3L 7s
| ↖2L 6s | ↑3L 6s | 4L 6s↗ |
| ←2L 7s | 3L 7s | 4L 7s→ |
| ↙2L 8s | ↓3L 8s | 4L 8s↘ |
┌╥┬┬╥┬┬╥┬┬┬┐ │║││║││║││││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sssLssLssL
3L 10s
3L 7s, named sephiroid in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 7 small steps, repeating every octave. Generators that produce this scale range from 360¢ to 400¢, or from 800¢ to 840¢.
Name
TAMNAMS suggests the temperament-agnostic name sephiroid for this scale, in reference to Kosmorsky's Tracatum de Modi Sephiratorum.
Intervals
- This article assumes TAMNAMS for naming step ratios, mossteps, and mosdegrees.
| Intervals (with relation to root) | Size | Abbrev. | ||
|---|---|---|---|---|
| Generic | Specific | L's and s's | Range in cents | |
| 0-sephstep (root) | Perfect 0-sephstep | 0 | 0.0¢ | P0ms |
| 1-sephstep | Minor 1-sephstep | s | 0.0¢ to 120.0¢ | m1ms |
| Major 1-sephstep | L | 120.0¢ to 400.0¢ | M1ms | |
| 2-sephstep | Minor 2-sephstep | 2s | 0.0¢ to 240.0¢ | m2ms |
| Major 2-sephstep | L + s | 240.0¢ to 400.0¢ | M2ms | |
| 3-sephstep | Diminished 3-sephstep | 3s | 0.0¢ to 360.0¢ | d3ms |
| Perfect 3-sephstep | L + 2s | 360.0¢ to 400.0¢ | P3ms | |
| 4-sephstep | Minor 4-sephstep | L + 3s | 400.0¢ to 480.0¢ | m4ms |
| Major 4-sephstep | 2L + 2s | 480.0¢ to 800.0¢ | M4ms | |
| 5-sephstep | Minor 5-sephstep | L + 4s | 400.0¢ to 600.0¢ | m5ms |
| Major 5-sephstep | 2L + 3s | 600.0¢ to 800.0¢ | M5ms | |
| 6-sephstep | Minor 6-sephstep | L + 5s | 400.0¢ to 720.0¢ | m6ms |
| Major 6-sephstep | 2L + 4s | 720.0¢ to 800.0¢ | M6ms | |
| 7-sephstep | Perfect 7-sephstep | 2L + 5s | 800.0¢ to 840.0¢ | P7ms |
| Augmented 7-sephstep | 3L + 4s | 840.0¢ to 1200.0¢ | A7ms | |
| 8-sephstep | Minor 8-sephstep | 2L + 6s | 800.0¢ to 960.0¢ | m8ms |
| Major 8-sephstep | 3L + 5s | 960.0¢ to 1200.0¢ | M8ms | |
| 9-sephstep | Minor 9-sephstep | 2L + 7s | 800.0¢ to 1080.0¢ | m9ms |
| Major 9-sephstep | 3L + 6s | 1080.0¢ to 1200.0¢ | M9ms | |
| 10-sephstep (octave) | Perfect 10-sephstep | 3L + 7s | 1200.0¢ | P10ms |
Theory
The modi sephiratorum
This MOS can represent tempered-flat chains of the 13th harmonic, which approximates phi (~833 cents).
With sephiroid scales with a soft-of-basic step ratio (around L:s = 3:2, or 23edo), 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.
Scales approaching an equalized step ratio (L:s = 1:1, or 10edo) contain a 13th harmonic that's nearly perfect. 121edo seems to be the first to 'accurately' represent the comma[clarification needed]. Scales approaching a collapsed step ratio (L:s = 1:0, or 3edo) have the comma 65/64 liable to be tempered out, thus equating 8/5 and 13/8. Edos include 13edo, 16edo, 19edo, 22edo, 29edo, and others.
Harmonically, the arrangement forming a chord (degrees 0, 1, 4, 7, 10)[clarification needed] 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.
There are MODMOS as well, but Kosmorsky has not explored them yet, as "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
Mode names are described by Kosmorsky, which use names from the Sefirot (or sephiroth). Kosmorsky describes the mode Keter to be akin to the lydian mode of 5L 2s, and the mode Malkuth like the locrian mode.
| UDP | Rotational order | Step pattern | Mode names |
|---|---|---|---|
| 9|0 | 1 | LssLssLsss | Malkuth |
| 8|1 | 4 | LssLsssLss | Yesod |
| 7|2 | 7 | LsssLssLss | Hod |
| 6|3 | 10 | sLssLssLss | Netzach |
| 5|4 | 3 | sLssLsssLs | Tiferet |
| 4|5 | 6 | sLsssLssLs | Gevurah |
| 3|6 | 9 | ssLssLssLs | Chesed |
| 2|7 | 2 | ssLssLsssL | Binah |
| 1|8 | 5 | ssLsssLssL | Chokmah |
| 0|9 | 8 | sssLssLssL | Keter |
Scale tree
| Generator (in steps of edo) | Cents | Step ratio | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | |||||||
| 3\10 | 360.000 | 840.000 | 1:1 | 1.000 | Equalized 3L 7s | |||||
| 16\53 | 362.264 | 837.736 | 6:5 | 1.200 | Submajor | |||||
| 13\43 | 362.791 | 837.209 | 5:4 | 1.250 | ||||||
| 23\76 | 363.158 | 836.842 | 9:7 | 1.286 | ||||||
| 10\33 | 363.636 | 836.364 | 4:3 | 1.333 | Supersoft 3L 7s | |||||
| 27\89 | 364.045 | 835.955 | 11:8 | 1.375 | ||||||
| 17\56 | 364.286 | 835.714 | 7:5 | 1.400 | ||||||
| 24\79 | 364.557 | 835.443 | 10:7 | 1.429 | ||||||
| 7\23 | 365.217 | 834.783 | 3:2 | 1.500 | Soft 3L 7s | |||||
| 25\82 | 365.854 | 834.146 | 11:7 | 1.571 | ||||||
| 18\59 | 366.102 | 833.898 | 8:5 | 1.600 | ||||||
| 29\95 | 366.316 | 833.684 | 13:8 | 1.625 | Unnamed golden tuning | |||||
| 11\36 | 366.667 | 833.333 | 5:3 | 1.667 | Semisoft 3L 7s | |||||
| 26\85 | 367.059 | 832.941 | 12:7 | 1.714 | ||||||
| 15\49 | 367.347 | 832.653 | 7:4 | 1.750 | ||||||
| 19\62 | 367.742 | 832.258 | 9:5 | 1.800 | ||||||
| 4\13 | 369.231 | 830.769 | 2:1 | 2.000 | Basic 3L 7s Scales with tunings softer than this are proper | |||||
| 17\55 | 370.909 | 829.091 | 9:4 | 2.250 | ||||||
| 13\42 | 371.429 | 828.571 | 7:3 | 2.333 | ||||||
| 22\71 | 371.831 | 828.169 | 12:5 | 2.400 | ||||||
| 9\29 | 372.414 | 827.586 | 5:2 | 2.500 | Semihard 3L 7s Sephiroth | |||||
| 23\74 | 372.973 | 827.027 | 13:5 | 2.600 | Golden sephiroth | |||||
| 14\45 | 373.333 | 826.667 | 8:3 | 2.667 | ||||||
| 19\61 | 373.770 | 826.230 | 11:4 | 2.750 | ||||||
| 5\16 | 375.000 | 825.000 | 3:1 | 3.000 | Hard 3L 7s | |||||
| 16\51 | 376.471 | 823.529 | 10:3 | 3.333 | ||||||
| 11\35 | 377.143 | 822.857 | 7:2 | 3.500 | ||||||
| 17\54 | 377.778 | 822.222 | 11:3 | 3.667 | Muggles | |||||
| 6\19 | 378.947 | 821.053 | 4:1 | 4.000 | Superhard 3L 7s Magic/horcrux | |||||
| 13\41 | 380.488 | 819.512 | 9:2 | 4.500 | Magic/witchcraft | |||||
| 7\22 | 381.818 | 818.182 | 5:1 | 5.000 | Magic/telepathy | |||||
| 8\25 | 384.000 | 816.000 | 6:1 | 6.000 | Würschmidt↓ | |||||
| 1\3 | 400.000 | 800.000 | 1:0 | → ∞ | Collapsed 3L 7s | |||||
External links
- Tractatum de Modi Sephiratorum by Kosmorsky