4L 3s: Difference between revisions

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There are two notable harmonic entropy minima: [[Kleismic_family|kleismic]], in which the generator is 6/5 and 6 of them make a 3/1, and [[Starling_temperaments|myna]], in which the generator is also 6/5 but now '''10''' of them make a 6/1 (so no 4/3's or 3/2's appear in this scale).
== Tuning ranges ==
== Tuning ranges ==
=== Sixix ===
=== Parasoft ===
Sixix tunings (with generator a supraminor third sharper than 5\18 and flatter than 9\32) have step ratios between 5/4 and 3/2.
[[Parasoft]] smitonic tunings have step ratios between 5/4 and 3/2, which implies a generator sharper than 5\18 = 333.33¢ and flatter than 9\32 = 337.5¢.


Sixix can be considered "meantone smitonic". This is because sixix tunings share the following features with [[meantone]] diatonic tunings:  
Parasoft smitonic can be considered "meantone smitonic". This is because these tunings share the following features with [[meantone]] diatonic tunings:  
* The large step is a "meantone", somewhere between near-10/9 (as in [[32edo]]) and near-9/8 (as in [[18edo]]). Thus sixix tempers out [[81/80]] like meantone does.
* The large step is a "meantone", somewhere between near-10/9 (as in [[32edo]]) and near-9/8 (as in [[18edo]]).
* The major mosthird (made of two large steps) is a roughly [[meantone]]-sized major third, thus is a stand-in for the classical diatonic major third.
* The major mosthird (made of two large steps) is a roughly [[meantone]]-sized major third, thus is a stand-in for the classical diatonic major third.


EDOs that support sixix include [[18edo]], [[25edo]], [[32edo]], and [[43edo]].
Parasoft smitonic EDOs include [[18edo]], [[25edo]], [[32edo]], and [[43edo]].
* 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted [[19edo]] diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted [[19edo]] diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* [[25edo]] can be used to make the major mosthird a good [[5/4]] (384¢).
* [[25edo]] can be used to make the major mosthird a good [[5/4]] (384¢).


The sizes of the generator, large step and small step of smitonic are as follows in various sixix tunings.  
The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.  
{| class="wikitable right-2 right-3 right-4 right-5"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
Line 517: Line 516:
|}
|}
=== Hyposoft smitonic ===
=== Hyposoft smitonic ===
These tunings (with generator a supraminor third sharper than 3\11 and flatter than 5\18) have [[step ratio]]s between 3/2 and 2/1.
[[Hyposoft]] tunings of smitonic  have [[step ratio]]s between 3/2 and 2/1 which implies that the generator is a supraminor third sharper than 3\11 = 327.27¢ and flatter than 5\18 = 333.33¢.


The large step is a sharper major second in these tunings than in sixix tunings. These tunings could be considered "[[parapyth]] smitonic" or "[[archy]] smitonic", in analogy to sixix being meantone smitonic.
The large step is a sharper major second in these tunings than in parasoft tunings. These tunings could be considered "[[parapyth]] smitonic" or "[[archy]] smitonic", in analogy to parasoft smitonic being meantone smitonic.


{| class="wikitable right-2 right-3 right-4 right-5"
{| class="wikitable right-2 right-3 right-4 right-5"
Line 543: Line 542:
| 3\29, 124.14
| 3\29, 124.14
|}
|}
=== Orgone ===
=== Hypohard ===
[[Orgone]] tunings (with generator a minor third sharper than 4\15 and flatter than 3\11) have step ratios between 2/1 and 3/1. It nominally approximates the 2.7.11 subgroup, on which the [[26edo]] tuning is very accurate and pretty much optimal. The large step approximates [[8/7]], and the major smifourth (2 large steps + 1 small step) approximates [[11/8]].
[[Hypohard]] tunings have [[step ratio]]s between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate [[8/7]], and the major smifourth (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations.


EDOs that support orgone include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
Hypohard smitonic edos include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
The sizes of the generator, large step and small step of smitonic are as follows in various orgone tunings.  
The sizes of the generator, large step and small step of smitonic are as follows in various hypohard smitonic tunings.  
{| class="wikitable right-2 right-3 right-4 right-5"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
Line 575: Line 574:
|}
|}


=== Kleismic ===
=== Parahard ===
[[Kleismic]] (aka hanson or keemun) tunings (with generator a minor third sharper than 5\19 and flatter than 4\15) have step ratios between 3/1 and 4/1. Kleismic is a [[5-limit]] microtemperament that tempers out the [[kleisma]] 15625/15552. The generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach [[3/2]]. The 7-note MOS only has one perfect fifth, so bigger MOSes, such as the [[4L 7s]] 11-note MOS, are suggested for getting 5-limit harmony.
In parahard smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure [[6/5]] minor third, and 6 minor thirds are used to reach a perfect fifth. The 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the [[4L 7s]] 11-note MOS, is suggested for getting 5-limit harmony.


EDOs that support kleismic include [[15edo]], [[19edo]], [[34edo]], [[53edo]], [[72edo]], and [[87edo]].
EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]].


The sizes of the generator, large step and small step of smitonic are as follows in various kleismic tunings.  
The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings.  
{| class="wikitable right-2 right-3 right-4 right-5"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
Line 587: Line 586:
! [[19edo]]
! [[19edo]]
! [[34edo]]
! [[34edo]]
! 2.3.5 [[POTE]] tuning
! [[53edo]]
! JI intervals represented
! JI intervals represented
|-
|-
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| 5\19, 315.79
| 5\19, 315.79
| 9\34, 317.65
| 9\34, 317.65
| 317.01
| 316.98
| 6/5
| 6/5
|-
|-
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| 4\19, 252.63
| 4\19, 252.63
| 7\34, 247.06
| 7\34, 247.06
| 248.98
| 249.06
| 15/13, 23/20
| 15/13, 23/20
|-
|-
Line 608: Line 607:
| 1\19, 63.16  
| 1\19, 63.16  
| 2\34, 70.59
| 2\34, 70.59
| 68.03
| 67.92
| 25/24
| 25/24
|}
|}
-->
 
== Intervals ==
== Intervals ==
{| class="wikitable center-all"
{| class="wikitable center-all"

Revision as of 00:30, 28 March 2021

User:IlL/Template:RTT restriction

↖ 3L 2s ↑ 4L 2s 5L 2s ↗
← 3L 3s 4L 3s 5L 3s →
↙ 3L 4s ↓ 4L 4s 5L 4s ↘
Scale structure
Step pattern LLsLsLs
sLsLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\7 to 3\4 (857.1 ¢ to 900.0 ¢)
Dark 1\4 to 2\7 (300.0 ¢ to 342.9 ¢)
TAMNAMS information
Name smitonic
Prefix smi-
Abbrev. smi
Related MOS scales
Parent 3L 1s
Sister 3L 4s
Daughters 7L 4s, 4L 7s
Neutralized 1L 6s
2-Flought 11L 3s, 4L 10s
Equal tunings
Equalized (L:s = 1:1) 5\7 (857.1 ¢)
Supersoft (L:s = 4:3) 18\25 (864.0 ¢)
Soft (L:s = 3:2) 13\18 (866.7 ¢)
Semisoft (L:s = 5:3) 21\29 (869.0 ¢)
Basic (L:s = 2:1) 8\11 (872.7 ¢)
Semihard (L:s = 5:2) 19\26 (876.9 ¢)
Hard (L:s = 3:1) 11\15 (880.0 ¢)
Superhard (L:s = 4:1) 14\19 (884.2 ¢)
Collapsed (L:s = 1:0) 3\4 (900.0 ¢)
View • Talk • Edit

4L 3s refers to the structure of MOS scales with generators ranging from 1\4edo (one degree of 4edo, 300¢) to 2\7edo (two degrees of 7edo, or approx. 342.857¢). The name smitonic smy-TON-ik /smaɪˈtɒnɪk/ has been proposed (derived from 'sharp minor third', taking sharp to mean sharp of the 12edo minor third).

4L 3s is a distorted diatonic, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).

Notation

The notation used in this article is LsLsLsL = JKLMNOPJ unless specified otherwise. We denote raising and lowering by a chroma (L − s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)

Thus the 11edo gamut is as follows:

J/Q& J&/K@ K/L@ L/K& L&/M@ M/N@ N/M& N&/O@ O/P@ P/O@ P&/J@ J

Scale tree

The spectrum looks like this:

Generator Tetrachord g in cents 2g 3g 4g Comments
1\4 1 0 1 300 600 900 0
9\35 8 1 8 308.571 617.143 925.714 34.286
8\31 7 1 7 309.677 619.355 929.023 38.71
7\27 6 1 6 311.111 622.222 933.333 44.444
6\23 5 1 5 313.043 626.087 939.13 52.174
5\19 4 1 4 315.789 631.579 947.368 63.158
9\34 7 2 7 317.647 634.294 951.941 70.588
4\15 3 1 3 320 640 960 80 L/s = 3.
11\41 8 3 8 321.951 643.902 965.854 87.805
29\108 21 8 21 322.222 644.444 966.667 88.889
18\67 13 5 13 322.388 644.776 967.364 89.522
7\26 5 2 5 323.077 646.154 969.231 92.308
31/115 22 9 22 323.478 646.956 970.434 93.913
2.44 1 2.44 323.501 647.002 970.003 94.004
24/89 17 7 17 323.595 647.191 970.786 94.382
17/63 12 5 12 323.809 647.619 971.428 95.238
10/37 7 3 7 324.324 648.648 972.972 97.297
3\11 2 1 2 327.273 654.545 981.818 109.091 Boundary of propriety (generators
larger than this are proper)
8\29 5 3 5 331.034 662.069 993.013 124.138
21\76 13 8 13 331.579 663.158 994.739 126.316
34\123 21 13 21 331.707 663.415 995.122 126.829 Golden smitonic
13\47 8 5 8 331.915 663.83 995.745 127.66
5\18 3 2 3 333.333 666.667 1000 133.333 Optimum rank range (L/s=3/2)
7\25 4 3 4 336 672 1008 144
9\32 5 4 5 337.5 675 1012.5 150
11\39 6 5 6 338.462 676.923 1015.385 153.846
13\46 7 6 7 339.13 678.261 1017.391 156.522
15\53 8 7 8 339.623 679.245 1018.868 158.491
2\7 1 1 1 342.857 685.714 1028.571 171.429

Tuning ranges

Parasoft

Parasoft smitonic tunings have step ratios between 5/4 and 3/2, which implies a generator sharper than 5\18 = 333.33¢ and flatter than 9\32 = 337.5¢.

Parasoft smitonic can be considered "meantone smitonic". This is because these tunings share the following features with meantone diatonic tunings:

  • The large step is a "meantone", somewhere between near-10/9 (as in 32edo) and near-9/8 (as in 18edo).
  • The major mosthird (made of two large steps) is a roughly meantone-sized major third, thus is a stand-in for the classical diatonic major third.

Parasoft smitonic EDOs include 18edo, 25edo, 32edo, and 43edo.

  • 18edo can be used to make large and small steps more distinct (the step ratio is 3/2, thus 18edo smitonic is distorted 19edo diatonic), or for its nearly pure 9/8. It also makes rising fifths (733.3c, a perfect mossixth) and falling fifths (666.7c, a major mosfifth) almost equally off from a just perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
  • 25edo can be used to make the major mosthird a good 5/4 (384¢).

The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.

18edo 25edo 32edo Optimal (2.9.5 POTE) tuning
generator (g) 5\18, 333.33 7\25, 336.00 9\32, 337.50 335.84
L (octave - 3g) 3\18, 200.00 4\25, 192.00 5\32, 187.50 193.16
s (4g - octave) 2\18, 133.33 3\25, 144.00 4\32, 150.00 143.36

Hyposoft smitonic

Hyposoft tunings of smitonic have step ratios between 3/2 and 2/1 which implies that the generator is a supraminor third sharper than 3\11 = 327.27¢ and flatter than 5\18 = 333.33¢.

The large step is a sharper major second in these tunings than in parasoft tunings. These tunings could be considered "parapyth smitonic" or "archy smitonic", in analogy to parasoft smitonic being meantone smitonic.

11edo 18edo 29edo
generator (g) 3\11, 327.27 5\18, 333.33 8\29, 331.03
L (octave - 3g) 2\11, 218.18 3\18, 200.00 5\29, 206.90
s (4g - octave) 1\11, 109.09 2\18, 133.33 3\29, 124.14

Hypohard

Hypohard tunings have step ratios between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate 8/7, and the major smifourth (2 large steps + 1 small step) tends to approximate 11/8; 26edo is stellar in both of these approximations.

Hypohard smitonic edos include 11edo, 15edo, 26edo, and 37edo. The sizes of the generator, large step and small step of smitonic are as follows in various hypohard smitonic tunings.

11edo 15edo 26edo JI intervals represented
generator (g) 3\11, 327.27 4\15, 320.00 7\26, 323.08 77/64
L (octave - 3g) 2\11, 218.18 3\15, 240.00 5\26, 230.77 8/7
s (4g - octave) 1\11, 109.09 1\15, 80.00 2\26, 92.31 128/121, (16/15)

Parahard

In parahard smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure 6/5 minor third, and 6 minor thirds are used to reach a perfect fifth. The 7-note MOS only has one perfect fifth, so extending the chain to bigger MOSes, such as the 4L 7s 11-note MOS, is suggested for getting 5-limit harmony.

EDOs that have parahard smitonic include 15edo, 19edo, 34edo, and 53edo.

The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings.

15edo 19edo 34edo 53edo JI intervals represented
generator (g) 4\15, 320.00 5\19, 315.79 9\34, 317.65 316.98 6/5
L (octave - 3g) 3\15, 240.00 4\19, 252.63 7\34, 247.06 249.06 15/13, 23/20
s (4g - octave) 1\15, 80.00 1\19, 63.16 2\34, 70.59 67.92 25/24

Intervals

Generators Notation (1/1 = J) Heptatonic interval category name Generators Notation of 2/1 inverse Heptatonic interval category name
The 7-note MOS has the following intervals (from some root):
0 J perfect unison 0 J octave
1 L perfect smithird -1 O perfect smisixth
2 N minor smififth (aka minor fifth) -2 M major smifourth (aka major fourth)
3 P minor smiseventh -3 K major smisecond
4 K@ minor smisecond -4 Q& major smiseventh
5 M@ minor smifourth (aka minor fourth) -5 N& major smififth (aka major fifth)
6 O@ diminished smisixth -6 L& augmented smithird
The chromatic 11-note MOS (either 7L 4s or 4L 7s) also has the following intervals (from some root):
7 J@ diminished octave -7 J& augmented unison
8 L@ diminished smithird -8 O& augmented smisixth
9 N@ diminished smififth -9 M& augmented smifourth
10 P@ diminished smiseventh -10 K& augmented smisecond

Modes

Pseudo-diatonic theory

Samples

A fugue in 18edo smitonic (WIP)

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