4L 3s: Difference between revisions
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== Approaches == | == Approaches == | ||
* [[4L 3s/Inthar's approach]] | |||
== Temperaments == | == Temperaments == | ||
{{main|4L 3s/Temperaments}} | {{main|4L 3s/Temperaments}} | ||
Revision as of 03:44, 28 May 2021
| ↖ 3L 2s | ↑ 4L 2s | 5L 2s ↗ |
| ← 3L 3s | 4L 3s | 5L 3s → |
| ↙ 3L 4s | ↓ 4L 4s | 5L 4s ↘ |
sLsLsLL
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¢).
4L 3s can be thought of as a warped diatonic scale, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).
Standing assumptions
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".)
Thus the 11edo gamut is as follows:
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
| 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 |
|---|---|---|---|---|---|---|---|
| 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 |
| 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
There are two notable harmonic entropy minima:
- Kleismic temperament, in which the generator is 6/5 and 6 of them make a 3/1 (making the diminished mossixth 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
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 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.
- 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.
- 25edo can be used to make the augmented mosthird a good 5/4 (384¢).
The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.
| 18edo (soft) | 25edo (supersoft) | 43edo | Optimized (2.9.5 POTE 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.
| Degree | 18edo (soft) | 25edo (supersoft) | 43edo | Note name on J | Approximate ratios | #Gens up |
|---|---|---|---|---|---|---|
| unison | 0\18, 0.0 | 0\25, 0.0 | 0\43, 0.0 | J | 1/1 | 0 |
| 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 | |
| dim. mos3rd | 4\18, 266.7 | 6\25, 288.0 | 10\43, 279.1 | L@ | +8 | |
| perf. mos3rd | 5\18, 333.3 | 7\25, 336.0 | 12\43, 334.9 | L | 17/14, 40/33 | +1 |
| aug. mos3rd | 6\18, 400.0 | 8\25, 384.4 | 14\43, 390.7 | L& | 5/4 | -6 |
| 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 |
| dim. mos5th | 9\18, 600.0 | 13\25, 624.0 | 22\43, 614.0 | N@ | 10/7 | +9 |
| minor mos5th | 10\18, 666.7 | 14\25, 672.0 | 24\43, 669.8 | N | 28/19, 25/17 | +2 |
| major mos5th | 11\18, 733.3 | 15\25, 720.0 | 26\43, 725.6 | N& | 32/21 | -5 |
| 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 | |
| dim. mos7th | 14\18, 933.3 | 20\25, 960.0 | 34\34, 948.8 | P@ | +10 | |
| minor mos7th | 15\18, 1000.0 | 21\25, 1008.0 | 36\43, 1004.7 | P | 16/9, 9/5 | +3 |
| major mos7th | 16\18, 1066.7 | 22\25, 1056.0 | 38\43, 1060.5 | P& | 24/13 | -4 |
| 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
Hyposoft tunings of smitonic have step ratios between 3/2 and 2/1 which implies that the generator is a supraminor third sharper than 3\11 = 327.27¢ and flatter than 5\18 = 333.33¢.
The large step is a sharper major second in these tunings than in parasoft tunings. These tunings could be considered "neogothic smitonic" or "archy smitonic", in analogy to parasoft smitonic being meantone smitonic.
| 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):
| Degree | 29edo (semisoft) | Note name on J | Approximate ratios (for 29edo) | #Gens up |
|---|---|---|---|---|
| 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 |
| perf. mos3rd | 8\29, 331.0 | L | 23/19, 40/33 | +1 |
| 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 |
| minor mos5th | 16\29, 662.1 | N | 19/13, 22/15 | +2 |
| 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 |
| minor mos7th | 24\29, 993.1 | P | 16/9 | +3 |
| major mos7th | 26\28, 1075.9 | P& | 13/7 | -4 |
Hypohard
Hypohard tunings have step ratios between 2 and 3, implying a generator sharper than 4\15 = 320¢ and flatter than 3\11 = 327.27¢. The large step tends to approximate 8/7, and the major 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.
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) | 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:
| Degree | 11edo (basic) | 15edo (hard) | 26edo (semihard) | Note name on J | Approximate ratios | #Gens up |
|---|---|---|---|---|---|---|
| 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 |
| perf. mos3rd | 3\11, 327.3 | 4\15, 320.0 | 7\26, 323.1 | L | 77/64, 6/5 | +1 |
| 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 |
| minor mos5th | 6\11, 656.6 | 8\15, 640.0 | 14\26, 646.2 | N | 16/11 | +2 |
| 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 |
| minor mos7th | 9\11, 981.8 | 12\15, 960.0 | 21\26, 969.2 | P | 7/4 | +3 |
| major mos7th | 10\11, 1090.9 | 14\15, 1120.0 | 24\26, 1107.7 | P& | -4 |
Parahard
In parahard smitonic (step ratio between 3 and 4, thus with generator between 5\19, 315.79¢ and 4\15, 320¢), the generator is close to a pure 6/5 minor third, and 6 minor thirds are used to reach a perfect fifth. The 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 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).
| 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
Sortable table of major and minor intervals in parahard smitonic tunings:
| Degree | 19edo (superhard) | 34edo | 53edo | Note name on J | Approximate ratios | #Gens up |
|---|---|---|---|---|---|---|
| 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 |
| perf. mos3rd | 5\19, 315.8 | 9\34, 317.6 | 14\53, 317.0 | L | 6/5 | +1 |
| 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 |
| minor mos5th | 10\19, 631.6 | 18\34, 635.3 | 28\53, 634.0 | N | 13/9 | +2 |
| 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 |
| minor mos7th | 15\19, 947.4 | 27\34, 952.9 | 42\53, 950.9 | P | 26/15 | +3 |
| major mos7th | 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)[1], relating to the Almsivi in Morrowind (TES):
| Mode | UDP | Name |
|---|---|---|
| LLsLsLs | 6|0 | Nerevarine |
| LsLLsLs | 5|1 | Vivecan |
| LsLsLLs | 4|2 | Lorkhanic |
| LsLsLsL | 3|3 | Sothic |
| sLLsLsL | 2|4 | Kagrenacan |
| sLsLLsL | 1|5 | Almalexian |
| sLsLsLL | 0|6 | Dagothic |
Approaches
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.
Scales
Music
- City of the Asleep, "An Amputated Elliptic Knob of the Cryptocurve Regenerates" (Various orgone edos)
- ks26, Ghost Bridge (11edo)
- Alexandru Ianu, Sylvian Moon Dance (11edo) (sheet music)
- A fugue in 18edo smitonic functional harmony (WIP)
Scale tree
The spectrum looks like this:
| Generator | Cents | L | s | L/s | 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 (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
- ↑ 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.