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{{Infobox MOS
{{Infobox MOS}}
| Name = semiquartal
{{MOS intro}} It is also equal to a degenerate form of [[diasem]].
| Periods = 1
 
| nLargeSteps = 5
{{TAMNAMS name}} Another attested name is '''hemifourths'''.
| nSmallSteps = 4
 
| Equalized = 2
== Scale properties ==
| Paucitonic = 1
{{TAMNAMS use}}
| Pattern = LLsLsLsLs
 
}}
=== Intervals ===
{{MOS intervals}}


'''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).
=== Generator chain ===
{{MOS genchain}}


5L 4s tunings can be divided into two major ranges:
=== Modes ===
# [[hard]]-of-[[basic]] 5L 4s, generated by semifourths flatter than 3\14 (257.14¢). This implies a diatonic fifth.
{{MOS mode degrees}}
#: 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]].
# [[soft]]-of-basic 5L 4s, generated by semifourths sharper than 3\14 (257.14¢). This implies a "[[mavila]]" or superdiatonic fifth.
== Names ==
The [[TAMNAMS]] convention, used by this article, uses '''semiquartal''' (derived from 'half a fourth') for the 5L 4s pattern.


== Notation ==
Note that the darkest two modes have no diatonic or [[armotonic]] fifth on the root in nonextreme semiquartal tunings.
This article uses the convention JKLMNOPQR = LSLSLSLSL. The accidentals & and @ are used for raising and lowering by the chroma = L − S, respectively.


== Low harmonic entropy scales ==
== Theory ==
The familiar harmonic entropy minimum with this MOS pattern is [[Meantone family#Godzilla|godzilla]], in which a generator is [[8/7]] or [[7/6]] (tempered to be the same interval) so two of them make a [[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]] 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 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 ==
== Tuning ranges ==
=== Hard-of-basic ===
=== Hard-of-basic ===
These tunings satisfy the property that 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¢).
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"
|-
!
! [[14edo]] ({{nowrap|L/s {{=}} 2/1}})
! [[47edo]] ({{nowrap|L/s {{=}} 7/3}})
! [[33edo]] ({{nowrap|L/s {{=}} 5/2}})
! [[52edo]] ({{nowrap|L/s {{=}} 8/3}})
! [[19edo]] ({{nowrap|L/s {{=}} 3/1}})
|-
| Generator (g)
| 3\14, 257.14
| 10\47, 255.32
| 7\33, 254.54
| 11\52, 253.85
| 4\19, 252.63
|-
| L ({{nowrap|octave − 4g}})
| 171.43
| 178.72
| 181.81
| 184.62
| 189.47
|-
| 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):
* 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.


The sizes of the generator, large step and small step of 5L 4s are as follows in various hard-of-basic tunings.
==== 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, 250.00
| 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
| 41.38
| 41.38
|}
|}
==== Parahard ====
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). This range has an RTT interpretation known as [[godzilla]].


=== Soft-of-basic ===
=== Soft-of-basic ===
These are tunings where two [[semifourth]] generators make a ''superdiatonic'' ([[7L 2s]]) fourth (i.e. 514.29¢ to 533.33¢), 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 this 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 soft-of-basic 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 69: Line 102:
! [[37edo]]
! [[37edo]]
|-
|-
| generator (g)
| Generator (g)
| 5\23, 260.87
| 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 85: Line 118:
|}
|}


== Intervals ==
=== Tuning examples ===
Note: In TAMNAMS, a k-step interval class in semiquartal may be called a "k-step", "k-mosstep", or "k-sequarstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.
An example in the Diasem Lydian mode LSLSLMLSLM with M and S equated. ([[:File:Diasem Lydian Example Score.pdf|score]])


== Modes ==
[[File:Diasem Lydian Example 14edo.mp3]] [[14edo]], [[basic]] semiquartal
== Approaches ==
* [[5L 4s/Inthar's approach]]


== Samples ==
[[File:Diasem Lydian Example 19edo.mp3]] [[19edo]], [[hard]] semiquartal
* [https://soundcloud.com/starshine99/rins-ufo-ride Rin's UFO Ride] by Starshine
 
* ''[[:File:Dream EP 14edo Sketch.mp3]]'' is a short swing ditty in [[14edo]] semiquartal, in the 212121221 mode.
[[File:Diasem Lydian Example 23edo.mp3]] [[23edo]], [[soft]] semiquartal
* ''[[:File:19edo Semaphore Fugue.mp3]]'' is an unfinished fugue in [[19edo]] semiquartal, in the 212121221 mode.
 
[[File:Diasem Lydian Example 24edo.mp3]] [[24edo]], [[superhard]] semiquartal
 
[[File:Diasem Lydian Example 33edo semiquartal.mp3]] [[33edo]], [[semihard]] semiquartal


== Scale tree ==
== 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
}}


{| class="wikitable center-all"
== Gallery ==
! colspan="6" rowspan="2" | Generator
[[File:Hemifourths.png|thumb|An alternative diagram with branch depth = 5|alt=|none|507x507px]]
! colspan="2" | Cents
 
! rowspan="2" | L
A voice-leading sketch in [[24edo]] by [[Jacob Barton]]:
! rowspan="2" | s
 
! rowspan="2" | L/s
[[File:qt_mode_chord_prog.mp3|qt mode chord prog]]
! rowspan="2" | Comments
 
|-
== Music ==
! Chroma-positive
* [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}}
! Chroma-negative
 
|-
; [[Frédéric Gagné]]
| 7\9 || || || || || || 933.333 || 266.667 || 1 || 1 || 1.000 ||  
* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – In [[51edo]], 4|4 mode
|-
 
| || || || || || 39\50 || 936.000 || 264.000 || 6 || 5 || 1.200 ||
; [[Inthar]]
|-
* [[:File:Dream EP 14edo Sketch.mp3|''Dream EP 14edo Sketch'']] (2021) – A short swing ditty in [[14edo]], in the 212121221 mode
| || || || || 32\41 || || 936.585 || 263.415 || 5 || 4 || 1.250 || Septimin
* [[:File:19edo Semaphore Fugue.mp3|''19edo Semaphore Fugue'']] (2021) – An unfinished fugue in [[19edo]], in the 313131331 mode
|-
 
| || || || || || 57\73 || 936.986 || 263.014 || 9 || 7 || 1.286 ||
; [[Starshine]]
|-
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] (2020) – Semaphore[9] in [[19edo]]
| || || || 25\32 || || || 937.500 || 262.500 || 4 || 3 || 1.333 || Beep
 
|-
; [[Sevish]]
| || || || || || 68\87 || 937.931 || 262.069 || 11 || 8 || 1.375 ||
* [http://www.youtube.com/watch?v=Gcgawrr2xao ''Desert Island Rain''] – Semaphore[9] in [[313edo]] using 65\313 as the generator
|-
| || || || || 43\55 || || 938.182 || 261.818 || 7 || 5 || 1.400 ||
|-
| || || || || || 61\78 || 938.462 || 261.538 || 10 || 7 || 1.428 ||
|-
| || || 18\23 || || || || 939.130 || 260.870 || 3 || 2 || 1.500 || L/s = 3/2, bug
|-
| || || || || || 65\83 || 939.759 || 260.241 || 11 || 7 || 1.571 ||
|-
| || || || || 47\60 || || 940.000 || 260.000 || 8 || 5 || 1.600 ||  
|-
| || || || || || 76\97 || 940.206 || 259.794 || 13 || 8 || 1.625 || Golden bug
|-
| || || || 29\37 || || || 940.541 || 259.459 || 5 || 3 || 1.667 ||
|-
| || || || || || 69\88 || 940.909 || 259.091 || 12 || 7 || 1.714 ||
|-
| || || || || 40\51 || || 941.176 || 258.824 || 7 || 4 || 1.750 ||
|-
| || || || || || 51\65 || 941.538 || 258.462 || 9 || 5 || 1.800 ||
|-
| || 11\14 || || || || || 942.857 || 257.143 || 2 || 1 || 2.000 || Basic semiquartal<br>(Generators smaller than this are proper)
|-
| || || || || || 48\61 || 944.262 || 255.738 || 9 || 4 || 2.250 ||
|-
| || || || || 37\47 || || 944.681 || 255.319 || 7 || 3 || 2.333 ||
|-
| || || || || || 63\80 || 945.000 || 255.000 || 12 || 5 || 2.400 ||
|-
| || || || 26\33 || || || 945.455 || 254.545 || 5 || 2 || 2.500 ||
|-
| || || || || || 67\85 || 945.882 || 254.118 || 13 || 5 || 2.600 || Unnamed golden tuning
|-
| || || || || 41\52 || || 946.154 || 253.846 || 8 || 3 || 2.667 ||
|-
| || || || || || 56\71 || 946.479 || 253.521 || 11 || 4 || 2.750 ||
|-
| || || 15\19 || || || || 947.368 || 252.632 || 3 || 1 || 3.000 || L/s = 3/1, godzilla
|-
| || || || || || 49\62 || 948.387 || 251.613 || 10 || 3 || 3.333 ||
|-
| || || || || 34\43 || || 948.837 || 251.163 || 7 || 2 || 3.500 ||
|-
| || || || || || 53\67 || 949.254 || 250.746 || 11 || 3 || 3.667 || Semaphore
|-
| || || || 19\24 || || || 950.000 || 250.000 || 4 || 1 || 4.000 ||
|-
| || || || || || 42\53 || 950.943 || 249.057 || 9 || 2 || 4.500 ||
|-
| || || || || 23\29 || || 951.724 || 248.276 || 5 || 1 || 5.000 ||
|-
| || || || || || 27\34 || 952.941 || 247.059 || 6 || 1 || 6.000 ||
|-
| 4\5 || || || || || || 960.000 || 240.000 || 1 || 0 || → inf ||
|}


[[Category:Scales]]
[[Category:Semiquartal| ]] <!-- Main article -->
[[Category:9-tone scales]]
[[Category:Listen]]
[[Category:Abstract MOS patterns]]
[[Category:Semiquartal| ]] <!-- main article -->
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