5L 7s (3/1-equivalent)

From Xenharmonic Wiki
Jump to navigation Jump to search
↖ 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⟩ ↘ 
┌╥┬╥┬╥┬┬╥┬╥┬┬┐
│║│║│║││║│║│││
││││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┴┘
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 them unrecognizable: the 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. Hard-of-basic 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.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for diatonic interval categories.

Intervals

Intervals of 5L 7s⟨3/1⟩
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 158.5 ¢
Major 1-mosstep M1ms L 158.5 ¢ to 380.4 ¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0 ¢ to 317.0 ¢
Major 2-mosstep M2ms L + s 317.0 ¢ to 380.4 ¢
3-mosstep Minor 3-mosstep m3ms L + 2s 380.4 ¢ to 475.5 ¢
Major 3-mosstep M3ms 2L + s 475.5 ¢ to 760.8 ¢
4-mosstep Minor 4-mosstep m4ms L + 3s 380.4 ¢ to 634.0 ¢
Major 4-mosstep M4ms 2L + 2s 634.0 ¢ to 760.8 ¢
5-mosstep Perfect 5-mosstep P5ms 2L + 3s 760.8 ¢ to 792.5 ¢
Augmented 5-mosstep A5ms 3L + 2s 792.5 ¢ to 1141.2 ¢
6-mosstep Minor 6-mosstep m6ms 2L + 4s 760.8 ¢ to 951.0 ¢
Major 6-mosstep M6ms 3L + 3s 951.0 ¢ to 1141.2 ¢
7-mosstep Diminished 7-mosstep d7ms 2L + 5s 760.8 ¢ to 1109.5 ¢
Perfect 7-mosstep P7ms 3L + 4s 1109.5 ¢ to 1141.2 ¢
8-mosstep Minor 8-mosstep m8ms 3L + 5s 1141.2 ¢ to 1268.0 ¢
Major 8-mosstep M8ms 4L + 4s 1268.0 ¢ to 1521.6 ¢
9-mosstep Minor 9-mosstep m9ms 3L + 6s 1141.2 ¢ to 1426.5 ¢
Major 9-mosstep M9ms 4L + 5s 1426.5 ¢ to 1521.6 ¢
10-mosstep Minor 10-mosstep m10ms 4L + 6s 1521.6 ¢ to 1585.0 ¢
Major 10-mosstep M10ms 5L + 5s 1585.0 ¢ to 1902.0 ¢
11-mosstep Minor 11-mosstep m11ms 4L + 7s 1521.6 ¢ to 1743.5 ¢
Major 11-mosstep M11ms 5L + 6s 1743.5 ¢ to 1902.0 ¢
12-mosstep Perfect 12-mosstep P12ms 5L + 7s 1902.0 ¢

Generator chain

Generator chain of 5L 7s⟨3/1⟩
Bright gens Scale degree Abbrev.
16 Augmented 4-mosdegree A4md
15 Augmented 9-mosdegree A9md
14 Augmented 2-mosdegree A2md
13 Augmented 7-mosdegree A7md
12 Augmented 0-mosdegree A0md
11 Augmented 5-mosdegree A5md
10 Major 10-mosdegree M10md
9 Major 3-mosdegree M3md
8 Major 8-mosdegree M8md
7 Major 1-mosdegree M1md
6 Major 6-mosdegree M6md
5 Major 11-mosdegree M11md
4 Major 4-mosdegree M4md
3 Major 9-mosdegree M9md
2 Major 2-mosdegree M2md
1 Perfect 7-mosdegree P7md
0 Perfect 0-mosdegree
Perfect 12-mosdegree		 P0md
P12md
−1 Perfect 5-mosdegree P5md
−2 Minor 10-mosdegree m10md
−3 Minor 3-mosdegree m3md
−4 Minor 8-mosdegree m8md
−5 Minor 1-mosdegree m1md
−6 Minor 6-mosdegree m6md
−7 Minor 11-mosdegree m11md
−8 Minor 4-mosdegree m4md
−9 Minor 9-mosdegree m9md
−10 Minor 2-mosdegree m2md
−11 Diminished 7-mosdegree d7md
−12 Diminished 12-mosdegree d12md
−13 Diminished 5-mosdegree d5md
−14 Diminished 10-mosdegree d10md
−15 Diminished 3-mosdegree d3md
−16 Diminished 8-mosdegree d8md

Modes

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 (1119.463 ¢)
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 Mintra
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 Nekkar
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⟩
Minalzidar
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⟩
Retrieved from "https://en.xen.wiki/index.php?title=5L_7s_(3/1-equivalent)&oldid=184261"