4L 3s: Difference between revisions

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).

The [[TAMNAMS]] system is used in this article to name 4L 3s intervals and step size ratios and step ratio ranges.

 

 

Thus the [[11edo]] gamut is as follows:

 

 

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

! Notation (1/1 = J)

! [[TAMNAMS]] name

! In L's and s's

| J

| 1

| perfect 2-mosstep

| 1L + 1s

| -1

| 2

| minor 4-mosstep  

| 2L + 2s

| -2

| M

| major 3-mosstep

| 3

| minor 6-mosstep

| 3L + 3s

| -3

| K

| major 1-mosstep

| 4

| major 4-mosstep  

| 6

| O@

| diminished 5-mosstep

| -6

| 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

| diminished 7-mosstep

| 5L + 2s

| J&

| 4L + 1s

| 1L + 3s

| -9

== Low harmonic entropy scales ==

* [[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)

* [[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 ==

=== Simple tunings ===

! [[15edo]] (hard)

| 5/3

 

[[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 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.

! [[18edo]] (soft)

! Optimized (2.9.5 [[POTE]] [[Dual-fifth temperaments|dual-3 sixix]]) tuning

| 5\18, 333.3

| 7\25, 336.0

| s (4g - octave)

| 2\18, 133.3

 

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

|-bgcolor="#eaeaff"

|-bgcolor="#eaeaff"

|-bgcolor="#eaeaff"

|-bgcolor="#eaeaff"

|-bgcolor="#eaeaff"

|-bgcolor="#eaeaff"

|-bgcolor="#eaeaff"

|-bgcolor="#eaeaff"

 

[[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"

==== Intervals ====

{| class="wikitable right-2 sortable "

| 1/1

| 3\29, 124.1

| 14/13

| major 1-mosstep

| -3

| 23/19, 40/33

| aug. 2-mosstep

|}

 

=== 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.

! [[11edo]] (basic)

Sortable table of major and minor intervals in hypohard smitonic tunings:

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]]).

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 (not including 15edo).  

{| class="wikitable right-2 right-3 right-4"

! [[19edo]] (superhard)

! [[34edo]]  

| 5\19, 315.79

| 9\34, 317.65

| 14\53, 316.98

| 6/5

| 4\19, 252.63

| 7\34, 247.06

| 11\53, 249.06

| s (4g - octave)

| 1\19, 63.16

| 2\34, 70.59

| 3\53, 67.92

! [[19edo]] (superhard)

! [[34edo]]

! [[53edo]]

! class="unsortable"| Note name on J

! class="unsortable"| Approximate ratios

|-bgcolor="#eaeaff"

| 1/1

| 1\19, 63.2

| 3\53, 67.9

| K@

| -3

|-bgcolor="#eaeaff"

| perf. 2-mosstep

| +1

| 8\19, 505.3

| 14\34, 494.1

| 22\53, 498.1

| minor 3-mosstep

| 6\19, 378.9

| 11\34, 388.2

| 5/4

| major 3-mosstep

| 11\19, 694.7

| 3/2

| 18\19, 1136.8

| 32\34, 1129.4

 

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):

! [[Modal UDP Notation|UDP]]

== Approaches ==

* [[4L 3s/Inthar's approach]]

{{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.

# 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.

== Scales ==

== 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)

* [[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]])

* [[File:Sixix Fugue.mp3]] A fugue in [[18edo]] smitonic functional harmony (WIP)

== Scale tree ==

 

{| class="wikitable center-all"

! colspan="6" rowspan="2" | Generator <br><small>(Chroma-positive)</small>

! <small>Chroma-negative</small>

| 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]]

| || || 13\18 || || || || 866.667 || 333.333 || 3 || 2 || 1.500 ||

| || || || || 34\47 || || 868.085 || 331.915 || 8 || 5 || 1.600 ||

| || || || || || 55\76 || 868.421 || 331.579 || 13 || 8 || 1.625 || Golden smitonic (868.3282¢)

| || || || || || 50\69 || 869.565 || 330.435 || 12 || 7 || 1.714 ||

| || || || || || 37\51 || 870.588 || 329.422 || 9 || 5 || 1.800 ||

| || || || || || 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 ||

| || || || || || 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]]↓