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'''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 ==
The [[TAMNAMS]] convention, used by this article, uses '''semiquartal''' (derived from 'half a fourth') for the 5L 4s pattern. Another attested name is '''hemifourths'''.


There are not really "good" temperament interpretations for 5L 4s scales except for semaphore or godzilla, but 5L 4s tunings can be divided into two major ranges:
== Scale properties ==
# [[Semaphore]] generated by semifourths flatter than 3\14 (257.14¢). This implies a diatonic fifth.  
{{TAMNAMS use}}
#: The generator could be viewed as a 15/13, and the resulting "ultramajor" chords and "inframinor" triads could be viewed as approximating 10:13:15 and 26:30:39. See [[Arto and Tendo Theory]].
 
# [[Superpelog]], or [[bug]], generated by semifourths sharper than 3\14 (257.14¢). This implies a "[[mavila]]" or superdiatonic fifth.
=== Intervals ===
== Scale tree ==
{{MOS intervals}}
{| class="wikitable"
 
|-
=== Generator chain ===
! colspan="11" | Generator
{{MOS genchain}}
! | Cents
 
! | Comments
=== Modes ===
|-
{{MOS mode degrees}}
| | 1\5
 
| |
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.  
| |
| |
| |
| |
| |
| |
| | 240
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 12\59
| | 244.068
| style="text-align:center;" | Pseudo-semaphore is around here
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 11\54
| |
| | 244.444
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| | 10\49
| |
| |
| | 244.898
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| | 9\44
| |
| |
| |
| | 245.455
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| | 8\39
| |
| |
| |
| |
| | 246.154
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| | 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
|-
| |
| |
| |
| |
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| | 9\43
| |
| |
| |
| |
| |
| | 251.163
| |
|-
| |
| |
| | 4\19
| |
| |
| |
| |
| |
| |
| |
| |
| | 252.632
| style="text-align:center;" | Godzilla is around here


L/s = 3
== 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]]).
| |
| |
| |
| | 11\52
| |
| |
| |
| |
| |
| |
| | 253.813
| |
|-
| |
| |
| |
| |
| |
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| | 29\137
| |
| |
| |
| |
| | 254.015
| |
|-
| |
| |
| |
| |
| |
| |
| |
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| | 76\359
| |
| |
| | 254.039
| |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
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| | 199\940
| | 254.043
| |
|-
| |
| |
| |
| |
| |
| |
| |
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| | 123\581
| |
| | 254.045
| |
|-
| |
| |
| |
| |
| |
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| | 47\222
| |
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| | 254.054
| |
|-
| |
| |
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| | 18\85
| |
| |
| |
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| | 254.118
| |
|-
| |
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| | 7\33
| |
| |
| |
| |
| |
| |
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| | 254.5455
| |
|-
| |
| |
| |
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| | 10\47
| |
| |
| |
| |
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| | 255.319
| |
|-
| |
| |
| |
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| | 13\61
| |
| |
| |
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| | 255.734
| |
|-
| |
| |
| |
| |
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| | 16\75
| |
| |
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| | 256.000
| |
|-
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| | 3\14
| |
| |
| |
| |
| |
| |
| |
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| |
| | 257.143
| style="text-align:center;" | Boundary of propriety (generators


larger than this are proper)
==== 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"
|-
|-
| |
!
| |
! [[14edo]] ({{nowrap|L/s {{=}} 2/1}})
| |
! [[47edo]] ({{nowrap|L/s {{=}} 7/3}})
| |
! [[33edo]] ({{nowrap|L/s {{=}} 5/2}})
| | 11\51
! [[52edo]] ({{nowrap|L/s {{=}} 8/3}})
| |
! [[19edo]] ({{nowrap|L/s {{=}} 3/1}})
| |
| |
| |
| |
| |
| | 258.8235
| |
|-
|-
| |
| Generator (g)
| |
| 3\14, 257.14
| |
| 10\47, 255.32
| |
| 7\33, 254.54
| |
| 11\52, 253.85
| |
| 4\19, 252.63
| |
| |
| |
| |
| |
| | 258.957
| |
|-
|-
| |  
| L ({{nowrap|octave − 4g}})
| |
| 171.43
| |
| 178.72
| | 8\37
| 181.81
| |
| 184.62
| |
| 189.47
| |
| |
| |
| |
| |
| | 259.459
| style="text-align:center;" |
|-
|-
| |  
| s ({{nowrap|5g − octave}})
| |
| 85.71
| |
| 76.60
| |
| 72.73
| |
| 69.23
| | 21\97
| 63.16
| |
| |
| |
| |
| |
| | 259.794
| style="text-align:center;" |
|-
| |
| |
| |
| |
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| | 55\254
| |
| |
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| | 259.843
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
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| | 144\665
| |
| | 259.850
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
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| | 233\1076
| | 259.851
| style="text-align:center;" | Golden [[superpelog|superpelog]]
|-
| |
| |
| |
| |
| |
| |
| |
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| | 89\411
| |
| |
| | 259.854
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
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| | 34\157
| |
| |
| |
| |
| | 259.873
| style="text-align:center;" |
|-
| |
| |
| |
| |
| | 13\60
| |
| |
| |
| |
| |
| |
| | 260
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
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| | 260.246
| |
|-
| |
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| | 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;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| | 17\77
| |
| |
| | 264.935
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 19\86
| |
| | 265.116
| style="text-align:center;" |
|-
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 21\95
| | 265.263
| style="text-align:center;" |
|-
| | 2\9
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| | 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]].


The sizes of the generator, large step and small step of 5L 4s are as follows in various semaphore tunings.
This range is notable for having many simple tunings that are close to being "eigentunings" (tunings that tune a certain JI interval exactly):
* 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}})
* 19edo semiquartal has close 6/5 (error +0.15{{c}}) and 28/27 (error +0.20{{c}})
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"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
!  
!  
! [[14edo]]
! [[19edo]]
! [[19edo]]
! [[24edo]]
! [[24edo]]
! [[29edo]]
! [[29edo]]
|-
|-
| generator (g)
| Generator (g)
| 3\14, 257.14
| 4\19, 252.63
| 4\19, 252.63
| 5\24, 250.
| 5\24, 250.00
| 6\29, 248.28
| 6\29, 248.28
|-
|-
| L (octave - 4g)
| L ({{nowrap|octave − 4g}})
| 171.43
| 189.47
| 189.47
| 200.00
| 200.00
| 206.90
| 206.90
|-
|-
| s (5g - octave)
| s ({{nowrap|5g − octave}})
| 85.71
| 63.16
| 63.16
| 50.00
| 50.00
Line 680: Line 92:
|}
|}


=== Superpelog ===
=== Soft-of-basic ===
For convenience' sake, we can view [[superpelog]] 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 superpelog generator.
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.


The sizes of the generator, large step and small step of 5L 4s are as follows in various superpelog tunings.
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"
{| class="wikitable right-2 right-3 right-4 right-5"
|-
|-
Line 691: Line 103:
! [[37edo]]
! [[37edo]]
|-
|-
| generator (g)
| Generator (g)
| 5\23, 252.63
| 5\23, 260.87
| 7\32, 262.50
| 7\32, 262.50
| 8\37, 259.46
| 8\37, 259.46
|-
|-
| L (octave - 4g)
| L ({{nowrap|octave − 4g}})
| 156.52
| 156.52
| 150.00
| 150.00
| 162.16
| 162.16
|-
|-
| s (5g - octave)
| s ({{nowrap|5g − octave}})
| 104.35
| 104.35
| 112.50
| 112.50
Line 707: Line 119:
|}
|}


== Notation ==
=== Tuning examples ===
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}}
 
; [[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"