4L 5s
- For the tritave-equivalent 4L 5s pattern, see 4L 5s (tritave-equivalent).
| Brightest mode | LsLsLsLss | |
| Period | 2/1 | |
| Range for bright generator | 2\9 (266.7¢) to 1\4 (300.0¢) | |
| Range for dark generator | 3\4 (900.0¢) to 7\9 (933.3¢) | |
| Parent MOS | 4L 1s | |
| Daughter MOSes | 9L 4s, 4L 9s | |
| Sister MOS | 5L 4s | |
| TAMNAMS name | gramitonic | |
| Equal tunings | ||
| Supersoft (L:s = 4:3) | 7\31 (271.0¢) | |
| Soft (L:s = 3:2) | 5\22 (272.7¢) | |
| Semisoft (L:s = 5:3) | 8\35 (274.3¢) | |
| Basic (L:s = 2:1) | 3\13 (276.9¢) | |
| Semihard (L:s = 5:2) | 7\30 (280.0¢) | |
| Hard (L:s = 3:1) | 4\17 (282.4¢) | |
| Superhard (L:s = 4:1) | 5\21 (285.7¢) | |
4L 5s refers to the structure of MOS scales whose generator falls between 2\9 (two degrees of 9edo = approx. 266.667¢) and 1\4 (one degree of 4edo = 300¢).
Names
The TAMNAMS name for this pattern is gramitonic (from grave minor third).
Notation
The notation used in this article is LsLsLsLss = JKLMNOPQRJ 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 13edo gamut is as follows:
J/R& J&/[email protected] K/[email protected] L/K& L&/[email protected] M/[email protected] N/M& N&/[email protected] O/[email protected] P/O& P&/[email protected] Q/[email protected] R/Q&/[email protected] J
Intervals
Note: In TAMNAMS, a k-step interval class in 4L 5s may be called a "k-step", "k-mosstep", or "k-orstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
Tuning ranges
Parasoft
Parasoft tunings of 4L 5s have a step ratio between 4/3 and 3/2, implying a generator sharper than 7\31 = 270.97¢ and flatter than 5\22 = 272.73¢.
In parasoft 4L 5s, the generator (major mosthird) is an approximate 7/6, the major mosfifth is an approximate but rather flat 11/8, the minor mosfourth is an approximate 5/4, and the major mossixth is an approximate 3/2.
Parasoft 4L 5s EDOs include 22edo, 31edo, 53edo, and 84edo.
- 22edo can be used to make large and small steps more distinct (the step ratio is 3/2).
- 31edo can be used for its nearly pure 5/4.
- 53edo can be used for its nearly pure 3/2 and good 5/4.
The sizes of the generator, large step and small step of 4L 5s are as follows in various parasoft 4L 5s tunings.
| 22edo | 31edo | 53edo | 84edo | JI intervals represented | |
|---|---|---|---|---|---|
| generator (g) | 5\22, 272.73 | 7\31, 270.97 | 12\53, 271.70 | 19\84, 271.43 | 7/6 |
| L (5g - octave) | 3\22, 163.64 | 4\31, 154.84 | 7\53, 158.49 | 11\84, 157.14 | 12/11, 11/10 |
| s (octave - 4g) | 2\22, 109.09 | 3\31, 116.13 | 5\53, 113.21 | 8\84, 114.29 | 16/15, 15/14 |
This set of JI interpretations (g = 7/6, 2g = 11/8, 3g = 8/5, 7g = 3/2) is called 11-limit orwell temperament in regular temperament theory.
Scale tree
In the case of 9edo, L and s are the same size; in the case of 4edo, s is so small it disappears. The spectrum, then, goes something like:
| Generator | Cents | L | s | L/s | Comments | |||||
|---|---|---|---|---|---|---|---|---|---|---|
| 2\9 | 266.667 | 1 | 1 | 1.000 | ||||||
| 11\49 | 269.388 | 6 | 5 | 1.200 | ||||||
| 9\40 | 270.000 | 5 | 4 | 1.250 | ||||||
| 16\71 | 270.423 | 9 | 7 | 1.286 | ||||||
| 7\31 | 270.968 | 4 | 3 | 1.333 | ||||||
| 19\84 | 271.429 | 11 | 8 | 1.375 | Orwell is in this region | |||||
| 12\53 | 271.698 | 7 | 5 | 1.400 | ||||||
| 17\75 | 272.000 | 10 | 7 | 1.428 | ||||||
| 5\22 | 272.727 | 3 | 2 | 1.500 | L/s = 3/2 | |||||
| 18\79 | 273.418 | 11 | 7 | 1.571 | ||||||
| 13\57 | 273.684 | 8 | 5 | 1.600 | ||||||
| 21\92 | 273.913 | 13 | 8 | 1.625 | Unnamed golden tuning | |||||
| 8\35 | 274.286 | 5 | 3 | 1.667 | ||||||
| 19\83 | 274.699 | 12 | 7 | 1.714 | ||||||
| 11\48 | 275.000 | 7 | 4 | 1.750 | ||||||
| 14\61 | 275.410 | 9 | 5 | 1.800 | ||||||
| 3\13 | 276.923 | 2 | 1 | 2.000 | Basic orwelloid (Generators smaller than this are proper) | |||||
| 13\56 | 278.571 | 9 | 4 | 2.250 | ||||||
| 10\43 | 279.070 | 7 | 3 | 2.333 | ||||||
| 17\73 | 279.452 | 12 | 5 | 2.400 | Lovecraft is around here | |||||
| 7\30 | 280.000 | 5 | 2 | 2.500 | ||||||
| 18\77 | 280.519 | 13 | 5 | 2.600 | Golden Lovecraft | |||||
| 11\47 | 280.851 | 8 | 3 | 2.667 | ||||||
| 15\64 | 281.250 | 11 | 4 | 2.750 | ||||||
| 4\17 | 282.353 | 3 | 1 | 3.000 | L/s = 3/1 | |||||
| 13\55 | 283.636 | 10 | 3 | 3.333 | ||||||
| 9\38 | 284.211 | 7 | 2 | 3.500 | ||||||
| 14\59 | 284.746 | 11 | 3 | 3.667 | ||||||
| 5\21 | 285.714 | 4 | 1 | 4.000 | ||||||
| 11\46 | 286.957 | 9 | 2 | 4.500 | ||||||
| 6\25 | 288.000 | 5 | 1 | 5.000 | ||||||
| 7\29 | 289.655 | 6 | 1 | 6.000 | Gariberttet | |||||
| 1\4 | 300.000 | 1 | 0 | → inf | ||||||
Note that between 7\31 and 5\22, g approximates frequency ratio 7:6, 2g approximates 11:8, and 3g approximates 8:5. This defines the range of Orwell Temperament, which is the only notable harmonic entropy minimum with this MOS pattern. 4L 5s scales outside of that range are not suitable for Orwell.