12L 5s

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↖ 11L 4s↑ 12L 4s 13L 4s ↗
← 11L 5s12L 5s13L 5s →
↙ 11L 6s↓ 12L 6s 13L 6s ↘
 
┌╥╥╥┬╥╥┬╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║║│║║│║║││
│││││││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLLsLLsLLs
sLLsLLsLLLsLLsLLL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 7\17 to 5\12 (494.1¢ to 500.0¢)
Dark 7\12 to 10\17 (700.0¢ to 705.9¢)
TAMNAMS information
Descends from 5L 2s (diatonic)
Ancestor's step ratio range 2:1 to 3:1 (hypohard)
Related MOS scales
Parent 5L 7s
Sister 5L 12s
Daughters 17L 12s, 12L 17s
Neutralized 7L 10s
2-Flought 29L 5s, 12L 22s
Equal tunings
Equalized (L:s = 1:1) 7\17 (494.1¢)
Supersoft (L:s = 4:3) 26\63 (495.2¢)
Soft (L:s = 3:2) 19\46 (495.7¢)
Semisoft (L:s = 5:3) 31\75 (496.0¢)
Basic (L:s = 2:1) 12\29 (496.6¢)
Semihard (L:s = 5:2) 29\70 (497.1¢)
Hard (L:s = 3:1) 17\41 (497.6¢)
Superhard (L:s = 4:1) 22\53 (498.1¢)
Collapsed (L:s = 1:0) 5\12 (500.0¢)

12L 5s, also called p-enharmonic, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 12 large steps and 5 small steps, repeating every octave. 12L 5s is a grandchild scale of 5L 2s, expanding it by 10 tones. Generators that produce this scale range from 494.1¢ to 500¢, or from 700¢ to 705.9¢. Temperaments supported by this scale include those under the Pythagorean and schismic families, characterized by a diesis (the difference between a large step and two small steps) that is smaller than the chroma.

The leapday/leapweek version is proper, but the Pythagorean/schismic version is improper (it does not become proper until you add 12 more notes to form the schismic 29-note scale).

Intervals

Intervals of 12L 5s
Intervals Steps
subtended		 Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0¢
1-mosstep Minor 1-mosstep m1ms s 0.0¢ to 70.6¢
Major 1-mosstep M1ms L 70.6¢ to 100.0¢
2-mosstep Minor 2-mosstep m2ms L + s 100.0¢ to 141.2¢
Major 2-mosstep M2ms 2L 141.2¢ to 200.0¢
3-mosstep Minor 3-mosstep m3ms 2L + s 200.0¢ to 211.8¢
Major 3-mosstep M3ms 3L 211.8¢ to 300.0¢
4-mosstep Minor 4-mosstep m4ms 2L + 2s 200.0¢ to 282.4¢
Major 4-mosstep M4ms 3L + s 282.4¢ to 300.0¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 300.0¢ to 352.9¢
Major 5-mosstep M5ms 4L + s 352.9¢ to 400.0¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 400.0¢ to 423.5¢
Major 6-mosstep M6ms 5L + s 423.5¢ to 500.0¢
7-mosstep Diminished 7-mosstep d7ms 4L + 3s 400.0¢ to 494.1¢
Perfect 7-mosstep P7ms 5L + 2s 494.1¢ to 500.0¢
8-mosstep Minor 8-mosstep m8ms 5L + 3s 500.0¢ to 564.7¢
Major 8-mosstep M8ms 6L + 2s 564.7¢ to 600.0¢
9-mosstep Minor 9-mosstep m9ms 6L + 3s 600.0¢ to 635.3¢
Major 9-mosstep M9ms 7L + 2s 635.3¢ to 700.0¢
10-mosstep Perfect 10-mosstep P10ms 7L + 3s 700.0¢ to 705.9¢
Augmented 10-mosstep A10ms 8L + 2s 705.9¢ to 800.0¢
11-mosstep Minor 11-mosstep m11ms 7L + 4s 700.0¢ to 776.5¢
Major 11-mosstep M11ms 8L + 3s 776.5¢ to 800.0¢
12-mosstep Minor 12-mosstep m12ms 8L + 4s 800.0¢ to 847.1¢
Major 12-mosstep M12ms 9L + 3s 847.1¢ to 900.0¢
13-mosstep Minor 13-mosstep m13ms 9L + 4s 900.0¢ to 917.6¢
Major 13-mosstep M13ms 10L + 3s 917.6¢ to 1000.0¢
14-mosstep Minor 14-mosstep m14ms 9L + 5s 900.0¢ to 988.2¢
Major 14-mosstep M14ms 10L + 4s 988.2¢ to 1000.0¢
15-mosstep Minor 15-mosstep m15ms 10L + 5s 1000.0¢ to 1058.8¢
Major 15-mosstep M15ms 11L + 4s 1058.8¢ to 1100.0¢
16-mosstep Minor 16-mosstep m16ms 11L + 5s 1100.0¢ to 1129.4¢
Major 16-mosstep M16ms 12L + 4s 1129.4¢ to 1200.0¢
17-mosstep Perfect 17-mosstep P17ms 12L + 5s 1200.0¢

Modes

Scale degrees of the modes of 12L 5s 
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
16|0 1 LLLsLLsLLLsLLsLLs Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Aug. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
15|1 8 LLLsLLsLLsLLLsLLs Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
14|2 15 LLsLLLsLLsLLLsLLs Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Perf.
13|3 5 LLsLLLsLLsLLsLLLs Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf.
12|4 12 LLsLLsLLLsLLsLLLs Perf. Maj. Maj. Min. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Maj. Maj. Perf.
11|5 2 LLsLLsLLLsLLsLLsL Perf. Maj. Maj. Min. Maj. Maj. Min. Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Maj. Min. Perf.
10|6 9 LLsLLsLLsLLLsLLsL Perf. Maj. Maj. Min. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Maj. Maj. Min. Perf.
9|7 16 LsLLLsLLsLLLsLLsL Perf. Maj. Min. Min. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Maj. Min. Maj. Maj. Min. Perf.
8|8 6 LsLLLsLLsLLsLLLsL Perf. Maj. Min. Min. Maj. Maj. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Maj. Maj. Min. Perf.
7|9 13 LsLLsLLLsLLsLLLsL Perf. Maj. Min. Min. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Maj. Maj. Min. Perf.
6|10 3 LsLLsLLLsLLsLLsLL Perf. Maj. Min. Min. Maj. Min. Min. Perf. Maj. Min. Perf. Maj. Min. Min. Maj. Min. Min. Perf.
5|11 10 LsLLsLLsLLLsLLsLL Perf. Maj. Min. Min. Maj. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Maj. Min. Min. Perf.
4|12 17 sLLLsLLsLLLsLLsLL Perf. Min. Min. Min. Maj. Min. Min. Perf. Min. Min. Perf. Maj. Min. Min. Maj. Min. Min. Perf.
3|13 7 sLLLsLLsLLsLLLsLL Perf. Min. Min. Min. Maj. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Perf.
2|14 14 sLLsLLLsLLsLLLsLL Perf. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Maj. Min. Min. Perf.
1|15 4 sLLsLLLsLLsLLsLLL Perf. Min. Min. Min. Min. Min. Min. Perf. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Perf.
0|16 11 sLLsLLsLLLsLLsLLL Perf. Min. Min. Min. Min. Min. Min. Dim. Min. Min. Perf. Min. Min. Min. Min. Min. Min. Perf.

Proposed tuning-specific names

Declan Paul Boushy has proposed names for these modes corresponding to step ratios 3:1 and 4:1.

Scales

Scale tree

Scale Tree and Tuning Spectrum of 12L 5s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
7\17 494.118 705.882 1:1 1.000 Equalized 12L 5s
40\97 494.845 705.155 6:5 1.200
33\80 495.000 705.000 5:4 1.250
59\143 495.105 704.895 9:7 1.286
26\63 495.238 704.762 4:3 1.333 Supersoft 12L 5s
Leapfrog
71\172 495.349 704.651 11:8 1.375
45\109 495.413 704.587 7:5 1.400 Leapweek
64\155 495.484 704.516 10:7 1.429
19\46 495.652 704.348 3:2 1.500 Soft 12L 5s
69\167 495.808 704.192 11:7 1.571 Leapday / polypyth
50\121 495.868 704.132 8:5 1.600
81\196 495.918 704.082 13:8 1.625 Golden neogothic (495.9044¢)
31\75 496.000 704.000 5:3 1.667 Semisoft 12L 5s
74\179 496.089 703.911 12:7 1.714
43\104 496.154 703.846 7:4 1.750
55\133 496.241 703.759 9:5 1.800
12\29 496.552 703.448 2:1 2.000 Basic 12L 5s
Scales with tunings softer than this are proper
53\128 496.875 703.125 9:4 2.250
41\99 496.970 703.030 7:3 2.333 Undecental
70\169 497.041 702.959 12:5 2.400 Argent tuning (497.0563¢)
29\70 497.143 702.857 5:2 2.500 Semihard 12L 5s
75\181 497.238 702.762 13:5 2.600
46\111 497.297 702.703 8:3 2.667
63\152 497.368 702.632 11:4 2.750
17\41 497.561 702.439 3:1 3.000 Hard 12L 5s
Garibaldi / andromeda
56\135 497.778 702.222 10:3 3.333
39\94 497.872 702.128 7:2 3.500 Garibaldi / cassandra
61\147 497.959 702.041 11:3 3.667
22\53 498.113 701.887 4:1 4.000 Superhard 12L 5s
Garibaldi / helenus, Pythagorean tuning (498.0450¢)
49\118 498.305 701.695 9:2 4.500 Pontiac
27\65 498.462 701.538 5:1 5.000 Photia
32\77 498.701 701.299 6:1 6.000 ↓ Grackle
5\12 500.000 700.000 1:0 → ∞ Collapsed 12L 5s
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