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 or smitonic smy-TON-ik /smaɪˈtɒnɪk/ 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¢). Smitonic 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.
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 J&/K@ K L L&/M@ M N N&/O@ O P 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, 4L 7s, or 11edo) also has the following intervals (from some root): | |||||
| 7 | J@ | diminished smioctave | -7 | J& | augmented smiunison; smichroma; smicomma (in parasoft smitonic contexts) |
| 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 augmented smithird (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 smichromas. 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 smisixth) and falling fifths (666.7c, a major smififth) 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 smithird 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) 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 smichroma 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 |
| smichroma | 1\18, 66.7 | 1\25, 48.0 | 2\43, 55.8 | J& | -7 | |
| dim. smi2nd | 1\18, 66.7 | 2\25, 96.0 | 3\43, 83.7 | K@@ | +11 | |
| min. smi2nd | 2\18, 133.3 | 3\25, 144.0 | 5\43, 139.5 | K@ | 13/12 | +4 |
| maj. smi2nd | 3\18, 200.0 | 4\25, 192.0 | 7\43, 195.3 | K | 9/8, 10/9 | -3 |
| aug. smi2nd | 4\18, 266.7 | 5\25, 240.0 | 9\43, 251.2 | K& | -10 | |
| dim. smi3rd | 4\18, 266.7 | 6\25, 288.0 | 10\43, 279.1 | L@ | +8 | |
| perf. smi3rd | 5\18, 333.3 | 7\25, 336.0 | 12\43, 334.9 | L | 17/14, 40/33 | +1 |
| aug. smi3rd | 6\18, 400.0 | 8\25, 384.4 | 14\43, 390.7 | L& | 5/4 | -6 |
| doubly aug. smi3rd | 7\18, 466.7 | 9\25, 432.0 | 16\43, 446.5 | L&& | -13 | |
| dim. smi4th | 6\18, 400.0 | 9\25, 432.0 | 15\43, 418.6 | M@@ | +12 | |
| min. smi4th | 7\18, 466.7 | 10\25, 480.0 | 17\43, 474.4 | M@ | 21/16 | +5 |
| maj. smi4th | 8\18, 533.3 | 11\25, 528.0 | 19\43, 530.2 | M | 19/14, 34/25 | -2 |
| aug. smi4th | 9\18, 600.0 | 12\25, 576.0 | 21\43, 586.0 | M& | 7/5 | -9 |
| dim. smi5th | 9\18, 600.0 | 13\25, 624.0 | 22\43, 614.0 | N@ | 10/7 | +9 |
| min. smi5th | 10\18, 666.7 | 14\25, 672.0 | 24\43, 669.8 | N | 28/19, 25/17 | +2 |
| maj. smi5th | 11\18, 733.3 | 15\25, 720.0 | 26\43, 725.6 | N& | 32/21 | -5 |
| aug. smi5th | 12\18, 800.0 | 16\25, 768.0 | 28\43, 781.4 | N&& | -12 | |
| doubly dim. smi6th | 11\18, 733.3 | 16\25, 768.0 | 27\43, 753.5 | O@@ | +13 | |
| dim. smi6th | 12\18, 800.0 | 17\25, 816.0 | 29\43, 809.3 | O@ | 8/5 | +6 |
| perf. smi6th | 13\18, 866.7 | 18\25, 864.0 | 31\43, 865.1 | O | 28/17, 33/20 | -1 |
| aug. smi6th | 14\18, 933.3 | 19\25, 912.0 | 33\43, 920.9 | O& | -8 | |
| dim. smi7th | 14\18, 933.3 | 20\25, 960.0 | 34\34, 948.8 | P@ | +10 | |
| min. smi7th | 15\18, 1000.0 | 21\25, 1008.0 | 36\43, 1004.7 | P | 16/9, 9/5 | +3 |
| maj. smi7th | 16\18, 1066.7 | 22\25, 1056.0 | 38\43, 1060.5 | P& | 20/13 | -4 |
| aug. smi7th | 17\18, 1133.3 | 23\25, 1104.0 | 40\43, 1116.3 | P& | -11 | |
| dim. smioctave | 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 |
| min. smi2nd | 3\29, 124.1 | K@ | 14/13 | +4 |
| maj. smi2nd | 5\29, 206.9 | K | 9/8 | -3 |
| perf. smi3rd | 8\29, 331.0 | L | 23/19, 40/33 | +1 |
| aug. smi3rd | 10\29, 413.8 | L& | 14/11 | -6 |
| min. smi4th | 11\29, 455.2 | M@ | 13/10 | +5 |
| maj. smi4th | 13\29, 537.9 | M | 15/11 | -2 |
| min. smi5th | 16\29, 662.1 | N | 19/13, 22/15 | +2 |
| maj. smi5th | 18\26, 744.8 | N& | 20/13 | -5 |
| dim. smi6th | 19\29, 786.2 | O@ | 11/7 | +6 |
| perf. smi6th | 21\29, 869.0 | O | 33/20, 38/23 | -1 |
| min. smi7th | 24\29, 993.1 | P | 16/9 | +3 |
| maj. smi7th | 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 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 (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 |
| 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 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.
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 |
| min. smi2nd | 1\19, 63.2 | 2\34, 70.6 | 3\53, 67.9 | K@ | 25/24, 26/25 | +4 |
| maj. smi2nd | 4\19, 252.6 | 7\34, 247.1 | 11\53, 249.1 | K | 15/13 | -3 |
| perf. smi3rd | 5\19, 315.8 | 9\34, 317.6 | 14\53, 317.0 | L | 6/5 | +1 |
| aug. smi3rd | 8\19, 505.3 | 14\34, 494.1 | 22\53, 498.1 | L& | 4/3 | -6 |
| min. smi4th | 6\19, 378.9 | 11\34, 388.2 | 17\53, 384.9 | M@ | 5/4 | +5 |
| maj. smi4th | 9\19, 568.4 | 16\34, 564.7 | 25\53, 566.0 | M | 18/13 | -2 |
| min. smi5th | 10\19, 631.6 | 18\34, 635.3 | 28\53, 634.0 | N | 13/9 | +2 |
| maj. smi5th | 16\19, 821.1 | 23\34, 811.8 | 39\53, 815.0 | N& | 8/5 | -5 |
| dim. smi6th | 11\19, 694.7 | 20\34, 705.9 | 31\53, 701.9 | O@ | 3/2 | +6 |
| perf. smi6th | 14\19, 884.2 | 25\34, 882.4 | 39\53, 883.0 | O | 5/3 | -1 |
| min. smi7th | 15\19, 947.4 | 27\34, 952.9 | 42\53, 950.9 | P | 26/15 | +3 |
| maj. smi7th | 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 |
Pseudo-diatonic theory
Hypohard
Parasoft
Primodal theory
Primodal chords
Nejis
Temperaments
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 | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| 5\7 | 857.143 | 1 | 1 | 1.000 | ||||||
| 28\39 | 861.538 | 6 | 5 | 1.200 | Amity↑ | |||||
| 23\32 | 862.500 | 5 | 4 | 1.250 | Sixix | |||||
| 41\57 | 863.158 | 9 | 7 | 1.286 | ||||||
| 18\25 | 864.000 | 4 | 3 | 1.333 | ||||||
| 49\68 | 864.706 | 11 | 8 | 1.375 | ||||||
| 31\43 | 865.116 | 7 | 5 | 1.400 | ||||||
| 17\58 | 865.574 | 10 | 7 | 1.428 | ||||||
| 13\18 | 866.667 | 3 | 2 | 1.500 | L/s = 3/2 | |||||
| 47\65 | 867.692 | 11 | 7 | 1.571 | ||||||
| 34\47 | 868.085 | 8 | 5 | 1.600 | ||||||
| 55\76 | 868.421 | 13 | 8 | 1.625 | Unnamed golden tuning | |||||
| 21\29 | 868.966 | 5 | 3 | 1.667 | ||||||
| 50\69 | 869.565 | 12 | 7 | 1.714 | ||||||
| 29\40 | 870.000 | 7 | 4 | 1.750 | ||||||
| 37\51 | 870.588 | 9 | 5 | 1.800 | ||||||
| 8\11 | 872.727 | 2 | 1 | 2.000 | Basic smitonic (Generators smaller than this are proper) | |||||
| 35\48 | 875.000 | 9 | 4 | 2.250 | ||||||
| 27\37 | 875.676 | 7 | 3 | 2.333 | ||||||
| 46\63 | 876.190 | 12 | 5 | 2.400 | ||||||
| 19\26 | 876.923 | 5 | 2 | 2.500 | ||||||
| 49\67 | 877.612 | 13 | 5 | 2.600 | Golden superkleismic | |||||
| 30\41 | 878.049 | 8 | 3 | 2.667 | Superkleismic | |||||
| 41\56 | 878.571 | 11 | 4 | 2.750 | ||||||
| 11\15 | 880.000 | 3 | 1 | 3.000 | L/s = 3/1 | |||||
| 36\49 | 881.633 | 10 | 3 | 3.333 | ||||||
| 25\34 | 882.353 | 7 | 2 | 3.500 | ||||||
| 39\53 | 883.019 | 11 | 3 | 3.667 | Hanson is in this region | |||||
| 14\19 | 884.211 | 4 | 1 | 4.000 | ||||||
| 31\42 | 885.714 | 9 | 2 | 4.500 | ||||||
| 17\23 | 886.957 | 5 | 1 | 5.000 | ||||||
| 20\27 | 888.889 | 6 | 1 | 6.000 | Oolong, myna↓ | |||||
| 3\4 | 900.000 | 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.