3L 7s

From Xenharmonic Wiki
Jump to navigation Jump to search
↖2L 6s↑3L 6s 4L 6s↗
←2L 7s3L 7s4L 7s→
↙2L 8s↓3L 8s 4L 8s↘
Brightest mode LssLssLsss
Period 2/1
Range for bright generator 3\10 (360¢) to 1\3 (400¢)
Range for dark generator 2\3 (800¢) to 7\10 (840¢)
TAMNAMS name sephiroid
TAMNAMS prefix seph-
Parent MOS 3L 4s
Sister MOS 7L 3s
Daughter MOSes 10L 3s, 3L 10s
Equal tunings
Supersoft (L:s = 4:3) 10\33 (363.6¢)
Soft (L:s = 3:2) 7\23 (365.2¢)
Semisoft (L:s = 5:3) 11\36 (366.7¢)
Basic (L:s = 2:1) 4\13 (369.2¢)
Semihard (L:s = 5:2) 9\29 (372.4¢)
Hard (L:s = 3:1) 5\16 (375¢)
Superhard (L:s = 4:1) 6\19 (378.9¢)
Brightest-mode tunings on xenpaper
Supersoft Soft Semisoft Basic Semihard Hard Superhard

3L 7s, named sephiroid in TAMNAMS, is an 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 of 3L 7s
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.

Modes of 3L 7s
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

Scale tree and tuning spectrum of 3L 7s
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
Bright generators smaller 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