Xenharmonic Wiki

4L 5s: Difference between revisions

← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled
15,004 edits
Undo revision 94244 by Inthar (talk)
Tag: Undo
ArrowHead294 (talk | contribs)
14,512 edits
 
(32 intermediate revisions by 9 users not shown)
Line 1: Line 1:
: ''For the tritave-equivalent 4L 5s pattern, see [[4L 5s (3/1-equivalent)]].''
{{Infobox MOS
{{Infobox MOS
| Name = gramitonic
| Name = gramitonic
Line 5: Line 7:
| nSmallSteps = 5
| nSmallSteps = 5
| Equalized = 2
| Equalized = 2
| Paucitonic = 1
| Collapsed = 1
| Pattern = LsLsLsLss
| Pattern = LsLsLsLss
}}
}}
'''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¢).
{{MOS intro}}
 
== Names ==
== Names ==
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 ==


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".)
== Scale properties ==
{{TAMNAMS use}}
 
=== Intervals ===
{{MOS intervals}}
 
=== Generator chain ===
{{MOS genchain}}


Thus the 13edo gamut is as follows:
=== Modes ===
{{MOS mode degrees}}


'''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'''
==== Proposed names ====
[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 $
}}


== Intervals ==
== Theory ==
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.
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 52: 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 59: 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 67: 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.
 
== Scales ==
* [[Guanyintet9]] – [[311edo|70\311]] tuning
* [[Orwell9]] – [[84edo|19\84]] tuning
* [[Lovecraft9]] – [[116edo|27\116]] tuning


== Scale tree ==
== Scale tree ==
{{MOS tuning spectrum
| 6/5 = Lower range of [[Orwell]]
| 5/3 = Upper range of Orwell
| 13/8 = Unnamed golden tuning
| 12/5 = [[Lovecraft]]
| 13/5 = Golden lovecraft
| 6/1 = [[Gariberttet]]/[[Quasitemp]]/[[Kleiboh]] ↓
}}


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 center-all"
! colspan="6" | 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<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 ||
|-
| || || || || 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.
[[Category:MOS Scales]]
[[Category:9-tone scales]]
[[Category:Gramitonic]] <!-- main article -->
[[Category:Gramitonic]] <!-- main article -->
Retrieved from "https://en.xen.wiki/w/4L_5s"