5L 7s (3/1-equivalent)

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↖ 4L 6s⟨3/1⟩↑ 5L 6s⟨3/1⟩ 6L 6s⟨3/1⟩ ↗
← 4L 7s⟨3/1⟩5L 7s (3/1-equivalent)6L 7s⟨3/1⟩ →
↙ 4L 8s⟨3/1⟩↓ 5L 8s⟨3/1⟩ 6L 8s⟨3/1⟩ ↘
 
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└┴┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLssLsLss
ssLsLssLsLsL
Equave 3/1 (1902.0¢)
Period 3/1 (1902.0¢)
Generator size(edt)
Bright 7\12 to 3\5 (1109.5¢ to 1141.2¢)
Dark 2\5 to 5\12 (760.8¢ to 792.5¢)
Related MOS scales
Parent 5L 2s⟨3/1⟩
Sister 7L 5s⟨3/1⟩
Daughters 12L 5s⟨3/1⟩, 5L 12s⟨3/1⟩
Neutralized 10L 2s⟨3/1⟩
2-Flought 17L 7s⟨3/1⟩, 5L 19s⟨3/1⟩
Equal tunings(edt)
Equalized (L:s = 1:1) 7\12 (1109.5¢)
Supersoft (L:s = 4:3) 24\41 (1113.3¢)
Soft (L:s = 3:2) 17\29 (1114.9¢)
Semisoft (L:s = 5:3) 27\46 (1116.4¢)
Basic (L:s = 2:1) 10\17 (1118.8¢)
Semihard (L:s = 5:2) 23\39 (1121.7¢)
Hard (L:s = 3:1) 13\22 (1123.9¢)
Superhard (L:s = 4:1) 16\27 (1127.1¢)
Collapsed (L:s = 1:0) 3\5 (1141.2¢)

5L 7s⟨3/1⟩ is a 3/1-equivalent (tritave-equivalent) moment of symmetry scale containing 5 large steps and 7 small steps, repeating every interval of 3/1 (1902.0¢). Generators that produce this scale range from 1109.5¢ to 1141.2¢, or from 760.8¢ to 792.5¢.

Theory

As a macrochromatic scale

It is a stretched p-chromatic scale, and an extension of a macrodiatonic scale (5L 2s (3/1-equivalent)) with the period of a tritave. This means it is a chromatic scale, but has octaves stretched out to the size of a tritave. Other intervals are also stretched in a way that makes the unrecognizable–them diatonic fifth is now the size of a major seventh. Interestingly, 27edt, an approximation of 17edo, has a tuning of this scale, meaning it contains both an octave-equivalent and tritave-equivalent p-chromatic.

Temperament interpretations

It is possible to construct no-twos rank-2 temperament interpretations of this scale, but it is difficult to interpret within commonly-studied no-twos subgroups like the 3.5.7 subgroup used for Bohlen-Pierce. Harder scales can be interpreted in Mintaka temperament in the 3.7.11 subgroup (or its extensions such as Mintra), which tempers out 1331/1323 so that the generator (the stretched counterpart of the fourth) is ~11/7, a stack of 2 generators (equivalent to the minor seventh) is ~27/11, and a stack of three generators (equivalent to the minor third) is ~9/7.

Modes

The modes have step patterns which are the same as the modes of the diatonic scale.

Modes of 5L 7s⟨3/1⟩
UDP Cyclic
Order		 Step
Pattern
11|0 1 LsLsLssLsLss
10|1 8 LsLssLsLsLss
9|2 3 LsLssLsLssLs
8|3 10 LssLsLsLssLs
7|4 5 LssLsLssLsLs
6|5 12 sLsLsLssLsLs
5|6 7 sLsLssLsLsLs
4|7 2 sLsLssLsLssL
3|8 9 sLssLsLsLssL
2|9 4 sLssLsLssLsL
1|10 11 ssLsLsLssLsL
0|11 6 ssLsLssLsLsL

Scale degrees

Scale degrees of the modes of 5L 7s⟨3/1⟩ 
UDP Cyclic
Order		 Step
Pattern		 Scale Degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11 12
11|0 1 LsLsLssLsLss Perf. Maj. Maj. Maj. Maj. Aug. Maj. Perf. Maj. Maj. Maj. Maj. Perf.
10|1 8 LsLssLsLsLss Perf. Maj. Maj. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Maj. Maj. Perf.
9|2 3 LsLssLsLssLs Perf. Maj. Maj. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Min. Maj. Perf.
8|3 10 LssLsLsLssLs Perf. Maj. Maj. Min. Maj. Perf. Maj. Perf. Maj. Maj. Min. Maj. Perf.
7|4 5 LssLsLssLsLs Perf. Maj. Maj. Min. Maj. Perf. Maj. Perf. Min. Maj. Min. Maj. Perf.
6|5 12 sLsLsLssLsLs Perf. Min. Maj. Min. Maj. Perf. Maj. Perf. Min. Maj. Min. Maj. Perf.
5|6 7 sLsLssLsLsLs Perf. Min. Maj. Min. Maj. Perf. Min. Perf. Min. Maj. Min. Maj. Perf.
4|7 2 sLsLssLsLssL Perf. Min. Maj. Min. Maj. Perf. Min. Perf. Min. Maj. Min. Min. Perf.
3|8 9 sLssLsLsLssL Perf. Min. Maj. Min. Min. Perf. Min. Perf. Min. Maj. Min. Min. Perf.
2|9 4 sLssLsLssLsL Perf. Min. Maj. Min. Min. Perf. Min. Perf. Min. Min. Min. Min. Perf.
1|10 11 ssLsLsLssLsL Perf. Min. Min. Min. Min. Perf. Min. Perf. Min. Min. Min. Min. Perf.
0|11 6 ssLsLssLsLsL Perf. Min. Min. Min. Min. Perf. Min. Dim. Min. Min. Min. Min. Perf.

Notation

Being a descendant of a macrodiatonic scale, it can notated using the traditional diatonic notation, if all intervals are reinterpreted as their stretched versions (like octaves as tritaves). However, this approach involves 1-based indexing for a non-diatonic MOS which is generally discouraged. Alternatively, a generic MOS notation may be used like diamond MOS notation, which enables 0-based indexing at the cost of obscuring the connection to the standard diatonic scale.

Scale tree

Scale Tree and Tuning Spectrum of 5L 7s⟨3/1⟩
Generator(edt) Cents Step Ratio Comments
Bright Dark L:s Hardness
7\12 1109.474 792.481 1:1 1.000 Equalized 5L 7s⟨3/1⟩
38\65 1111.912 790.043 6:5 1.200
31\53 1112.464 789.491 5:4 1.250
55\94 1112.846 789.109 9:7 1.286
24\41 1113.340 788.615 4:3 1.333 Supersoft 5L 7s⟨3/1⟩
65\111 1113.757 788.198 11:8 1.375
41\70 1114.002 787.953 7:5 1.400
58\99 1114.277 787.678 10:7 1.429
17\29 1114.939 787.016 3:2 1.500 Soft 5L 7s⟨3/1⟩
61\104 1115.570 786.385 11:7 1.571
44\75 1115.814 786.141 8:5 1.600
71\121 1116.023 785.932 13:8 1.625
27\46 1116.365 785.590 5:3 1.667 Semisoft 5L 7s⟨3/1⟩
64\109 1116.744 785.211 12:7 1.714
37\63 1117.021 784.934 7:4 1.750
47\80 1117.399 784.556 9:5 1.800
10\17 1118.797 783.158 2:1 2.000 Basic 5L 7s⟨3/1⟩
Scales with tunings softer than this are proper
Just 21/11 generator (1119.463c)
43\73 1120.330 781.625 9:4 2.250
33\56 1120.795 781.160 7:3 2.333
56\95 1121.152 780.803 12:5 2.400
23\39 1121.666 780.289 5:2 2.500 Semihard 5L 7s⟨3/1⟩
59\100 1122.153 779.802 13:5 2.600
36\61 1122.465 779.490 8:3 2.667
49\83 1122.841 779.114 11:4 2.750
13\22 1123.883 778.073 3:1 3.000 Hard 5L 7s⟨3/1⟩
Mintaka is around here
42\71 1125.100 776.855 10:3 3.333
29\49 1125.647 776.308 7:2 3.500
45\76 1126.158 775.797 11:3 3.667
16\27 1127.084 774.871 4:1 4.000 Superhard 5L 7s⟨3/1⟩
35\59 1128.278 773.677 9:2 4.500
19\32 1129.286 772.669 5:1 5.000
22\37 1130.892 771.063 6:1 6.000
3\5 1141.173 760.782 1:0 → ∞ Collapsed 5L 7s⟨3/1⟩
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