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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"
|-
|-
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=== 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"
|-
|-
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=== 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"
|-
|-
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=== 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"
|-
|-
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== 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]]
Retrieved from "https://en.xen.wiki/w/4L_7s"