10L 1s (3/1-equivalent)

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← 9L 1s⟨3/1⟩ 10L 1s (3/1-equivalent) 11L 1s⟨3/1⟩ →
↙ 9L 2s⟨3/1⟩ ↓ 10L 2s⟨3/1⟩ 11L 2s⟨3/1⟩ ↘
Scale structure
Step pattern LLLLLLLLLLs
sLLLLLLLLLL
Equave 3/1 (1902.0 ¢)
Period 3/1 (1902.0 ¢)
Generator size(edt)
Bright 1\11 to 1\10 (172.9 ¢ to 190.2 ¢)
Dark 9\10 to 10\11 (1711.8 ¢ to 1729.1 ¢)
Related MOS scales
Parent 1L 9s⟨3/1⟩
Sister 1L 10s⟨3/1⟩
Daughters 11L 10s⟨3/1⟩, 10L 11s⟨3/1⟩
Neutralized 9L 2s⟨3/1⟩
2-Flought 21L 1s⟨3/1⟩, 10L 12s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 1\11 (172.9 ¢)
Supersoft (L:s = 4:3) 4\43 (176.9 ¢)
Soft (L:s = 3:2) 3\32 (178.3 ¢)
Semisoft (L:s = 5:3) 5\53 (179.4 ¢)
Basic (L:s = 2:1) 2\21 (181.1 ¢)
Semihard (L:s = 5:2) 5\52 (182.9 ¢)
Hard (L:s = 3:1) 3\31 (184.1 ¢)
Superhard (L:s = 4:1) 4\41 (185.6 ¢)
Collapsed (L:s = 1:0) 1\10 (190.2 ¢)
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10L 1s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 10 large steps and 1 small step, repeating every interval of 3/1 (1902.0 ¢). Generators that produce this scale range from 172.9 ¢ to 190.2 ¢, or from 1711.8 ¢ to 1729.1 ¢. Scales of this form are always proper because there is only one small step.

Theory

Modes

Modes of 10L 1s⟨3/1⟩
UDP Cyclic
order		 Step
pattern
10|0 1 LLLLLLLLLLs
9|1 2 LLLLLLLLLsL
8|2 3 LLLLLLLLsLL
7|3 4 LLLLLLLsLLL
6|4 5 LLLLLLsLLLL
5|5 6 LLLLLsLLLLL
4|6 7 LLLLsLLLLLL
3|7 8 LLLsLLLLLLL
2|8 9 LLsLLLLLLLL
1|9 10 LsLLLLLLLLL
0|10 11 sLLLLLLLLLL

Temperament interpretations

The temperament that joins 32edt and 75edt produces this MOS. Both 32edt and 75edt are no-twos zeta peaks.

In 32edt it is generated by 3\32 (~178.3¢) and has a hardness of exactly 1.5.

In 75edt it is generated by 7\75 (~177.5¢) and has a hardness of exactly 1.4.

In 32edt this MOS has the intervals (in cents):

  • 178.308
  • 356.617
  • 534.925
  • 713.233
  • 891.541
  • 1069.85
  • 1248.158
  • 1426.466
  • 1604.775
  • 1723.647
  • 1901.955

(And rotations thereof.)

That roughly approximates this 13-limit JI scale:

  • 10/9
  • 11/9
  • 11/8
  • 3/2
  • 5/3
  • 13/7
  • 25/12
  • 16/7
  • 5/2
  • 8/3
  • 3/1

(The intervals with multiple-of-3 denominators gain consonance in higher tritaves.)

Intervals

Intervals of 10L 1s⟨3/1⟩
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-mosstep Perfect 0-mosstep P0ms 0 0.0 ¢
1-mosstep Diminished 1-mosstep d1ms s 0.0 ¢ to 172.9 ¢
Perfect 1-mosstep P1ms L 172.9 ¢ to 190.2 ¢
2-mosstep Minor 2-mosstep m2ms L + s 190.2 ¢ to 345.8 ¢
Major 2-mosstep M2ms 2L 345.8 ¢ to 380.4 ¢
3-mosstep Minor 3-mosstep m3ms 2L + s 380.4 ¢ to 518.7 ¢
Major 3-mosstep M3ms 3L 518.7 ¢ to 570.6 ¢
4-mosstep Minor 4-mosstep m4ms 3L + s 570.6 ¢ to 691.6 ¢
Major 4-mosstep M4ms 4L 691.6 ¢ to 760.8 ¢
5-mosstep Minor 5-mosstep m5ms 4L + s 760.8 ¢ to 864.5 ¢
Major 5-mosstep M5ms 5L 864.5 ¢ to 951.0 ¢
6-mosstep Minor 6-mosstep m6ms 5L + s 951.0 ¢ to 1037.4 ¢
Major 6-mosstep M6ms 6L 1037.4 ¢ to 1141.2 ¢
7-mosstep Minor 7-mosstep m7ms 6L + s 1141.2 ¢ to 1210.3 ¢
Major 7-mosstep M7ms 7L 1210.3 ¢ to 1331.4 ¢
8-mosstep Minor 8-mosstep m8ms 7L + s 1331.4 ¢ to 1383.2 ¢
Major 8-mosstep M8ms 8L 1383.2 ¢ to 1521.6 ¢
9-mosstep Minor 9-mosstep m9ms 8L + s 1521.6 ¢ to 1556.1 ¢
Major 9-mosstep M9ms 9L 1556.1 ¢ to 1711.8 ¢
10-mosstep Perfect 10-mosstep P10ms 9L + s 1711.8 ¢ to 1729.1 ¢
Augmented 10-mosstep A10ms 10L 1729.1 ¢ to 1902.0 ¢
11-mosstep Perfect 11-mosstep P11ms 10L + s 1902.0 ¢

Scale degrees

Scale degrees of the modes of 10L 1s⟨3/1⟩
UDP Cyclic
order 		Step
pattern 		Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11
10|0 1 LLLLLLLLLLs Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Aug. Perf.
9|1 2 LLLLLLLLLsL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Perf. Perf.
8|2 3 LLLLLLLLsLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Maj. Min. Perf. Perf.
7|3 4 LLLLLLLsLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Maj. Min. Min. Perf. Perf.
6|4 5 LLLLLLsLLLL Perf. Perf. Maj. Maj. Maj. Maj. Maj. Min. Min. Min. Perf. Perf.
5|5 6 LLLLLsLLLLL Perf. Perf. Maj. Maj. Maj. Maj. Min. Min. Min. Min. Perf. Perf.
4|6 7 LLLLsLLLLLL Perf. Perf. Maj. Maj. Maj. Min. Min. Min. Min. Min. Perf. Perf.
3|7 8 LLLsLLLLLLL Perf. Perf. Maj. Maj. Min. Min. Min. Min. Min. Min. Perf. Perf.
2|8 9 LLsLLLLLLLL Perf. Perf. Maj. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
1|9 10 LsLLLLLLLLL Perf. Perf. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.
0|10 11 sLLLLLLLLLL Perf. Dim. Min. Min. Min. Min. Min. Min. Min. Min. Perf. Perf.

Scale tree

Scale tree and tuning spectrum of 10L 1s⟨3/1⟩
Generator(edt) Cents Step ratio Comments(always proper)
Bright Dark L:s Hardness
1\11 172.905 1729.050 1:1 1.000 Equalized 10L 1s⟨3/1⟩
6\65 175.565 1726.390 6:5 1.200
5\54 176.107 1725.848 5:4 1.250
9\97 176.470 1725.485 9:7 1.286
4\43 176.926 1725.029 4:3 1.333 Supersoft 10L 1s⟨3/1⟩
11\118 177.301 1724.654 11:8 1.375
7\75 177.516 1724.439 7:5 1.400
10\107 177.753 1724.202 10:7 1.429
3\32 178.308 1723.647 3:2 1.500 Soft 10L 1s⟨3/1⟩
11\117 178.816 1723.139 11:7 1.571
8\85 179.008 1722.947 8:5 1.600
13\138 179.170 1722.785 13:8 1.625
5\53 179.430 1722.525 5:3 1.667 Semisoft 10L 1s⟨3/1⟩
12\127 179.712 1722.243 12:7 1.714
7\74 179.915 1722.040 7:4 1.750
9\95 180.185 1721.770 9:5 1.800
2\21 181.139 1720.816 2:1 2.000 Basic 10L 1s⟨3/1⟩
9\94 182.102 1719.853 9:4 2.250
7\73 182.379 1719.576 7:3 2.333
12\125 182.588 1719.367 12:5 2.400
5\52 182.880 1719.075 5:2 2.500 Semihard 10L 1s⟨3/1⟩
13\135 183.151 1718.804 13:5 2.600
8\83 183.321 1718.634 8:3 2.667
11\114 183.522 1718.433 11:4 2.750
3\31 184.060 1717.895 3:1 3.000 Hard 10L 1s⟨3/1⟩
10\103 184.656 1717.299 10:3 3.333
7\72 184.912 1717.043 7:2 3.500
11\113 185.146 1716.809 11:3 3.667
4\41 185.557 1716.398 4:1 4.000 Superhard 10L 1s⟨3/1⟩
9\92 186.061 1715.894 9:2 4.500
5\51 186.466 1715.489 5:1 5.000
6\61 187.078 1714.877 6:1 6.000
1\10 190.196 1711.760 1:0 → ∞ Collapsed 10L 1s⟨3/1⟩
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