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5L 4s: Difference between revisions

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{{Infobox MOS}}
{{Infobox MOS}}
 
{{MOS intro}} It is also equal to a degenerate form of [[diasem]].
{{MOS intro}}
 
It is also equal to a degenerate form of [[diasem]].


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


== Scale properties ==
== Scale properties ==
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=== Intervals ===
=== Intervals ===
{{MOS intervals}}
{{MOS intervals}}
=== Generator chain ===
{{MOS genchain}}


=== Modes ===
=== Modes ===
{{MOS mode degrees}}
{{MOS mode degrees}}


==== Proposed Names ====
Inthar and cellularAutomaton have proposed mode names based on scientific names of various corvids.
{| class="wikitable"
|- style="text-align: center;"
! Mode !! [[Modal UDP Notation|UDP]] !! [[User:Inthar|Inthar]]'s names as of 5/2/23 !! Origin
|-
| LLsLsLsLs
| style="text-align: center;" | 8{{pipe}}0 || Cristatan || Bluejay (''Cyanocitta cristata'')
|-
| LsLLsLsLs
| style="text-align: center;" | 7{{pipe}}1 || Pican || Magpie (''Pica pica'')
|-
| LsLsLLsLs
| style="text-align: center;" | 6{{pipe}}2 || Stellerian || Steller's jay (''Cyanocitta stelleri'')
|-
| LsLsLsLLs
| style="text-align: center;" | 5{{pipe}}3 || Podocian || Ground jay (genus ''Podoces'')
|-
| LsLsLsLsL
| style="text-align: center;" | 4{{pipe}}4 || Nucifragan || Nutcracker (genus ''Nucifraga'')
|-
| sLLsLsLsL
| style="text-align: center;" | 3{{pipe}}5 || Coracian || Common raven (''Corvus corax'')
|-
| sLsLLsLsL
| style="text-align: center;" | 2{{pipe}}6 || Frugilegian || Rook (''Corvus frugilegus'')
|-
| sLsLsLLsL
| style="text-align: center;" | 1{{pipe}}7 || Temnurian || Ratchet-tailed treepie (genus ''Temnurus'')
|-
| sLsLsLsLL
| style="text-align: center;" | 0{{pipe}}8 || Pyrrhian || Chough (genus ''Pyrrhocorax'')
|}
Note that the darkest two modes have no diatonic or [[armotonic]] fifth on the root in nonextreme semiquartal tunings.
Note that the darkest two modes have no diatonic or [[armotonic]] fifth on the root in nonextreme semiquartal tunings.


== Theory ==
== Theory ==
The familiar harmonic entropy minimum with this MOS pattern is [[godzilla]], in which the generator [[8/7]] or [[7/6]] (tempered 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.  
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 ===
Hard-of-basic tunings have [[Semifourth|semifourths]] as generators, between 1\5 (240¢) and 3\14 (257.14¢), where two of them create a diatonic 4th. 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]].
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 ====
==== Hypohard ====
The sizes of the generator, large step and small step of 5L 4s are as follows in various hypohard (2/1 ≤ L/s ≤ 3/1) tunings.
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"
{| class="wikitable right-2 right-3 right-4 right-5 right-6"
|-
!  
!  
! [[14edo]] (L/s = 2/1)
! [[14edo]] ({{nowrap|L/s {{=}} 2/1}})
! [[47edo]] (L/s = 7/3)
! [[47edo]] ({{nowrap|L/s {{=}} 7/3}})
! [[33edo]] (L/s = 5/2)
! [[33edo]] ({{nowrap|L/s {{=}} 5/2}})
! [[52edo]] (L/s = 8/3)
! [[52edo]] ({{nowrap|L/s {{=}} 8/3}})
! [[19edo]] (L/s = 3/1)
! [[19edo]] ({{nowrap|L/s {{=}} 3/1}})
|-
|-
| generator (g)
| Generator (g)
| 3\14, 257.14
| 3\14, 257.14
| 10\47, 255.32
| 10\47, 255.32
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| 4\19, 252.63
| 4\19, 252.63
|-
|-
| L (octave - 4g)
| L ({{nowrap|octave − 4g}})
| 171.43
| 171.43
| 178.72
| 178.72
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| 189.47
| 189.47
|-
|-
| s (5g - octave)
| s ({{nowrap|5g − octave}})
| 85.71
| 85.71
| 76.60
| 76.60
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| 63.16
| 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):
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¢), 9/5 (error -0.59¢) and 9/7 (error +1.28¢), thus can be used for the close 5:7:9 in the two Locrian-like modes 1|7 and 0|8
* 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¢)
* 52edo semiquartal has close 22/19 (error +0.04{{c}})
* 19edo semiquartal has close 6/5 (error +0.15¢) and 28/27 (error +0.20¢)
* 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.
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.


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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]].
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 (2/1 ≤ L/s ≤ 3/1) tunings.
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"
|-
|-
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! [[29edo]]
! [[29edo]]
|-
|-
| generator (g)
| Generator (g)
| 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}})
| 189.47
| 189.47
| 200.00
| 200.00
| 206.90
| 206.90
|-
|-
| s (5g - octave)
| s ({{nowrap|5g − octave}})
| 63.16
| 63.16
| 50.00
| 50.00
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=== Soft-of-basic ===
=== Soft-of-basic ===
Soft-of-basic tunings have semifourths that are between 3\14 (257.14¢) and 2\9 (266.67¢), creating a "[[mavila]]" or "[[superdiatonic]]" 4th. [[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"
|-
|-
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! [[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
| 97.30
| 97.30
|}
|}
=== Tuning examples ===
=== Tuning examples ===
An example in the Diasem Lydian mode LSLSLMLSLM with M and S equated. ([[:File:Diasem Lydian Example Score.pdf|score]])
An example in the Diasem Lydian mode LSLSLMLSLM with M and S equated. ([[:File:Diasem Lydian Example Score.pdf|score]])
Line 163: Line 133:


== Scale tree ==
== Scale tree ==
{{Scale tree|Comments=5/4: Septimin;
{{MOS tuning spectrum
4/3: Beep;
| 5/4 = Septimin
3/2: Bug;
| 4/3 = Beep
13/8: Golden bug;
| 3/2 = Bug
13/5: Golden semaphore;
| 13/8 = Golden bug
3/1: Godzilla;
| 13/5 = Golden semaphore
11/3: Semaphore}}
| 3/1 = Godzilla
[[Category:9-tone scales]]
| 11/3 = Semaphore
[[Category:Semiquartal| ]]
}}


== Gallery ==
== Gallery ==
[[File:Hemifourths.png|thumb|An alternative diagram with branch depth = 5|alt=|none|507x507px]]
[[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 ==
== 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
; [[Frédéric Gagné]]
*[[:File:qt_mode_chord_prog.mp3|qt mode chord prog.mp3]]
* ''Whalectric'' (2022) – [https://youtu.be/_E6qvbJWYY8 YouTube] | [https://musescore.com/fredg999/whalectric score] – In [[51edo]], 4|4 mode
*[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}}<!-- main article -->
 
; [[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]]
 
; [[Sevish]]
* [http://www.youtube.com/watch?v=Gcgawrr2xao ''Desert Island Rain''] – Semaphore[9] in [[313edo]] using 65\313 as the generator
 
[[Category:Semiquartal| ]] <!-- Main article -->
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