4L 5s: Difference between revisions

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{{Infobox MOS
{{Infobox MOS
| Name = gramitonic
| Name = orwelloid
| Periods = 1
| Periods = 1
| nLargeSteps = 4
| nLargeSteps = 4
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'''4L 5s''' refers to the structure of [[MOS scales]] whose generator falls between 2\9 (two degrees of [[9edo|9edo]] = approx. 266.667¢) and 1\4 (one degree of [[4edo|4edo]] = 300¢).
'''4L 5s''' refers to the structure of [[MOS scales]] whose generator falls between 2\9 (two degrees of [[9edo|9edo]] = approx. 266.667¢) and 1\4 (one degree of [[4edo|4edo]] = 300¢).
== Names ==
== Names ==
The [[TAMNAMS]] name for this pattern is '''gramitonic''' (from ''grave minor third'').
The [[TAMNAMS]] name for this pattern is '''orwelloid''' (named after the abstract temperament [[orwell]]).
== Notation ==
== Notation ==


Line 20: Line 20:


== Intervals ==
== 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.
Note: In TAMNAMS, a k-step interval class in orwelloid 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 ==
== Tuning ranges ==
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=== Parasoft ===
=== 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¢.
Parasoft tunings of orwelloid 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]].
In parasoft orwelloid, 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]].
Parasoft orwelloid 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).
* [[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]].
* [[31edo]] can be used for its nearly pure [[5/4]].
* [[53edo]] can be used for its nearly pure [[3/2]] and good [[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.  
The sizes of the generator, large step and small step of orwelloid are as follows in various parasoft orwelloid tunings.  
{| class="wikitable right-2 right-3 right-4 right-5 right-6 right-7"
{| class="wikitable right-2 right-3 right-4 right-5 right-6 right-7"
|-
|-
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[[Category:MOS Scales]]
[[Category:MOS Scales]]
[[Category:9-tone scales]]
[[Category:9-tone scales]]
[[Category:Gramitonic]] <!-- main article -->
[[Category:Orwelloid]] <!-- main article -->

Revision as of 15:40, 14 August 2022

↖ 3L 4s ↑ 4L 4s 5L 4s ↗
← 3L 5s 4L 5s 5L 5s →
↙ 3L 6s ↓ 4L 6s 5L 6s ↘ 
┌╥┬╥┬╥┬╥┬┬┐
│║│║│║│║│││
│││││││││││
└┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLsLsLss
ssLsLsLsL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 2\9 to 1\4 (266.7 ¢ to 300.0 ¢)
Dark 3\4 to 7\9 (900.0 ¢ to 933.3 ¢)
TAMNAMS information
Name gramitonic
Prefix gram-
Abbrev. gm
Related MOS scales
Parent 4L 1s
Sister 5L 4s
Daughters 9L 4s, 4L 9s
Neutralized 8L 1s
2-Flought 13L 5s, 4L 14s
Equal tunings
Equalized (L:s = 1:1) 2\9 (266.7 ¢)
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 ¢)
Collapsed (L:s = 1:0) 1\4 (300.0 ¢)

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 orwelloid (named after the abstract temperament orwell).

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&/K@ K/L@ L/K& L&/M@ M/N@ N/M& N&/O@ O/P@ P/O& P&/Q@ Q/R@ R/Q&/J@ J

Intervals

Note: In TAMNAMS, a k-step interval class in orwelloid 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 orwelloid 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 orwelloid, 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 orwelloid 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 orwelloid are as follows in various parasoft orwelloid 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
7\30 280.000 5 2 2.500
18\77 280.519 13 5 2.600 Unnamed golden tuning
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
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.

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