4L 5s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older editNewer edit →
VisualWikitext
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
m →Tuning ranges: the commas (punctuation) aren't part of the formulas!
Line 35: Line 35:
Parasoft tunings of 4L 5s have a step ratio between 4/3 and 3/2, implying a generator sharper than {{nowrap|7\31 {{=}} 270.97¢}} and flatter than {{nowrap|5\22 {{=}} 272.73¢}}.
Parasoft tunings of 4L 5s have a step ratio between 4/3 and 3/2, implying a generator sharper than {{nowrap|7\31 {{=}} 270.97¢}} and flatter than {{nowrap|5\22 {{=}} 272.73¢}}.


Parasoft 4L 5s EDOs include [[22edo]], [[31edo]], [[53edo]], and [[84edo]].
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).
* [[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]].
Line 72: Line 72:
|}
|}


This set of JI interpretations ({{nowrap|g {{=}} 7/6|2g → 11/8|3g → 8/5|7g → 3/2}}) is called 11-limit [[orwell]] temperament in regular temperament theory.
This set of JI interpretations ({{nowrap|g → 7/6}}, {{nowrap|2g → 11/8}}, {{nowrap|3g → 8/5}}, {{nowrap|7g → 3/2}}) is called 11-limit [[orwell]] temperament in regular temperament theory.


== Scale tree ==
== Scale tree ==

Revision as of 17:07, 10 January 2025

For the tritave-equivalent 4L 5s pattern, see 4L 5s (3/1-equivalent).
↖ 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, named gramitonic in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 4 large steps and 5 small steps, repeating every octave. Generators that produce this scale range from 266.7 ¢ to 300 ¢, or from 900 ¢ to 933.3 ¢.

Names

The TAMNAMS name for this pattern is gramitonic (from grave minor third).

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 4L 5s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-gramstep Perfect 0-gramstep P0gms 0 0.0 ¢
1-gramstep Minor 1-gramstep m1gms s 0.0 ¢ to 133.3 ¢
Major 1-gramstep M1gms L 133.3 ¢ to 300.0 ¢
2-gramstep Diminished 2-gramstep d2gms 2s 0.0 ¢ to 266.7 ¢
Perfect 2-gramstep P2gms L + s 266.7 ¢ to 300.0 ¢
3-gramstep Minor 3-gramstep m3gms L + 2s 300.0 ¢ to 400.0 ¢
Major 3-gramstep M3gms 2L + s 400.0 ¢ to 600.0 ¢
4-gramstep Minor 4-gramstep m4gms L + 3s 300.0 ¢ to 533.3 ¢
Major 4-gramstep M4gms 2L + 2s 533.3 ¢ to 600.0 ¢
5-gramstep Minor 5-gramstep m5gms 2L + 3s 600.0 ¢ to 666.7 ¢
Major 5-gramstep M5gms 3L + 2s 666.7 ¢ to 900.0 ¢
6-gramstep Minor 6-gramstep m6gms 2L + 4s 600.0 ¢ to 800.0 ¢
Major 6-gramstep M6gms 3L + 3s 800.0 ¢ to 900.0 ¢
7-gramstep Perfect 7-gramstep P7gms 3L + 4s 900.0 ¢ to 933.3 ¢
Augmented 7-gramstep A7gms 4L + 3s 933.3 ¢ to 1200.0 ¢
8-gramstep Minor 8-gramstep m8gms 3L + 5s 900.0 ¢ to 1066.7 ¢
Major 8-gramstep M8gms 4L + 4s 1066.7 ¢ to 1200.0 ¢
9-gramstep Perfect 9-gramstep P9gms 4L + 5s 1200.0 ¢

Modes

Scale degrees of the modes of 4L 5s
UDP Cyclic
order 		Step
pattern 		Scale degree (gramdegree)
0 1 2 3 4 5 6 7 8 9
8|0 1 LsLsLsLss Perf. Maj. Perf. Maj. Maj. Maj. Maj. Aug. Maj. Perf.
7|1 3 LsLsLssLs Perf. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Perf.
6|2 5 LsLssLsLs Perf. Maj. Perf. Maj. Maj. Min. Maj. Perf. Maj. Perf.
5|3 7 LssLsLsLs Perf. Maj. Perf. Min. Maj. Min. Maj. Perf. Maj. Perf.
4|4 9 sLsLsLsLs Perf. Min. Perf. Min. Maj. Min. Maj. Perf. Maj. Perf.
3|5 2 sLsLsLssL Perf. Min. Perf. Min. Maj. Min. Maj. Perf. Min. Perf.
2|6 4 sLsLssLsL Perf. Min. Perf. Min. Maj. Min. Min. Perf. Min. Perf.
1|7 6 sLssLsLsL Perf. Min. Perf. Min. Min. Min. Min. Perf. Min. Perf.
0|8 8 ssLsLsLsL Perf. Min. Dim. Min. Min. Min. Min. Perf. Min. Perf.

Proposed names

Lilly Flores proposed using the Greek name relating to water as mode names. The names are in reference to the scale's former name orwelloid because the word Orwell comes from 'a spring situated near a promontory'.

Modes of 4L 5s
UDP Cyclic
order		 Step
pattern
8|0 1 LsLsLsLss
7|1 3 LsLsLssLs
6|2 5 LsLssLsLs
5|3 7 LssLsLsLs
4|4 9 sLsLsLsLs
3|5 2 sLsLsLssL
2|6 4 sLsLssLsL
1|7 6 sLssLsLsL
0|8 8 ssLsLsLsL

Theory

The only low harmonic entropy minimum corresponds to orwell temperament, where 1 generator approximates 7/6, 2 generators approximate 11/8, and 3 generators approximate 8/5.

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¢.

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

Template:Scale tree

Retrieved from "https://en.xen.wiki/index.php?title=4L_5s&oldid=175593"