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>
| {{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;" | 119.283+85.7795


<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">79.522+128.769</span>
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">79.522+128.769</span>
| style="text-align:center;" |238.566+257.3385
| style="text-align: center;" | 238.566+257.3385
| |
|  
|-
|-
| |
|  
| |(29+42)\121
| (29+42)\121
| style="text-align:center;" |287.603+416.529
| style="text-align: center;" | 287.603+416.529
| |&lt;<nowiki> 121 192 339~340 419 448|</nowiki>
| {{val| 121 192 339~340 419 448 }}
| style="text-align:center;" |''23\242''+9\121
| style="text-align: center;" | ''23\242''+9\121


''114.05''+89.256
''114.05''+89.256
Line 98: Line 98:


<span style="background-color: rgba(255,255,255,0);">''76.033+133.884''</span>
<span style="background-color: rgba(255,255,255,0);">''76.033+133.884''</span>
| style="text-align:center;" |23\121+27\121
| style="text-align: center;" | 23\121+27\121


228.099+267.769
228.099+267.769
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" |287.267+416.978
| style="text-align: center;" | 287.267+416.978
| |&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>
| {{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;" | 116.8205+87.323


<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">78.617+130.984</span>
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">78.617+130.984</span>
| style="text-align:center;" |235.85+261.969
| style="text-align: center;" | 235.85+261.969
| |
|  
|-
|-
| |(11+16)\46
| (11+16)\46
| |
|  
| style="text-align:center;" |286.9565+417.391
| style="text-align: center;" | 286.9565+417.391
| |&lt;<nowiki> 46 73 129 159 170|</nowiki>
| {{val| 46 73 129 159 170 }}
| style="text-align:center;" |''9\92+5\69''
| style="text-align: center;" | ''9\92+5\69''


''117.391+86.9565''
''117.391+86.9565''
Line 124: Line 124:


<span style="background-color: rgba(255,255,255,0);">78.261+130.435</span>
<span style="background-color: rgba(255,255,255,0);">78.261+130.435</span>
| style="text-align:center;" |9\46+5\23
| style="text-align: center;" | 9\46+5\23


234.783+260.87
234.783+260.87
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" |286.587+417.884
| style="text-align: center;" | 286.587+417.884
| |&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>
| {{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;" | 117.925+88.626


<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.635+132.9395</span>
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.635+132.9395</span>
| style="text-align:center;" |232.936+265.879
| style="text-align: center;" | 232.936+265.879
| |
|  
|-
|-
| |
|  
| | (48+70)\201
| (48+70)\201
| style="text-align:center;" |286.567+417.91
| style="text-align: center;" | 286.567+417.91
| |&lt;<nowiki> 201 319 564 695 703|</nowiki>
| {{val| 201 319 564 695 703 }}
| style="text-align:center;" |''13\134+44\603''
| style="text-align: center;" | ''13\134+44\603''


''116.418+87.56''
''116.418+87.56''
Line 150: Line 150:


<span style="background-color: rgba(255,255,255,0);">77.612+131.343</span>
<span style="background-color: rgba(255,255,255,0);">77.612+131.343</span>
| style="text-align:center;" |39\201+44\201
| style="text-align: center;" | 39\201+44\201


232.836+262.687
232.836+262.687
| |
|  
|-
|-
| |
|  
| |(37+54)\155
| (37+54)\155
| style="text-align:center;" |286.452+418.0645
| style="text-align: center;" | 286.452+418.0645
| |&lt;<nowiki> 155 246 435 536 573|</nowiki>
| {{val| 155 246 435 536 573 }}
| style="text-align:center;" | 3\31+''34\465''
| style="text-align: center;" | 3\31+''34\465''


116.129+''87.742''
116.129+''87.742''
Line 166: Line 166:


<span style="background-color: rgba(255,255,255,0);">77.419+131.613</span>
<span style="background-color: rgba(255,255,255,0);">77.419+131.613</span>
| style="text-align:center;" |30\155+34\155
| style="text-align: center;" | 30\155+34\155


232.258+263.226
232.258+263.226
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" |286.387+418.151
| style="text-align: center;" | 286.387+418.151
| |&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>
| {{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;" | 115.968+87.842


<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.312+131.7365</span>
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.312+131.7365</span>
| style="text-align:center;" |231.935+263.527
| style="text-align: center;" | 231.935+263.527
| |
|  
|-
|-
| |
|  
| |(63+92)\264
| (63+92)\264
| style="text-align:center;" |286.364+418.182
| style="text-align: center;" | 286.364+418.182
| |&lt;<nowiki> 264 419 741 913 976|</nowiki>
| {{val| 264 419 741 913 976 }}
| style="text-align:center;" |''51\528+29\396''
| style="text-align: center;" | ''51\528+29\396''


''115.909+87.87''
''115.909+87.87''
Line 192: Line 192:


<span style="background-color: rgba(255,255,255,0);">77.273+131.818</span>
<span style="background-color: rgba(255,255,255,0);">77.273+131.818</span>
| style="text-align:center;" | 51\264+58\264
| style="text-align: center;" | 51\264+58\264


231.818+263.636
231.818+263.636
| |
|  
|-
|-
| |(26+38)\109
| (26+38)\109
| |
|  
| style="text-align:center;" |286.2385+418.349
| style="text-align: center;" | 286.2385+418.349
| |&lt;<nowiki> 109 173 306 377 403|</nowiki>
| {{val| 109 173 306 377 403 }}
| style="text-align:center;" |''21\218''+8\109
| style="text-align: center;" | ''21\218''+8\109


''115.596''+88.07
''115.596''+88.07
Line 208: Line 208:


<span style="background-color: rgba(255,255,255,0);">77.064+132.11</span>
<span style="background-color: rgba(255,255,255,0);">77.064+132.11</span>
| style="text-align:center;" |21\109+24\109
| style="text-align: center;" | 21\109+24\109


231.192+264.22
231.192+264.22
| |Boundary of smaller "gentle region"
| Boundary of smaller "gentle region"
|-
|-
| |
|  
| |(67+98)\281
| (67+98)\281
| style="text-align:center;" |286.121+418.505
| style="text-align: center;" | 286.121+418.505
| |&lt;<nowiki> 281 446 789 972 1039|</nowiki>
| {{val| 281 446 789 972 1039 }}
| style="text-align:center;" |27\281+''62\843''
| style="text-align: center;" | 27\281+''62\843''


115.3025+''88.256''
115.3025+''88.256''
Line 224: Line 224:


<span style="background-color: rgba(255,255,255,0);">76.868+132.384</span>
<span style="background-color: rgba(255,255,255,0);">76.868+132.384</span>
| style="text-align:center;" |54\281+62\281
| style="text-align: center;" | 54\281+62\281


230.605+264.769
230.605+264.769
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" |286.101+418.533
| style="text-align: center;" | 286.101+418.533
| |&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>
| {{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;" | 116.526+89.264


<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.684+133.8965</span>
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">77.684+133.8965</span>
| style="text-align:center;" |233.052+267.793
| style="text-align: center;" | 233.052+267.793
| |
|  
|-
|-
| |
|  
| |(41+60)\172
| (41+60)\172
| style="text-align:center;" |286.0465+418.605
| style="text-align: center;" | 286.0465+418.605
| |&lt;<nowiki> 172 273 483 595 636|</nowiki>
| {{val| 172 273 483 595 636 }}
| style="text-align:center;" |''33\344+19\258''
| style="text-align: center;" | ''33\344+19\258''


''115.116+88.372''
''115.116+88.372''
Line 250: Line 250:


<span style="background-color: rgba(255,255,255,0);">76.744+132.558</span>
<span style="background-color: rgba(255,255,255,0);">76.744+132.558</span>
| style="text-align:center;" |33\172+38\172
| style="text-align: center;" | 33\172+38\172


230.232+265.116
230.232+265.116
| |
|  
|-
|-
| |
|  
| |(56+82)\235
| (56+82)\235
| style="text-align:center;" |285.957+418.723
| style="text-align: center;" | 285.957+418.723
| |&lt;<nowiki> 235 373 660 813 869|</nowiki>
| {{val| 235 373 660 813 869 }}
| style="text-align:center;" |''9\94+52\705''
| style="text-align: center;" | ''9\94+52\705''


114.894+81.511
114.894+81.511
Line 266: Line 266:


<span style="background-color: rgba(255,255,255,0);">76.596+132.766</span>
<span style="background-color: rgba(255,255,255,0);">76.596+132.766</span>
| style="text-align:center;" |9\47+52\235
| style="text-align: center;" | 9\47+52\235


229.787+265.532
229.787+265.532
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" |285.852+418.864
| style="text-align: center;" | 285.852+418.864
| |&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>
| {{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;" | 114.963+88.4675


<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">76.642+132.701</span>
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">76.642+132.701</span>
| style="text-align:center;" |229.926+265.402
| style="text-align: center;" | 229.926+265.402
| |
|  
|-
|-
| |<span style="display: block; text-align: center;">(15+22)\63</span>
| <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>
| style="text-align: center;" | <span style="display: block; text-align: center;">285.714+419.048</span>
| |&lt;<nowiki> 63 100 177 218 233|</nowiki>
| {{val| 63 100 177 218 233 }}
| style="text-align:center;" |2\21+''2\27''
| style="text-align: center;" | 2\21+''2\27''


114.286+''88.889''
114.286+''88.889''
Line 292: Line 292:


<span style="background-color: rgba(255,255,255,0); text-align: start;">76.1905+133.333</span>
<span style="background-color: rgba(255,255,255,0); text-align: start;">76.1905+133.333</span>
| style="text-align:center;" |4\21+2\9
| style="text-align: center;" | 4\21+2\9


228.571+266.667
228.571+266.667
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" |285.513+419.316
| style="text-align: center;" | 285.513+419.316
| |&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>
| {{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;" | <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
| style="text-align: center;" | 227.565+267.606
| |
|  
|-
|-
| |
|  
| |<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>
| |&lt;<nowiki> 206 327 578~579 713 762|</nowiki>
| {{val| 206 327 578~579 713 762 }}
| style="text-align:center;" |10\103+15\206
| style="text-align: center;" | 10\103+15\206


116.505+87.37
116.505+87.37
Line 316: Line 316:


<span style="background-color: rgba(255,255,255,0);">''77.67+131.068''</span>
<span style="background-color: rgba(255,255,255,0);">''77.67+131.068''</span>
| style="text-align:center;" |20\103+45\206
| style="text-align: center;" | 20\103+45\206


233.01+262.136
233.01+262.136
| |
|  
|-
|-
| |<span style="display: block; text-align: center;">(34+50)\143</span>
| <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>
| style="text-align: center;" | <span style="display: block; text-align: center;">285.315+419.58</span>
| |&lt;<nowiki> 143 227 401~402 495 529|</nowiki>
| {{val| 143 227 401~402 495 529 }}
| style="text-align:center;" |''14\143+31\429''
| style="text-align: center;" | ''14\143+31\429''


117.4825+86.71
117.4825+86.71
Line 332: Line 332:


<span style="background-color: rgba(255,255,255,0);">''78.322+130.07''</span>
<span style="background-color: rgba(255,255,255,0);">''78.322+130.07''</span>
| style="text-align:center;" |28\143+31\143
| style="text-align: center;" | 28\143+31\143


234.965+260.14
234.965+260.14
| |
|  
|-
|-
| |
|  
| |
|  
| style="text-align:center;" | 285.234+419.688
| style="text-align: center;" | 285.234+419.688
| |&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>
| {{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;" | 113.085+89.636


<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">75.39+134.454</span>
<span style="background-color: rgba(255,255,255,0); display: block; text-align: center;">75.39+134.454</span>
| style="text-align:center;" | 226.169+268.909
| style="text-align: center;" | 226.169+268.909
| |
|  
|-
|-
| |<span style="display: block; text-align: center;">(54+78)\223</span>
| <span style="display: block; text-align: center;">(54+78)\223</span>
| |
|  
| style="text-align:center;" |
| style="text-align: center;" |  
<span style="display: block; text-align: center;">285.202+419.731</span>
<span style="display: block; text-align: center;">285.202+419.731</span>
| |&lt;<nowiki> 223 354 626~627 771 825|</nowiki>
| {{val| 223 354 626~627 771 825 }}
| style="text-align:center;" |''43\446+49\669''
| style="text-align: center;" | ''43\446+49\669''


115.695+87.892
115.695+87.892
Line 359: Line 359:


<span style="background-color: rgba(255,255,255,0);">''77.13+131.839''</span>
<span style="background-color: rgba(255,255,255,0);">''77.13+131.839''</span>
| style="text-align:center;" |43\223+49\223
| style="text-align: center;" | 43\223+49\223


231.39+263.677
231.39+263.677
| |
|  
|-
|-
| |<span style="display: block; text-align: center;">(19+28)\80</span>
| <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>
| style="text-align: center;" | <span style="display: block; text-align: center;">285+420</span>
| |&lt;<nowiki> 80 127 225 277 296|</nowiki>
| {{val| 80 127 225 277 296 }}
| style="text-align:center;" |''3\32+3\40''
| style="text-align: center;" | ''3\32+3\40''


''112.5+90''
''112.5+90''
Line 375: Line 375:


<span style="background-color: rgba(255,255,255,0);">75+135</span>
<span style="background-color: rgba(255,255,255,0);">75+135</span>
| style="text-align:center;" |3\16+9\40
| style="text-align: center;" | 3\16+9\40


225+270
225+270
| |Boundary of larger "gentle region"
| Boundary of larger "gentle region"
|-
|-
| style="text-align:center;" | (4+6)\17
| style="text-align: center;" | (4+6)\17
|
|
| style="text-align:center;" |282.353+423.529
| style="text-align: center;" | 282.353+423.529
|<nowiki>< 17 27 48 60 63|</nowiki>
|<nowiki>< 17 27 48 60 63|</nowiki>
| style="text-align:center;" |1\17+''4\51''
| style="text-align: center;" | 1\17+''4\51''
70.588+''93.1765''
70.588+''93.1765''


Line 390: Line 390:


''105.882''+70.588
''105.882''+70.588
| style="text-align:center;" |3\17+4\17
| style="text-align: center;" | 3\17+4\17
211.765+282.353
211.765+282.353
|
|
|}
|}
[[Category:Gentle]]
[[Category:Gentle]]
[[Category:Interval region]]
[[Category:Interval region]]
[[Category:Tables]]
[[Category:Tables]]

Latest revision as of 15:16, 16 January 2025

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 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.

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 Notes
(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 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 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

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