4L 7s

From Xenharmonic Wiki
Jump to navigation Jump to search
↖ 3L 6s↑ 4L 6s 5L 6s ↗
← 3L 7s4L 7s5L 7s →
↙ 3L 8s↓ 4L 8s 5L 8s ↘
 
┌╥┬╥┬┬╥┬┬╥┬┬┐
│║│║││║││║│││
│││││││││││││
└┴┴┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLssLssLss
ssLssLssLsL
Equave 2/1 (1200.0¢)
Period 2/1 (1200.0¢)
Generator size
Bright 8\11 to 3\4 (872.7¢ to 900.0¢)
Dark 1\4 to 3\11 (300.0¢ to 327.3¢)
TAMNAMS information
Descends from 4L 3s (smitonic)
Ancestor's step ratio range 2:1 to 1:0 (hard-of-basic)
Related MOS scales
Parent 4L 3s
Sister 7L 4s
Daughters 11L 4s, 4L 11s
Neutralized 8L 3s
2-Flought 15L 7s, 4L 18s
Equal tunings
Equalized (L:s = 1:1) 8\11 (872.7¢)
Supersoft (L:s = 4:3) 27\37 (875.7¢)
Soft (L:s = 3:2) 19\26 (876.9¢)
Semisoft (L:s = 5:3) 30\41 (878.0¢)
Basic (L:s = 2:1) 11\15 (880.0¢)
Semihard (L:s = 5:2) 25\34 (882.4¢)
Hard (L:s = 3:1) 14\19 (884.2¢)
Superhard (L:s = 4:1) 17\23 (887.0¢)
Collapsed (L:s = 1:0) 3\4 (900.0¢)

4L 7s is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 7 small steps, repeating every octave. 4L 7s is a child scale of 4L 3s, expanding it by 4 tones. Generators that produce this scale range from 872.7¢ to 900¢, or from 300¢ to 327.3¢. One of the harmonic entropy minimums in this range is Kleismic/Hanson.

Name

TAMNAMS formerly used the name kleistonic for the name of this scale (prefix klei-). Other names include p-chro smitonic or smipechromic.

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 4L 7s
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 109.1¢
Major 1-mosstep M1ms L 109.1¢ to 300.0¢
2-mosstep Minor 2-mosstep m2ms 2s 0.0¢ to 218.2¢
Major 2-mosstep M2ms L + s 218.2¢ to 300.0¢
3-mosstep Perfect 3-mosstep P3ms L + 2s 300.0¢ to 327.3¢
Augmented 3-mosstep A3ms 2L + s 327.3¢ to 600.0¢
4-mosstep Minor 4-mosstep m4ms L + 3s 300.0¢ to 436.4¢
Major 4-mosstep M4ms 2L + 2s 436.4¢ to 600.0¢
5-mosstep Minor 5-mosstep m5ms L + 4s 300.0¢ to 545.5¢
Major 5-mosstep M5ms 2L + 3s 545.5¢ to 600.0¢
6-mosstep Minor 6-mosstep m6ms 2L + 4s 600.0¢ to 654.5¢
Major 6-mosstep M6ms 3L + 3s 654.5¢ to 900.0¢
7-mosstep Minor 7-mosstep m7ms 2L + 5s 600.0¢ to 763.6¢
Major 7-mosstep M7ms 3L + 4s 763.6¢ to 900.0¢
8-mosstep Diminished 8-mosstep d8ms 2L + 6s 600.0¢ to 872.7¢
Perfect 8-mosstep P8ms 3L + 5s 872.7¢ to 900.0¢
9-mosstep Minor 9-mosstep m9ms 3L + 6s 900.0¢ to 981.8¢
Major 9-mosstep M9ms 4L + 5s 981.8¢ to 1200.0¢
10-mosstep Minor 10-mosstep m10ms 3L + 7s 900.0¢ to 1090.9¢
Major 10-mosstep M10ms 4L + 6s 1090.9¢ to 1200.0¢
11-mosstep Perfect 11-mosstep P11ms 4L + 7s 1200.0¢

Modes

Scale degrees of the modes of 4L 7s 
UDP Cyclic
Order		 Step
Pattern		 Scale Degree (mosdegree)
0 1 2 3 4 5 6 7 8 9 10 11
10|0 1 LsLssLssLss Perf. Maj. Maj. Aug. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
9|1 9 LssLsLssLss Perf. Maj. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Maj. Perf.
8|2 6 LssLssLsLss Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Perf. Maj. Maj. Perf.
7|3 3 LssLssLssLs Perf. Maj. Maj. Perf. Maj. Maj. Min. Maj. Perf. Min. Maj. Perf.
6|4 11 sLsLssLssLs Perf. Min. Maj. Perf. Maj. Maj. Min. Maj. Perf. Min. Maj. Perf.
5|5 8 sLssLsLssLs Perf. Min. Maj. Perf. Min. Maj. Min. Maj. Perf. Min. Maj. Perf.
4|6 5 sLssLssLsLs Perf. Min. Maj. Perf. Min. Maj. Min. Min. Perf. Min. Maj. Perf.
3|7 2 sLssLssLssL Perf. Min. Maj. Perf. Min. Maj. Min. Min. Perf. Min. Min. Perf.
2|8 10 ssLsLssLssL Perf. Min. Min. Perf. Min. Maj. Min. Min. Perf. Min. Min. Perf.
1|9 7 ssLssLsLssL Perf. Min. Min. Perf. Min. Min. Min. Min. Perf. Min. Min. Perf.
0|10 4 ssLssLssLsL Perf. Min. Min. Perf. Min. Min. Min. Min. Dim. Min. Min. Perf.

Genchain

Generators -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11
Interval quality d11md d8md m5md m2md m10md m7md m4md m1md m9md m6md P3md P0md P8md M5md M2md M10md M7md M4md M1md M9md M6md A3md A0md

Tuning ranges

Soft range

The soft range for tunings of 4L 7s encompasses parasoft and hyposoft tunings. This implies step ratios smaller than 2/1, meaning a generator sharper than 4\15 = 320¢.

This is the range associated with extensions of Orgone[7]. The small step is recognizable as a near diatonic semitone, while the large step is in the ambiguous area of neutral seconds.

Soft edos include 15edo and 26edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various soft tunings:

15edo (basic) 26edo (soft) Some JI approximations
generator (g) 4\15, 320.00 7\26, 323.08 77/64, 6/5
L (octave - 3g) 2\15, 160.00 3\26, 138.46 12/11, 13/12
s (4g - octave) 1\15, 80.00 2\19, 92.31 21/20, 22/21, 20/19

Hypohard

Hypohard tunings of 4L 7s have step ratios between 2/1 and 3/1, implying a generator sharper than 5\19 = 315.79¢ and flatter than 4\15 = 320¢.

This range represents one of the harmonic entropy minimums, where 6 generators make a just diatonic fifth (3/2), an octave above. This is the range associated with the eponymous Kleismic (aka Hanson) temperament and its extensions.

Hypohard edos include 15edo, 19edo, and 34edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various hypohard tunings:

15edo (basic) 19edo (hard) 34edo (semihard) Some JI approximations
generator (g) 4\15, 320.00 5\19, 315.79 9\34, 317.65 6/5
L (octave - 3g) 2\15, 160.00 3\19, 189.47 5\34, 176.47 10/9, 11/10 (in 15edo)
s (4g - octave) 1\15, 80.00 1\19, 63.16 2\34, 70.59 25/24, 26/25 (in better kleismic tunings)

Parahard

Parahard tunings of 4L 7s have step ratios between 3/1 and 4/1, implying a generator sharper than 6\23 = 313.04¢ and flatter than 5\19 = 315.79¢.

The minor third is at its purest here, but the resulting scales tend to approximate intervals that employ a much higher limit harmony, especially in the case of the superhard 23edo. However, the large step is recognizable as a regular diatonic whole step, approximating both 10/9 and 9/8, while the small step is a slightly sharp of a quarter tone.

Parahard edos include 19edo, 23edo, and 42edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various parahard tunings:

19edo (hard) 23edo (superhard) 42edo (parahard) Some JI approximations
generator (g) 5\19, 315.79 6\23, 313.04 11\42, 314.29 6/5
L (octave - 3g) 3\19, 189.47 4\23, 208.70 7\42, 200.00 10/9, 9/8
s (4g - octave) 1\19, 63.16 1\23, 52.17 2\42, 57.14 28/27, 33/32

Hyperhard

Hyperhard tunings of 4L 7s have step ratios between 4/1 and 6/1, implying a generator sharper than 8\31 = 309.68¢ and flatter than 6\23 = 313.04¢.

The temperament known as Myna (a pun on "minor third") resides here, as this is the range where 10 generators make a just diatonic fifth (3/2), two octaves above. These scales are stacked with simple intervals, but are melodically difficult due to the extreme step size disparity, where the small step is generally flat of a quarter tone.

Hyperhard edos include 23edo, 31edo, and 27edo. The sizes of the generator, large step and small step of 4L 7s are as follows in various hyperhard tunings:

23edo (superhard) 31edo (extrahard) 27edo (pentahard) Some JI approximations
generator (g) 6\23, 313.04 8\31, 309.68 7\27, 311.11 6/5
L (octave - 3g) 4\23, 208.70 6\31, 232.26 5\27, 222.22 8/7, 9/8
s (4g - octave) 1\23, 52.17 1\31, 38.71 1\27, 44.44 36/35, 45/44

Temperaments

Scales

Scale tree

Scale Tree and Tuning Spectrum of 4L 7s
Generator(edo) Cents Step Ratio Comments
Bright Dark L:s Hardness
8\11 872.727 327.273 1:1 1.000 Equalized 4L 7s
43\59 874.576 325.424 6:5 1.200 Oregon
35\48 875.000 325.000 5:4 1.250
62\85 875.294 324.706 9:7 1.286
27\37 875.676 324.324 4:3 1.333 Supersoft 4L 7s
73\100 876.000 324.000 11:8 1.375
46\63 876.190 323.810 7:5 1.400
65\89 876.404 323.596 10:7 1.429 Orgone
19\26 876.923 323.077 3:2 1.500 Soft 4L 7s
68\93 877.419 322.581 11:7 1.571 Magicaltet
49\67 877.612 322.388 8:5 1.600
79\108 877.778 322.222 13:8 1.625 Golden superklesimic
30\41 878.049 321.951 5:3 1.667 Semisoft 4L 7s
Superkleismic
71\97 878.351 321.649 12:7 1.714
41\56 878.571 321.429 7:4 1.750
52\71 878.873 321.127 9:5 1.800
11\15 880.000 320.000 2:1 2.000 Basic 4L 7s
Scales with tunings softer than this are proper
47\64 881.250 318.750 9:4 2.250
36\49 881.633 318.367 7:3 2.333 Catalan
61\83 881.928 318.072 12:5 2.400
25\34 882.353 317.647 5:2 2.500 Semihard 4L 7s
64\87 882.759 317.241 13:5 2.600 Countercata
39\53 883.019 316.981 8:3 2.667 Hanson/cata
53\72 883.333 316.667 11:4 2.750
14\19 884.211 315.789 3:1 3.000 Hard 4L 7s
45\61 885.246 314.754 10:3 3.333 Parakleismic
31\42 885.714 314.286 7:2 3.500
48\65 886.154 313.846 11:3 3.667
17\23 886.957 313.043 4:1 4.000 Superhard 4L 7s
37\50 888.000 312.000 9:2 4.500 Oolong
20\27 888.889 311.111 5:1 5.000 Starlingtet
23\31 890.323 309.677 6:1 6.000 Myna
3\4 900.000 300.000 1:0 → ∞ Collapsed 4L 7s

Gallery

Cheat sheet for 19EDO, a hard tuning for 4L 7s (or kleistonic).
Retrieved from "https://en.xen.wiki/index.php?title=4L_7s&oldid=149023"