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{{MOS intro}}
{{MOS intro}}
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).
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.
== Name==
[[TAMNAMS]] suggests the temperament-agnostic name '''smitonic''' ''smy-TON-ik'' /smaɪˈtɒnɪk/ for this scale. The name is derived from 'sharp minor third', since the central range for the dark generator (320¢ to 333.3¢) is significantly sharp of [[6/5]] (just minor 3rd, 315.6¢).
== Notation==
:''This article assumes [[TAMNAMS]] for naming step ratios, intervals, and scale degrees, and [[Diamond-mos notation|diamond-MOS notation]] for note names.''
=== Intervals and degrees===
Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.


== Standing assumptions ==
Being a moment-of-symmetry scale, every [[interval class]] of 4L 3s, except for the unison and octave, has two [[Interval variety|varieties]] – large and small – whose [[Interval quality|relative qualities]] are denoted as major or minor, or augmented, perfect, and diminished for the generators.
The [[TAMNAMS]] system is used in this article to name 4L 3s intervals and step size ratios and step ratio ranges.
{| class="wikitable"
 
|+
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".)
! rowspan="2" |Interval class
 
! colspan="2" |Large variety
Thus the [[11edo]] gamut is as follows:
! colspan="2" |Small variety
 
'''J''' J&/K@ '''K''' '''L''' L&/M@ '''M''' '''N''' N&/O@ '''O''' '''P''' P&/J@ '''J'''
 
== Names ==
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.
 
== Intervals ==
Note: In TAMNAMS, a k-step interval class in smitonic may be called a "k-step", "k-mosstep", or "k-smistep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
 
{| class="wikitable center-all"
|-
|-
! Generators
!Size
! Notation (1/1 = J)
!Quality
! [[TAMNAMS]] name
!Size
! In L's and s's
! Quality
! 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-mosstep (unison)'''
|0
|Perfect
| 0
| 0
| J
|Perfect
| perfect unison
| 0L + 0s
| 0
| J
| octave
| 4L + 3s
|-
|-
| 1
|1-mosstep
| L
| L
| perfect 2-mosstep
|Major
| 1L + 1s
|s
| -1
|Minor
| O
| perfect 5-mosstep
| 3L + 2s
|-
|-
| 2
|2-mosstep
| N
|2L
| minor 4-mosstep  
| Augmented
| 2L + 2s
|L + s
| -2
| Perfect
| M
| major 3-mosstep
| 2L + 1s
|-
|-
| 3
|3-mosstep
| P
|2L + s
| minor 6-mosstep
|Major
| 3L + 3s
| L + 2s
| -3
|Minor
| K
| major 1-mosstep
| 1L + 0s
|-
|-
| 4
|4-mosstep
| K@
| minor 1-mosstep
| 0L + 1s
| -4
| Q&
| major 6-mosstep
| 4L + 2s
|-
| 5
| M@
| minor 3-mosstep
| 1L + 2s
| -5
| N&
| major 4-mosstep  
| 3L + 1s
| 3L + 1s
| Perfect
|2L + 2s
|Minor
|-
|-
| 6
|5-mosstep
| O@
|3L + 2s
| diminished 5-mosstep
|Perfect
| 2L + 3s
| 2L + 3s
| -6
|Diminished
| L&
| augmented 2-mosstep
| 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):
|6-mosstep
|3L + 3s
| Major
|2L + 3s
|Minor
|-
|-
| 7
|'''7-mosstep (octave)'''
| J@
|4L + 3s
| diminished 7-mosstep
|Perfect
| 5L + 2s
|4L + 3s
| -7
|Perfect
| J&
| augmented mosunison; chroma
| 1L - 1s
|-
| 8
| L@
| diminished 2-mosstep
| 0L + 2s
| -8
| O&
| augmented 5-mosstep
| 4L + 1s
|-
| 9
| N@
| diminished 4-mosstep
| 1L + 3s
| -9
| M&
| augmented 3-mosstep
| 3L + 0s
|-
| 10
| P@
| diminished 6-mosstep
| 2L + 4s
| -10
| K&
| augmented 1-mosstep
| 2L - 1s
|}
|}
===Note names===
For this article, note names are based on diamond-MOS notation, where the naturals JKLMNOP are applied to the step pattern LsLsLsL and the accidentals & (pronounced "am" or "amp") and @ (pronounced "at") are used to represent sharps and flats respectively. Thus, the basic gamut for 4L 3s is the following:


== Low harmonic entropy scales ==
{{MOS gamut|Scale Signature=4L 3s}}
==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 5-mosstep 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 10 of them make a 6/1, resulting in the intervals 4/3 and 3/2 being absent.
 
===Temperament interpretations===
== Tuning ranges ==
:''Main article: [[4L 3s/Temperaments]]''
=== Simple tunings ===
4L 3s has the following temperament interpretations:
{| class="wikitable right-2 right-3 right-4 sortable "
*[[Sixix]], with generators around 338.6¢.
|-
*[[Orgone]], with generators around 323.4¢.
! class="unsortable"|Degree
*[[Kleismic]], with generators around 317¢.
! [[11edo]] (basic)
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.
! [[15edo]] (hard)
==Tuning ranges==
! [[18edo]] (soft)
===Simple tunings===
! class="unsortable"| Note name on J
The basic tuning for 4L 3s has a large and small step size of 2 and 1 respectively, which is supported by 11edo. Other small edos include [[15edo]] and [[18edo]].{{MOS degrees|Scale Signature=4L 3s|Step Ratio=2/1; 3/1; 3/2}}
! class="unsortable"| Approximate ratios
===Parasoft tunings===
! #Gens up
Parasoft smitonic tunings (4:3 to 3:2) can be considered "meantone smitonic" since it has the following features of [[meantone]] diatonic tunings:
|-bgcolor="#eaeaff"
| unison
| 0\11, 0.0
| 0\15, 0.0
| 0\18, 0.0
| J
|
| 0
|-
| minor 1-mosstep
| 1\11, 109.1
| 1\15, 80.0
| 2\18, 133.3
| K@
|
| +4
|-
| major 1-mosstep
| 2\11, 218.2
| 3\15, 240.0
| 3\18, 200.0
| K
| 8/7
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 3\11, 327.3
| 4\15, 320.0
| 5\18, 333.3
| L
| 77/64, 6/5
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 4\11, 436.4
| 6\15, 480.0
| 6\18, 400.0
| L&
|
| -6
|-
| minor 3-mosstep
| 4\11, 436.4
| 5\15, 400.0
| 7\18, 466.7
| M@
| 14/11
| +5
|-
| major 3-mosstep
| 5\11, 545.5
| 7\15, 560.0
| 8\18, 533.3
| M
| 11/8
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 6\11, 656.6
| 8\15, 640.0
| 10\18, 666.7
| N
| 16/11
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 7\11, 763.6
| 10\15, 800.0
| 11\18, 733.3
| N&
| 11/7
| -5
|-
| dim. 5-mosstep
| 7\11, 763.6
| 9\15, 720.0
| 12\18, 800.0
| O@
|
| +6
|-
| perf. 5-mosstep
| 8\11, 872.7
| 11\15, 880.0
| 13\18, 866.7
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 9\11, 981.8
| 12\15, 960.0
| 15\18, 1000.0
| P
| 7/4
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 10\11, 1090.9
| 14\15, 1120.0
| 16\18, 1066.7
| P&
|
| -4
|}
 
=== Parasoft ===
[[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 2-mosstep (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]].
* 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 5-mosstep) and falling fifths (666.7c, a major 4-mosstep) almost equally off from a just fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
* [[25edo]] can be used to make the augmented 2-mosstep 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.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[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 ====
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.
{| 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. 1-mosstep
| 1\18, 66.7
| 2\25, 96.0
| 3\43, 83.7
| K@@
|
| +11
|-
| minor 1-mosstep
| 2\18, 133.3
| 3\25, 144.0
| 5\43, 139.5
| K@
| 13/12
| +4
|-
| major 1-mosstep
| 3\18, 200.0
| 4\25, 192.0
| 7\43, 195.3
| K
| 9/8, 10/9
| -3
|-
| aug. 1-mosstep
| 4\18, 266.7
| 5\25, 240.0
| 9\43, 251.2
| K&
|
| -10
|-bgcolor="#eaeaff"
| dim. 2-mosstep
| 4\18, 266.7
| 6\25, 288.0
| 10\43, 279.1
| L@
|
| +8
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 5\18, 333.3
| 7\25, 336.0
| 12\43, 334.9
| L
| 17/14, 40/33
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 6\18, 400.0
| 8\25, 384.4
| 14\43, 390.7
| L&
| 5/4
| -6
|-bgcolor="#eaeaff"
| doubly aug. 2-mosstep
| 7\18, 466.7
| 9\25, 432.0
| 16\43, 446.5
| L&&
|
| -13
|-
| dim. 3-mosstep
| 6\18, 400.0
| 9\25, 432.0
| 15\43, 418.6
| M@@
|
| +12
|-
| minor 3-mosstep
| 7\18, 466.7
| 10\25, 480.0
| 17\43, 474.4
| M@
| 21/16
| +5
|-
| major 3-mosstep
| 8\18, 533.3
| 11\25, 528.0
| 19\43, 530.2
| M
| 19/14, 34/25
| -2
|-
| aug. 3-mosstep
| 9\18, 600.0
| 12\25, 576.0
| 21\43, 586.0
| M&
| 7/5
| -9
|-bgcolor="#eaeaff"
| dim. 4-mosstep
| 9\18, 600.0
| 13\25, 624.0
| 22\43, 614.0
| N@
| 10/7
| +9
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 10\18, 666.7
| 14\25, 672.0
| 24\43, 669.8
| N
| 28/19, 25/17
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 11\18, 733.3
| 15\25, 720.0
| 26\43, 725.6
| N&
| 32/21
| -5
|-bgcolor="#eaeaff"
| aug. 4-mosstep
| 12\18, 800.0
| 16\25, 768.0
| 28\43, 781.4
| N&&
|
| -12
|-
| doubly dim. 5-mosstep
| 11\18, 733.3
| 16\25, 768.0
| 27\43, 753.5
| O@@
|
| +13
|-
| dim. 5-mosstep
| 12\18, 800.0
| 17\25, 816.0
| 29\43, 809.3
| O@
| 8/5
| +6
|-
| perf. 5-mosstep
| 13\18, 866.7
| 18\25, 864.0
| 31\43, 865.1
| O
| 28/17, 33/20
| -1
|-
| aug. 5-mosstep
| 14\18, 933.3
| 19\25, 912.0
| 33\43, 920.9
| O&
|
| -8
|-bgcolor="#eaeaff"
| dim. 6-mosstep
| 14\18, 933.3
| 20\25, 960.0
| 34\34, 948.8
| P@
|
| +10
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 15\18, 1000.0
| 21\25, 1008.0
| 36\43, 1004.7
| P
| 16/9, 9/5
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 16\18, 1066.7
| 22\25, 1056.0
| 38\43, 1060.5
| P&
| 24/13
| -4
|-bgcolor="#eaeaff"
| aug. 6-mosstep
| 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 ===
[[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.
 
{| class="wikitable right-2 right-3 right-4 right-5"
|-
!
! [[11edo]] (basic)
! [[18edo]] (soft)
! [[29edo]] (semisoft)
|-
| 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 1-mosstep
| 3\29, 124.1
| K@
| 14/13
| +4
|-
| major 1-mosstep
| 5\29, 206.9
| K
| 9/8
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 8\29, 331.0
| L
| 23/19, 40/33
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 10\29, 413.8
| L&
| 14/11
| -6
|-
| minor 3-mosstep
| 11\29, 455.2
| M@
| 13/10
| +5
|-
| major 3-mosstep
| 13\29, 537.9
| M
| 15/11
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 16\29, 662.1
| N
| 19/13, 22/15
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 18\26, 744.8
| N&
| 20/13
| -5
|-
| dim. 5-mosstep
| 19\29, 786.2
| O@
| 11/7
| +6
|-
| perf. 5-mosstep
| 21\29, 869.0
| O
| 33/20, 38/23
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 24\29, 993.1
| P
| 16/9
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 26\28, 1075.9
| P&
| 13/7
| -4
|}
 
=== Hypohard ===
[[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 3-mosstep (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.
 
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.
{| 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 1-mosstep
| 1\11, 109.1
| 1\15, 80.0
| 2\26, 92.3
| K@
|
| +4
|-
| major 1-mosstep
| 2\11, 218.2
| 3\15, 240.0
| 5\26, 230.8
| K
| 8/7
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 3\11, 327.3
| 4\15, 320.0
| 7\26, 323.1
| L
| 77/64, 6/5
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 4\11, 436.4
| 6\15, 480.0
| 10\26, 461.5
| L&
|
| -6
|-
| minor 3-mosstep
| 4\11, 436.4
| 5\15, 400.0
| 9\26, 415.4
| M@
| 14/11
| +5
|-
| major 3-mosstep
| 5\11, 545.5
| 7\15, 560.0
| 12\26, 553.9
| M
| 11/8
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 6\11, 656.6
| 8\15, 640.0
| 14\26, 646.2
| N
| 16/11
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 7\11, 763.6
| 10\15, 800.0
| 17\26, 784.62
| N&
| 11/7
| -5
|-
| dim. 5-mosstep
| 7\11, 763.6
| 9\15, 720.0
| 16\26, 738.5
| O@
|
| +6
|-
| perf. 5-mosstep
| 8\11, 872.7
| 11\15, 880.0
| 19\26, 876.9
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 9\11, 981.8
| 12\15, 960.0
| 21\26, 969.2
| P
| 7/4
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 10\11, 1090.9
| 14\15, 1120.0
| 24\26, 1107.7
| P&
|
| -4
|}


=== Parahard ===
*The major 1-mosstep, or large step, is around [[10/9]] to [[9/8]], thus making it a "meantone".
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.
*The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.


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]]).
These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702¢), 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.


EDOs that have parahard smitonic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]].
Edos include [[18edo]], [[25edo]], and [[43edo]]. Some key considerations include:


The sizes of the generator, large step and small step of smitonic are as follows in various parahard smitonic tunings (not including 15edo).  
*18edo can be used to make the large and small steps more distinct, or can be considered a distorted 19edo diatonic.
{| class="wikitable right-2 right-3 right-4"
**18edo has a major 1-mosstep that is close to 9/8 (203¢).
|-
**18edo's major and minor 4-mossteps are both equally off from 12edo's diatonic perfect 5th (700¢) by 33.3¢.
!
**18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
! [[19edo]] (superhard)
*The augmented 2-mosstep of 25edo is very close to 5/4 (386¢).
! [[34edo]]  
*The various interval flavors separated by a chroma shows that parasoft smitonic is a useful [[cluster MOS]]. However, many of these intervals lack simple JI interpretations.
! [[53edo]]
{{MOS degrees|Scale Signature=4L 3s|Step Ratio=3/2; 4/3; 7/5|Genchain Extend=9}}
! JI intervals represented
===Hyposoft tunings===
|-
Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327¢ and 333¢. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "[[Gentle region|neogothic]] smitonic" or "[[archy]] smitonic".
| 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 ====
Edos include [[11edo]] (not shown), [[18edo]], and [[29edo]].
Sortable table of major and minor intervals in parahard smitonic tunings:
{{MOS degrees|Scale Signature=4L 3s|Step Ratio=3/2; 5/3|Genchain Extend=3}}
{| class="wikitable right-2 right-3 right-4 sortable "
=== Hypohard tunings===
|-
Hypohard smitonic tunings (2:1 to 3:1) have generators between 320¢ and 327¢. The major 1-mosstep, or large step, tends to approximate [[8/7]] (231¢) and the major 3-mosstep tends to approximate [[11/8]] (551¢). [[26edo]] approximates these two intervals very well. These JI approximations are associated with [[orgone]] temperament.
! 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 1-mosstep
| 1\19, 63.2
| 2\34, 70.6
| 3\53, 67.9
| K@
| 25/24, 26/25
| +4
|-
| major 1-mosstep
| 4\19, 252.6
| 7\34, 247.1
| 11\53, 249.1
| K
| 15/13
| -3
|-bgcolor="#eaeaff"
| perf. 2-mosstep
| 5\19, 315.8
| 9\34, 317.6
| 14\53, 317.0
| L
| 6/5
| +1
|-bgcolor="#eaeaff"
| aug. 2-mosstep
| 8\19, 505.3
| 14\34, 494.1
| 22\53, 498.1
| L&
| 4/3
| -6
|-
| minor 3-mosstep
| 6\19, 378.9
| 11\34, 388.2
| 17\53, 384.9
| M@
| 5/4
| +5
|-
| major 3-mosstep
| 9\19, 568.4
| 16\34, 564.7
| 25\53, 566.0
| M
| 18/13
| -2
|-bgcolor="#eaeaff"
| minor 4-mosstep
| 10\19, 631.6
| 18\34, 635.3
| 28\53, 634.0
| N
| 13/9
| +2
|-bgcolor="#eaeaff"
| major 4-mosstep
| 16\19, 821.1
| 23\34, 811.8
| 39\53, 815.0
| N&
| 8/5
| -5
|-
| dim. 5-mosstep
| 11\19, 694.7
| 20\34, 705.9
| 31\53, 701.9
| O@
| 3/2
| +6
|-
| perf. 5-mosstep
| 14\19, 884.2
| 25\34, 882.4
| 39\53, 883.0
| O
| 5/3
| -1
|-bgcolor="#eaeaff"
| minor 6-mosstep
| 15\19, 947.4
| 27\34, 952.9
| 42\53, 950.9
| P
| 26/15
| +3
|-bgcolor="#eaeaff"
| major 6-mosstep
| 18\19, 1136.8
| 32\34, 1129.4
| 50\53, 1132.1
| P&
| 25/13
| -4
|}
 
== Modes ==
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 ==
Other hypohard edos include [[11edo]] (not shown), [[15edo]] and [[37edo]].
* [[4L 3s/Inthar's approach]]
{{MOS degrees|Scale Signature=4L 3s|Step Ratio=3/1; 5/2; 7/3|Genchain Extend=3}}
===Parahard tunings===
Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9¢ and 320¢, putting it close to a pure 6/5 (316¢). Stacking six generators and octave-reducing approximates 3/2 (702¢), a diatonic perfect 5th, represented by the diminished 5-mosstep.


== Temperaments ==
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.
{{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.
These JI approximations are associated with [[kleismic]] temperament, though the 2.3.5.13 extension described here is called [[Chromatic pairs#Cata|cata]].
# 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 concordant chords optimized by these temperaments; you would need to use a [[MODMOS]] or use a larger MOS gamut, if you restrict to a rank-2 approach.
Parahard edos smaller than 53edo include [[15edo]] (not shown), [[19edo]], and [[34edo]].
{{MOS degrees|Scale Signature=4L 3s|Step Ratio=4/1; 7/2; 11/3; 15/4|Genchain Extend=3|Prefix=smi}}
==Modes==
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):


== Scales ==
{{MOS modes|Scale Signature=4L 3s|Mode Names=Nerevarine; Vivecan; Lorkhanic; Sothic; Kagrenacan; Almalexian; Dagothic}}
* [[Orgone7]]
==Scales==
* [[Cata7]]
* [[Myna7]]


== Music ==
===Scala files===
* [[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)
*[[Orgone7]]
* [[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
*[[Cata7]]
* [[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
*[[Myna7]]
* [[File:Sixix Fugue.mp3]] A fugue in [[18edo]] smitonic functional harmony (WIP)


== Scale tree ==
== Scale tree==
Generator ranges:
{{Scale tree|4L 3s|Comments=6/5: [[Amity]]/[[hitchcock]]↑;
* Chroma-positive generator: 857.1429 cents (5\7) to 900 cents (3\4)
5/4: [[Sixix]];
* Chroma-negative generator: 300 cents (1\4) to 342.8571 cents (2\7)
13/8: Golden 4L 3s (868.3282¢);
 
12/5: [[Hyperkleismic]];
{| class="wikitable center-all"
5/2 [[Orgone]];
! colspan="6" rowspan="2" | Generator <br><small>(Chroma-positive)</small>
13/5: Golden superkleismic;
! colspan="2" | Cents
8/3: [[Superkleismic]];
! rowspan="2" | L
11/3: [[Hanson]]/[[keemun]];
! rowspan="2" | s
6/1: [[Oolong]]/[[myna]]↓;
! rowspan="2" | L/s
2/1: (Generators smaller than this are proper)}}
! rowspan="2" | Comments
==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)
! <small>Chroma-positive</small>
*[[User:Ks26|ks26]], [https://www.youtube.com/watch?v=AEnEYk3X1as Ghost Bridge] (11edo)
! <small>Chroma-negative</small>
*[[User:Ayceman|Alexandru Ianu]], [https://youtu.be/81uZbsmbet8 Sylvian Moon Dance] (11edo) ([[:File:Sylvian_Moon_Dance.pdf|sheet music]])
|-
*[[File:Sixix Fugue.mp3|256x256px]] A fugue in [[18edo]] smitonic functional harmony (WIP)
| 5\7 || || || || || || 857.143 || 342.857 || 1 || 1 || 1.000 ||
==See also==
|-
Approaches:
| || || || || || 28\39 || 861.538 || 338.462 || 6 || 5 || 1.200 || [[Amity]]/[[hitchcock]]↑
*[[4L 3s/Inthar's approach]]
|-
==References==
| || || || || 23\32 || || 862.500 || 337.500 || 5 || 4 || 1.250 || [[Sixix]]
<references />
|-
| || || || || || 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 || 1.429 ||
|-
| || || 13\18 || || || || 866.667 || 333.333 || 3 || 2 || 1.500 ||
|-
| || || || || || 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 (868.3282¢)
|-
| || || || 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 || [[Hyperkleismic]]
|-
| || || || 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 (877.7318¢)
|-
| || || || || 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 ||
|-
| || || || || || 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:Smitonic|*]]<!--Main article-->
[[Category:7-tone scales]]
[[Category:7-tone scales]]

Revision as of 00:24, 3 July 2023

↖ 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, named smitonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 3 small steps, repeating every octave. Generators that produce this scale range from 857.1 ¢ to 900 ¢, or from 300 ¢ to 342.9 ¢. 4L 3s can be seen as a warped diatonic scale, where one large step of diatonic (5L 2s) is replaced with a small step.

Name

TAMNAMS suggests the temperament-agnostic name smitonic smy-TON-ik /smaɪˈtɒnɪk/ for this scale. The name is derived from 'sharp minor third', since the central range for the dark generator (320¢ to 333.3¢) is significantly sharp of 6/5 (just minor 3rd, 315.6¢).

Notation

This article assumes TAMNAMS for naming step ratios, intervals, and scale degrees, and diamond-MOS notation for note names.

Intervals and degrees

Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.

Being a moment-of-symmetry scale, every interval class of 4L 3s, except for the unison and octave, has two varieties – large and small – whose relative qualities are denoted as major or minor, or augmented, perfect, and diminished for the generators.

Interval class Large variety Small variety
Size Quality Size Quality
0-mosstep (unison) 0 Perfect 0 Perfect
1-mosstep L Major s Minor
2-mosstep 2L Augmented L + s Perfect
3-mosstep 2L + s Major L + 2s Minor
4-mosstep 3L + 1s Perfect 2L + 2s Minor
5-mosstep 3L + 2s Perfect 2L + 3s Diminished
6-mosstep 3L + 3s Major 2L + 3s Minor
7-mosstep (octave) 4L + 3s Perfect 4L + 3s Perfect

Note names

For this article, note names are based on diamond-MOS notation, where the naturals JKLMNOP are applied to the step pattern LsLsLsL and the accidentals & (pronounced "am" or "amp") and @ (pronounced "at") are used to represent sharps and flats respectively. Thus, the basic gamut for 4L 3s is the following:

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

Theory

Low harmonic entropy scales

There are two notable harmonic entropy minima:

  • Kleismic temperament, in which the generator is 6/5 and 6 of them make a 3/1.
  • Myna temperament, in which the generator is also 6/5 but 10 of them make a 6/1, resulting in the intervals 4/3 and 3/2 being absent.

Temperament interpretations

Main article: 4L 3s/Temperaments

4L 3s has the following temperament interpretations:

  • Sixix, with generators around 338.6¢.
  • Orgone, with generators around 323.4¢.
  • Kleismic, with generators around 317¢.

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

Simple tunings

The basic tuning for 4L 3s has a large and small step size of 2 and 1 respectively, which is supported by 11edo. Other small edos include 15edo and 18edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 4L 3s
Scale degree 11edo (Basic, L:s = 2:1) 15edo (Hard, L:s = 3:1) 18edo (Soft, L:s = 3:2) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-smidegree 1 109.1 1 80 2 133.3
Major 1-smidegree 2 218.2 3 240 3 200
Perfect 2-smidegree 3 327.3 4 320 5 333.3
Augmented 2-smidegree 4 436.4 6 480 6 400
Minor 3-smidegree 4 436.4 5 400 7 466.7
Major 3-smidegree 5 545.5 7 560 8 533.3
Minor 4-smidegree 6 654.5 8 640 10 666.7
Major 4-smidegree 7 763.6 10 800 11 733.3
Diminished 5-smidegree 7 763.6 9 720 12 800
Perfect 5-smidegree 8 872.7 11 880 13 866.7
Minor 6-smidegree 9 981.8 12 960 15 1000
Major 6-smidegree 10 1090.9 14 1120 16 1066.7
Perfect 7-smidegree (octave) 11 1200 15 1200 18 1200 2/1 (exact)

Parasoft tunings

Parasoft smitonic tunings (4:3 to 3:2) can be considered "meantone smitonic" since it has the following features of meantone diatonic tunings:

  • The major 1-mosstep, or large step, is around 10/9 to 9/8, thus making it a "meantone".
  • The augmented 2-mosstep is around the size of a meantone-sized major 3rd and can be used as a stand-in for such.

These tunings have a major 4-mosstep and minor 4-mosstep that are about equally off a just 3/2 (702¢), 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.

Edos include 18edo, 25edo, and 43edo. Some key considerations include:

  • 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¢).
    • 18edo's major and minor 4-mossteps are both equally off from 12edo's diatonic perfect 5th (700¢) by 33.3¢.
    • 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¢).
  • The various interval flavors separated by a chroma shows that parasoft smitonic is a useful cluster MOS. However, many of these intervals lack simple JI interpretations.
MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 4L 3s
Scale degree 18edo (Soft, L:s = 3:2) 25edo (Supersoft, L:s = 4:3) 43edo (L:s = 7:5) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-smidegree 2 133.3 3 144 5 139.5
Major 1-smidegree 3 200 4 192 7 195.3
Perfect 2-smidegree 5 333.3 7 336 12 334.9
Augmented 2-smidegree 6 400 8 384 14 390.7
Minor 3-smidegree 7 466.7 10 480 17 474.4
Major 3-smidegree 8 533.3 11 528 19 530.2
Minor 4-smidegree 10 666.7 14 672 24 669.8
Major 4-smidegree 11 733.3 15 720 26 725.6
Diminished 5-smidegree 12 800 17 816 29 809.3
Perfect 5-smidegree 13 866.7 18 864 31 865.1
Minor 6-smidegree 15 1000 21 1008 36 1004.7
Major 6-smidegree 16 1066.7 22 1056 38 1060.5
Perfect 7-smidegree (octave) 18 1200 25 1200 43 1200 2/1 (exact)

Hyposoft tunings

Hyposoft smitonic tunings (3:2 to 2:1) are characterized by generators that are a supraminor 3rd, between 327¢ and 333¢. By analogy of parasoft tunings being called "meantone smitonic", these tunings can be considered "neogothic smitonic" or "archy smitonic".

Edos include 11edo (not shown), 18edo, and 29edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 4L 3s
Scale degree 18edo (Soft, L:s = 3:2) 29edo (Semisoft, L:s = 5:3) Approx. JI Ratios
Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 1/1 (exact)
Minor 1-smidegree 2 133.3 3 124.1
Major 1-smidegree 3 200 5 206.9
Perfect 2-smidegree 5 333.3 8 331
Augmented 2-smidegree 6 400 10 413.8
Minor 3-smidegree 7 466.7 11 455.2
Major 3-smidegree 8 533.3 13 537.9
Minor 4-smidegree 10 666.7 16 662.1
Major 4-smidegree 11 733.3 18 744.8
Diminished 5-smidegree 12 800 19 786.2
Perfect 5-smidegree 13 866.7 21 869
Minor 6-smidegree 15 1000 24 993.1
Major 6-smidegree 16 1066.7 26 1075.9
Perfect 7-smidegree (octave) 18 1200 29 1200 2/1 (exact)

Hypohard tunings

Hypohard smitonic tunings (2:1 to 3:1) have generators between 320¢ and 327¢. The major 1-mosstep, or large step, tends to approximate 8/7 (231¢) and the major 3-mosstep tends to approximate 11/8 (551¢). 26edo approximates these two intervals very well. These JI approximations are associated with orgone temperament.

Other hypohard edos include 11edo (not shown), 15edo and 37edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 4L 3s
Scale degree 15edo (Hard, L:s = 3:1) 26edo (Semihard, L:s = 5:2) 37edo (L:s = 7:3) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 1/1 (exact)
Minor 1-smidegree 1 80 2 92.3 3 97.3
Major 1-smidegree 3 240 5 230.8 7 227
Perfect 2-smidegree 4 320 7 323.1 10 324.3
Augmented 2-smidegree 6 480 10 461.5 14 454.1
Minor 3-smidegree 5 400 9 415.4 13 421.6
Major 3-smidegree 7 560 12 553.8 17 551.4
Minor 4-smidegree 8 640 14 646.2 20 648.6
Major 4-smidegree 10 800 17 784.6 24 778.4
Diminished 5-smidegree 9 720 16 738.5 23 745.9
Perfect 5-smidegree 11 880 19 876.9 27 875.7
Minor 6-smidegree 12 960 21 969.2 30 973
Major 6-smidegree 14 1120 24 1107.7 34 1102.7
Perfect 7-smidegree (octave) 15 1200 26 1200 37 1200 2/1 (exact)

Parahard tunings

Parahard smitonic tunings (3:1 to 4:1) have generators between 315.9¢ and 320¢, putting it close to a pure 6/5 (316¢). Stacking six generators and octave-reducing approximates 3/2 (702¢), 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, though the 2.3.5.13 extension described here is called cata.

Parahard edos smaller than 53edo include 15edo (not shown), 19edo, and 34edo.

MOS degrees is deprecated. Please use Template:MOS tunings instead.
Scale degree of 4L 3s
Scale degree 19edo (Superhard, L:s = 4:1) 34edo (L:s = 7:2) 53edo (L:s = 11:3) 72edo (L:s = 15:4) Approx. JI Ratios
Steps Cents Steps Cents Steps Cents Steps Cents
Perfect 0-smidegree (unison) 0 0 0 0 0 0 0 0 1/1 (exact)
Minor 1-smidegree 1 63.2 2 70.6 3 67.9 4 66.7
Major 1-smidegree 4 252.6 7 247.1 11 249.1 15 250
Perfect 2-smidegree 5 315.8 9 317.6 14 317 19 316.7
Augmented 2-smidegree 8 505.3 14 494.1 22 498.1 30 500
Minor 3-smidegree 6 378.9 11 388.2 17 384.9 23 383.3
Major 3-smidegree 9 568.4 16 564.7 25 566 34 566.7
Minor 4-smidegree 10 631.6 18 635.3 28 634 38 633.3
Major 4-smidegree 13 821.1 23 811.8 36 815.1 49 816.7
Diminished 5-smidegree 11 694.7 20 705.9 31 701.9 42 700
Perfect 5-smidegree 14 884.2 25 882.4 39 883 53 883.3
Minor 6-smidegree 15 947.4 27 952.9 42 950.9 57 950
Major 6-smidegree 18 1136.8 32 1129.4 50 1132.1 68 1133.3
Perfect 7-smidegree (octave) 19 1200 34 1200 53 1200 72 1200 2/1 (exact)

Modes

Alexandru Ianu (Ayceman)[1] has proposed the following mode names relating to the Almsivi in Morrowind (TES):


Modes of 4L 3s
UDP Cyclic
order		 Step
pattern
6|0 1 LLsLsLs
5|1 6 LsLLsLs
4|2 4 LsLsLLs
3|3 2 LsLsLsL
2|4 7 sLLsLsL
1|5 5 sLsLLsL
0|6 3 sLsLsLL

Scales

Scala files

Scale tree

Template: Scale tree is deprecated. Please use Template: MOS tuning spectrum instead. Details:
Use of a single Comments parameter has become unmaintainable. Existing scale trees should be migrated to the new template, where comments are entered using a step ratio p/q as a parameter: 
{{MOS tuning spectrum
| 3/2 = Example comment
| 4/3 = Another example comment
}}


The parameters tuning and depth have been replaced with Scale Signature and Depth, respectively.


Scale tree and tuning spectrum of 4L 3s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
5\7 857.143 342.857 1:1 1.000 Equalized 4L 3s
28\39 861.538 338.462 6:5 1.200
23\32 862.500 337.500 5:4 1.250
41\57 863.158 336.842 9:7 1.286
18\25 864.000 336.000 4:3 1.333 Supersoft 4L 3s
49\68 864.706 335.294 11:8 1.375
31\43 865.116 334.884 7:5 1.400
44\61 865.574 334.426 10:7 1.429
13\18 866.667 333.333 3:2 1.500 Soft 4L 3s
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
21\29 868.966 331.034 5:3 1.667 Semisoft 4L 3s
50\69 869.565 330.435 12:7 1.714
29\40 870.000 330.000 7:4 1.750
37\51 870.588 329.412 9:5 1.800
8\11 872.727 327.273 2:1 2.000 Basic 4L 3s
Scales with tunings softer 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 Semihard 4L 3s
49\67 877.612 322.388 13:5 2.600
30\41 878.049 321.951 8:3 2.667
41\56 878.571 321.429 11:4 2.750
11\15 880.000 320.000 3:1 3.000 Hard 4L 3s
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
14\19 884.211 315.789 4:1 4.000 Superhard 4L 3s
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
3\4 900.000 300.000 1:0 → ∞ Collapsed 4L 3s

Music

See also

Approaches:

References

  1. ↑ 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.
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