4L 7s: Difference between revisions

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| nSmallSteps = 7
| nSmallSteps = 7
| Equalized = 3
| Equalized = 3
| Paucitonic = 1
| Collapsed = 1
| Pattern = LssLssLssLs
| Pattern = LssLssLssLs
}}
}}
{{MOS intro}}
One of the [[harmonic entropy]] minimums in this range is [[Kleismic family|Kleismic/Hanson]].


'''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.<u>27</u>¢), representing approximate diatonic minor thirds ([[6/5]]). One of the harmonic entropy minimums in this range is [[Kleismic family|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'''.


4L 7s has a heptatonic subset, which is the [[hard]] end of the spectrum of the [[smitonic]] scale (4L 3s).
== Scale properties ==
{{TAMNAMS use}}


A proposed name for this scale is '''kleistonic''', based on an extension of [[TAMNAMS]] to bigger MOS scales.
=== Intervals ===
{{MOS intervals}}


== Notation ==
=== Generator chain ===
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.
{{MOS genchain}}


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'''
=== Modes ===
{{MOS mode degrees}}


==== Letter names ====
== Tuning ranges==
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 ==
{| class="wikitable center-all"
|-
! Generators
! Notation (1/1 = А҃)
! Interval category name
! Generators
! Notation of 2/1 inverse
! Interval category name
|-
| colspan="6" style="text-align:left" | 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
|-
| colspan="6" style="text-align:left" | 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:
{| class="wikitable center-all"
|-
| 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 ===
=== 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¢.
The soft range for tunings of 4L&nbsp;7s encompasses parasoft and hyposoft tunings. This implies step ratios smaller than 2/1, meaning a generator sharper than {{nowrap|4\15 {{=}} 320{{c}}}}.


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 kleistonic edos include [[15edo]] and [[26edo]].
Soft 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:
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"
|-
|-
!
!
![[15edo]] (basic)
! [[15edo]] (basic)
! [[26edo]] (soft)
! [[26edo]] (soft)
! Some JI approximations
! Some JI approximations
Line 247: Line 57:


=== Hypohard ===
=== Hypohard ===
[[File:19EDO_Kleistonic_cheat_sheet.png|400px|thumb|right|Cheat sheet for 19EDO kleistonic, a hard kleistonic tuning]]
Hypohard tunings of 4L&nbsp;7s have step ratios between 2/1 and 3/1, implying a generator sharper than {{nowrap|5\19 {{=}} 315.79{{c}}}} and flatter than {{nowrap|4\15 {{=}} 320{{c}}}}.
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.
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]].
Hypohard 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:  
The sizes of the generator, large step and small step of 4L&nbsp;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 268: Line 77:
| 6/5
| 6/5
|-
|-
| L (octave - 3g)
| L ({{nowrap|octave − 3g}})
| 2\15, 160.00
| 2\15, 160.00
| 3\19, 189.47
| 3\19, 189.47
Line 274: Line 83:
| 10/9, 11/10 (in 15edo)
| 10/9, 11/10 (in 15edo)
|-
|-
| s (4g - octave)
| s ({{nowrap|4g − octave}})
| 1\15, 80.00
| 1\15, 80.00
| 1\19, 63.16
| 1\19, 63.16
Line 282: Line 91:


=== Parahard ===
=== 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¢.
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 kleistonic edos include [[19edo]], [[32edo]], and [[42edo]].
Parahard edos include [[19edo]], 23[[23edo|edo]], and [[42edo]].
The sizes of the generator, large step and small step of kleistonic are as follows in various parahard kleistonic 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"
|-
|-
!
!
![[19edo]] (hard)
! [[19edo]] (hard)
![[23edo]] (superhard)
! [[23edo]] (superhard)
! [[42edo]] (parahard)
! [[42edo]] (parahard)
! Some JI approximations
! Some JI approximations
Line 302: Line 111:
| 6/5
| 6/5
|-
|-
| L (octave - 3g)
| L ({{nowrap|octave − 3g}})
| 3\19, 189.47
| 3\19, 189.47
| 4\23, 208.70
| 4\23, 208.70
Line 308: Line 117:
| 10/9, 9/8
| 10/9, 9/8
|-
|-
| s (4g - octave)
| s ({{nowrap|4g − octave}})
| 1\19, 63.16
| 1\19, 63.16
| 1\23, 52.17
| 1\23, 52.17
Line 315: Line 124:
|}
|}


=== Hyperhard ===
=== 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¢.
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 kleistonic edos include [[23edo]], [[31edo]], and [[27edo]].
Hyperhard 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:
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 337: Line 146:
| 6/5
| 6/5
|-
|-
| L (octave - 3g)
| L ({{nowrap|octave − 3g}})
| 4\23, 208.70
| 4\23, 208.70
| 6\31, 232.26
| 6\31, 232.26
Line 343: Line 152:
| 8/7, 9/8
| 8/7, 9/8
|-
|-
| s (4g - octave)
| s ({{nowrap|4g − octave}})
| 1\23, 52.17
| 1\23, 52.17
| 1\31, 38.71
| 1\31, 38.71
| 1\27, 44.44
| 1\27, 44.44
| 36/35, 45/44
| 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.
{| class="wikitable center-all"
|-
! Mode
! [[Modal UDP Notation|UDP]]
! Name
|-
| LsLssLssLss
| <nowiki>10|0</nowiki>
| Supernerevarine
|-
| LssLsLssLss
| <nowiki>9|1</nowiki>
| Supervivecan
|-
| LssLssLsLss
| <nowiki>8|2</nowiki>
| Superbaardauan
|-
| LssLssLssLs
| <nowiki>7|3</nowiki>
| Superlorkhanic
|-
| sLsLssLssLs
| <nowiki>6|4</nowiki>
| Supervvardenic
|-
| sLssLsLssLs
| <nowiki>5|5</nowiki>
| Supersothic
|-
| sLssLssLsLs
| <nowiki>4|6</nowiki>
| Supernumidian
|-
| sLssLssLssL
| <nowiki>3|7</nowiki>
| Superkagrenacan
|-
| ssLsLssLssL
| <nowiki>2|8</nowiki>
| Supernecromic
|-
| ssLssLsLssL
| <nowiki>1|9</nowiki>
| Superalmalexian
|-
| ssLssLssLsL
| <nowiki>0|10</nowiki>
| Superdagothic
|}
|}


== Temperaments ==
== Temperaments ==
== Scales ==
* [[Oregon11]]
* [[Orgone11]]
* [[Magicaltet11]]
* [[Cata11]]
* [[Starlingtet11]]
* [[Myna11]]


== Scale tree ==
== Scale tree ==
The spectrum looks like this:
{{MOS tuning spectrum
{| class="wikitable center-all"
| 6/5 = [[Oregon]]
! colspan="6" rowspan="2" | Generator
| 10/7 = [[Orgone]]
! colspan="2" | Cents
| 11/7 = [[Magicaltet]]
! rowspan="2" | L
| 13/8 = Golden superklesimic
! rowspan="2" | s
| 5/3 = [[Superkleismic]]
! rowspan="2" | L/s
| 7/3 = [[Catalan]]
! rowspan="2" | Comments
| 13/5 = [[Countercata]]
|-
| 8/3 = [[Hanson]]/[[cata]]
! Chroma-positive
| 11/4 = [[Catakleismic]]
! Chroma-negative
| 10/3 = [[Parakleismic]]
|-
| 9/2 = [[Oolong]]
| 8\11 || || || || || || 872.727 || 327.273 || 1 || 1 || 1.000 ||
| 5/1 = [[Starlingtet]]
|-
| 6/1 = [[Myna]]
| || || || || || 43\59 || 874.576 || 325.424 || 6 || 5 || 1.200 || Oregon
}}
|-
 
| || || || || 35\48 || || 875.000 || 325.000 || 5 || 4 || 1.250 ||
== Gallery ==
|-
[[File:19EDO_Kleistonic_cheat_sheet.png|825x825px|thumb|Cheat sheet for 19EDO, a hard tuning for 4L&nbsp;7s (or kleistonic).|alt=|left]]
| || || || || || 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<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:Scales]]
[[Category:11-tone scales]]
[[Category:Abstract MOS patterns]]
[[Category:Kleistonic]] <!-- main article -->

Latest revision as of 19:30, 5 August 2025

↖ 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.

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 interval regions.

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 ¢

Generator chain

Generator chain of 4L 7s
Bright gens Scale degree Abbrev.
14 Augmented 2-mosdegree A2md
13 Augmented 5-mosdegree A5md
12 Augmented 8-mosdegree A8md
11 Augmented 0-mosdegree A0md
10 Augmented 3-mosdegree A3md
9 Major 6-mosdegree M6md
8 Major 9-mosdegree M9md
7 Major 1-mosdegree M1md
6 Major 4-mosdegree M4md
5 Major 7-mosdegree M7md
4 Major 10-mosdegree M10md
3 Major 2-mosdegree M2md
2 Major 5-mosdegree M5md
1 Perfect 8-mosdegree P8md
0 Perfect 0-mosdegree
Perfect 11-mosdegree		 P0md
P11md
−1 Perfect 3-mosdegree P3md
−2 Minor 6-mosdegree m6md
−3 Minor 9-mosdegree m9md
−4 Minor 1-mosdegree m1md
−5 Minor 4-mosdegree m4md
−6 Minor 7-mosdegree m7md
−7 Minor 10-mosdegree m10md
−8 Minor 2-mosdegree m2md
−9 Minor 5-mosdegree m5md
−10 Diminished 8-mosdegree d8md
−11 Diminished 11-mosdegree d11md
−12 Diminished 3-mosdegree d3md
−13 Diminished 6-mosdegree d6md
−14 Diminished 9-mosdegree d9md

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.

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 Catakleismic
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).
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