User:Moremajorthanmajor/Greater sephiroid: Difference between revisions

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Line 38: Line 38:
!Cents
!Cents
!Normalized Cents
!Normalized Cents
!''ed13\12''
!L
!L
!s
!s
Line 46: Line 45:
|3\10|| || || || ||
|3\10|| || || || ||
| ||360.000
| ||360.000
|360.000
|360.000||1||1||1.000||
|''390.000''||1||1||1.000||
|-
|-
|
|
Line 58: Line 56:
|361.905
|361.905
|367.742
|367.742
|''392.0635''
|7
|7
|6
|6
Line 66: Line 63:
| || || || || ||16\53
| || || || || ||16\53
| ||362.264
| ||362.264
|369.231
|369.231||6||5||1.200||Submajor
|''392.453''||6||5||1.200||Submajor
|-
|-
|
|
Line 78: Line 74:
|362.500
|362.500
|370.213
|370.213
|''392.708''
|11
|11
|9
|9
Line 86: Line 81:
| || || || ||13\43||
| || || || ||13\43||
| ||362.791
| ||362.791
|371.429
|371.429||5||4||1.250||
|''393.023''||5||4||1.250||
|-
|-
| || || || || ||23\76
| || || || || ||23\76
| ||363.158
| ||363.158
|372.973
|372.973||9||7||1.286||
|''393.421''||9||7||1.286||
|-
|-
|
|
Line 103: Line 96:
|363.303
|363.303
|373.585
|373.585
|''393.578''
|13
|13
|10
|10
Line 111: Line 103:
| || || ||10\33|| ||
| || || ||10\33|| ||
| ||363.636
| ||363.636
|375.000
|375.000||4||3||1.333||
|''393.939''||4||3||1.333||
|-
|-
| || || || || ||27\89
| || || || || ||27\89
| ||364.045
| ||364.045
|376.744
|376.744||11||8||1.375||
|''394.382''||11||8||1.375||
|-
|-
| || || || ||17\56||
| || || || ||17\56||
| ||364.286
| ||364.286
|377.778
|377.778||7||5||1.400||
|''394.643''||7||5||1.400||
|-
|-
| || || || || ||24\79
| || || || || ||24\79
| ||364.557
| ||364.557
|378.947
|378.947||10||7||1.428||
|''394.937''||10||7||1.428||
|-
|-
|
|
Line 138: Line 126:
|364.706
|364.706
|379.592
|379.592
|''395.098''
|13
|13
|9
|9
Line 146: Line 133:
| || ||7\23|| || ||
| || ||7\23|| || ||
| ||365.217
| ||365.217
|381.818
|381.818||3||2||1.500||L/s = 3/2
|''395.652''||3||2||1.500||L/s = 3/2
|-
|-
|
|
Line 158: Line 144:
|365.714
|365.714
|384.000
|384.000
|''396.1905''
|14
|14
|9
|9
Line 166: Line 151:
| || || || || ||25\82
| || || || || ||25\82
| ||365.854
| ||365.854
|384.615
|384.615||11||7||1.571||
|''396.3415''||11||7||1.571||
|-
|-
| || || || ||18\59||
| || || || ||18\59||
| ||366.102
| ||366.102
|385.714
|385.714||8||5||1.600||
|''396.610''||8||5||1.600||
|-
|-
| || || || || ||29\95
| || || || || ||29\95
| ||366.316
| ||366.316
|386.667
|386.667||13||8||1.625||Unnamed golden tuning
|''396.842''||13||8||1.625||Unnamed golden tuning
|-
|-
|
|
Line 188: Line 170:
|366.412
|366.412
|387.097
|387.097
|''396.947''
|18
|18
|11
|11
Line 196: Line 177:
| || || ||11\36|| ||
| || || ||11\36|| ||
| ||366.667
| ||366.667
|388.235
|388.235||5||3||1.667||
|''397.222''||5||3||1.667||
|-
|-
|
|
Line 208: Line 188:
|366.942
|366.942
|389.474
|389.474
|''397.521''
|17
|17
|10
|10
Line 216: Line 195:
| || || || || ||26\85
| || || || || ||26\85
| ||367.059
| ||367.059
|390.000
|390.000||12||7||1.714||
|''397.647''||12||7||1.714||
|-
|-
| || || || ||15\49||
| || || || ||15\49||
| ||367.347
| ||367.347
|391.304
|391.304||7||4||1.750||
|''397.959''||7||4||1.750||
|-
|-
| || || || || ||19\62
| || || || || ||19\62
| ||367.742
| ||367.742
|393.103
|393.103||9||5||1.800||
|''398.387''||9||5||1.800||
|-
|-
|
|
Line 238: Line 214:
|368.000
|368.000
|394.286
|394.286
|''398.667''
|11
|11
|6
|6
Line 246: Line 221:
| ||4\13|| || || ||
| ||4\13|| || || ||
| ||369.231
| ||369.231
|400.000
|400.000||2||1||2.000||Basic sephiroid<br>(Generators smaller than this are proper)
|''400.000''||2||1||2.000||Basic sephiroid<br>(Generators smaller than this are proper)
|-
|-
|
|
Line 258: Line 232:
|370.588
|370.588
|406.452
|406.452
|''401.471''
|11
|11
|5
|5
Line 266: Line 239:
| || || || || ||17\55
| || || || || ||17\55
| ||370.909
| ||370.909
|408.000
|408.000||9||4||2.250||
|''401.818''||9||4||2.250||
|-
|-
|
|
Line 278: Line 250:
|371.134
|371.134
|409.091
|409.091
|''402.062''
|16
|16
|7
|7
Line 286: Line 257:
| || || || ||13\42||
| || || || ||13\42||
| ||371.429
| ||371.429
|410.526
|410.526||7||3||2.333||
|''402.381''||7||3||2.333||
|-
|-
|
|
Line 298: Line 268:
|371.681
|371.681
|411.765
|411.765
|''402.655''
|19
|19
|8
|8
Line 306: Line 275:
| || || || || ||22\71
| || || || || ||22\71
| ||371.831
| ||371.831
|412.500
|412.500||12||5||2.400||
|''402.817''||12||5||2.400||
|-
|-
|
|
Line 318: Line 286:
|372.000
|372.000
|413.333
|413.333
|''403.000''
|17
|17
|7
|7
Line 326: Line 293:
| || || ||9\29|| ||
| || || ||9\29|| ||
| ||372.414
| ||372.414
|415.385
|415.385||5||2||2.500||Sephiroth
|''403.448''||5||2||2.500||Sephiroth
|-
|-
|
|
Line 338: Line 304:
|372.8155
|372.8155
|417.391
|417.391
|''403.8835''
|18
|18
|7
|7
Line 346: Line 311:
| || || || || ||23\74
| || || || || ||23\74
| ||372.973
| ||372.973
|418.182
|418.182||13||5||2.600||
|''404.054''||13||5||2.600||
|-
|-
|
|
Line 358: Line 322:
|373.109
|373.109
|418.868
|418.868
|''404.202''
|21
|21
|8
|8
Line 366: Line 329:
| || || || ||14\45||
| || || || ||14\45||
| ||373.333
| ||373.333
|420.000
|420.000||8||3||2.667||
|''404.444''||8||3||2.667||
|-
|-
|
|
Line 378: Line 340:
|373.585
|373.585
|421.277
|421.277
|''404.717''
|19
|19
|7
|7
Line 386: Line 347:
| || || || || ||19\61
| || || || || ||19\61
| ||373.770
| ||373.770
|422.222
|422.222||11||4||2.750||
|''404.981''||11||4||2.750||
|-
|-
|
|
Line 398: Line 358:
|374.000
|374.000
|423.529
|423.529
|''405.195''
|14
|14
|5
|5
Line 406: Line 365:
| || ||5\16|| || ||
| || ||5\16|| || ||
| ||375.000
| ||375.000
|428.571
|428.571||3||1||3.000||L/s = 3/1
|''406.250''||3||1||3.000||L/s = 3/1
|-
|-
|
|
Line 418: Line 376:
|376.119
|376.119
|434.483
|434.483
|''407.463''
|13
|13
|4
|4
Line 426: Line 383:
| || || || || ||16\51
| || || || || ||16\51
| ||376.471
| ||376.471
|436.364
|436.364||10||3||3.333||
|''407.843''||10||3||3.333||
|-
|-
|
|
Line 438: Line 394:
|376.744
|376.744
|437.838
|437.838
|''408.1395''
|17
|17
|5
|5
Line 446: Line 401:
| || || || ||11\35||
| || || || ||11\35||
| ||377.143
| ||377.143
|440.000
|440.000||7||2||3.500||
|''408.571''||7||2||3.500||
|-
|-
|
|
Line 458: Line 412:
|377.528
|377.528
|442.105
|442.105
|''408.989''
|18
|18
|5
|5
Line 466: Line 419:
| || || || || ||17\54
| || || || || ||17\54
| ||377.778
| ||377.778
|443.478
|443.478||11||3||3.667||Muggles
|''409.259''||11||3||3.667||Muggles
|-
|-
|
|
Line 478: Line 430:
|378.082
|378.082
|445.161
|445.161
|''409.589''
|15
|15
|4
|4
Line 486: Line 437:
| || || ||6\19|| ||
| || || ||6\19|| ||
| ||378.947
| ||378.947
|450.000
|450.000||4||1||4.000||Magic/horcrux
|''410.526''||4||1||4.000||Magic/horcrux
|-
|-
|
|
Line 498: Line 448:
|380.000
|380.000
|456.000
|456.000
|''411.667''
|13
|13
|3
|3
Line 506: Line 455:
| || || || || ||13\41
| || || || || ||13\41
| ||380.488
| ||380.488
|458.8235
|458.8235||9||2||4.500||Magic/witchcraft
|''412.195''||9||2||4.500||Magic/witchcraft
|-
|-
|
|
Line 518: Line 466:
|380.952
|380.952
|461.5385
|461.5385
|''412.698''
|14
|14
|3
|3
Line 526: Line 473:
| || || || ||7\22||
| || || || ||7\22||
| ||381.818
| ||381.818
|466.667
|466.667||5||1||5.000||Magic/telepathy
|''413.636''||5||1||5.000||Magic/telepathy
|-
|-
|
|
Line 538: Line 484:
|382.979
|382.979
|473.684
|473.684
|''414.894''
|11
|11
|2
|2
Line 546: Line 491:
| || || || || ||8\25
| || || || || ||8\25
| ||384.000
| ||384.000
|480.000
|480.000||6||1||6.000||Würschmidt↓
|''416.000''||6||1||6.000||Würschmidt↓
|-
|-
|
|
Line 558: Line 502:
|385.714
|385.714
|490.909
|490.909
|''417.857''
|7
|7
|1
|1
Line 566: Line 509:
|1\3|| || || || ||
|1\3|| || || || ||
| ||400.000
| ||400.000
|600.000
|600.000||1||0||→ inf||
|''433.333''||1||0||→ inf||
|}
|}


Revision as of 00:06, 6 June 2023

3L 7s(<5/2>) occupies the spectrum from 10edo (L = s) to 3edo (s = 0).

TAMNAMS calls this MOS pattern sephiroid (named after the abstract temperament sephiroth).

This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents). In the region of the spectrum around 23edo (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 sephiratorum. Temperament using phi directly approximates the higher Fibonacci harmonics best.

If L = s, i.e. multiples of 10edo, the 13th harmonic becomes nearly perfect. 121edo seems to be the first to 'accurately' represent the comma (which might as well be represented accurately as it is 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 8/5 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 see Kosmorsky's Tractatum de Modi Sephiratorum (Kosmorsky knows it should be "tractatus", but considers changing it is nothing but a bother.)

There are MODMOS as well, but Kosmorsky has not explored them yet. There's enough undiscovered harmonic resources already in these to last me a while! Taking this approach to the 13th harmonic also yields heptatonic MOS with similar properties: 4s+3L "mish" in the form of modes of ssLsLsL "led".

Modes

s s s L s s L s s L - 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

Scale tree

Generator Cents Normalized Cents L s L/s Comments
3\10 360.000 360.000 1 1 1.000
19\63 361.905 367.742 7 6 1.167
16\53 362.264 369.231 6 5 1.200 Submajor
29\96 362.500 370.213 11 9 1.222
13\43 362.791 371.429 5 4 1.250
23\76 363.158 372.973 9 7 1.286
33\109 363.303 373.585 13 10 1.300
10\33 363.636 375.000 4 3 1.333
27\89 364.045 376.744 11 8 1.375
17\56 364.286 377.778 7 5 1.400
24\79 364.557 378.947 10 7 1.428
31\102 364.706 379.592 13 9 1.444
7\23 365.217 381.818 3 2 1.500 L/s = 3/2
32\105 365.714 384.000 14 9 1.556
25\82 365.854 384.615 11 7 1.571
18\59 366.102 385.714 8 5 1.600
29\95 366.316 386.667 13 8 1.625 Unnamed golden tuning
40\131 366.412 387.097 18 11 1.636
11\36 366.667 388.235 5 3 1.667
37\121 366.942 389.474 17 10 1.700
26\85 367.059 390.000 12 7 1.714
15\49 367.347 391.304 7 4 1.750
19\62 367.742 393.103 9 5 1.800
23\75 368.000 394.286 11 6 1.833
4\13 369.231 400.000 2 1 2.000 Basic sephiroid
(Generators smaller than this are proper)
21\68 370.588 406.452 11 5 2.200
17\55 370.909 408.000 9 4 2.250
30\97 371.134 409.091 16 7 2.286
13\42 371.429 410.526 7 3 2.333
35\113 371.681 411.765 19 8 2.375
22\71 371.831 412.500 12 5 2.400
31\100 372.000 413.333 17 7 2.429
9\29 372.414 415.385 5 2 2.500 Sephiroth
32\103 372.8155 417.391 18 7 2.571
23\74 372.973 418.182 13 5 2.600
37\119 373.109 418.868 21 8 2.625 Golden sephiroth
14\45 373.333 420.000 8 3 2.667
33\106 373.585 421.277 19 7 2.714
19\61 373.770 422.222 11 4 2.750
24\77 374.000 423.529 14 5 2.800
5\16 375.000 428.571 3 1 3.000 L/s = 3/1
21\67 376.119 434.483 13 4 3.250
16\51 376.471 436.364 10 3 3.333
27\86 376.744 437.838 17 5 3.400
11\35 377.143 440.000 7 2 3.500
28\89 377.528 442.105 18 5 3.600
17\54 377.778 443.478 11 3 3.667 Muggles
23\73 378.082 445.161 15 4 3.750
6\19 378.947 450.000 4 1 4.000 Magic/horcrux
19\60 380.000 456.000 13 3 4.333
13\41 380.488 458.8235 9 2 4.500 Magic/witchcraft
20\63 380.952 461.5385 14 3 4.667
7\22 381.818 466.667 5 1 5.000 Magic/telepathy
15\47 382.979 473.684 11 2 5.500
8\25 384.000 480.000 6 1 6.000 Würschmidt↓
9\28 385.714 490.909 7 1 7.000
1\3 400.000 600.000 1 0 → inf

See also

Parents of Greater Luachoid

3L 7s (33/16-equivalent) - harmonic subminor ninth tuning

3L 7s (21/10-equivalent) - septimal chromatic minor ninth tuning

3L 7s (19/9-equivalent) - simplest ratio near median edo interval (13\12)

Upper tunings

3L 7s (15/7-equivalent) - septimal diatonic minor ninth tuning

3L 7s (11/5-equivalent) - undecimal neutral ninth tuning

3L 7s (9/4-equivalent) - major ninth tuning

3L 7s (7/3-equivalent) - septimal minor tenth tuning

3L 7s (5/2-equivalent) - major tenth tuning

3L 7s (8/3-equivalent) - Anti-Choralic

3L 7s (11/4-equivalent) - undecimal augmented eleventh/diminished twelfth tuning

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