Xenharmonic Wiki

5L 4s: Difference between revisions

← Older edit
VisualWikitext
Inthar (talk | contribs)
autopatrolled
15,004 edits
ArrowHead294 (talk | contribs)
14,512 edits
mNo edit summary
 
(148 intermediate revisions by 9 users not shown)
Line 1: Line 1:
'''5L 4s''' refers to the structure of [[MOS]] scales with generators ranging from 1\5 (one degree of [[5edo]] = 240¢) to 2\9 (two degrees of [[9edo]] = 266.7¢). In the case of 9edo, L and s are the same size; in the case of 5edo, s becomes so small it disappears (and all that remains are the five equal L's).
{{Infobox MOS}}
{{MOS intro}} It is also equal to a degenerate form of [[diasem]].


The familiar harmonic entropy minimum with this MOS pattern is [[Meantone_family#Godzilla|godzilla]], in which a generator is [[8/7|8/7]] or [[7/6|7/6]] (tempered to be the same interval, or even 37/32 if you like) so two of them make a [[4/3|4/3]]. However, in addition to godzilla (tempering out 81/80) and the 2.3.7 temperament [[Chromatic_pairs#semaphore|semaphore]], there is also a weird scale called "[[Pseudo-semaphore|pseudo-semaphore]]", in which two different flavors of [[3/2|3/2]] exist in the same scale: an octave minus two generators makes a sharp 3/2, and two octaves minus seven generators makes a flat 3/2.
== Names ==
== Scale tree ==
The [[TAMNAMS]] convention, used by this article, uses '''semiquartal''' (derived from 'half a fourth') for the 5L 4s pattern. Another attested name is '''hemifourths'''.
{| class="wikitable"
 
== Scale properties ==
{{TAMNAMS use}}
 
=== Intervals ===
{{MOS intervals}}
 
=== Generator chain ===
{{MOS genchain}}
 
=== Modes ===
{{MOS mode degrees}}
 
Note that the darkest two modes have no diatonic or [[armotonic]] fifth on the root in nonextreme semiquartal tunings.
 
== Theory ==
The harmonic entropy minimum with this MOS pattern is [[godzilla]], in which the generator tempers [[8/7]] or [[7/6]] to be the same interval, and two generators is [[4/3]]. However, in addition to godzilla (tempering out 81/80) and the 2.3.7 temperament [[semaphore]], there is also a weird scale called "[[pseudo-semaphore]]", in which two different flavors of [[3/2]] exist in the same scale: an octave minus two generators makes a sharp 3/2, and two octaves minus seven generators makes a flat 3/2. The 2.3.13/5 [[barbados]] temperament is another possible interpretation.
 
== Tuning ranges ==
=== Hard-of-basic ===
Hard-of-basic tunings have [[semifourth]]s as generators, between 1\5 (240{{c}}) and 3\14 (257.14{{c}}), where two of them create a diatonic 4th. The generator could be viewed as a 15/13, and the resulting "inframinor" and "ultramajor" chords and triads could be viewed as approximating, respectively, 26:30:39 and 10:13:15 (see [[Arto and tendo theory]]).
 
==== Hypohard ====
The sizes of the generator, large step and small step of 5L 4s are as follows in various hypohard ({{nowrap|2/1 ≤ L/s ≤ 3/1}}) tunings.
{| class="wikitable right-2 right-3 right-4 right-5 right-6"
|-
|-
! colspan="11" | Generator
!  
! | Cents
! [[14edo]] ({{nowrap|L/s {{=}} 2/1}})
! | Comments
! [[47edo]] ({{nowrap|L/s {{=}} 7/3}})
! [[33edo]] ({{nowrap|L/s {{=}} 5/2}})
! [[52edo]] ({{nowrap|L/s {{=}} 8/3}})
! [[19edo]] ({{nowrap|L/s {{=}} 3/1}})
|-
|-
| | 1\5
| Generator (g)
| |
| 3\14, 257.14
| |
| 10\47, 255.32
| |
| 7\33, 254.54
| |
| 11\52, 253.85
| |
| 4\19, 252.63
| |
| |
| |
| |
| |
| | 240
| style="text-align:center;" |
|-
|-
| |  
| L ({{nowrap|octave − 4g}})
| |
| 171.43
| |
| 178.72
| |
| 181.81
| |
| 184.62
| |
| 189.47
| |
| |
| |
| |
| | 12\59
| | 244.068
| style="text-align:center;" | Pseudo-semaphore is around here
|-
|-
| |  
| s ({{nowrap|5g − octave}})
| |  
| 85.71
| |  
| 76.60
| |  
| 72.73
| |  
| 69.23
| |
| 63.16
| |
|}
| |
 
| |
This range is notable for having many simple tunings that are close to being "eigentunings" (tunings that tune a certain JI interval exactly):
| | 11\54
* 33edo semiquartal has close 7/5 (error −0.69{{c}}), 9/5 (error −0.59{{c}}) and 9/7 (error +1.28{{c}}), thus can be used for the close 5:7:9 in the two Locrian-like modes 1|7 and 0|8
| |
* 52edo semiquartal has close 22/19 (error +0.04{{c}})
| | 244.444
* 19edo semiquartal has close 6/5 (error +0.15{{c}}) and 28/27 (error +0.20{{c}})
| style="text-align:center;" |
However, for the more complex intervals such as 22/19 and 28/27, you might want to use the exact eigentuning for the full effect, unless you specifically need an edo for modulatory purposes.
 
==== Parahard and ultrahard ====
One important sub-range is given by stipulating that two semifourth generators must make a ''meantone'' fourth; i.e. that four fifths should approximate a [[5/4]] major third. This can be considered the [[19edo]] (4\19)-to-[[24edo]] (5\24) range, i.e. parahard semiquartal, which also contains [[43edo]] (9\43) and [[62edo]] (13\62). Parahard semiquartal can be given an RTT interpretation known as [[godzilla]].
 
The sizes of the generator, large step and small step of 5L 4s are as follows in various hypohard ({{nowrap|2/1 ≤ L/s ≤ 3/1}}) tunings.
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
| |
!
| |
! [[19edo]]
| |
! [[24edo]]
| |
! [[29edo]]
| |
| |
| |
| |
| | 10\49
| |
| |
| | 244.898
| style="text-align:center;" |
|-
|-
| |
| Generator (g)
| |
| 4\19, 252.63
| |
| 5\24, 250.00
| |
| 6\29, 248.28
| |
| |
| |
| | 9\44
| |
| |
| |
| | 245.455
| style="text-align:center;" |
|-
|-
| |  
| L ({{nowrap|octave − 4g}})
| |
| 189.47
| |
| 200.00
| |
| 206.90
| |
| |
| | 8\39
| |
| |
| |
| |
| | 246.154
| style="text-align:center;" |
|-
|-
| |  
| s ({{nowrap|5g − octave}})
| |
| 63.16
| |
| 50.00
| |
| 41.38
| |
|}
| | 7\34
| |
| |
| |
| |
| |
| | 247.059
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 6\29
| |
| |
| |
| |
| |
| |
| | 248.276
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 11\53
| |
| |
| |
| |
| |
| | 249.057
| style="text-align:center;" | Semaphore is around here
|-
| |
| |
| |
| | 5\24
| |
| |
| |
| |
| |
| |
| |
| | 250
| style="text-align:center;" | L/s = 4
|-
| |
| |
| |
| |
| |
| | 9\43
| |
| |
| |
| |
| |
| | 251.163
| |
|-
| |
| |
| | 4\19
| |
| |
| |
| |
| |
| |
| |
| |
| | 252.632
| style="text-align:center;" | Godzilla is around here


L/s = 3
=== Soft-of-basic ===
|-
Soft-of-basic tunings have semifourths that are between 3\14 (257.14{{c}}) and 2\9 (266.67{{c}}), creating a "[[mavila]]" or "[[superdiatonic]]" 4th. [[23edo]]'s 5\23 (260.87{{c}}) is an example of this generator.
| |
| |
| |
| |
| | 11\52
| |
| |
| |
| |
| |
| |
| | 253.813
| |
|-
| |
| |
| |
| |
| |
| |
| | 29\137
| |
| |
| |
| |
| | 254.015
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| | 76\359
| |
| |
| | 254.039
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 199\940
| | 254.043
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 123\581
| |
| | 254.045
| |
|-
| |
| |
| |
| |
| |
| |
| |
| | 47\222
| |
| |
| |
| | 254.054
| |
|-
| |
| |
| |
| |
| |
| | 18\85
| |
| |
| |
| |
| |
| | 254.118
| |
|-
| |
| |
| |
| | 7\33
| |
| |
| |
| |
| |
| |
| |
| | 254.5455
| |
|-
| |
| |
| |
| |
| | 10\47
| |
| |
| |
| |
| |
| |
| | 255.319
| |
|-
| |
| |
| |
| |
| |
| | 13\61
| |
| |
| |
| |
| |
| | 255.734
| |
|-
| |
| |
| |
| |
| |
| |
| | 16\75
| |
| |
| |
| |
| | 256.000
| |
|-
| |
| | 3\14
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 257.143
| style="text-align:center;" | Boundary of propriety (generators


larger than this are proper)
The sizes of the generator, large step and small step of 5L 4s are as follows in various soft-of-basic tunings.
|-
{| class="wikitable right-2 right-3 right-4 right-5"
| |
| |
| |
| |
| | 11\51
| |
| |
| |
| |
| |
| |
| | 258.8235
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 258.957
| |
|-
| |
| |
| |
| | 8\37
| |
| |
| |
| |
| |
| |
| |
| | 259.459
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 21\97
| |
| |
| |
| |
| |
| | 259.794
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| | 55\254
| |
| |
| |
| | 259.843
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 144\665
| |
| | 259.850
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 233\1076
| | 259.851
| style="text-align:center;" | Golden [[superpelog|superpelog]]
|-
| |
| |
| |
| |
| |
| |
| |
| |
| | 89\411
| |
| |
| | 259.854
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 34\157
| |
| |
| |
| |
| | 259.873
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 13\60
| |
| |
| |
| |
| |
| |
| | 260
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 260.246
| |
|-
| |
| |
| | 5\23
| |
| |
| |
| |
| |
| |
| |
| |
| | 260.870
| style="text-align:center;" | Optimum rank range (L/s=3/2) superpelog
|-
| |
| |
| |
| | 7\32
| |
| |
| |
| |
| |
| |
| |
| | 262.5
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 9\41
| |
| |
| |
| |
| |
| |
| | 263.415
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 11\50
| |
| |
| |
| |
| |
| | 264
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 13\59
| |
| |
| |
| |
| | 264.407
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| | 15\68
| |
| |
| |
| | 264.706
| style="text-align:center;" |
|-
|-
| |
!
| |
! [[23edo]]
| |
! [[32edo]]
| |
! [[37edo]]
| |
| |
| |
| |
| | 17\77
| |
| |
| | 264.935
| style="text-align:center;" |
|-
|-
| |
| Generator (g)
| |
| 5\23, 260.87
| |
| 7\32, 262.50
| |
| 8\37, 259.46
| |
| |
| |
| |
| |
| | 19\86
| |
| | 265.116
| style="text-align:center;" |
|-
|-
| |  
| L ({{nowrap|octave − 4g}})
| |
| 156.52
| |
| 150.00
| |
| 162.16
| |
| |
| |
| |
| |
| |
| | 21\95
| | 265.263
| style="text-align:center;" |
|-
|-
| | 2\9
| s ({{nowrap|5g − octave}})
| |
| 104.35
| |
| 112.50
| |
| 97.30
| |
| |
| |
| |
| |
| |
| |
| | 266.667
| style="text-align:center;" |
|}
|}
== Tuning ranges ==
=== Semaphore ===
We can view [[semaphore]] as any 5L 4s tuning where two [[semifourth]] generators make a ''diatonic'' ([[5L 2s]]) fourth, i.e. any tuning where the semifourth is between 1\5 (240¢) or 3\14 (257.14¢). One important sub-range of semaphore is given by stipulating that two semifourth generators must make a ''meantone'' fourth; i.e. that four fifths should approximate a [[5/4]] major third. This results in [[godzilla]] temperament, which is supported by [[19edo]] and [[24edo]].


=== Bug ===
=== Tuning examples ===
For convenience' sake, we can view [[bug]] as any 5L 4s tuning where two [[semifourth]] generators make a ''superdiatonic'' ([[7L 2s]]) fourth, i.e. any tuning where the semifourth is between 3\14 (257.14¢) and 2\9 (266.67¢). [[23edo]]'s 5\23 (260.87¢) is an example of a bug generator.
An example in the Diasem Lydian mode LSLSLMLSLM with M and S equated. ([[:File:Diasem Lydian Example Score.pdf|score]])
 
[[File:Diasem Lydian Example 14edo.mp3]] [[14edo]], [[basic]] semiquartal
 
[[File:Diasem Lydian Example 19edo.mp3]] [[19edo]], [[hard]] semiquartal
 
[[File:Diasem Lydian Example 23edo.mp3]] [[23edo]], [[soft]] semiquartal
 
[[File:Diasem Lydian Example 24edo.mp3]] [[24edo]], [[superhard]] semiquartal
 
[[File:Diasem Lydian Example 33edo semiquartal.mp3]] [[33edo]], [[semihard]] semiquartal
 
== Scale tree ==
{{MOS tuning spectrum
| 5/4 = Septimin
| 4/3 = Beep
| 3/2 = Bug
| 13/8 = Golden bug
| 13/5 = Golden semaphore
| 3/1 = Godzilla
| 11/3 = Semaphore
}}
 
== Gallery ==
[[File:Hemifourths.png|thumb|An alternative diagram with branch depth = 5|alt=|none|507x507px]]
 
A voice-leading sketch in [[24edo]] by [[Jacob Barton]]:
 
[[File:qt_mode_chord_prog.mp3|qt mode chord prog]]  
 
== Music ==
* [https://www.soundclick.com/bands/songInfo.cfm?bandID=376205&songID=5327098 ''Entropy, the Grandfather of Wind''] (broken link. 2011-03-04) In [[14edo]]{{dead link}}


== Notation ==
; [[Frédéric Gagné]]
* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – In [[51edo]], 4|4 mode


== Intervals ==
; [[Inthar]]
== Modes == 
* [[:File:Dream EP 14edo Sketch.mp3|''Dream EP 14edo Sketch'']] (2021) – A short swing ditty in [[14edo]], in the 212121221 mode
TODO: names
* [[:File:19edo Semaphore Fugue.mp3|''19edo Semaphore Fugue'']] (2021) – An unfinished fugue in [[19edo]], in the 212121221 mode
* LLsLsLsLs
* LsLLsLsLs
* LsLsLLsLs
* LsLsLsLLs
* LsLsLsLsL
* sLLsLsLsL
* sLsLLsLsL
* sLsLsLLsL
* sLsLsLsLL


One can think of 5L 4s modes as being built from two pentachords (division of the perfect fourth into four intervals) plus a whole tone. The possible pentachords are LsLs, sLLs, and sLsL.
; [[Starshine]]
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] (2020) – Semaphore[9] in [[19edo]]


== Chords ==
; [[Sevish]]
== Primodal theory ==
* [http://www.youtube.com/watch?v=Gcgawrr2xao ''Desert Island Rain''] – Semaphore[9] in [[313edo]] using 65\313 as the generator
=== Nejis ===
==== 14nejis ====
# 95:100:105:110:116:122:128:135:141:148:156:164:172:180:190 (uses /19 prime family intervals while being pretty close to equal)
== Samples ==
[[File:Dream EP 14edo Sketch.mp3]]
''[[:File:Dream EP 14edo Sketch.mp3]]'' is a short swing ditty in [[14edo]] semaphore[9], in the 212121221 mode.


[[Category:Abstract MOS patterns]]
[[Category:Semiquartal| ]] <!-- Main article -->
[[Category:Scales]]
Retrieved from "https://en.xen.wiki/w/5L_4s"