3L 7s: Difference between revisions
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Wikispaces>JosephRuhf **Imported revision 343609764 - Original comment: ** |
Wikispaces>keenanpepper **Imported revision 370398288 - Original comment: ** |
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Line 1: | Line 1: | ||
<h2>IMPORTED REVISION FROM WIKISPACES</h2> | <h2>IMPORTED REVISION FROM WIKISPACES</h2> | ||
This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | This is an imported revision from Wikispaces. The revision metadata is included below for reference:<br> | ||
: This revision was by author [[User: | : This revision was by author [[User:keenanpepper|keenanpepper]] and made on <tt>2012-10-04 20:30:15 UTC</tt>.<br> | ||
: The original revision id was <tt> | : The original revision id was <tt>370398288</tt>.<br> | ||
: The revision comment was: <tt></tt><br> | : The revision comment was: <tt></tt><br> | ||
The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | The revision contents are below, presented both in the original Wikispaces Wikitext format, and in HTML exactly as Wikispaces rendered it.<br> | ||
Line 32: | Line 32: | ||
-- | -- | ||
|| | ||||||||||||||||||||~ Generator ||~ Cents ||~ L ||~ s ||~ Comments || | ||
|| | || 3\10 || || || || || || || || || || 360 || 120 || 120 ||= || | ||
|| || || || | || || || || || 13\43 || || || || || || 362.791 || 139.535 || 111.628 ||= || | ||
|| || || || || | || || || || 10\33 || || || || || || || 363.636 || 145.455 || 109.091 ||= || | ||
|| || | || || || 7\23 || || || || || || || || 365.217 || 156.522 || 104.348 ||= || | ||
|| || || || | || || || || 11\36 || || || || || || || 366.667 || 166.667 || 100 ||= || | ||
|| || || || || 15\49 || || || || || || 367.347 || 171.429 || 97.959 ||= || | |||
|| || 4\13 || || || || || || || || || 369.231 || 184.615 || 92.308 ||= Boundary of propriety | |||
|| || | |||
|| || || | |||
|| | |||
|| || || || | |||
|| || | |||
|| || || || || | |||
|| || | |||
|| || | |||
(smaller generators are proper) || | (smaller generators are proper) || | ||
|| || || || || | || || || || || 13\42 || || || || || || 371.429 || 200 || 85.714 ||= || | ||
|| || || || || | || || || || 9\29 || || || || || || || 372.414 || 206.897 || 82.759 ||= || | ||
|| || || || | || || || || || 14\45 || || || || || || 373.333 || 213.333 || 80 ||= || | ||
|| || || | || || || 5\16 || || || || || || || || 375 || 225 || 75 ||= || | ||
|| || || || || | || || || || || 11\35 || || || || || || 377.143 || 240 || 68.571 ||= || | ||
|| || || || || | || || || || 6\19 || || || || || || || 378.947 || 252.632 || 63.158 ||= || | ||
|| || || | || || || || || || || 19\60 || || || || 380 || 260 || 60 ||= Magic is around here || | ||
|| || | || || || || || || 13\41 || || || || || 380.488 || 263.415 || 58.537 ||= || | ||
|| || || || || | || || || || || || || 20\63 || || || || 380.952 || 266.667 || 57.143 ||= || | ||
|| || || || || | || || || || || 7\22 || || || || || || 381.818 || 272.72 || 54.545 ||= || | ||
|| || || || | || || || || || || 8\25 || || || || || 384 || 288 || 48 ||= || | ||
|| || || | || || || || || || || 9\28 || || || || 385.714 || 300 || 42.857 ||= || | ||
|| || || || || | || || || || || || || || 10\31 || || || 387.097 || 309.677 || 38.710 ||= || | ||
|| || | || || || || || || || || || || 21\65 || 387.692 || 313.846 || 36.923 ||= Würschmidt is around here || | ||
|| || || || || || | || || || || || || || || || 11\34 || || 388.235 || 317.647 || 35.294 ||= || | ||
|| | || 1\3 || || || || || || || || || || 400 || 400 || 0 ||= || | ||
|| || || || || | |||
|| || || || | |||
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|| || || || || || | |||
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L=1 s=1 [[10edo]] | L=1 s=1 [[10edo]] | ||
Line 151: | Line 110: | ||
<table class="wiki_table"> | <table class="wiki_table"> | ||
<tr> | <tr> | ||
< | <th colspan="10">Generator<br /> | ||
</th> | |||
<th>Cents<br /> | |||
</th> | |||
<th>L<br /> | |||
</th> | |||
<th>s<br /> | |||
</th> | |||
<th>Comments<br /> | |||
</th> | |||
</ | |||
< | |||
</ | |||
< | |||
</ | |||
< | |||
</ | |||
< | |||
</ | |||
</tr> | </tr> | ||
<tr> | <tr> | ||
<td> | <td>3\10<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 182: | Line 129: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 202: | Line 139: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>360<br /> | ||
</td> | </td> | ||
<td> | <td>120<br /> | ||
</td> | </td> | ||
<td> | <td>120<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 225: | Line 160: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>13\43<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 257: | Line 178: | ||
<td>111.628<br /> | <td>111.628<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 267: | Line 188: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>10\33<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 345: | Line 208: | ||
<td>109.091<br /> | <td>109.091<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 485: | Line 216: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>7\23<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 521: | Line 238: | ||
<td>104.348<br /> | <td>104.348<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 619: | Line 248: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>11\36<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 697: | Line 268: | ||
<td>100<br /> | <td>100<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 753: | Line 280: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>15\49<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 785: | Line 298: | ||
<td>97.959<br /> | <td>97.959<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 791: | Line 304: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>4\13<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 873: | Line 328: | ||
<td>92.308<br /> | <td>92.308<br /> | ||
</td> | </td> | ||
<td>Boundary of propriety<br /> | <td style="text-align: center;">Boundary of propriety<br /> | ||
(smaller generators are proper)<br /> | (smaller generators are proper)<br /> | ||
</td> | </td> | ||
Line 886: | Line 341: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>13\42<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 908: | Line 353: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>371.429<br /> | ||
</td> | </td> | ||
<td><br /> | <td>200<br /> | ||
</td> | </td> | ||
<td> | <td>85.714<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 928: | Line 369: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>9\29<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 949: | Line 380: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>372.414<br /> | ||
</td> | </td> | ||
<td> | <td>206.897<br /> | ||
</td> | </td> | ||
<td> | <td>82.759<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 974: | Line 401: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>14\45<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 996: | Line 413: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>373.333<br /> | ||
</td> | </td> | ||
<td><br /> | <td>213.333<br /> | ||
</td> | </td> | ||
<td> | <td>80<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,014: | Line 427: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>5\16<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 1,042: | Line 443: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>375<br /> | ||
</td> | </td> | ||
<td> | <td>225<br /> | ||
</td> | </td> | ||
<td> | <td>75<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,062: | Line 461: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>11\35<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 1,084: | Line 473: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>377.143<br /> | ||
</td> | </td> | ||
<td><br /> | <td>240<br /> | ||
</td> | </td> | ||
<td> | <td>68.571<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,104: | Line 489: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>6\19<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 1,125: | Line 500: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>378.947<br /> | ||
</td> | </td> | ||
<td> | <td>252.632<br /> | ||
</td> | </td> | ||
<td> | <td>63.158<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,152: | Line 523: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>19\60<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>380<br /> | ||
</td> | </td> | ||
<td> | <td>260<br /> | ||
</td> | </td> | ||
<td> | <td>60<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;">Magic is around here<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,196: | Line 553: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>13\41<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 1,216: | Line 563: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>380.488<br /> | ||
</td> | </td> | ||
<td><br /> | <td>263.415<br /> | ||
</td> | </td> | ||
<td> | <td>58.537<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,239: | Line 582: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td>20\63<br /> | |||
<td><br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 1,260: | Line 593: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>380.952<br /> | ||
</td> | </td> | ||
<td><br /> | <td>266.667<br /> | ||
</td> | </td> | ||
<td> | <td>57.143<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,282: | Line 611: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>7\22<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 1,301: | Line 620: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>381.818<br /> | ||
</td> | </td> | ||
<td> | <td>272.72<br /> | ||
</td> | </td> | ||
<td> | <td>54.545<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,328: | Line 643: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>8\25<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 1,347: | Line 650: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>384<br /> | ||
</td> | </td> | ||
<td> | <td>288<br /> | ||
</td> | </td> | ||
<td> | <td>48<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,371: | Line 672: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td>9\28<br /> | |||
<td><br /> | |||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
Line 1,394: | Line 683: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>385.714<br /> | ||
</td> | </td> | ||
<td> | <td>300<br /> | ||
</td> | </td> | ||
<td> | <td>42.857<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,416: | Line 703: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>10\31<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>387.097<br /> | ||
</td> | </td> | ||
<td> | <td>309.677<br /> | ||
</td> | </td> | ||
<td> | <td>38.710<br /> | ||
</td> | </td> | ||
<td><br /> | <td style="text-align: center;"><br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,459: | Line 732: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
Line 1,478: | Line 741: | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>21\65<br /> | ||
</td> | </td> | ||
<td><br /> | <td>387.692<br /> | ||
</td> | </td> | ||
<td><br /> | <td>313.846<br /> | ||
</td> | </td> | ||
<td> | <td>36.923<br /> | ||
</td> | </td> | ||
<td | <td style="text-align: center;">Würschmidt is around here<br /> | ||
</td> | </td> | ||
</tr> | </tr> | ||
Line 1,503: | Line 762: | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td><br /> | <td>11\34<br /> | ||
</td> | </td> | ||
<td><br /> | <td><br /> | ||
</td> | </td> | ||
<td> | <td>388.235<br /> | ||
</td> | </td> | ||
<td> | <td>317.647<br /> | ||
</td> | </td> | ||
<td> | <td>35.294<br /> | ||
</td> | </td> | ||
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Revision as of 20:30, 4 October 2012
IMPORTED REVISION FROM WIKISPACES
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Original Wikitext content:
=3L+7s "Fair Mosh" (Modi Sephirotorum)= = = Fair Mosh is found in [[Magic|magic]] (chains of the 5th harmonic). It occupies the spectrum from 10edo (L=s) to 3edo (s=0). This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents). In the region of the spectrum around 23 edo (L=3 s=2) , the 17th and 21st harmonics are tempered toward most accurately, which together are a stable harmony. This is the major chord of the modi sephirotorum. Temperament using phi directly approximates the higher Fibonacci harmonics best. If L=s, ie. multiples of 10edo, the 13th harmonic becomes nearly perfect. 121 edo seems to be the first to 'accurately' represent the comma (which might as well be represented accurately as it's quite small). Towards the other end, where the large and small steps are more contrasted, the comma 65/64 is liable to be tempered out, equating 5/4 and 13/8. In this category fall [[13edo]], [[16edo]], [[19edo]], [[22edo]], [[29edo]], and so on. This ends at s=0 which gives multiples of [[3edo]]. Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10) is symmetrical - not ascending but rather descending, and so reminiscent of ancient Greek practice. These scales, and their truncated heptatonic forms referenced below, are strikingly linear in several ways and so seem suited to a similar outlook as traditional western music (modality, baroque tonality, classical tonality, etc. progressing to today) but with higher harmonics. For more details [[http://ia600706.us.archive.org/23/items/TractatumDeModiSephiratorum/ModiSephiratorum.pdf]] (I know it should be "tractatus", changing it is just a bother) There are MODMOS as well, but I haven't explored them yet. There's enough undiscovered harmonic resource already in these to last me a while! Taking this approach to the 13th harmonic also yields heptatonic MOS with similar properties: [[3L 4s|4s+3L "mish"]] in the form of modes of ssLsLsL "led". (ascending) s s s L s s L s s L - Mode Keter s s L s s L s s L s - Chesed s L s s L s s L s s - Netzach L s s L s s L s s s - Malkuth s s L s s L s s s L - Binah s L s s L s s s L s - Tiferet L s s L s s s L s s - Yesod s s L s s s L s s L - Chokmah s L s s s L s s L s - Gevurah L s s s L s s L s s - Hod -- ||||||||||||||||||||~ Generator ||~ Cents ||~ L ||~ s ||~ Comments || || 3\10 || || || || || || || || || || 360 || 120 || 120 ||= || || || || || || 13\43 || || || || || || 362.791 || 139.535 || 111.628 ||= || || || || || 10\33 || || || || || || || 363.636 || 145.455 || 109.091 ||= || || || || 7\23 || || || || || || || || 365.217 || 156.522 || 104.348 ||= || || || || || 11\36 || || || || || || || 366.667 || 166.667 || 100 ||= || || || || || || 15\49 || || || || || || 367.347 || 171.429 || 97.959 ||= || || || 4\13 || || || || || || || || || 369.231 || 184.615 || 92.308 ||= Boundary of propriety (smaller generators are proper) || || || || || || 13\42 || || || || || || 371.429 || 200 || 85.714 ||= || || || || || 9\29 || || || || || || || 372.414 || 206.897 || 82.759 ||= || || || || || || 14\45 || || || || || || 373.333 || 213.333 || 80 ||= || || || || 5\16 || || || || || || || || 375 || 225 || 75 ||= || || || || || || 11\35 || || || || || || 377.143 || 240 || 68.571 ||= || || || || || 6\19 || || || || || || || 378.947 || 252.632 || 63.158 ||= || || || || || || || || 19\60 || || || || 380 || 260 || 60 ||= Magic is around here || || || || || || || 13\41 || || || || || 380.488 || 263.415 || 58.537 ||= || || || || || || || || 20\63 || || || || 380.952 || 266.667 || 57.143 ||= || || || || || || 7\22 || || || || || || 381.818 || 272.72 || 54.545 ||= || || || || || || || 8\25 || || || || || 384 || 288 || 48 ||= || || || || || || || || 9\28 || || || || 385.714 || 300 || 42.857 ||= || || || || || || || || || 10\31 || || || 387.097 || 309.677 || 38.710 ||= || || || || || || || || || || || 21\65 || 387.692 || 313.846 || 36.923 ||= Würschmidt is around here || || || || || || || || || || 11\34 || || 388.235 || 317.647 || 35.294 ||= || || 1\3 || || || || || || || || || || 400 || 400 || 0 ||= || L=1 s=1 [[10edo]] L=2 s=1 [[13edo]] (L=3 s=1 [[16edo]]) L=3 s=2 [[23edo]] (L=4 s=1 [[19edo]]) L=4 s=3 [[33edo]] (L=5 s=1 [[22edo]]) (L=5 s=2 [[29edo]]) L=5 s=3 [[36edo]] L=5 s=4 [[43edo]] (L=6 s=1 [[25edo|25edo)]] L=6 s=5 [[53edo]] L=7 s=6 [[63edo]] L=7 s=5 [[56edo]] L=7 s=4 [[49edo]] etc.
Original HTML content:
<html><head><title>3L 7s</title></head><body><!-- ws:start:WikiTextHeadingRule:0:<h1> --><h1 id="toc0"><a name="x3L+7s "Fair Mosh" (Modi Sephirotorum)"></a><!-- ws:end:WikiTextHeadingRule:0 -->3L+7s "Fair Mosh" (Modi Sephirotorum)</h1> <!-- ws:start:WikiTextHeadingRule:2:<h1> --><h1 id="toc1"><!-- ws:end:WikiTextHeadingRule:2 --> </h1> Fair Mosh is found in <a class="wiki_link" href="/Magic">magic</a> (chains of the 5th harmonic). It occupies the spectrum from 10edo (L=s) to 3edo (s=0).<br /> <br /> This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents). In the region of the spectrum around 23 edo (L=3 s=2) , the 17th and 21st harmonics are tempered toward most accurately, which together are a stable harmony. This is the major chord of the modi sephirotorum. Temperament using phi directly approximates the higher Fibonacci harmonics best.<br /> <br /> If L=s, ie. multiples of 10edo, the 13th harmonic becomes nearly perfect. 121 edo seems to be the first to 'accurately' represent the comma (which might as well be represented accurately as it's quite small). Towards the other end, where the large and small steps are more contrasted, the comma 65/64 is liable to be tempered out, equating 5/4 and 13/8. In this category fall <a class="wiki_link" href="/13edo">13edo</a>, <a class="wiki_link" href="/16edo">16edo</a>, <a class="wiki_link" href="/19edo">19edo</a>, <a class="wiki_link" href="/22edo">22edo</a>, <a class="wiki_link" href="/29edo">29edo</a>, and so on. This ends at s=0 which gives multiples of <a class="wiki_link" href="/3edo">3edo</a>.<br /> <br /> Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10) is symmetrical - not ascending but rather descending, and so reminiscent of ancient Greek practice. These scales, and their truncated heptatonic forms referenced below, are strikingly linear in several ways and so seem suited to a similar outlook as traditional western music (modality, baroque tonality, classical tonality, etc. progressing to today) but with higher harmonics. For more details <a class="wiki_link_ext" href="http://ia600706.us.archive.org/23/items/TractatumDeModiSephiratorum/ModiSephiratorum.pdf" rel="nofollow">http://ia600706.us.archive.org/23/items/TractatumDeModiSephiratorum/ModiSephiratorum.pdf</a><br /> (I know it should be "tractatus", changing it is just a bother)<br /> <br /> There are MODMOS as well, but I haven't explored them yet. There's enough undiscovered harmonic resource already in these to last me a while! Taking this approach to the 13th harmonic also yields heptatonic MOS with similar properties: <a class="wiki_link" href="/3L%204s">4s+3L "mish"</a> in the form of modes of ssLsLsL "led".<br /> <br /> (ascending)<br /> s s s L s s L s s L - Mode Keter<br /> s s L s s L s s L s - Chesed<br /> s L s s L s s L s s - Netzach<br /> L s s L s s L s s s - Malkuth<br /> s s L s s L s s s L - Binah<br /> s L s s L s s s L s - Tiferet<br /> L s s L s s s L s s - Yesod<br /> s s L s s s L s s L - Chokmah<br /> s L s s s L s s L s - Gevurah<br /> L s s s L s s L s s - Hod<br /> <br /> --<br /> <table class="wiki_table"> <tr> <th colspan="10">Generator<br /> </th> <th>Cents<br /> </th> <th>L<br /> </th> <th>s<br /> </th> <th>Comments<br /> </th> </tr> <tr> <td>3\10<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>360<br /> </td> <td>120<br /> </td> <td>120<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>13\43<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>362.791<br /> </td> <td>139.535<br /> </td> <td>111.628<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>10\33<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>363.636<br /> </td> <td>145.455<br /> </td> <td>109.091<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td>7\23<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>365.217<br /> </td> <td>156.522<br /> </td> <td>104.348<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>11\36<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>366.667<br /> </td> <td>166.667<br /> </td> <td>100<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>15\49<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>367.347<br /> </td> <td>171.429<br /> </td> <td>97.959<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td>4\13<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>369.231<br /> </td> <td>184.615<br /> </td> <td>92.308<br /> </td> <td style="text-align: center;">Boundary of propriety<br /> (smaller generators are proper)<br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>13\42<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>371.429<br /> </td> <td>200<br /> </td> <td>85.714<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>9\29<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>372.414<br /> </td> <td>206.897<br /> </td> <td>82.759<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>14\45<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>373.333<br /> </td> <td>213.333<br /> </td> <td>80<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td>5\16<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>375<br /> </td> <td>225<br /> </td> <td>75<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>11\35<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>377.143<br /> </td> <td>240<br /> </td> <td>68.571<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>6\19<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>378.947<br /> </td> <td>252.632<br /> </td> <td>63.158<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>19\60<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>380<br /> </td> <td>260<br /> </td> <td>60<br /> </td> <td style="text-align: center;">Magic is around here<br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>13\41<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>380.488<br /> </td> <td>263.415<br /> </td> <td>58.537<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>20\63<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>380.952<br /> </td> <td>266.667<br /> </td> <td>57.143<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>7\22<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>381.818<br /> </td> <td>272.72<br /> </td> <td>54.545<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>8\25<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>384<br /> </td> <td>288<br /> </td> <td>48<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>9\28<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>385.714<br /> </td> <td>300<br /> </td> <td>42.857<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>10\31<br /> </td> <td><br /> </td> <td><br /> </td> <td>387.097<br /> </td> <td>309.677<br /> </td> <td>38.710<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>21\65<br /> </td> <td>387.692<br /> </td> <td>313.846<br /> </td> <td>36.923<br /> </td> <td style="text-align: center;">Würschmidt is around here<br /> </td> </tr> <tr> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>11\34<br /> </td> <td><br /> </td> <td>388.235<br /> </td> <td>317.647<br /> </td> <td>35.294<br /> </td> <td style="text-align: center;"><br /> </td> </tr> <tr> <td>1\3<br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td><br /> </td> <td>400<br /> </td> <td>400<br /> </td> <td>0<br /> </td> <td style="text-align: center;"><br /> </td> </tr> </table> <br /> L=1 s=1 <a class="wiki_link" href="/10edo">10edo</a><br /> L=2 s=1 <a class="wiki_link" href="/13edo">13edo</a><br /> <br /> (L=3 s=1 <a class="wiki_link" href="/16edo">16edo</a>)<br /> L=3 s=2 <a class="wiki_link" href="/23edo">23edo</a><br /> <br /> (L=4 s=1 <a class="wiki_link" href="/19edo">19edo</a>)<br /> L=4 s=3 <a class="wiki_link" href="/33edo">33edo</a><br /> <br /> (L=5 s=1 <a class="wiki_link" href="/22edo">22edo</a>)<br /> (L=5 s=2 <a class="wiki_link" href="/29edo">29edo</a>)<br /> L=5 s=3 <a class="wiki_link" href="/36edo">36edo</a><br /> L=5 s=4 <a class="wiki_link" href="/43edo">43edo</a><br /> <br /> (L=6 s=1 <a class="wiki_link" href="/25edo">25edo)</a><br /> L=6 s=5 <a class="wiki_link" href="/53edo">53edo</a><br /> <br /> L=7 s=6 <a class="wiki_link" href="/63edo">63edo</a><br /> L=7 s=5 <a class="wiki_link" href="/56edo">56edo</a><br /> L=7 s=4 <a class="wiki_link" href="/49edo">49edo</a><br /> etc.</body></html>