4L 3s
User:IlL/Template:RTT restriction
| ↖ 3L 2s | ↑ 4L 2s | 5L 2s ↗ |
| ← 3L 3s | 4L 3s | 5L 3s → |
| ↙ 3L 4s | ↓ 4L 4s | 5L 4s ↘ |
sLsLsLL
4L 3s refers to the structure of MOS scales with generators ranging from 1\4edo (one degree of 4edo, 300¢) to 2\7edo (two degrees of 7edo, or approx. 342.857¢). The name smitonic smy-TON-ik /smaɪˈtɒnɪk/ has been proposed (derived from 'sharp minor third', since the central range of the spectrum, 4\15 = 320¢ to 7\18 = 333.33¢ have minor third generators that are significantly sharp of 6/5).
4L 3s is a distorted diatonic, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).
Notation
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/Q& J&/K@ K/L@ L/K& L&/M@ M/N@ N/M& N&/O@ O/P@ P/O@ P&/J@ J
Intervals
| Generators | Notation (1/1 = J) | Interval category name | Generators | Notation of 2/1 inverse | Interval category name |
|---|---|---|---|---|---|
| The 7-note MOS has the following intervals (from some root): | |||||
| 0 | J | perfect unison | 0 | J | octave |
| 1 | L | perfect smithird | -1 | O | perfect smisixth |
| 2 | N | minor smififth (aka minor fifth) | -2 | M | major smifourth (aka major fourth) |
| 3 | P | minor smiseventh | -3 | K | major smisecond |
| 4 | K@ | minor smisecond | -4 | Q& | major smiseventh |
| 5 | M@ | minor smifourth (aka minor fourth) | -5 | N& | major smififth (aka major fifth) |
| 6 | O@ | diminished smisixth | -6 | L& | augmented smithird |
| The chromatic 11-note MOS (either 7L 4s or 4L 7s) also has the following intervals (from some root): | |||||
| 7 | J@ | diminished smioctave | -7 | J& | augmented smiunison; smichroma; smicomma |
| 8 | L@ | diminished smithird | -8 | O& | augmented smisixth |
| 9 | N@ | diminished smififth | -9 | M& | augmented smifourth |
| 10 | P@ | diminished smiseventh | -10 | K& | augmented smisecond |
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 major mosthird (made of two large steps) is a roughly meantone-sized major third, thus is a stand-in for the classical diatonic major third.
Parasoft smitonic tunings have both minor fifths and major fifths about equally off a just perfect fifth, and they approximate diatonic structure and 5-limit harmony fairly well otherwise. For this reason, parasoft might be the most accessible smitonic tuning range.
Parasoft smitonic EDOs include 18edo, 25edo, 32edo, 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 perfect fifth. 18edo is also more suited for conventionally jazz styles due to its 6-fold symmetry.
- 25edo can be used to make the major mosthird a good 5/4 (384¢).
The sizes of the generator, large step and small step of smitonic are as follows in various parasoft smitonic tunings.
| 18edo | 25edo | Optimal (2.9.5 POTE) tuning | |
|---|---|---|---|
| generator (g) | 5\18, 333.3 | 7\25, 336.0 | 335.84 |
| L (octave - 3g) | 3\18, 200.0 | 4\25, 192.0 | 193.16 |
| s (4g - octave) | 2\18, 133.3 | 3\25, 144.0 | 143.36 |
Intervals
Sortable table of major and minor intervals in parasoft smitonic tunings:
| Degree | 18edo | 25edo | Note name on J | Approximate ratios | #Gens up |
|---|---|---|---|---|---|
| unison | 0\18, 0.0 | 0\25, 0.0 | J | 1/1 | 0 |
| min. smi2nd | 2\18, 133.3 | 3\18, 144.0 | K@ | +4 | |
| maj. smi2nd | 3\18, 200.0 | 4\25, 192.0 | K | 9/8, 10/9 | -3 |
| perf. smi3rd | 5\18, 333.3 | 7\25, 336.0 | L | 17/14, 40/33 | +1 |
| aug. smi3rd | 4\13, 400.0 | 8\25, 384.4 | L& | 5/4 | -6 |
| min. smi4th | 7\18, 466.7 | 10\25, 480.0 | M@ | +5 | |
| maj. smi4th | 8\18, 533.3 | 11\25, 528.0 | M | 19/14, 34/25 | -2 |
| min. smi5th | 10\18, 666.7 | 14\25, 672.0 | N | 28/19, 25/17 | +2 |
| maj. smi5th | 11\18, 733.3 | 15\25, 720.0 | N& | -5 | |
| dim. smi6th | 12\18, 800.0 | 17\25, 816.0 | O@ | 8/5 | +6 |
| perf. smi6th | 13\18, 866.7 | 18\25, 864.0 | O | 28/17, 33/20 | -1 |
| min. smi7th | 15\18, 1000.0 | 21\25, 1008.0 | P | 16/9, 9/5 | +3 |
| maj. smi7th | 16\18, 1066.7 | 22\25, 1056.0 | P& | 12/7 | -4 |
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 "parapyth smitonic" or "archy smitonic", in analogy to parasoft smitonic being meantone smitonic.
| 11edo | 18edo | 29edo | |
|---|---|---|---|
| generator (g) | 3\11, 327.27 | 5\18, 333.33 | 8\29, 331.03 |
| L (octave - 3g) | 2\11, 218.18 | 3\18, 200.00 | 5\29, 206.90 |
| s (4g - octave) | 1\11, 109.09 | 2\18, 133.33 | 3\29, 124.14 |
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 smifourth (2 large steps + 1 small step) tends to approximate 11/8; 26edo is stellar in both of these approximations.
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 | 15edo | 26edo | 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 | 15edo | 26edo | 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 |
| min. smi2nd | 1\11, 109.1 | 1\15, 80.0 | 2\26, 92.3 | K@ | +4 | |
| maj. smi2nd | 2\11, 218.2 | 3\15, 240.0 | 5\26, 230.8 | K | 8/7 | -3 |
| perf. smi3rd | 3\11, 327.3 | 4\15, 320.0 | 7\26, 323.1 | L | 77/64, 6/5 | +1 |
| aug. smi3rd | 4\11, 436.4 | 6\15, 480.0 | 10\26, 461.5 | L& | -6 | |
| min. smi4th | 4\11, 436.4 | 5\15, 400.0 | 9\26, 415.4 | M@ | 14/11 | +5 |
| maj. smi4th | 5\11, 545.5 | 7\15, 560.0 | 12\26, 553.9 | M | 11/8 | -2 |
| min. smi5th | 6\11, 656.6 | 8\15, 640.0 | 14\26, 646.2 | N | 16/11 | +2 |
| maj. smi5th | 7\11, 763.6 | 10\15, 800.0 | 17\26, 784.62 | N& | 11/7 | -5 |
| dim. smi6th | 7\11, 763.6 | 9\15, 720.0 | 16\26, 738.5 | O@ | +6 | |
| perf. smi6th | 8\11, 872.7 | 11\15, 880.0 | 19\26, 876.9 | O | 5/3 | -1 |
| min. smi7th | 9\11, 981.8 | 12\15, 960.0 | 21\26, 969.2 | P | 7/4 | +3 |
| maj. smi7th | 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 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.
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 | 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 | |
| s (4g - octave) | 1\19, 63.16 | 2\34, 70.59 | 3\53, 67.92 | 25/24 |
Modes
The mode names are temporary and impressionistic.
- LLsLsLs: Smimajor
- LsLLsLs: Smiharmonic minor
- LsLsLLs: Smimelodic minor
- LsLsLsL: Smisymmetric
- sLLsLsL: Smilocrian
- sLsLLsL: Smikatadorian
- sLsLsLL: Smifreygish
Pseudo-diatonic theory
Hypohard
Parasoft
Primodal theory
Primodal chords
Nejis
Temperaments
- Main article: 4L 3s/Temperaments
Samples
A fugue in 18edo smitonic (WIP)
Scale tree
The spectrum looks like this:
| Generator | Tetrachord | g in cents | 2g | 3g | 4g | Comments | |||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 1\4 | 1 0 1 | 300 | 600 | 900 | 0 | ||||||||
| 9\35 | 8 1 8 | 308.571 | 617.143 | 925.714 | 34.286 | ||||||||
| 8\31 | 7 1 7 | 309.677 | 619.355 | 929.023 | 38.71 | ||||||||
| 7\27 | 6 1 6 | 311.111 | 622.222 | 933.333 | 44.444 | ||||||||
| 6\23 | 5 1 5 | 313.043 | 626.087 | 939.13 | 52.174 | ||||||||
| 5\19 | 4 1 4 | 315.789 | 631.579 | 947.368 | 63.158 | ||||||||
| 9\34 | 7 2 7 | 317.647 | 634.294 | 951.941 | 70.588 | ||||||||
| 4\15 | 3 1 3 | 320 | 640 | 960 | 80 | L/s = 3. | |||||||
| 11\41 | 8 3 8 | 321.951 | 643.902 | 965.854 | 87.805 | ||||||||
| 29\108 | 21 8 21 | 322.222 | 644.444 | 966.667 | 88.889 | ||||||||
| 18\67 | 13 5 13 | 322.388 | 644.776 | 967.364 | 89.522 | ||||||||
| 7\26 | 5 2 5 | 323.077 | 646.154 | 969.231 | 92.308 | ||||||||
| 31/115 | 22 9 22 | 323.478 | 646.956 | 970.434 | 93.913 | ||||||||
| 2.44 1 2.44 | 323.501 | 647.002 | 970.003 | 94.004 | |||||||||
| 24/89 | 17 7 17 | 323.595 | 647.191 | 970.786 | 94.382 | ||||||||
| 17/63 | 12 5 12 | 323.809 | 647.619 | 971.428 | 95.238 | ||||||||
| 10/37 | 7 3 7 | 324.324 | 648.648 | 972.972 | 97.297 | ||||||||
| 3\11 | 2 1 2 | 327.273 | 654.545 | 981.818 | 109.091 | Boundary of propriety (generators larger than this are proper) | |||||||
| 8\29 | 5 3 5 | 331.034 | 662.069 | 993.013 | 124.138 | ||||||||
| 21\76 | 13 8 13 | 331.579 | 663.158 | 994.739 | 126.316 | ||||||||
| 34\123 | 21 13 21 | 331.707 | 663.415 | 995.122 | 126.829 | Golden smitonic | |||||||
| 13\47 | 8 5 8 | 331.915 | 663.83 | 995.745 | 127.66 | ||||||||
| 5\18 | 3 2 3 | 333.333 | 666.667 | 1000 | 133.333 | Optimum rank range (L/s=3/2) | |||||||
| 7\25 | 4 3 4 | 336 | 672 | 1008 | 144 | ||||||||
| 9\32 | 5 4 5 | 337.5 | 675 | 1012.5 | 150 | ||||||||
| 11\39 | 6 5 6 | 338.462 | 676.923 | 1015.385 | 153.846 | ||||||||
| 13\46 | 7 6 7 | 339.13 | 678.261 | 1017.391 | 156.522 | ||||||||
| 15\53 | 8 7 8 | 339.623 | 679.245 | 1018.868 | 158.491 | ||||||||
| 2\7 | 1 1 1 | 342.857 | 685.714 | 1028.571 | 171.429 | ||||||||