4L 3s: Difference between revisions
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'''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''' has been | '''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 the obsolete temperament name ''smite'' for [[sixix]], from 'sharp minor third', taking sharp to mean sharp of the 12edo minor third). | ||
4L 3s is a distorted diatonic, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs). | 4L 3s is a distorted diatonic, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs). | ||
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# 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 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 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): [[ | # 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 most concordant chords in these temperaments; you would need to use a [[MODMOS]] or use a larger MOS gamut. | 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 most concordant chords in these temperaments; you would need to use a [[MODMOS]] or use a larger MOS gamut. | ||
== 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".) Ultharian has been chosen as the default mode because we want to carry over the diatonic idea of sharpening the second-to-last degree to get the leading tone for minor keys and the sharpened "Vmaj", and we also have the "sharp V" for the oneiromajor tonality by default. | |||
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''' | |||
== Scale tree == | == Scale tree == | ||
| Line 137: | Line 143: | ||
| | 951.941 | | | 951.941 | ||
| | 70.588 | | | 70.588 | ||
| style="text-align:center;" | | | style="text-align:center;" | Kleismic is around here | ||
|- | |- | ||
| | | | | | ||
| Line 469: | Line 475: | ||
| style="text-align:center;" | | | style="text-align:center;" | | ||
|} | |} | ||
There are two notable harmonic entropy minima: [[Kleismic_family| | There are two notable harmonic entropy minima: [[Kleismic_family|kleismic]], in which the generator is 6/5 and 6 of them make a 3/1, and [[Starling_temperaments|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 == | == Tuning ranges == | ||
=== Sixix === | === Sixix === | ||
Sixix tunings (with generator a supraminor third sharper than 5\18 and flatter than 9\32) have step ratios between 5/4 and 3/2. | |||
Sixix can be considered "meantone smitonic". This is because sixix 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]]). Thus sixix tempers out [[81/80]] like meantone does. | |||
* 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. | |||
EDOs that support sixix 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 for 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 sixix tunings. | |||
{| class="wikitable right-2 right-3 right-4 right-5" | |||
|- | |||
! | |||
! [[18edo]] | |||
! [[25edo]] | |||
! Optimal (2.9.5 [[POTE]]) tuning | |||
|- | |||
| generator (g) | |||
| 5\18, 333.33 | |||
| 7\25, 336.00 | |||
| 335.84 | |||
|- | |||
| L (octave - 3g) | |||
| 3\18, 200.00 | |||
| 4\25, 192.00 | |||
| 193.16 | |||
|- | |||
| s (4g - octave) | |||
| 2\18, 133.33 | |||
| 3\25, 144.00 | |||
| 143.36 | |||
|} | |||
=== Orgone === | === Orgone === | ||
=== | [[Orgone]] tunings (with generator a minor third sharper than 4\15 and flatter than 3\11) have step ratios between 2/1 and 3/1. It nominally approximates the 2.7.11 subgroup, on which the [[26edo]] tuning is pretty much optimal. The large step approximates [[8/7]], and the major smifourth (2 large steps + 1 small step) approximates [[11/8]]. | ||
EDOs that support orgone include [[11edo]], [[15edo]], [[26edo]], and [[37edo]]. | |||
The sizes of the generator, large step and small step of smitonic are as follows in various orgone tunings. | |||
{| class="wikitable right-2 right-3 right-4 right-5" | |||
|- | |||
! | |||
! [[11edo]] | |||
! [[15edo]] | |||
! [[26edo]] | |||
! JI intervals represented | |||
|- | |||
| generator (g) | |||
| 3\11, 327.27 | |||
| 4\15, 320.00 | |||
| 7\26, 323.08 | |||
| 77/64 | |||
|- | |||
| 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) | |||
|} | |||
=== Kleismic === | |||
[[Kleismic]] (aka hanson or keemun) tunings (with generator a minor third sharper than 5\19 and flatter than 4\15) have step ratios between 3/1 and 4/1. It is a [[5-limit]] microtemperament supported by the likes of [[53edo]]. The generator is close to a [[6/5]], and 6 of them are used to reach [[3/2]]; hence kleismic tempers out the [[kleisma]] 15625/15552. | |||
EDOs that support kleismic include [[15edo]], [[19edo]], [[34edo]], and [[53edo]]. | |||
The sizes of the generator, large step and small step of smitonic are as follows in various kleismic tunings. | |||
{| class="wikitable right-2 right-3 right-4 right-5" | |||
|- | |||
! | |||
! [[15edo]] | |||
! [[19edo]] | |||
! [[34edo]] | |||
! 2.3.5 [[POTE]] tuning | |||
! JI intervals represented | |||
|- | |||
| generator (g) | |||
| 4\15, 320.00 | |||
| 5\19, 315.79 | |||
| 7\26, 317.65 | |||
| 317.01 | |||
| 6/5 | |||
|- | |||
| L (octave - 3g) | |||
| 3\15, 240.00 | |||
| 4\19, 252.63 | |||
| 7\34, 247.06 | |||
| 248.98 | |||
| 15/13, 23/20 | |||
|- | |||
| s (4g - octave) | |||
| 1\15, 80.00 | |||
| 1\19, 63.16 | |||
| 2\34, 70.59 | |||
| 68.03 | |||
| 25/24 | |||
|} | |||
== Intervals == | == Intervals == | ||
{| class="wikitable center-all" | |||
|- | |||
! Generators | |||
! Notation (1/1 = J) | |||
! Heptatonic interval category name | |||
! Generators | |||
! Notation of 2/1 inverse | |||
! Heptatonic interval category name | |||
|- | |||
| colspan="6" style="text-align:left" | 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 | |||
|- | |||
| colspan="6" style="text-align:left" | The chromatic 11-note MOS also has the following intervals (from some root): | |||
|- | |||
| 7 | |||
| J@ | |||
| diminished octave | |||
| -7 | |||
| J& | |||
| augmented unison | |||
|- | |||
| 8 | |||
| L@ | |||
| diminished smithird | |||
| -8 | |||
| O& | |||
| augmented smisixth | |||
|- | |||
| 9 | |||
| N@ | |||
| diminished smififth | |||
| -9 | |||
| M& | |||
| augmented smifourth | |||
|- | |||
| 10 | |||
| P@ | |||
| diminished smiseventh | |||
| -10 | |||
| K& | |||
| augmented smisecond | |||
|} | |||
== Modes == | == Modes == | ||
| Line 483: | Line 676: | ||
=== Orgone === | === Orgone === | ||
=== Sixix === | === Sixix === | ||
== Primodal theory == | == Primodal theory == | ||
=== Primodal chords === | === Primodal chords === | ||
| Line 494: | Line 682: | ||
== Rank-2 temperaments == | == Rank-2 temperaments == | ||
=== Myna (27&31) === | === Myna (27&31) === | ||
=== | === Kleismic (19&15, 2.3.5.7) === | ||
=== Orgone (15&11, 2.7.11) === | === Orgone (15&11, 2.7.11) === | ||
=== Sixix (18&25, 2.3.5.7) | === Sixix (18&25) === | ||
[[Sixix]] can be viewed as a [[dual-fifth]] temperament, i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3): | |||
* It has both a sharp fifth and a flat fifth but no near-just 3/2. | |||
* Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup. | |||
* The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in [[porcupine]] harmony. Tempering out this comma leads to [[7edo]]. | |||
== Samples == | == Samples == | ||
[[File:Sixix Fugue.mp3]] [[Sixix]] Fugue in [[18edo]] (WIP) | [[File:Sixix Fugue.mp3]] [[Sixix]] Fugue in [[18edo]] (WIP) | ||
Revision as of 05:45, 26 March 2021
| ↖ 3L 2s | ↑ 4L 2s | 5L 2s ↗ |
| ← 3L 3s | 4L 3s | 5L 3s → |
| ↙ 3L 4s | ↓ 4L 4s | 5L 4s ↘ |
Scale structure
sLsLsLL
Generator size
TAMNAMS information
Related MOS scales
Equal tunings
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 the obsolete temperament name smite for sixix, from 'sharp minor third', taking sharp to mean sharp of the 12edo minor third).
4L 3s is a distorted diatonic, because it has one large step of diatonic (5L 2s, LLsLLLs) replaced with a small step (yielding LLsLsLs).
4L 3s has several temperament interpretations:
- 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 most concordant chords in these temperaments; you would need to use a MODMOS or use a larger MOS gamut.
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".) Ultharian has been chosen as the default mode because we want to carry over the diatonic idea of sharpening the second-to-last degree to get the leading tone for minor keys and the sharpened "Vmaj", and we also have the "sharp V" for the oneiromajor tonality by default.
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
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 | Myna is around here | |||||||
| 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 | L/s = 4 | |||||||
| 9\34 | 7 2 7 | 317.647 | 634.294 | 951.941 | 70.588 | Kleismic is around here | |||||||
| 4\15 | 3 1 3 | 320 | 640 | 960 | 80 | L/s = 3. Orgone starts here | |||||||
| 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 | Orgone minmax tuning | ||||||||
| 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) Orgone ends here. | |||||||
| 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) Sixix | |||||||
| 7\25 | 4 3 4 | 336 | 672 | 1008 | 144 | Sixix | |||||||
| 9\32 | 5 4 5 | 337.5 | 675 | 1012.5 | 150 | Sixix | |||||||
| 11\39 | 6 5 6 | 338.462 | 676.923 | 1015.385 | 153.846 | Sixix | |||||||
| 13\46 | 7 6 7 | 339.13 | 678.261 | 1017.391 | 156.522 | (17/14)^3=9/5 | |||||||
| 15\53 | 8 7 8 | 339.623 | 679.245 | 1018.868 | 158.491 | Amity is around here | |||||||
| 2\7 | 1 1 1 | 342.857 | 685.714 | 1028.571 | 171.429 | ||||||||
There are two notable harmonic entropy minima: kleismic, in which the generator is 6/5 and 6 of them make a 3/1, and 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
Sixix
Sixix tunings (with generator a supraminor third sharper than 5\18 and flatter than 9\32) have step ratios between 5/4 and 3/2.
Sixix can be considered "meantone smitonic". This is because sixix 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). Thus sixix tempers out 81/80 like meantone does.
- 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.
EDOs that support sixix 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 for 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 sixix tunings.
| 18edo | 25edo | Optimal (2.9.5 POTE) tuning | |
|---|---|---|---|
| generator (g) | 5\18, 333.33 | 7\25, 336.00 | 335.84 |
| L (octave - 3g) | 3\18, 200.00 | 4\25, 192.00 | 193.16 |
| s (4g - octave) | 2\18, 133.33 | 3\25, 144.00 | 143.36 |
Orgone
Orgone tunings (with generator a minor third sharper than 4\15 and flatter than 3\11) have step ratios between 2/1 and 3/1. It nominally approximates the 2.7.11 subgroup, on which the 26edo tuning is pretty much optimal. The large step approximates 8/7, and the major smifourth (2 large steps + 1 small step) approximates 11/8.
EDOs that support orgone include 11edo, 15edo, 26edo, and 37edo. The sizes of the generator, large step and small step of smitonic are as follows in various orgone tunings.
| 11edo | 15edo | 26edo | JI intervals represented | |
|---|---|---|---|---|
| generator (g) | 3\11, 327.27 | 4\15, 320.00 | 7\26, 323.08 | 77/64 |
| 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) |
Kleismic
Kleismic (aka hanson or keemun) tunings (with generator a minor third sharper than 5\19 and flatter than 4\15) have step ratios between 3/1 and 4/1. It is a 5-limit microtemperament supported by the likes of 53edo. The generator is close to a 6/5, and 6 of them are used to reach 3/2; hence kleismic tempers out the kleisma 15625/15552.
EDOs that support kleismic include 15edo, 19edo, 34edo, and 53edo.
The sizes of the generator, large step and small step of smitonic are as follows in various kleismic tunings.
| 15edo | 19edo | 34edo | 2.3.5 POTE tuning | JI intervals represented | |
|---|---|---|---|---|---|
| generator (g) | 4\15, 320.00 | 5\19, 315.79 | 7\26, 317.65 | 317.01 | 6/5 |
| L (octave - 3g) | 3\15, 240.00 | 4\19, 252.63 | 7\34, 247.06 | 248.98 | 15/13, 23/20 |
| s (4g - octave) | 1\15, 80.00 | 1\19, 63.16 | 2\34, 70.59 | 68.03 | 25/24 |
Intervals
| Generators | Notation (1/1 = J) | Heptatonic interval category name | Generators | Notation of 2/1 inverse | Heptatonic 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 also has the following intervals (from some root): | |||||
| 7 | J@ | diminished octave | -7 | J& | augmented unison |
| 8 | L@ | diminished smithird | -8 | O& | augmented smisixth |
| 9 | N@ | diminished smififth | -9 | M& | augmented smifourth |
| 10 | P@ | diminished smiseventh | -10 | K& | augmented smisecond |
Modes
Pseudo-diatonic theory
Orgone
Sixix
Primodal theory
Primodal chords
Nejis
Rank-2 temperaments
Myna (27&31)
Kleismic (19&15, 2.3.5.7)
Orgone (15&11, 2.7.11)
Sixix (18&25)
Sixix can be viewed as a dual-fifth temperament, i.e. a temperament on the 2.3+.3-.5 "subgroup" (3+ = sharp 3, 3- = flat 3):
- It has both a sharp fifth and a flat fifth but no near-just 3/2.
- Combining the sharp fifth and the flat fifth yields a good approximation of 9/8; two 9/8's make a 5/4, so it tempers out 81/80 in the underlying 2.9.5 subgroup.
- The chroma of sixix[7] is the difference between the sharp fifth and the flat fifth, and functions much like a(n untempered) comma in sixix harmony, giving two slightly different flavors of fifths, minor thirds, major thirds, etc, much like in porcupine harmony. Tempering out this comma leads to 7edo.