3L 7s
↖ 2L 6s | ↑3L 6s | 4L 6s ↗ |
← 2L 7s | 3L 7s | 4L 7s → |
↙ 2L 8s | ↓3L 8s | 4L 8s ↘ |
┌╥┬┬╥┬┬╥┬┬┬┐ │║││║││║││││ ││││││││││││ └┴┴┴┴┴┴┴┴┴┴┘
sssLssLssL
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 | Steps subtended | Range in cents | Average of HE (from HE Calc) |
Min of HE | ||
---|---|---|---|---|---|---|
Generic[1] | Specific[2] | Abbrev.[3] | ||||
0-sephstep | Perfect 0-sephstep | P0ms | 0 | 0.0¢ | ~2.4654 nats | ~2.4654 nats |
1-sephstep | Minor 1-sephstep | m1ms | s | 0.0¢ to 120.0¢ | ~4.7039 nats | ~4.6644 nats |
Major 1-sephstep | M1ms | L | 120.0¢ to 400.0¢ | ~4.5991 nats | ~4.5794 nats | |
2-sephstep | Minor 2-sephstep | m2ms | 2s | 0.0¢ to 240.0¢ | ~4.6036 nats | ~4.5842 nats |
Major 2-sephstep | M2ms | L + s | 240.0¢ to 400.0¢ | ~4.5661 nats | ~4.5379 nats | |
3-sephstep | Diminished 3-sephstep | d3ms | 3s | 0.0¢ to 360.0¢ | ~4.5705 nats | ~4.5385 nats |
Perfect 3-sephstep | P3ms | L + 2s | 360.0¢ to 400.0¢ | ~4.5561 nats | ~4.5066 nats | |
4-sephstep | Minor 4-sephstep | m4ms | L + 3s | 400.0¢ to 480.0¢ | ~4.6041 nats | ~4.5769 nats |
Major 4-sephstep | M4ms | 2L + 2s | 480.0¢ to 800.0¢ | ~4.5670 nats | ~4.4205 nats | |
5-sephstep | Minor 5-sephstep | m5ms | L + 4s | 400.0¢ to 600.0¢ | ~4.5634 nats | ~4.3884 nats |
Major 5-sephstep | M5ms | 2L + 3s | 600.0¢ to 800.0¢ | ~4.5520 nats | ~4.1942 nats | |
6-sephstep | Minor 6-sephstep | m6ms | L + 5s | 400.0¢ to 720.0¢ | ~4.5470 nats | ~4.2657 nats |
Major 6-sephstep | M6ms | 2L + 4s | 720.0¢ to 800.0¢ | ~4.6051 nats | ~4.5145 nats | |
7-sephstep | Perfect 7-sephstep | P7ms | 2L + 5s | 800.0¢ to 840.0¢ | ~4.5912 nats | ~4.5754 nats |
Augmented 7-sephstep | A7ms | 3L + 4s | 840.0¢ to 1200.0¢ | ~4.5302 nats | ~4.4212 nats | |
8-sephstep | Minor 8-sephstep | m8ms | 2L + 6s | 800.0¢ to 960.0¢ | ~4.5619 nats | ~4.4208 nats |
Major 8-sephstep | M8ms | 3L + 5s | 960.0¢ to 1200.0¢ | ~4.5863 nats | ~4.5502 nats | |
9-sephstep | Minor 9-sephstep | m9ms | 2L + 7s | 800.0¢ to 1080.0¢ | ~4.5845 nats | ~4.5338 nats |
Major 9-sephstep | M9ms | 3L + 6s | 1080.0¢ to 1200.0¢ | ~4.6360 nats | ~4.6123 nats | |
10-sephstep | Perfect 10-sephstep | P10ms | 3L + 7s | 1200.0¢ | ~3.3273 nats | ~3.3273 nats |
- Generic intervals are denoted solely by the number of steps they subtend.
- Specific intervals denote whether an interval is major, minor, augmented, perfect, or diminished.
- Abbreviations can be further shortened to 'ms' if context allows.
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
UDP | Rotational Order | Step pattern | Scale degree (sephdegree) | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |||
9|0 | 1 | LssLssLsss | Perf. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Aug. | Maj. | Maj. | Perf. |
8|1 | 4 | LssLsssLss | Perf. | Maj. | Maj. | Perf. | Maj. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
7|2 | 7 | LsssLssLss | Perf. | Maj. | Maj. | Perf. | Min. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
6|3 | 10 | sLssLssLss | Perf. | Min. | Maj. | Perf. | Min. | Maj. | Maj. | Perf. | Maj. | Maj. | Perf. |
5|4 | 3 | sLssLsssLs | Perf. | Min. | Maj. | Perf. | Min. | Maj. | Maj. | Perf. | Min. | Maj. | Perf. |
4|5 | 6 | sLsssLssLs | Perf. | Min. | Maj. | Perf. | Min. | Min. | Maj. | Perf. | Min. | Maj. | Perf. |
3|6 | 9 | ssLssLssLs | Perf. | Min. | Min. | Perf. | Min. | Min. | Maj. | Perf. | Min. | Maj. | Perf. |
2|7 | 2 | ssLssLsssL | Perf. | Min. | Min. | Perf. | Min. | Min. | Maj. | Perf. | Min. | Min. | Perf. |
1|8 | 5 | ssLsssLssL | Perf. | Min. | Min. | Perf. | Min. | Min. | Min. | Perf. | Min. | Min. | Perf. |
0|9 | 8 | sssLssLssL | Perf. | Min. | Min. | Dim. | Min. | Min. | Min. | Perf. | Min. | Min. | Perf. |
Proposed Names
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 |
---|---|---|---|
8|0 | 1 | LssLssLsss | Malkuth |
7|1 | 4 | LssLsssLss | Yesod |
6|2 | 7 | LsssLssLss | Hod |
5|3 | 10 | sLssLssLss | Netzach |
4|4 | 3 | sLssLsssLs | Tiferet |
3|5 | 6 | sLsssLssLs | Gevurah |
2|6 | 9 | ssLssLssLs | Chesed |
1|7 | 2 | ssLssLsssL | Binah |
0|8 | 5 | ssLsssLssL | Chokmah |
-1|9 | 8 | sssLssLssL | Keter |
Scale tree
Generator(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