4L 7s: Difference between revisions

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{| class="wikitable center-all"
{| class="wikitable center-all"
|-
|-
| -11
! Generators
| -10
| −11 || −10 || −9 || −8 || −7 || −6 || −5 || −4 || −3 || −2 || −1 || 0 || +1 || +2 || +3 || +4 || +5 || +6 || +7 || +8 || +9 || +10 || +11
| -9
| -8
| -7
| -6
| -5
| -4
| -3
| -2
| -1
| 0
| +1
| +2
| +3
| +4
| +5
| +6
| +7
| +8
| +9
| +10
| +11
|-
|-
| d12
! Interval quality
| d9
| d12 || d9 || m6 || m3 || m11 || m8 || m5 || m2 || m10 || m7 || P4 || P1 || P9 || M6 || M3 || M11 || M8 || M5 || M2 || M10 || M7 || A4 || A1
| m6
| m3
| m11
| m8
| m5
| m2
| m10
| m7
| P4
| P1
| P9
| M6
| M3
| M11
| M8
| M5
| M2
| M10
| M7
| A4
| A1
|}
|}


Revision as of 12:34, 26 July 2024

↖ 3L 6s ↑ 4L 6s 5L 6s ↗
← 3L 7s 4L 7s 5L 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
Related to 4L 3s (smitonic)
With tunings 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.

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 ¢

Genchain

The generator chain for this scale is as follows:

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 d12 d9 m6 m3 m11 m8 m5 m2 m10 m7 P4 P1 P9 M6 M3 M11 M8 M5 M2 M10 M7 A4 A1

Tuning ranges

Soft range

The soft range for tunings of p-chro smitonic 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 p-chro smitonic edos include 15edo and 26edo. The sizes of the generator, large step and small step of p-chro smitonic 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

Cheat sheet for 19EDO p-chro smitonic, a hard p-chro smitonic tuning

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

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.

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 p-chro smitonic
(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
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