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4L 5s: Difference between revisions

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4L 5s refers to the structure of [[MOSScales|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¢). 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:
: ''For the tritave-equivalent 4L 5s pattern, see [[4L 5s (3/1-equivalent)]].''


{| class="wikitable"
{{Infobox MOS
|-
| Name = gramitonic
! colspan="5" | Generator
| Periods = 1
! | Scale
| nLargeSteps = 4
! | Generator in cents
| nSmallSteps = 5
! | Comments
| Equalized = 2
|-
| Collapsed = 1
| | 2\9
| Pattern = LsLsLsLss
| |
}}
| |
{{MOS intro}}
| |
 
| |
== Names ==
| style="text-align:center;" | 1 1 1 1 1 1 1 1 1
The [[TAMNAMS]] name for this pattern is '''gramitonic''' (from ''grave minor third'').
| | 266.667
 
| style="text-align:center;" |
== Scale properties ==
|-
{{TAMNAMS use}}
| |
 
| |
=== Intervals ===
| |
{{MOS intervals}}
| |
 
| | 9\40
=== Generator chain ===
| style="text-align:center;" | 4 5 4 5 4 5 4 5 4
{{MOS genchain}}
| | 270
 
| |
=== Modes ===
|-
{{MOS mode degrees}}
| |
 
| |
==== 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'.
| | 7\31
{{MOS modes
| |
| Mode Names=
| style="text-align:center;" | 3 4 3 4 3 4 3 4 3
Roi $
| | 270.968
Steno $
| style="text-align:center;" |
Limni $
|-
Telma $
| |
Krini $
| |
Elos $
| |
Mychos $
| |
Akti $
| | 12\53
Dini $
| style="text-align:center;" | 5 7 5 7 5 7 5 7 5
}}
| | 271.698
 
| style="text-align:center;" | Orwell is around here
== 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 ==
| | 5\22
=== 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}}}}.
| |
 
| style="text-align:center;" | 2 3 2 3 2 3 2 3 2
Parasoft 4L 5s edos include [[22edo]], [[31edo]], [[53edo]], and [[84edo]].
| | 272.727
* [[22edo]] can be used to make large and small steps more distinct (the step ratio is 3/2).
| style="text-align:center;" | Optimum rank range (L/s=3/2) orwell
* [[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.
| |
| |
| |
| style="text-align:center;" | <span style="display: block; text-align: center;">2 pi 2 pi 2 pi 2 pi 2</span>
| | 273.412
| |
|-
| |
| |
| |
| |
| | 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)
|-
| |
| |
| |
| | 8\35
| |
| style="text-align:center;" | 3 5 3 5 3 5 3 5 3
| | 274.286
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | <span style="background-color: #ffffff;">1 √3 1 √3 1 √3 1 √3 1</span>
| | 274.85
| |
|-
| |
| |
| |
| |
| | 11\48
| style="text-align:center;" | 4 7 4 7 4 7 4 7 4
| | 275
| |
|-
| |
| | 3\13
| |
| |
| |
| style="text-align:center;" | 1 2 1 2 1 2 1 2 1
| | 276.923
| style="text-align:center;" | Boundary of propriety:


generators smaller than this are proper
{| class="wikitable right-2 right-3 right-4 right-5 right-6 right-7"
|-
| |
| |
| |
| |
| | 10\43
| style="text-align:center;" | 3 7 3 7 3 7 3 7 3
| | 279.07
| |
|-
|-
| |
!
| |
! [[22edo]]
| |
! [[31edo]]
| | 7\30
! [[53edo]]
| |
! [[84edo]]
| style="text-align:center;" | 2 5 2 5 2 5 2 5 2
! JI intervals represented
| | 280.000
| style="text-align:center;" |
|-
|-
| |
| generator (g)
| |
| 5\22, 272.73
| |
| 7\31, 270.97
| |
| 12\53, 271.70
| | 11\47
| 19\84, 271.43
| style="text-align:center;" | 3 8 3 8 3 8 3 8 3
| [[7/6]]
| | 280.851
| |
|-
|-
| |
| L (5g − octave)
| |
| 3\22, 163.64
| |
| 4\31, 154.84
| |
| 7\53, 158.49
| |
| 11\84, 157.14
| style="text-align:center;" | 1 e 1 e 1 e 1 e 1
| [[12/11]], [[11/10]]
| | 281.100
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = e</span>
|-
|-
| |
| s (octave − 4g)
| |
| 2\22, 109.09
| | 4\17
| 3\31, 116.13
| |
| 5\53, 113.21
| |
| 8\84, 114.29
| style="text-align:center;" | 1 3 1 3 1 3 1 3 1
| [[16/15]], [[15/14]]
| | 282.353
| style="text-align:center;" | L/s = 3
|-
| |
| |
| |
| |
| |
| style="text-align:center;" | 1 pi 1 pi 1 pi 1 pi 1
| | 282.922
| style="text-align:center;" | <span style="display: block; text-align: center;">L/s = pi</span>
|-
| |
| |
| |
| |
| | 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
| |
|-
| |
| |
| |
| | 5\21
| |
| style="text-align:center;" | 1 4 1 4 1 4 1 4 1
| | 285.714
| style="text-align:center;" | L/s = 4
|-
| |
| |
| |
| |
| | 6\25
| style="text-align:center;" | 1 5 1 5 1 5 1 5 1
| | 288
| |
|-
| | 1\4
| |
| |
| |
| |
| 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.
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.
[[Category:mos]]
 
[[Category:scales]]
== Scales ==
* [[Guanyintet9]] – [[311edo|70\311]] tuning
* [[Orwell9]] – [[84edo|19\84]] tuning
* [[Lovecraft9]] – [[116edo|27\116]] tuning
 
== 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]]&nbsp;↓
}}
 
[[Category:Gramitonic]] <!-- main article -->
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