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{{Infobox MOS
{{Interwiki
| Name = smitonic
|en=4L 3s
| Periods = 1
|es=
| nLargeSteps = 4
|de=
| nSmallSteps = 3
|ja=4L 3s
| Equalized = 2
| Paucitonic = 1
| Pattern = LLsLsLs
}}
}}
{{Infobox MOS}}


'''4L 3s''' refers to an [[MOS]] scale with four large steps and three small steps, one mode of which is '''LLsLsLs'''. It is generated by any interval between 1\4edo (one degree of [[4edo]], or 300¢) and 2\7edo (two degrees of [[7edo]], or approx. 342.857¢).
{{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 can be thought of as a [[Warped diatonic|warped diatonic scale]], because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).
== Name ==
== Standing assumptions ==
{{TAMNAMS name}}
The [[TAMNAMS]] system is used in this article to name 4L 3s intervals and step size ratios and step ratio ranges.


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".)
== Scale properties ==
{{TAMNAMS use}}


Thus the [[11edo]] gamut is as follows:
=== Intervals ===
{{MOS intervals}}


'''J''' J&/K@ '''K''' '''L''' L&/M@ '''M''' '''N''' N&/O@ '''O''' '''P''' P&/J@ '''J'''
=== Generator chain ===
{{MOS genchain}}


== Names ==
=== Modes ===
The [[TAMNAMS]] MOS naming system (used in this article) uses the name '''smitonic''' ''smy-TON-ik'' /smaɪˈtɒnɪk/ for this pattern. The name is derived from 'sharp minor third', since the central range of the spectrum, 4\15 = 320¢ to 7\18 = 333.33¢, has minor third generators that are significantly sharp of 6/5.
{{MOS mode degrees}}
== Intervals ==
{| class="wikitable center-all"
|-
! Generators
! Notation (1/1 = J)
! [[TAMNAMS]] name
! In L's and s's
! Generators
! Notation of 2/1 inverse
! [[TAMNAMS]] name
! In L's and s's
|-
| colspan="8" style="text-align:left" | The 7-note MOS has the following intervals (from some root):
|-
| 0
| J
| perfect unison
| 0L + 0s
| 0
| J
| octave
| 4L + 3s
|-
| 1
| L
| perfect mosthird
| 1L + 1s
| -1
| O
| perfect mossixth
| 3L + 2s
|-
| 2
| N
| minor mosfifth
| 2L + 2s
| -2
| M
| major mosfourth
| 2L + 1s
|-
| 3
| P
| minor mosseventh
| 3L + 3s
| -3
| K
| major mossecond
| 1L + 0s
|-
| 4
| K@
| minor mossecond
| 0L + 1s
| -4
| Q&
| major mosseventh
| 4L + 2s
|-
| 5
| M@
| minor mosfourth
| 1L + 2s
| -5
| N&
| major mosfifth
| 3L + 1s
|-
| 6
| O@
| diminished mossixth
| 2L + 3s
| -6
| L&
| augmented mosthird
| 2L + 0s
|-
| colspan="8" style="text-align:left" | The chromatic 11-note MOS (either [[7L 4s]], [[4L 7s]], or [[11edo]]) also has the following intervals (from some root):
|-
| 7
| J@
| diminished mosoctave
| 5L + 2s
| -7
| J&
| augmented mosunison; chroma
| 1L - 1s
|-
| 8
| L@
| diminished mosthird
| 0L + 2s
| -8
| O&
| augmented mossixth
| 4L + 1s
|-
| 9
| N@
| diminished mosfifth
| 1L + 3s
| -9
| M&
| augmented mosfourth
| 3L + 0s
|-
| 10
| P@
| diminished mosseventh
| 2L + 4s
| -10
| K&
| augmented mossecond
| 2L - 1s
|}


== Low harmonic entropy scales ==
==== 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:
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 (making the diminished mossixth 3/2)
* [[Kleismic family|Kleismic temperament]], in which the generator is 6/5 and 6 of them make a 3/1.
* [[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).
* [[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.


== Tuning ranges ==
== Tuning ranges ==
=== Parasoft ===
{{Todo|Populate|comment=Populate with JI ratios from prior edits of this page.|inline=1}}
[[Parasoft]] smitonic tunings have step ratios between 4/3 and 3/2, which implies a generator sharper than 5\18 = 333.3¢ and flatter than 7\25 = 336.0¢.
 
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 augmented 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 tunings have both minor fifths and major fifths about equally off a just fifth, 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.


Parasoft smitonic EDOs include [[18edo]], [[25edo]], and [[43edo]].
=== Simple tunings ===
* 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 fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
The simplest tunings are those with step ratios 2:1, 3:1, and 3:2, producing 11edo, 15edo, and 18edo, respectively.
* [[25edo]] can be used to make the augmented mosthird a good [[5/4]] (384¢).
{{MOS tunings}}


The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.
=== Parasoft tunings ===
{| class="wikitable right-2 right-3 right-4 right-5"
Parasoft smitonic tunings can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:
|-
!
! [[18edo]] (soft)
! [[25edo]] (supersoft)
! [[43edo]]
! Optimized (2.9.5 [[POTE]] [[Dual-fifth temperaments|dual-3 sixix]]) tuning
|-
| generator (g)
| 5\18, 333.3
| 7\25, 336.0
| 12\43, 334.9
| 335.84
|-
| L (octave - 3g)
| 3\18, 200.0
| 4\25, 192.0
| 7\43, 195.3
| 193.16
|-
| s (4g - octave)
| 2\18, 133.3
| 3\25, 144.0
| 5\43, 139.5
| 143.36
|}


==== Intervals ====
* The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
Sortable table of the extended generator chain (-13 to 13 generators) in parasoft smitonic tunings. The several different interval flavors separated by the chroma shows that parasoft smitonic is a useful [[cluster MOS]], though many of the intervals lack simple JI interpretations.
* The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable"|Degree
! [[18edo]] (soft)
! [[25edo]] (supersoft)
! [[43edo]]
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-bgcolor="#eaeaff"
| | unison
| 0\18, 0.0
| 0\25, 0.0
| 0\43, 0.0
| J
| 1/1
| 0
|-bgcolor="#eaeaff"
| chroma
| 1\18, 66.7
| 1\25, 48.0
| 2\43, 55.8
| J&
|
| -7
|-
| dim. mos2nd
| 1\18, 66.7
| 2\25, 96.0
| 3\43, 83.7
| K@@
|
| +11
|-
| minor mos2nd
| 2\18, 133.3
| 3\25, 144.0
| 5\43, 139.5
| K@
| 13/12
| +4
|-
| major mos2nd
| 3\18, 200.0
| 4\25, 192.0
| 7\43, 195.3
| K
| 9/8, 10/9
| -3
|-
| aug. mos2nd
| 4\18, 266.7
| 5\25, 240.0
| 9\43, 251.2
| K&
|
| -10
|-bgcolor="#eaeaff"
| dim. mos3rd
| 4\18, 266.7
| 6\25, 288.0
| 10\43, 279.1
| L@
|
| +8
|-bgcolor="#eaeaff"
| perf. mos3rd
| 5\18, 333.3
| 7\25, 336.0
| 12\43, 334.9
| L
| 17/14, 40/33
| +1
|-bgcolor="#eaeaff"
| aug. mos3rd
| 6\18, 400.0
| 8\25, 384.4
| 14\43, 390.7
| L&
| 5/4
| -6
|-bgcolor="#eaeaff"
| doubly aug. mos3rd
| 7\18, 466.7
| 9\25, 432.0
| 16\43, 446.5
| L&&
|
| -13
|-
| dim. mos4th
| 6\18, 400.0
| 9\25, 432.0
| 15\43, 418.6
| M@@
|
| +12
|-
| minor mos4th
| 7\18, 466.7
| 10\25, 480.0
| 17\43, 474.4
| M@
| 21/16
| +5
|-
| major mos4th
| 8\18, 533.3
| 11\25, 528.0
| 19\43, 530.2
| M
| 19/14, 34/25
| -2
|-
| aug. mos4th
| 9\18, 600.0
| 12\25, 576.0
| 21\43, 586.0
| M&
| 7/5
| -9
|-bgcolor="#eaeaff"
| dim. mos5th
| 9\18, 600.0
| 13\25, 624.0
| 22\43, 614.0
| N@
| 10/7
| +9
|-bgcolor="#eaeaff"
| minor mos5th
| 10\18, 666.7
| 14\25, 672.0
| 24\43, 669.8
| N
| 28/19, 25/17
| +2
|-bgcolor="#eaeaff"
| major mos5th
| 11\18, 733.3
| 15\25, 720.0
| 26\43, 725.6
| N&
| 32/21
| -5
|-bgcolor="#eaeaff"
| aug. mos5th
| 12\18, 800.0
| 16\25, 768.0
| 28\43, 781.4
| N&&
|
| -12
|-
| doubly dim. mos6th
| 11\18, 733.3
| 16\25, 768.0
| 27\43, 753.5
| O@@
|
| +13
|-
| dim. mos6th
| 12\18, 800.0
| 17\25, 816.0
| 29\43, 809.3
| O@
| 8/5
| +6
|-
| perf. mos6th
| 13\18, 866.7
| 18\25, 864.0
| 31\43, 865.1
| O
| 28/17, 33/20
| -1
|-
| aug. mos6th
| 14\18, 933.3
| 19\25, 912.0
| 33\43, 920.9
| O&
|
| -8
|-bgcolor="#eaeaff"
| dim. mos7th
| 14\18, 933.3
| 20\25, 960.0
| 34\34, 948.8
| P@
|
| +10
|-bgcolor="#eaeaff"
| minor mos7th
| 15\18, 1000.0
| 21\25, 1008.0
| 36\43, 1004.7
| P
| 16/9, 9/5
| +3
|-bgcolor="#eaeaff"
| major mos7th
| 16\18, 1066.7
| 22\25, 1056.0
| 38\43, 1060.5
| P&
| 24/13
| -4
|-bgcolor="#eaeaff"
| aug. mos7th
| 17\18, 1133.3
| 23\25, 1104.0
| 40\43, 1116.3
| P&
|
| -11
|-
| dim. mosoctave
| 17\18, 1133.3
| 24\25, 1152.0
| 41\43, 1144.2
| J@
|
| +7
|}


=== Hyposoft ===
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.
[[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 parasoft tunings. These tunings could be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic", in analogy to parasoft smitonic being meantone smitonic.
Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:


{| class="wikitable right-2 right-3 right-4 right-5"
* 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}}.
! [[11edo]] (basic)
** 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
! [[18edo]] (soft)
* The augmented 2-mosstep of 25edo is very close to 5/4 (386{{c}}).
! [[29edo]] (semisoft)
{{MOS tunings|Step Ratios=3/2; 7/5; 4/3}}
|-
| 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
|}
==== Intervals ====
Sortable table of major and minor intervals in hyposoft smitonic tunings (11edo and 18edo not shown):
{| class="wikitable right-2 sortable "
|-
! class="unsortable"|Degree
! [[29edo]] (semisoft)
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios (for 29edo)
! #Gens up
|-bgcolor="#eaeaff"
| unison
| 0\29, 0.0
| J
| 1/1
| 0
|-
| minor mos2nd
| 3\29, 124.1
| K@
| 14/13
| +4
|-
| major mos2nd
| 5\29, 206.9
| K
| 9/8
| -3
|-bgcolor="#eaeaff"
| perf. mos3rd
| 8\29, 331.0
| L
| 23/19, 40/33
| +1
|-bgcolor="#eaeaff"
| aug. mos3rd
| 10\29, 413.8
| L&
| 14/11
| -6
|-
| minor mos4th
| 11\29, 455.2
| M@
| 13/10
| +5
|-
| major mos4th
| 13\29, 537.9
| M
| 15/11
| -2
|-bgcolor="#eaeaff"
| minor mos5th
| 16\29, 662.1
| N
| 19/13, 22/15
| +2
|-bgcolor="#eaeaff"
| major mos5th
| 18\26, 744.8
| N&
| 20/13
| -5
|-
| dim. mos6th
| 19\29, 786.2
| O@
| 11/7
| +6
|-
| perf. mos6th
| 21\29, 869.0
| O
| 33/20, 38/23
| -1
|-bgcolor="#eaeaff"
| minor mos7th
| 24\29, 993.1
| P
| 16/9
| +3
|-bgcolor="#eaeaff"
| major mos7th
| 26\28, 1075.9
| P&
| 13/7
| -4
|}


=== Hypohard ===
=== Hyposoft tunings ===
[[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 mosfourth (2 large steps + 1 small step) tends to approximate [[11/8]]; [[26edo]] is stellar in both of these approximations. This set of JI approximations is associated with [[orgone]] temperament.
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".


Hypohard smitonic edos 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 hypohard smitonic tunings.
{| class="wikitable right-2 right-3 right-4"
|-
!
! [[11edo]] (basic)
! [[15edo]] (hard)
! [[26edo]] (semihard)
! Some JI approximations
|-
| generator (g)
| 3\11, 327.27
| 4\15, 320.00
| 7\26, 323.08
| 77/64, 6/5
|-
| 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)
|}
==== Intervals ====
Sortable table of major and minor intervals in hypohard smitonic tunings:
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable"|Degree
! [[11edo]] (basic)
! [[15edo]] (hard)
! [[26edo]] (semihard)
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-bgcolor="#eaeaff"
| unison
| 0\11, 0.0
| 0\15, 0.0
| 0\26, 0.0
| J
| 1/1
| 0
|-
| minor mos2nd
| 1\11, 109.1
| 1\15, 80.0
| 2\26, 92.3
| K@
|
| +4
|-
| major mos2nd
| 2\11, 218.2
| 3\15, 240.0
| 5\26, 230.8
| K
| 8/7
| -3
|-bgcolor="#eaeaff"
| perf. mos3rd
| 3\11, 327.3
| 4\15, 320.0
| 7\26, 323.1
| L
| 77/64, 6/5
| +1
|-bgcolor="#eaeaff"
| aug. mos3rd
| 4\11, 436.4
| 6\15, 480.0
| 10\26, 461.5
| L&
|
| -6
|-
| minor mos4th
| 4\11, 436.4
| 5\15, 400.0
| 9\26, 415.4
| M@
| 14/11
| +5
|-
| major mos4th
| 5\11, 545.5
| 7\15, 560.0
| 12\26, 553.9
| M
| 11/8
| -2
|-bgcolor="#eaeaff"
| minor mos5th
| 6\11, 656.6
| 8\15, 640.0
| 14\26, 646.2
| N
| 16/11
| +2
|-bgcolor="#eaeaff"
| major mos5th
| 7\11, 763.6
| 10\15, 800.0
| 17\26, 784.62
| N&
| 11/7
| -5
|-
| dim. mos6th
| 7\11, 763.6
| 9\15, 720.0
| 16\26, 738.5
| O@
|
| +6
|-
| perf. mos6th
| 8\11, 872.7
| 11\15, 880.0
| 19\26, 876.9
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor mos7th
| 9\11, 981.8
| 12\15, 960.0
| 21\26, 969.2
| P
| 7/4
| +3
|-bgcolor="#eaeaff"
| major mos7th
| 10\11, 1090.9
| 14\15, 1120.0
| 24\26, 1107.7
| P&
|
| -4
|}


=== Parahard ===
{{MOS tunings|Step Ratios=3/2; 5/3; 7/4}}
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 parahard 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, 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.


This set of JI approximations is associated with [[kleismic]] temperament (we're specifically describing the 2.3.5.13 extension of it called [[Chromatic pairs#Cata|cata]]).
=== 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.


EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]].
Other hypohard edos include [[11edo]] (not shown), [[15edo]] and [[37edo]].


The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings (not including 15edo).
{{MOS tunings|Step Ratios=3/1; 5/2; 7/3}}
{| class="wikitable right-2 right-3 right-4"
|-
!
! [[19edo]] (superhard)
! [[34edo]]
! [[53edo]]
! JI intervals represented
|-
| generator (g)
| 5\19, 315.79
| 9\34, 317.65
| 14\53, 316.98
| 6/5
|-
| L (octave - 3g)
| 4\19, 252.63
| 7\34, 247.06
| 11\53, 249.06
| 15/13
|-
| s (4g - octave)
| 1\19, 63.16
| 2\34, 70.59
| 3\53, 67.92
| 25/24, 26/25
|}


==== Intervals ====
=== Parahard tunings ===
Sortable table of major and minor intervals in parahard smitonic 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.
{| class="wikitable right-2 right-3 right-4 sortable "
|-
! class="unsortable"|Degree
! [[19edo]] (superhard)
! [[34edo]]
! [[53edo]]
! class="unsortable"| Note name on J
! class="unsortable"| Approximate ratios
! #Gens up
|-bgcolor="#eaeaff"
| unison
| 0\19, 0.0
| 0\34, 0.0
| 0\53, 0.0
| J
| 1/1
| 0
|-
| minor mos2nd
| 1\19, 63.2
| 2\34, 70.6
| 3\53, 67.9
| K@
| 25/24, 26/25
| +4
|-
| major mos2nd
| 4\19, 252.6
| 7\34, 247.1
| 11\53, 249.1
| K
| 15/13
| -3
|-bgcolor="#eaeaff"
| perf. mos3rd
| 5\19, 315.8
| 9\34, 317.6
| 14\53, 317.0
| L
| 6/5
| +1
|-bgcolor="#eaeaff"
| aug. mos3rd
| 8\19, 505.3
| 14\34, 494.1
| 22\53, 498.1
| L&
| 4/3
| -6
|-
| minor mos4th
| 6\19, 378.9
| 11\34, 388.2
| 17\53, 384.9
| M@
| 5/4
| +5
|-
| major mos4th
| 9\19, 568.4
| 16\34, 564.7
| 25\53, 566.0
| M
| 18/13
| -2
|-bgcolor="#eaeaff"
| minor mos5th
| 10\19, 631.6
| 18\34, 635.3
| 28\53, 634.0
| N
| 13/9
| +2
|-bgcolor="#eaeaff"
| major mos5th
| 16\19, 821.1
| 23\34, 811.8
| 39\53, 815.0
| N&
| 8/5
| -5
|-
| dim. mos6th
| 11\19, 694.7
| 20\34, 705.9
| 31\53, 701.9
| O@
| 3/2
| +6
|-
| perf. mos6th
| 14\19, 884.2
| 25\34, 882.4
| 39\53, 883.0
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor mos7th
| 15\19, 947.4
| 27\34, 952.9
| 42\53, 950.9
| P
| 26/15
| +3
|-bgcolor="#eaeaff"
| major mos7th
| 18\19, 1136.8
| 32\34, 1129.4
| 50\53, 1132.1
| P&
| 25/13
| -4
|}


== Modes ==
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.
A naming scheme proposed by Alexandru Ianu ([[User: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>, relating to the Almsivi in Morrowind (TES):
{| class="wikitable center-all"
|-
! Mode
! [[Modal UDP Notation|UDP]]
! Name
|-
| LLsLsLs
| <nowiki>6|0</nowiki>
| Nerevarine
|-
| LsLLsLs
| <nowiki>5|1</nowiki>
| Vivecan
|-
| LsLsLLs
| <nowiki>4|2</nowiki>
| Lorkhanic
|-
| LsLsLsL
| <nowiki>3|3</nowiki>
| Sothic
|-
| sLLsLsL
| <nowiki>2|4</nowiki>
| Kagrenacan
|-
| sLsLLsL
| <nowiki>1|5</nowiki>
| Almalexian
|-
| sLsLsLL
| <nowiki>0|6</nowiki>
| Dagothic
|}


== Approaches ==
These JI approximations are associated with [[kleismic]] temperament, through the 2.3.5.13 extension known as [[Kleismic family#Cata|cata]].
== Temperaments ==
{{main|4L 3s/Temperaments}}
4L 3s has several temperament interpretations (see main article for mappings and optimal generator tunings):


# 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.
Parahard edos smaller than 53edo include [[15edo]] (not shown), [[19edo]], and [[34edo]].
# With generator size between 4\15 (320.0c) and 3\11 (327.3c): [[Orgone]], corresponding to a L/s ratio between 3 and 2.
# 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.
{{MOS tunings|Step Ratios=4/1; 11/3; 7/2}}


== Scales ==
== Scales ==
Line 882: Line 105:
* [[Cata7]]
* [[Cata7]]
* [[Myna7]]
* [[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;↓
}}


== Music ==
== Music ==
Line 887: Line 124:
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
* [[File:Sixix Fugue.mp3]] A fugue in [[18edo]] smitonic functional harmony (WIP)


== Scale tree ==
== References ==
The spectrum looks like this:
<references />


{| class="wikitable center-all"
[[Category:Smitonic|*]] <!--Main article-->
! colspan="6" rowspan="2" | Generator
[[Category:7-tone scales]]
! colspan="2" | Cents
! rowspan="2" | L
! rowspan="2" | s
! rowspan="2" | L/s
! rowspan="2" | Comments
|-
!Chroma-positive
!Chroma-negative
|-
| 5\7 || || || || || || 857.143 || 342.857 || 1 || 1 || 1.000 ||
|-
| || || || || || 28\39 || 861.538 || 338.462 || 6 || 5 || 1.200 || Amity/hitchcock↑
|-
| || || || || 23\32 || || 862.500 || 337.500 || 5 || 4 || 1.250 || Sixix
|-
| || || || || || 41\57 || 863.158 || 336.842 || 9 || 7 || 1.286 ||
|-
| || || || 18\25 || || || 864.000 || 336.000 || 4 || 3 || 1.333 ||
|-
| || || || || || 49\68 || 864.706 || 335.294 || 11 || 8 || 1.375 ||
|-
| || || || || 31\43 || || 865.116 || 334.884 || 7 || 5 || 1.400 ||
|-
| || || || || || 17\58 || 865.574 || 334.426 || 10 || 7 || 1.428 ||
|-
| || || 13\18 || || || || 866.667 || 333.333 || 3 || 2 || 1.500 || L/s = 3/2
|-
| || || || || || 47\65 || 867.692 || 332.308 || 11 || 7 || 1.571 ||
|-
| || || || || 34\47 || || 868.085 || 331.915 || 8 || 5 || 1.600 ||
|-
| || || || || || 55\76 || 868.421 || 331.579 || 13 || 8 || 1.625 || Golden smitonic (?)
|-
| || || || 21\29 || || || 868.966 || 331.034 || 5 || 3 || 1.667 ||
|-
| || || || || || 50\69 || 869.565 || 330.435 || 12 || 7 || 1.714 ||
|-
| || || || || 29\40 || || 870.000 || 330.000 || 7 || 4 || 1.750 ||
|-
| || || || || || 37\51 || 870.588 || 329.422 || 9 || 5 || 1.800 ||
|-
| || 8\11 || || || || || 872.727 || 327.273 || 2 || 1 || 2.000 || Basic smitonic<br>(Generators smaller than this are proper)
|-
| || || || || || 35\48 || 875.000 || 325.000 || 9 || 4 || 2.250 ||
|-
| || || || || 27\37 || || 875.676 || 324.324 || 7 || 3 || 2.333 ||
|-
| || || || || || 46\63 || 876.190 || 323.810 || 12 || 5 || 2.400 ||
|-
| || || || 19\26 || || || 876.923 || 323.077 || 5 || 2 || 2.500 || Orgone is in this region
|-
| || || || || || 49\67 || 877.612 || 322.388 || 13 || 5 || 2.600 || Golden superkleismic
|-
| || || || || 30\41 || || 878.049 || 321.951 || 8 || 3 || 2.667 || Superkleismic
|-
| || || || || || 41\56 || 878.571 || 321.429 || 11 || 4 || 2.750 ||
|-
| || || 11\15 || || || || 880.000 || 320.000 || 3 || 1 || 3.000 || L/s = 3/1
|-
| || || || || || 36\49 || 881.633 || 318.367 || 10 || 3 || 3.333 ||
|-
| || || || || 25\34 || || 882.353 || 317.647 || 7 || 2 || 3.500 ||
|-
| || || || || || 39\53 || 883.019 || 316.981 || 11 || 3 || 3.667 || Hanson/keemun is in this region
|-
| || || || 14\19 || || || 884.211 || 315.789 || 4 || 1 || 4.000 ||
|-
| || || || || || 31\42 || 885.714 || 314.286 || 9 || 2 || 4.500 ||
|-
| || || || || 17\23 || || 886.957 || 313.043 || 5 || 1 || 5.000 ||
|-
| || || || || || 20\27 || 888.889 || 311.111 || 6 || 1 || 6.000 || Oolong, myna↓
|-
| 3\4 || || || || || || 900.000 || 300.000 || 1 || 0 || → inf ||
|}
 
== References ==
[[Category:Smitonic|*]]<!--Main article-->
[[Category:Scales]]
[[Category:MOS scales]]
[[Category:Abstract MOS patterns]]
Retrieved from "https://en.xen.wiki/w/4L_3s"