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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
== Names ==
| nSmallSteps = 4
The [[TAMNAMS]] convention, used by this article, uses '''semiquartal''' (derived from 'half a fourth') for the 5L 4s pattern. Another attested name is '''hemifourths'''.
| Equalized = 2
 
| Paucitonic = 1
== Scale properties ==
| Pattern = LsLsLsLsL
{{TAMNAMS use}}
}}


'''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).
=== Intervals ===
{{MOS intervals}}


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


== 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.


== Temperaments ==
== Theory ==
The familiar harmonic entropy minimum with this MOS pattern is [[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 [[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.  
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¢) and 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]]).


The sizes of the generator, large step and small step of 5L 4s are as follows in various hard-of-basic tunings.
==== 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.
 
==== 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 70: Line 103:
! [[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 86: Line 119:
|}
|}


== 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]])
 
[[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]]


== Modes ==
A voice-leading sketch in [[24edo]] by [[Jacob Barton]]:  
{| class="wikitable"
|-
| style="text-align:center;" |'''Mode'''
| style="text-align:center;" |[[Modal UDP Notation|'''UDP''']]
|-
| |LLsLsLsLs
| style="text-align:center;" |<nowiki>8|0</nowiki>
|-
| |LsLLsLsLs
| style="text-align:center;" |<nowiki>7|1</nowiki>
|-
| |LsLsLLsLs
| style="text-align:center;" |<nowiki>6|2</nowiki>
|-
| |LsLsLsLLs
| style="text-align:center;" |<nowiki>5|3</nowiki>
|-
| |LsLsLsLsL
| style="text-align:center;" |<nowiki>4|4</nowiki>
|-
| |sLLsLsLsL
| style="text-align:center;" |<nowiki>3|5</nowiki>
|-
| |sLsLLsLsL
| style="text-align:center;" |<nowiki>2|6</nowiki>
|-
| |sLsLsLLsL
| style="text-align:center;" |<nowiki>1|7</nowiki>
|-
| |sLsLsLsLL
| style="text-align:center;" |<nowiki>0|8</nowiki>
|}
Note that the darkest two modes have no fifth on the root in nonextreme semiquartal tunings.


== Approaches ==
[[File:qt_mode_chord_prog.mp3|qt mode chord prog]]  
* [[5L 4s/Inthar's approach]]


== Music ==
== Music ==
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] by Starshine, in [[19edo]]
* [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}}
* [[:File:Dream EP 14edo Sketch.mp3|''Dream EP 14edo Sketch'']] by [[Inthar]], a short swing ditty in [[14edo]], in the 212121221 mode
* [[:File:19edo Semaphore Fugue.mp3|''19edo Semaphore Fugue'']] by [[Inthar]], an unfinished fugue in [[19edo]], in the 212121221 mode
* [[:File:qt_mode_chord_prog.mp3|qt mode chord prog.mp3]]
* [http://www.soundclick.com/bands/songInfo.cfm?bandID=376205&songID=5327098 ''Entropy, the Grandfather of Wind''] (broken link. 2011-03-04) in [[14edo]] {{dead link}}


== Scale tree ==
; [[Frédéric Gagné]]
[[File:Hemifourths.png|thumb|An alternative diagram with branch depth = 5]]
* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – In [[51edo]], 4|4 mode
 
; [[Inthar]]
* [[:File:Dream EP 14edo Sketch.mp3|''Dream EP 14edo Sketch'']] (2021) – A short swing ditty in [[14edo]], in the 212121221 mode
* [[:File:19edo Semaphore Fugue.mp3|''19edo Semaphore Fugue'']] (2021) – An unfinished fugue in [[19edo]], in the 212121221 mode
 
; [[Starshine]]
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] (2020) – Semaphore[9] in [[19edo]]


{| class="wikitable center-all"
; [[Sevish]]
! colspan="6" rowspan="2" | Generator
* [http://www.youtube.com/watch?v=Gcgawrr2xao ''Desert Island Rain''] – Semaphore[9] in [[313edo]] using 65\313 as the generator
! colspan="2" | Cents
! rowspan="2" | L
! rowspan="2" | s
! rowspan="2" | L/s
! rowspan="2" | Comments
|-
! Chroma-positive
! Chroma-negative
|-
| 7\9 || || || || || || 933.333 || 266.667 || 1 || 1 || 1.000 ||
|-
| || || || || || 39\50 || 936.000 || 264.000 || 6 || 5 || 1.200 ||
|-
| || || || || 32\41 || || 936.585 || 263.415 || 5 || 4 || 1.250 || Septimin
|-
| || || || || || 57\73 || 936.986 || 263.014 || 9 || 7 || 1.286 ||
|-
| || || || 25\32 || || || 937.500 || 262.500 || 4 || 3 || 1.333 || Beep
|-
| || || || || || 68\87 || 937.931 || 262.069 || 11 || 8 || 1.375 ||
|-
| || || || || 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:Scale theory]]
[[Category:Semiquartal| ]] <!-- Main article -->
[[Category:9-tone scales]]
[[Category:Abstract MOS patterns]]
[[Category:Semiquartal| ]] <!-- main article -->
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