Xenharmonic Wiki

4L 3s: Difference between revisions

← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled, emailconfirmed
15,668 edits
Overthink (talk | contribs)
autopatrolled
2,730 edits
→Scale tree: add this
 
(163 intermediate revisions by 14 users not shown)
Line 1: Line 1:
{{User:IlL/Template:RTT_restriction}}
{{Interwiki
{{Infobox MOS
|en=4L 3s
| Name = smitonic
|es=
| Periods = 1
|de=
| nLargeSteps = 4
|ja=4L 3s
| nSmallSteps = 3
| Equalized = 2
| Paucitonic = 1
| Pattern = LLsLsLs
}}
}}
{{Infobox MOS}}


'''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).
{{MOS intro}}
4L 3s can be seen as a [[Warped diatonic|warped diatonic scale]], where one large step of diatonic ([[5L 2s]]) is replaced with a small step.


4L 3s is a distorted diatonic, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).
== Name ==
<!--
{{TAMNAMS name}}
4L 3s has several temperament interpretations:


# With generator size between 5\18 (333.3c) and 11\39 (338.5c): [[Sixix]], corresponding to a L/s ratio between 3/2 and 6/5.
== Scale properties ==
# With generator size between 4\15 (320.0c) and 3\11 (327.3c): [[Orgone]], corresponding to a L/s ratio between 3 and 2.
{{TAMNAMS use}}
# With generator size between 5\19 (315.8c) and 4\15 (320.0c): [[Kleismic]], corresponding to a L/s ratio between 4 and 3.


There are also other temperaments in the 4L 3s range, particularly [[amity]] and [[myna]], but 7 notes in the generator chain are not enough to contain the most concordant chords in these temperaments; you would need to use a [[MODMOS]] or use a larger MOS gamut.-->
=== Intervals ===
== Notation==
{{MOS intervals}}
The notation used in this article is LsLsLsL = JKLMNOPJ unless specified otherwise. We denote raising and lowering by a chroma (L &minus; s) by & "amp" and @ "at". (Mnemonics: & "and" means additional pitch. @ "at" rhymes with "flat".)


Thus the [[11edo]] gamut is as follows:
=== Generator chain ===
{{MOS genchain}}


'''J/Q&''' J&/K@ '''K/L@''' '''L/K&''' L&/M@ '''M/N@''' '''N/M&''' N&/O@ '''O/P@''' '''P/O@''' P&/J@ '''J'''
=== Modes ===
== Scale tree ==
{{MOS mode degrees}}
The spectrum looks like this:
 
==== Proposed names ====
Alexandru Ianu ([[User:Ayceman|Ayceman]])<ref>Description of ''Sylvian Moon Dance'' mentioning the naming proposal https://musescore.com/user/36772625/scores/6700443 – The theme relates to the mystical nature of the Tribunal and TES lore, which fits smitonic.</ref> has proposed the following mode names relating to the Almsivi in Morrowind (TES):
{{MOS modes
| Mode Names=Nerevarine $
Vivecan $
Lorkhanic $
Sothic $
Kagrenacan $
Almalexian $
Dagothic $
}}
 
== Theory ==
=== Low harmonic entropy scales ===
There are two notable harmonic entropy minima:
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1.
* [[myna|Myna temperament]], in which the generator is also 6/5 but it takes 10 of them to make a 6/1, meaning that a larger MOS than 4L&nbsp;3s is required to reach 3/2 or 4/3.
 
=== Temperament interpretations ===
{{main|4L&nbsp;3s/Temperaments}}
4L&nbsp;3s has the following temperament interpretations:
* [[Sixix]], with generators around 338.6{{c}}.
* [[Orgone]], with generators around 323.4{{c}}.
* [[Kleismic]], with generators around 317{{c}}.
 
Other temperaments, such as [[amity]] and [[myna]], require more than 7 pitches to contain the concordant chords optimized by these temperaments. If restricted to a rank-2 approach, a [[MODMOS]] or a larger MOS gamut is necessary to access these pitches.


{| class="wikitable"
|-
! colspan="8" | Generator
! | Tetrachord
! | g in cents
! | 2g
! | 3g
! | 4g
! | Comments
|-
| | 1\4
| |
| |
| |
| |
| |
| |
| |
| | 1 0 1
| | 300
| | 600
| | 900
| | 0
| style="text-align:center;" |
|-
|
|
|
|
|
|
|
|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
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 7\27
| |
| |
| | 6 1 6
| | 311.111
| | 622.222
| | 933.333
| | 44.444
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 6\23
| |
| |
| |
| | 5 1 5
| | 313.043
| | 626.087
| | 939.13
| | 52.174
| style="text-align:center;" |
|-
| |
| |
| |
| | 5\19
| |
| |
| |
| |
| | 4 1 4
| | 315.789
| | 631.579
| | 947.368
| | 63.158
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 9\34
| |
| |
| |
| | 7 2 7
| | 317.647
| | 634.294
| | 951.941
| | 70.588
| style="text-align:center;" |
|-
| |
| |
| | 4\15
| |
| |
| |
| |
| |
| | 3 1 3
| | 320
| | 640
| | 960
| | 80
| style="text-align:center;" | 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
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| | 31/115
| | 22 9 22
| | 323.478
| | 646.956
| | 970.434
| | 93.913
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| | 2.44 1 2.44
| | 323.501
| | 647.002
| | 970.003
| | 94.004
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 24/89
| |
| | 17 7 17
| | 323.595
| | 647.191
| | 970.786
| | 94.382
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 17/63
| |
| |
| | 12 5 12
| | 323.809
| | 647.619
| | 971.428
| | 95.238
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 10/37
| |
| |
| |
| | 7 3 7
| | 324.324
| | 648.648
| | 972.972
| | 97.297
| style="text-align:center;" |
|-
| |
| | 3\11
| |
| |
| |
| |
| |
| |
| | 2 1 2
| | 327.273
| | 654.545
| | 981.818
| | 109.091
| style="text-align:center;" | Boundary of propriety (generators <br>larger than this are proper)
|-
| |
| |
| |
| | 8\29
| |
| |
| |
| |
| | 5 3 5
| | 331.034
| | 662.069
| | 993.013
| | 124.138
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 21\76
| |
| |
| | 13 8 13
| | 331.579
| | 663.158
| | 994.739
| | 126.316
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| | 34\123
| | 21 13 21
| | 331.707
| | 663.415
| | 995.122
| | 126.829
| style="text-align:center;" | Golden smitonic
|-
| |
| |
| |
| |
| | 13\47
| |
| |
| |
| | 8 5 8
| | 331.915
| | 663.83
| | 995.745
| | 127.66
| style="text-align:center;" |
|-
| |
| |
| | 5\18
| |
| |
| |
| |
| |
| | 3 2 3
| | 333.333
| | 666.667
| | 1000
| | 133.333
| style="text-align:center;" | Optimum rank range (L/s=3/2)
|-
| |
| |
| |
| | 7\25
| |
| |
| |
| |
| | 4 3 4
| | 336
| | 672
| | 1008
| | 144
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 9\32
| |
| |
| |
| | 5 4 5
| | 337.5
| | 675
| | 1012.5
| | 150
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 11\39
| |
| |
| | 6 5 6
| | 338.462
| | 676.923
| | 1015.385
| | 153.846
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 13\46
| |
| | 7 6 7
| | 339.13
| | 678.261
| | 1017.391
| | 156.522
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| | 15\53
| | 8 7 8
| | 339.623
| | 679.245
| | 1018.868
| | 158.491
| style="text-align:center;" |
|-
| | 2\7
| |
| |
| |
| |
| |
| |
| |
| | 1 1 1
| | 342.857
| | 685.714
| | 1028.571
| | 171.429
| style="text-align:center;" |
|}
<!--
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 ===
{{Todo|Populate|comment=Populate with JI ratios from prior edits of this page.|inline=1}}
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.
 
=== Simple tunings ===
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 11edo, 15edo, and 18edo, respectively.
{{MOS tunings}}
 
=== Parasoft tunings ===
Parasoft smitonic tunings can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:


Sixix can be considered "meantone smitonic". This is because sixix tunings share the following features with [[meantone]] diatonic tunings:
* The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
* 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 augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.
* 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]].
These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702{{c}}), and they have otherwise fairly convincing versions of both diatonic structure and tertian harmony, provided you frequently modify using the comma-like chromas. For this reason, parasoft might be the most accessible smitonic tuning range.
* 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 sixix tunings.
Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[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 ===
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.


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.
* 18edo can be used to make the large and small steps more distinct, or can be considered a distorted 19edo diatonic.
** 18edo has a major 1-mosstep that is close to 9/8 (203{{c}}).
** 18edo's major and minor 4-mossteps are both equally off from 12edo's diatonic perfect 5th (700{{c}}) by 33.3{{c}}.
** 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* The augmented 2-mosstep of 25edo is very close to 5/4 (386{{c}}).
{{MOS tunings|Step Ratios=3/2; 7/5; 4/3}}


{| class="wikitable right-2 right-3 right-4 right-5"
=== Hyposoft tunings ===
|-
Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327{{c}} and 333{{c}}. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] 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
|}
=== Orgone ===
[[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]].


EDOs that support orgone include [[11edo]], [[15edo]], [[26edo]], and [[37edo]].
Edos include [[11edo]] (not shown), [[18edo]], and [[29edo]].
The sizes of the generator, large step and small step of smitonic are as follows in various orgone tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[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)
|}


=== Kleismic ===
{{MOS tunings|Step Ratios=3/2; 5/3; 7/4}}
[[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.


EDOs that support kleismic include [[15edo]], [[19edo]], [[34edo]], [[53edo]], [[72edo]], and [[87edo]].
=== Hypohard tunings===
Hypohard smitonic tunings (2:1 to 3:1) have generators between 320{{c}} and 327{{c}}. The major 1-mosstep, or large step, tends to approximate [[8/7]] (231{{c}}) and the major 3-mosstep tends to approximate [[11/8]] (551{{c}}). [[26edo]] approximates these two intervals very well. These JI approximations are associated with [[orgone]] temperament.


The sizes of the generator, large step and small step of smitonic are as follows in various kleismic tunings.
Other hypohard edos include [[11edo]] (not shown), [[15edo]] and [[37edo]].
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[15edo]]
! [[19edo]]
! [[34edo]]
! 2.3.5 [[POTE]] tuning
! JI intervals represented
|-
| generator (g)
| 4\15, 320.00
| 5\19, 315.79
| 9\34, 317.65
| 317.01
| 6/5
|-
| L (octave - 3g)
| 3\15, 240.00
| 4\19, 252.63
| 7\34, 247.06
| 248.98
| 15/13, 23/20
|-
| s (4g - octave)
| 1\15, 80.00
| 1\19, 63.16
| 2\34, 70.59
| 68.03
| 25/24
|}
-->
== Intervals ==
{| class="wikitable center-all"
|-
! Generators
! Notation (1/1 = J)
! Heptatonic interval category name
! Generators
! Notation of 2/1 inverse
! Heptatonic interval category name
|-
| colspan="6" style="text-align:left" | 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
|-
| colspan="6" style="text-align:left" | 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 ==
{{MOS tunings|Step Ratios=3/1; 5/2; 7/3}}
 
=== Parahard tunings ===
Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9{{c}} and 320{{c}}, putting it close to a pure 6/5 (316{{c}}). Stacking six generators and octave-reducing approximates 3/2 (702{{c}}), a diatonic perfect 5th, represented by the diminished 5-mosstep.
 
This range contains very accurate edos such as [[53edo]] and [[72edo]], and has very accurate approximations to many [[low-overtone JI]] intervals, namely basic [[5-limit]] ratios and some ratios involving 13. However, 4L 3s only has one interval of 3/2, so it's suggested to use a larger MOS, such as [[4L 7s]], to achieve 5-limit harmony.
 
These JI approximations are associated with [[kleismic]] temperament, through the 2.3.5.13 extension known as [[Kleismic family#Cata|cata]].
 
Parahard edos smaller than 53edo include [[15edo]] (not shown), [[19edo]], and [[34edo]].
 
{{MOS tunings|Step Ratios=4/1; 11/3; 7/2}}
 
== Scales ==
* [[Orgone7]]
* [[Cata7]]
* [[Myna7]]
 
== Scale tree==
{{MOS tuning spectrum
| 6/5 = [[Amity]]/[[hitchcock]]&nbsp;↑
| 5/4 = [[Sixix]]
| 4/3 = [[Supramin]]
| 13/8 = Golden 4L&nbsp;3s (868.3282{{c}})
| 12/5 = [[Hyperkleismic]]
| 5/2 = [[Orgone]]
| 13/5 = Golden superkleismic
| 8/3 = [[Superkleismic]]
| 11/3 = [[Hanson]]/[[keemun]]
| 6/1 = [[Oolong]]/[[myna]]&nbsp;↓
}}


== Pseudo-diatonic theory ==
== Music ==
<!--
* [[City of the Asleep]], [https://cityoftheasleep.bandcamp.com/album/an-amputated-elliptic-knob-of-the-cryptocurve-regenerates "An Amputated Elliptic Knob of the Cryptocurve Regenerates"] (Various orgone edos)
=== Orgone ===
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
=== Sixix ===
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
== Primodal theory ==
=== Primodal chords ===
=== Nejis ===


== Rank-2 temperaments ==
== References ==
=== Myna (27&31) ===
<references />
=== Kleismic (19&15, 2.3.5.7) ===
=== Orgone (15&11, 2.7.11) ===
=== Sixix (18&25) ===
[[Sixix]] can be viewed as a [[dual-fifth]] temperament, i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3):
* It has both a sharp fifth and a flat fifth but no near-just 3/2.
* Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup.
* The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in [[porcupine]] harmony. Tempering out this comma leads to [[7edo]].


-->
[[Category:Smitonic|*]] <!--Main article-->
== Samples ==
[[Category:7-tone scales]]
[[File:Sixix Fugue.mp3]] A fugue in [[18edo]] smitonic (WIP)
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]
Retrieved from "https://en.xen.wiki/w/4L_3s"