4L 5s: Difference between revisions

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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:
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:


{| class="wikitable"
{| class="wikitable center-all"
! colspan="6" | Generator
! Cents
! L
! s
! L/s
! Comments
|-
|-
! colspan="5" | Generator
| 2\9 || || || || || || 266.667 || 1 || 1 || 1.000 ||
! | Scale
! | Generator in cents
! | Comments
|-
|-
| | 2\9
| || || || || || 11\49 || 269.388 || 6 || 5 || 1.200 ||
| |  
| |  
| |  
| |  
| style="text-align:center;" | 1 1 1 1 1 1 1 1 1
| | 266.667
| style="text-align:center;" |  
|-
|-
| |  
| || || || || 9\40 || || 270.000 || 5 || 4 || 1.250 ||
| |  
| |  
| |
| | 9\40
| style="text-align:center;" | 4 5 4 5 4 5 4 5 4
| | 270
| |  
|-
|-
| |  
| || || || || || 16\71 || 270.423 || 9 || 7 || 1.286 ||
| |  
| |  
| | 7\31
| |  
| style="text-align:center;" | 3 4 3 4 3 4 3 4 3
| | 270.968
| style="text-align:center;" |  
|-
|-
| |  
| || || || 7\31 || || || 270.968 || 4 || 3 || 1.333 ||
| |  
| |  
| |  
| | 12\53
| style="text-align:center;" | 5 7 5 7 5 7 5 7 5
| | 271.698
| style="text-align:center;" | <!--Orwell is around here-->
|-
|-
| |  
| || || || || || 19\84 || 271.429 || 11 || 8 || 1.375 || Orwell is in this region
| |  
| | 5\22
| |  
| |  
| style="text-align:center;" | 2 3 2 3 2 3 2 3 2
| | 272.727
| style="text-align:center;" | <!--Optimum rank range (L/s=3/2) orwell-->
|-
|-
| |  
| || || || || 12\53 || || 271.698 || 7 || 5 || 1.400 ||
| |  
| |  
| |  
| | 13\57
| style="text-align:center;" | 5 8 5 8 5 8 5 8 5
| | 273.684
| style="text-align:center;" | <!--Golden orwell (bad tuning)-->
|-
|-
| |  
| || || || || || 17\75 || 272.000 || 10 || 7 || 1.428 ||
| |  
| |  
| | 8\35
| |  
| style="text-align:center;" | 3 5 3 5 3 5 3 5 3
| | 274.286
| style="text-align:center;" |  
|-
|-
| |  
| || || 5\22 || || || || 272.727 || 3 || 2 || 1.500 || L/s = 3/2
| |  
| |  
| |  
| | 11\48
| style="text-align:center;" | 4 7 4 7 4 7 4 7 4
| | 275
| |  
|-
|-
| |  
| || || || || || 18\79 || 273.418 || 11 || 7 || 1.571 ||
| | 3\13
|-
| |  
| || || || || 13\57 || || 273.684 || 8 || 5 || 1.600 ||
| |  
|-
| |  
| || || || || || 21\92 || 273.913 || 13 || 8 || 1.625 || Unnamed golden tuning
| style="text-align:center;" | 1 2 1 2 1 2 1 2 1
|-
| | 276.923
| || || || 8\35 || || || 274.286 || 5 || 3 || 1.667 ||
| style="text-align:center;" | Boundary of propriety:
|-
 
| || || || || || 19\83 || 274.699 || 12 || 7 || 1.714 ||
generators smaller than this are proper
|-
| || || || || 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<br>(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 ||
| |  
| |  
| |  
| | 10\43
| style="text-align:center;" | 3 7 3 7 3 7 3 7 3
| | 279.07
| |  
|-
|-
| |  
| || || || || 9\38 || || 284.211 || 7 || 2 || 3.500 ||
| |  
| |  
| | 7\30
| |  
| style="text-align:center;" | 2 5 2 5 2 5 2 5 2
| | 280.000
| style="text-align:center;" |  
|-
|-
| |  
| || || || || || 14\59 || 284.746 || 11 || 3 || 3.667 ||
| |  
| |  
| |  
| | 11\47
| style="text-align:center;" | 3 8 3 8 3 8 3 8 3
| | 280.851
| |  
|-
|-
| |  
| || || || 5\21 || || || 285.714 || 4 || 1 || 4.000 ||
| |  
| | 4\17
| |  
| |  
| style="text-align:center;" | 1 3 1 3 1 3 1 3 1
| | 282.353
| style="text-align:center;" | L/s = 3
|-
|-
| |  
| || || || || || 11\46 || 286.957 || 9 || 2 || 4.500 ||
| |  
| |  
| |  
| | 9\38
| style="text-align:center;" | 2 7 2 <span style="font-size: 12.8000001907349px; line-height: 1.5;">7 </span><span style="font-size: 13px; line-height: 1.5;">2 7 2 7 2 </span>
| | 284.2105
| |  
|-
|-
| |  
| || || || || 6\25 || || 288.000 || 5 || 1 || 5.000 ||
| |  
| |  
| | 5\21
| |  
| style="text-align:center;" | 1 4 1 4 1 4 1 4 1
| | 285.714
| style="text-align:center;" | L/s = 4
|-
|-
| |  
| || || || || || 7\29 || 289.655 || 6 || 1 || 6.000 ||
| |  
| |  
| |  
| | 6\25
| style="text-align:center;" | 1 5 1 5 1 5 1 5 1
| | 288
| |  
|-
|-
| | 1\4
| 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.-->
| |  
| style="text-align:center;" | 0 1 0 1 0 1 0 1 0
| | 300.000
| style="text-align:center;" |  
|}<!--


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.-->
[[Category:Scales]]
[[Category:MOS Scales]]
[[Category:9-tone scales]]
[[Category:Abstract MOS patterns]]
[[Category:Abstract MOS patterns]]
[[Category:scales]]
[[Category:Orwelloid]] <!-- main article -->

Revision as of 09:35, 12 April 2021

User:IlL/Template:RTT restriction

↖ 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 or orwelloid (named after the abstract temperament orwell) 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¢).

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

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

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
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