7L 3s (4/1-equivalent)

From Xenharmonic Wiki
Jump to navigation Jump to search
↖ 6L 2s⟨4/1⟩ ↑ 7L 2s⟨4/1⟩ 8L 2s⟨4/1⟩ ↗
← 6L 3s⟨4/1⟩ 7L 3s (4/1-equivalent) 8L 3s⟨4/1⟩ →
↙ 6L 4s⟨4/1⟩ ↓ 7L 4s⟨4/1⟩ 8L 4s⟨4/1⟩ ↘ 
┌╥╥╥┬╥╥┬╥╥┬┐
│║║║│║║│║║││
││││││││││││
└┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLLsLLsLLs
sLLsLLsLLL
Equave 4/1 (2400.0 ¢)
Period 4/1 (2400.0 ¢)
Generator size(ed4/1)
Bright 7\10 to 5\7 (1680.0 ¢ to 1714.3 ¢)
Dark 2\7 to 3\10 (685.7 ¢ to 720.0 ¢)
Related MOS scales
Parent 3L 4s⟨4/1⟩
Sister 3L 7s⟨4/1⟩
Daughters 10L 7s⟨4/1⟩, 7L 10s⟨4/1⟩
Neutralized 4L 6s⟨4/1⟩
2-Flought 17L 3s⟨4/1⟩, 7L 13s⟨4/1⟩
Equal tunings(ed4/1)
Equalized (L:s = 1:1) 7\10 (1680.0 ¢)
Supersoft (L:s = 4:3) 26\37 (1686.5 ¢)
Soft (L:s = 3:2) 19\27 (1688.9 ¢)
Semisoft (L:s = 5:3) 31\44 (1690.9 ¢)
Basic (L:s = 2:1) 12\17 (1694.1 ¢)
Semihard (L:s = 5:2) 29\41 (1697.6 ¢)
Hard (L:s = 3:1) 17\24 (1700.0 ¢)
Superhard (L:s = 4:1) 22\31 (1703.2 ¢)
Collapsed (L:s = 1:0) 5\7 (1714.3 ¢)

7L 3s⟨4/1⟩ is a 4/1-equivalent (non-octave) moment of symmetry scale containing 7 large steps and 3 small steps, repeating every interval of 4/1 (2400.0 ¢). Generators that produce this scale range from 1680 ¢ to 1714.3 ¢, or from 685.7 ¢ to 720 ¢.

These scales are in some ways a double octave clone of diatonic scales because they are generated from perfect fifths and have the 7edo/7ed4 685 ¢ flat fifth and 5edo/10ed4's 720 ¢ sharp fifth at the ends of the tuning spectrum. However, ~3/2 is now the chroma-negative rather than chroma-positive generator, equalized and collapsed tunings are now swapped (equalized is 720 ¢ fifth and collapsed is 685 ¢ fifth), and the basic tuning is now the 705-cent gentle fifth from 17edo/17ed4 instead of 12edo's fifth. Diaquadic scales are also more widely spaced than diatonic scales, closer melodically to an octave-period pentatonic scale.

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 7L 3s⟨4/1⟩
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 240.0 ¢
Major 1-mosstep M1ms L 240.0 ¢ to 342.9 ¢
2-mosstep Minor 2-mosstep m2ms L + s 342.9 ¢ to 480.0 ¢
Major 2-mosstep M2ms 2L 480.0 ¢ to 685.7 ¢
3-mosstep Perfect 3-mosstep P3ms 2L + s 685.7 ¢ to 720.0 ¢
Augmented 3-mosstep A3ms 3L 720.0 ¢ to 1028.6 ¢
4-mosstep Minor 4-mosstep m4ms 2L + 2s 685.7 ¢ to 960.0 ¢
Major 4-mosstep M4ms 3L + s 960.0 ¢ to 1028.6 ¢
5-mosstep Minor 5-mosstep m5ms 3L + 2s 1028.6 ¢ to 1200.0 ¢
Major 5-mosstep M5ms 4L + s 1200.0 ¢ to 1371.4 ¢
6-mosstep Minor 6-mosstep m6ms 4L + 2s 1371.4 ¢ to 1440.0 ¢
Major 6-mosstep M6ms 5L + s 1440.0 ¢ to 1714.3 ¢
7-mosstep Diminished 7-mosstep d7ms 4L + 3s 1371.4 ¢ to 1680.0 ¢
Perfect 7-mosstep P7ms 5L + 2s 1680.0 ¢ to 1714.3 ¢
8-mosstep Minor 8-mosstep m8ms 5L + 3s 1714.3 ¢ to 1920.0 ¢
Major 8-mosstep M8ms 6L + 2s 1920.0 ¢ to 2057.1 ¢
9-mosstep Minor 9-mosstep m9ms 6L + 3s 2057.1 ¢ to 2160.0 ¢
Major 9-mosstep M9ms 7L + 2s 2160.0 ¢ to 2400.0 ¢
10-mosstep Perfect 10-mosstep P10ms 7L + 3s 2400.0 ¢

Generator chain

Generator chain of 7L 3s⟨4/1⟩
Bright gens Scale degree Abbrev.
16 Augmented 2-mosdegree A2md
15 Augmented 5-mosdegree A5md
14 Augmented 8-mosdegree A8md
13 Augmented 1-mosdegree A1md
12 Augmented 4-mosdegree A4md
11 Augmented 7-mosdegree A7md
10 Augmented 0-mosdegree A0md
9 Augmented 3-mosdegree A3md
8 Major 6-mosdegree M6md
7 Major 9-mosdegree M9md
6 Major 2-mosdegree M2md
5 Major 5-mosdegree M5md
4 Major 8-mosdegree M8md
3 Major 1-mosdegree M1md
2 Major 4-mosdegree M4md
1 Perfect 7-mosdegree P7md
0 Perfect 0-mosdegree
Perfect 10-mosdegree		 P0md
P10md
−1 Perfect 3-mosdegree P3md
−2 Minor 6-mosdegree m6md
−3 Minor 9-mosdegree m9md
−4 Minor 2-mosdegree m2md
−5 Minor 5-mosdegree m5md
−6 Minor 8-mosdegree m8md
−7 Minor 1-mosdegree m1md
−8 Minor 4-mosdegree m4md
−9 Diminished 7-mosdegree d7md
−10 Diminished 10-mosdegree d10md
−11 Diminished 3-mosdegree d3md
−12 Diminished 6-mosdegree d6md
−13 Diminished 9-mosdegree d9md
−14 Diminished 2-mosdegree d2md
−15 Diminished 5-mosdegree d5md
−16 Diminished 8-mosdegree d8md

Modes

Scale degrees of the modes of 7L 3s⟨4/1⟩
UDP Cyclic
order		 Step
pattern		 Scale degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10
9|0 1 LLLsLLsLLs Perf. Maj. Maj. Aug. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
8|1 8 LLsLLLsLLs Perf. Maj. Maj. Perf. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
7|2 5 LLsLLsLLLs Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Maj. Perf.
6|3 2 LLsLLsLLsL Perf. Maj. Maj. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
5|4 9 LsLLLsLLsL Perf. Maj. Min. Perf. Maj. Maj. Min. Perf. Maj. Min. Perf.
4|5 6 LsLLsLLLsL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Maj. Min. Perf.
3|6 3 LsLLsLLsLL Perf. Maj. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
2|7 10 sLLLsLLsLL Perf. Min. Min. Perf. Maj. Min. Min. Perf. Min. Min. Perf.
1|8 7 sLLsLLLsLL Perf. Min. Min. Perf. Min. Min. Min. Perf. Min. Min. Perf.
0|9 4 sLLsLLsLLL Perf. Min. Min. Perf. Min. Min. Min. Dim. Min. Min. Perf.

Scale tree

Scale tree and tuning spectrum of 7L 3s⟨4/1⟩
Generator(ed4/1) Cents Step ratio Comments
Bright Dark L:s Hardness
7\10 1680.000 720.000 1:1 1.000 Equalized 7L 3s⟨4/1⟩
40\57 1684.211 715.789 6:5 1.200
33\47 1685.106 714.894 5:4 1.250
59\84 1685.714 714.286 9:7 1.286
26\37 1686.486 713.514 4:3 1.333 Supersoft 7L 3s⟨4/1⟩
71\101 1687.129 712.871 11:8 1.375
45\64 1687.500 712.500 7:5 1.400
64\91 1687.912 712.088 10:7 1.429
19\27 1688.889 711.111 3:2 1.500 Soft 7L 3s⟨4/1⟩
69\98 1689.796 710.204 11:7 1.571
50\71 1690.141 709.859 8:5 1.600
81\115 1690.435 709.565 13:8 1.625
31\44 1690.909 709.091 5:3 1.667 Semisoft 7L 3s⟨4/1⟩
Quarchy is around here
74\105 1691.429 708.571 12:7 1.714
43\61 1691.803 708.197 7:4 1.750
55\78 1692.308 707.692 9:5 1.800
12\17 1694.118 705.882 2:1 2.000 Basic 7L 3s⟨4/1⟩
Scales with tunings softer than this are proper
53\75 1696.000 704.000 9:4 2.250
41\58 1696.552 703.448 7:3 2.333
70\99 1696.970 703.030 12:5 2.400
29\41 1697.561 702.439 5:2 2.500 Semihard 7L 3s⟨4/1⟩
75\106 1698.113 701.887 13:5 2.600
46\65 1698.462 701.538 8:3 2.667
63\89 1698.876 701.124 11:4 2.750
17\24 1700.000 700.000 3:1 3.000 Hard 7L 3s⟨4/1⟩
Tetrominant is around here
56\79 1701.266 698.734 10:3 3.333
39\55 1701.818 698.182 7:2 3.500
61\86 1702.326 697.674 11:3 3.667
22\31 1703.226 696.774 4:1 4.000 Superhard 7L 3s⟨4/1⟩
Meanquad is around here
49\69 1704.348 695.652 9:2 4.500
27\38 1705.263 694.737 5:1 5.000
32\45 1706.667 693.333 6:1 6.000
5\7 1714.286 685.714 1:0 → ∞ Collapsed 7L 3s⟨4/1⟩

Temperaments

There are two major rank-2 temperament interpretations of diaquadic, corresponding to double octave clones of meantone and archy.

Meanquad

Main article: Meanquad

Tetrominant

Quarchy

Retrieved from "https://en.xen.wiki/index.php?title=7L_3s_(4/1-equivalent)&oldid=184225"