4L 7s: Difference between revisions

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== Intervals ==
== Intervals ==
{{MOS intervals}}
{{MOS intervals}}
== Genchain ==
The generator chain for this scale is as follows:
{| class="wikitable center-all"
|-
! 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 ==
== Tuning ranges ==
=== Soft range ===
=== 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¢.
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|Orgone[7]]]. The small step is recognizable as a near diatonic semitone, while the large step is in the ambiguous area of neutral seconds.
This is the range associated with extensions of [[Orgone|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]].
Soft 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:
The sizes of the generator, large step and small step of 4L 7s are as follows in various soft tunings:
{| class="wikitable right-2 right-3 right-4"
{| class="wikitable right-2 right-3 right-4"
|-
|-
Line 59: Line 48:


=== Hypohard ===
=== Hypohard ===
[[File:19EDO_Kleistonic_cheat_sheet.png|400px|thumb|right|Cheat sheet for 19EDO p-chro smitonic, a hard p-chro smitonic tuning]]
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¢.
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.
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]].
Hypohard 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:  
The sizes of the generator, large step and small step of 4L 7s are as follows in various hypohard tunings:  
{| class="wikitable right-2 right-3 right-4"
{| class="wikitable right-2 right-3 right-4"
|-
|-
Line 94: Line 82:


=== Parahard ===
=== 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¢.
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.
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]], 23[[23edo|edo]], and [[42edo]].
Parahard edos include [[19edo]], 23[[23edo|edo]], 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:
The sizes of the generator, large step and small step of 4L 7s are as follows in various parahard tunings:
{| class="wikitable right-2 right-3 right-4"
{| class="wikitable right-2 right-3 right-4"
|-
|-
Line 128: Line 116:


=== Hyperhard ===
=== 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¢.
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.
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.
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]].
Hyperhard 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:
The sizes of the generator, large step and small step of 4L 7s are as follows in various hyperhard tunings:
{| class="wikitable right-2 right-3 right-4"
{| class="wikitable right-2 right-3 right-4"
|-
|-
Line 175: Line 163:


== Scale tree ==
== Scale tree ==
The spectrum looks like this:
{{Scale tree|Comments=6/5: Oregon;
{| class="wikitable center-all"
10/7: Orgone;
! colspan="6" rowspan="2" | Generator
11/7: Magicaltet;
! colspan="2" | Cents
13/8: Golden superklesimic;
! rowspan="2" | L
5/3: Superkleismic;
! rowspan="2" | s
7/3: Catalan;
! rowspan="2" | L/s
13/5: Countercata;
! rowspan="2" | Comments
8/3: Hanson/cata;
|-
10/3: Parakleismic;
! Chroma-positive
9/2: Oolong;
! Chroma-negative
5/1: Starlingtet;
|-
6/1: Myna}}
| 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<br>(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 ||
|}
 
[[Category:11-tone scales]]
[[Category:11-tone scales]]
[[Category:Kleistonic]] <!-- main article -->
[[Category:Kleistonic]] <!-- main article -->
== Gallery ==
[[File:19EDO_Kleistonic_cheat_sheet.png|825x825px|thumb|Cheat sheet for 19EDO, a hard tuning for 4L 7s (or kleistonic).|alt=|left]]

Revision as of 20:52, 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 ¢

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

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

Template:Scale tree

Gallery

Cheat sheet for 19EDO, a hard tuning for 4L 7s (or kleistonic).
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