4L 5s: Difference between revisions

From Xenharmonic Wiki
Jump to navigation Jump to search
← Older edit
VisualWikitext
CompactStar (talk | contribs)
5,020 edits
40edo is a forgotten orwell tuning
ArrowHead294 (talk | contribs)
14,512 edits
 
(20 intermediate revisions by 6 users not shown)
Line 1: Line 1:
: ''For the tritave-equivalent 4L 5s pattern, see [[4L 5s (3/1-equivalent)]].''
: ''For the tritave-equivalent 4L 5s pattern, see [[4L 5s (3/1-equivalent)]].''


{{Infobox MOS
{{Infobox MOS
Line 15: Line 15:
The [[TAMNAMS]] name for this pattern is '''gramitonic''' (from ''grave minor third'').
The [[TAMNAMS]] name for this pattern is '''gramitonic''' (from ''grave minor third'').


== Notation ==
== Scale properties ==
{{TAMNAMS use}}


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".)
=== Intervals ===
{{MOS intervals}}


Thus the 13edo gamut is as follows:
=== Generator chain ===
{{MOS genchain}}


'''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'''
=== Modes ===
{{MOS mode degrees}}


== Intervals ==
==== Proposed names ====
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.
[http://twitter.com/Lilly__Flores/status/1640779893108805632 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'.
{{MOS modes
| Mode Names=
Roi $
Steno $
Limni $
Telma $
Krini $
Elos $
Mychos $
Akti $
Dini $
}}
 
== 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 ==
== Tuning ranges ==
=== Parasoft ===
=== Parasoft ===
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{{c}}}} and flatter than {{nowrap|5\22 {{=}} 272.73{{c}}}}.


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]] and having a better approximation of [[13/8]] than 22edo.
* [[53edo]] can be used for its nearly pure [[3/2]] and [[5/4]] and having much more accurate approximations of 13-limit intervals than 22edo or 31edo.


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]].
The sizes of the generator, large step and small step of 4L 5s are as follows in various parasoft 4L 5s tunings.


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.
{| 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"
|-
|-
Line 56: Line 72:
| [[7/6]]
| [[7/6]]
|-
|-
| L (5g - octave)
| L (5g − octave)
| 3\22, 163.64
| 3\22, 163.64
| 4\31, 154.84
| 4\31, 154.84
Line 63: Line 79:
| [[12/11]], [[11/10]]
| [[12/11]], [[11/10]]
|-
|-
| s (octave - 4g)
| s (octave − 4g)
| 2\22, 109.09
| 2\22, 109.09
| 3\31, 116.13
| 3\31, 116.13
Line 71: Line 87:
|}
|}


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.
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.


== Modes ==
== Scales ==
The names have been proposed for these Modes of 2/1 by [https://twitter.com/Lilly__Flores/status/1640779893108805632 Lilly Flores].
* [[Guanyintet9]] – [[311edo|70\311]] tuning
 
* [[Orwell9]] – [[84edo|19\84]] tuning
He told us that he assigned the Greek name relating to water because the word Orwell comes from 'a spring situated near a promontory'.
* [[Lovecraft9]] – [[116edo|27\116]] tuning
{{MOS modes|Scale Signature=4L 5s|Mode Names=Roi; Steno; Limni; Telma; Krini; Elos; Mychos; Akti; Dini}}


== Scale tree ==
== Scale tree ==
 
{{MOS tuning spectrum
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:
| 6/5 = Lower range of [[Orwell]]
 
| 5/3 = Upper range of Orwell
{| class="wikitable center-all"
| 13/8 = Unnamed golden tuning
! colspan="6" | Generator
| 12/5 = [[Lovecraft]]
! Cents
| 13/5 = Golden lovecraft
! L
| 6/1 = [[Gariberttet]]/[[Quasitemp]]/[[Kleiboh]] ↓
! 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 gramitonic<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 || [https://en.xen.wiki/w/Chromatic_pairs#Lovecraft 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 9\40 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:Gramitonic]] <!-- main article -->
[[Category:Gramitonic]] <!-- main article -->

Latest revision as of 12:41, 1 June 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 ¢

Generator chain

Generator chain of 4L 5s
Bright gens Scale degree Abbrev.
12 Augmented 6-gramdegree A6gmd
11 Augmented 4-gramdegree A4gmd
10 Augmented 2-gramdegree A2gmd
9 Augmented 0-gramdegree A0gmd
8 Augmented 7-gramdegree A7gmd
7 Major 5-gramdegree M5gmd
6 Major 3-gramdegree M3gmd
5 Major 1-gramdegree M1gmd
4 Major 8-gramdegree M8gmd
3 Major 6-gramdegree M6gmd
2 Major 4-gramdegree M4gmd
1 Perfect 2-gramdegree P2gmd
0 Perfect 0-gramdegree
Perfect 9-gramdegree		 P0gmd
P9gmd
−1 Perfect 7-gramdegree P7gmd
−2 Minor 5-gramdegree m5gmd
−3 Minor 3-gramdegree m3gmd
−4 Minor 1-gramdegree m1gmd
−5 Minor 8-gramdegree m8gmd
−6 Minor 6-gramdegree m6gmd
−7 Minor 4-gramdegree m4gmd
−8 Diminished 2-gramdegree d2gmd
−9 Diminished 9-gramdegree d9gmd
−10 Diminished 7-gramdegree d7gmd
−11 Diminished 5-gramdegree d5gmd
−12 Diminished 3-gramdegree d3gmd

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		 Mode names
8|0 1 LsLsLsLss Roi
7|1 3 LsLsLssLs Steno
6|2 5 LsLssLsLs Limni
5|3 7 LssLsLsLs Telma
4|4 9 sLsLsLsLs Krini
3|5 2 sLsLsLssL Elos
2|6 4 sLsLssLsL Mychos
1|7 6 sLssLsLsL Akti
0|8 8 ssLsLsLsL Dini

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 and having a better approximation of 13/8 than 22edo.
  • 53edo can be used for its nearly pure 3/2 and 5/4 and having much more accurate approximations of 13-limit intervals than 22edo or 31edo.

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.

Scales

Scale tree

Scale tree and tuning spectrum of 4L 5s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
2\9 266.667 933.333 1:1 1.000 Equalized 4L 5s
11\49 269.388 930.612 6:5 1.200 Lower range of Orwell
9\40 270.000 930.000 5:4 1.250
16\71 270.423 929.577 9:7 1.286
7\31 270.968 929.032 4:3 1.333 Supersoft 4L 5s
19\84 271.429 928.571 11:8 1.375
12\53 271.698 928.302 7:5 1.400
17\75 272.000 928.000 10:7 1.429
5\22 272.727 927.273 3:2 1.500 Soft 4L 5s
18\79 273.418 926.582 11:7 1.571
13\57 273.684 926.316 8:5 1.600
21\92 273.913 926.087 13:8 1.625 Unnamed golden tuning
8\35 274.286 925.714 5:3 1.667 Semisoft 4L 5s
Upper range of Orwell
19\83 274.699 925.301 12:7 1.714
11\48 275.000 925.000 7:4 1.750
14\61 275.410 924.590 9:5 1.800
3\13 276.923 923.077 2:1 2.000 Basic 4L 5s
Scales with tunings softer than this are proper
13\56 278.571 921.429 9:4 2.250
10\43 279.070 920.930 7:3 2.333
17\73 279.452 920.548 12:5 2.400 Lovecraft
7\30 280.000 920.000 5:2 2.500 Semihard 4L 5s
18\77 280.519 919.481 13:5 2.600 Golden lovecraft
11\47 280.851 919.149 8:3 2.667
15\64 281.250 918.750 11:4 2.750
4\17 282.353 917.647 3:1 3.000 Hard 4L 5s
13\55 283.636 916.364 10:3 3.333
9\38 284.211 915.789 7:2 3.500
14\59 284.746 915.254 11:3 3.667
5\21 285.714 914.286 4:1 4.000 Superhard 4L 5s
11\46 286.957 913.043 9:2 4.500
6\25 288.000 912.000 5:1 5.000
7\29 289.655 910.345 6:1 6.000 Gariberttet/Quasitemp/Kleiboh ↓
1\4 300.000 900.000 1:0 → ∞ Collapsed 4L 5s
Retrieved from "https://en.xen.wiki/index.php?title=4L_5s&oldid=198354"