3L 7s
| ↖2L 6s | ↑3L 6s | 4L 6s↗ |
| ←2L 7s | 3L 7s | 4L 7s→ |
| ↙2L 8s | ↓3L 8s | 4L 8s↘ |
3L 7s occupies the spectrum from 10edo (L = s) to 3edo (s = 0).
TAMNAMS calls this MOS pattern sephiroid.
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
Scales tree
| Generator | Cents | L | s | L/s | Comments | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| 3\10 | 360.000 | 1 | 1 | 1.000 | ||||||
| 16\53 | 362.264 | 6 | 5 | 1.200 | Submajor | |||||
| 13\43 | 362.791 | 5 | 4 | 1.250 | ||||||
| 23\76 | 363.158 | 9 | 7 | 1.286 | ||||||
| 10\33 | 363.636 | 4 | 3 | 1.333 | ||||||
| 27\89 | 364.045 | 11 | 8 | 1.375 | ||||||
| 17\56 | 364.286 | 7 | 5 | 1.400 | ||||||
| 24\79 | 364.557 | 10 | 7 | 1.428 | ||||||
| 7\23 | 365.217 | 3 | 2 | 1.500 | L/s = 3/2 | |||||
| 25\82 | 365.854 | 11 | 7 | 1.571 | ||||||
| 18\59 | 366.102 | 8 | 5 | 1.600 | ||||||
| 29\95 | 366.316 | 13 | 8 | 1.625 | Unnamed golden tuning | |||||
| 11\36 | 366.667 | 5 | 3 | 1.667 | ||||||
| 26\85 | 367.059 | 12 | 7 | 1.714 | ||||||
| 15\49 | 367.347 | 7 | 4 | 1.750 | ||||||
| 19\62 | 367.742 | 9 | 5 | 1.800 | ||||||
| 4\13 | 369.231 | 2 | 1 | 2.000 | Basic sephiroid (Generators smaller than this are proper) | |||||
| 17\55 | 370.909 | 9 | 4 | 2.250 | ||||||
| 13\42 | 371.429 | 7 | 3 | 2.333 | ||||||
| 22\71 | 371.831 | 12 | 5 | 2.400 | ||||||
| 9\29 | 372.414 | 5 | 2 | 2.500 | Sephiroth | |||||
| 23\74 | 372.973 | 13 | 5 | 2.600 | Golden sephiroth | |||||
| 14\45 | 373.333 | 8 | 3 | 2.667 | ||||||
| 19\61 | 373.770 | 11 | 4 | 2.750 | ||||||
| 5\16 | 375.000 | 3 | 1 | 3.000 | L/s = 3/1 | |||||
| 16\51 | 376.471 | 10 | 3 | 3.333 | ||||||
| 11\35 | 377.143 | 7 | 2 | 3.500 | ||||||
| 17\54 | 377.778 | 11 | 3 | 3.667 | Muggles | |||||
| 6\19 | 378.947 | 4 | 1 | 4.000 | Magic/horcrux | |||||
| 13\41 | 380.488 | 9 | 2 | 4.500 | Magic/witchcraft | |||||
| 7\22 | 381.818 | 5 | 1 | 5.000 | Magic/telepathy | |||||
| 8\25 | 384.000 | 6 | 1 | 6.000 | Würschmidt↓ | |||||
| 1\3 | 400.000 | 1 | 0 | → inf | ||||||