4L 7s

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4L 7s
Pattern LssLssLssLs
Period 2/1
Generator range 1\4 (300.0¢) to 3\11 (327.3¢)
Parent MOS 4L 3s
Daughter MOSes 11L 4s, 4L 11s
Sister MOS 7L 4s
TAMNAMS name kleistonic
Equal tunings
Supersoft (L:s = 4:3) 10\37 (324.3¢)
Soft (L:s = 3:2) 7\26 (323.1¢)
Semisoft (L:s = 5:3) 11\41 (322.0¢)
Basic (L:s = 2:1) 4\15 (320.0¢)
Semihard (L:s = 5:2) 9\34 (317.6¢)
Hard (L:s = 3:1) 5\19 (315.8¢)
Superhard (L:s = 4:1) 6\23 (313.0¢)

4L 7s refers to the structure of MOS scales with generators ranging from 1\4edo (one degree of 4edo, 300¢) to 3\11edo (three degrees of 11edo, 327.27¢), representing approximate diatonic minor thirds (6/5). One of the harmonic entropy minimums in this range is Kleismic/Hanson.

4L 7s has a heptatonic subset, which is the hard end of the spectrum of the smitonic scale (4L 3s).

The TAMNAMS name for this scale is kleistonic.

Notation

The notation used in this article is LssLsLssLss = АВГДЕЅЗИѲІѦА, based on old Cyrillic numerals 1-10, and the addition of the small yus (Ѧ) for 11 (old "ya" symbolically representing І҃А҃=11). A titlo can be optionally used as a numeric sign (А҃), depending on font rendering, clarity, and style. Chromas are represented by regular sharps and flats.

Thus the 15edo gamut is as follows: А А#/Вb В Г Д Д#/Еb Е Ѕ Ѕ#/Зb З И Ѳ Ѳ#/Іb І Ѧ А

Letter names

The letters can be named in English as such: Az, Vede, Glagol, Dobro, Yest, Dzelo, Zemlya, Izhe, Thita, I (Ee), Yas. They can also be named as numbers 1-11.

Intervals

Generators Notation (1/1 = А҃) Interval category name Generators Notation of 2/1 inverse Interval category name
The 11-note MOS has the following intervals (from some root):
0 А perfect unison 0 А dodecave (same as octave)
1 Д perfect kleifourth (minor third) -1 Ѳ perfect kleininth (major sixth)
2 Зb minor kleiseventh -2 Ѕ major kleisixth
3 Іb minor kleitenth -3 Г major kleithird
4 Вb minor kleisecond -4 Ѧ major kleieleventh
5 Еb minor kleififth -5 И major kleieighth
6 Иb minor kleieighth -6 Е major kleififth
7 Ѧb minor kleieleventh -7 В major kleisecond
8 Гb minor kleithird -8 І major kleitenth
9 Ѕb minor kleisixth -9 З major kleiseventh
10 Ѳb diminished kleininth -10 Д# augmented kleithird
The chromatic 15-note MOS (either 4L 11s, 11L 4s, or 15edo) also has the following intervals (from some root):
11 Аb diminished dodecave -11 А# augmented unison (chroma)
12 Дb diminished kleifourth -12 Ѳ# augmented kleininth
13 Зbb diminished kleiseventh -13 Ѕ# augmented kleisixth
14 Іbb diminished kleitenth -14 Г# augmented kleithird

Genchain

The generator chain for this scale is as follows:

Дb Аb Ѳb Ѕb Гb Ѧb Иb Еb Вb Іb Зb Д А Ѳ Ѕ Г Ѧ И Е В І З Д# А# Ѳ# Ѕ# Г# Ѧ# И# Е# В# І# З#
d4 d12 d9 m6 m3 m11 m8 m5 m2 m10 m7 P4 P1 P9 M6 M3 M11 M8 M5 M2 M10 M7 A4 A1 A9 A6 A3 A11 A8 A5 A2 A10 A7

Tuning ranges

Soft range

The soft range for tunings of kleistonic 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 kleistonic edos include 15edo and 26edo. The sizes of the generator, large step and small step of kleistonic are as follows in various soft kleistonic 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

Cheat sheet for 19EDO kleistonic, a hard kleistonic tuning

Hypohard tunings of kleistonic 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 kleistonic edos include 15edo, 19edo, and 34edo. The sizes of the generator, large step and small step of kleistonic are as follows in various hypohard kleistonic 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 kleistonic 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 kleistonic edos include 19edo, 23edo, and 42edo. The sizes of the generator, large step and small step of kleistonic are as follows in various parahard kleistonic 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 kleistonic 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 kleistonic edos include 23edo, 31edo, and 27edo. The sizes of the generator, large step and small step of kleistonic are as follows in various hyperhard kleistonic 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

Modes

The names are based on smitonic modes, modified with the "super-" prefix, with thematic additions, as there are an extra 4 modes available.

Mode UDP Name
LsLssLssLss 10|0 Supernerevarine
LssLsLssLss 9|1 Supervivecan
LssLssLsLss 8|2 Superbaardauan
LssLssLssLs 7|3 Superlorkhanic
sLsLssLssLs 6|4 Supervvardenic
sLssLsLssLs 5|5 Supersothic
sLssLssLsLs 4|6 Supernumidian
sLssLssLssL 3|7 Superkagrenacan
ssLsLssLssL 2|8 Supernecromic
ssLssLsLssL 1|9 Superalmalexian
ssLssLssLsL 0|10 Superdagothic

Temperaments

Scales

Scale tree

The spectrum looks like this:

Generator Cents L s L/s Comments
Chroma-positive Chroma-negative
8\11 872.727 327.273 1 1 1.000
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
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 1.428 Orgone
19\26 876.923 323.077 3 2 1.500 L/s = 3/2
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 superkleismic
30\41 878.049 321.951 5 3 1.667 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 kleistonic
(Generators smaller 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
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 Catakleismic
14\19 884.211 315.789 3 1 3.000 L/s = 3/1
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
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 → inf