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5L 4s: Difference between revisions

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{{MOS intro}}
{{MOS intro}}
5L 4s tunings can be divided into two major ranges:
# [[hard]]-of-[[basic]] 5L 4s, generated by semifourths flatter than 3\14 (257.14¢). This implies a diatonic fifth.
#: 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.


It is also equal to a degenerate form of [[diasem]].
It is also equal to a degenerate form of [[diasem]].
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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'''.


== Notation ==
== Scale properties ==
This article uses the convention JKLMNOPQR = LsLsLsLsL. The accidentals & and @ are used for raising and lowering by the chroma = L − s, respectively.
{{TAMNAMS use}}


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


== 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|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]].
 
==== 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 (2/1 ≤ L/s ≤ 3/1) tunings.
Line 87: Line 127:


=== 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¢) and 2\9 (266.67¢), creating a "[[mavila]]" or "[[superdiatonic]]" 4th. [[23edo]]'s 5\23 (260.87¢) 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.
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[[File:Diasem Lydian Example 33edo semiquartal.mp3]] [[33edo]], [[semihard]] semiquartal
[[File:Diasem Lydian Example 33edo semiquartal.mp3]] [[33edo]], [[semihard]] semiquartal


== Intervals ==
== Scale tree ==
Note: In TAMNAMS, a k-step interval class in semiquartal may be called a "k-step", "k-mosstep", or "k-thonstep". TAMNAMS discourages 1-indexed terms such as "mos(k+1)th" for non-diatonic mosses.
{{Scale tree|Comments=5/4: Septimin;
4/3: Beep;
3/2: Bug;
13/8: Golden bug;
13/5: Golden semiquartal (unnamed temperament interpretation);
3/1: Godzilla;
11/3: Semaphore}}
[[Category:9-tone scales]]
[[Category:Semiquartal| ]]


== Modes ==
== Gallery ==
{{MOS mode degrees}}
[[File:Hemifourths.png|thumb|An alternative diagram with branch depth = 5|alt=|none|507x507px]]
===Proposed Names===
Inthar and cellularAutomaton have named 5L 4s modes after scientific names of various corvids.
{| class="wikitable"
|-
! style="text-align:center;" |Mode
! style="text-align:center;" |[[Modal UDP Notation|UDP]]
! style="text-align:center;" |[[User:Inthar|Inthar]]'s names as of 5/2/23
! style="text-align:center;" |Origin
|-
| |LLsLsLsLs
| style="text-align:center;" |<nowiki>8|0</nowiki> || Cristatan || Bluejay (''Cyanocitta cristata'')
|-
| |LsLLsLsLs
| style="text-align:center;" |<nowiki>7|1</nowiki> || Pican || Magpie (''Pica pica'')
|-
| |LsLsLLsLs
| style="text-align:center;" |<nowiki>6|2</nowiki> || Stellerian || Steller's jay (''Cyanocitta stelleri'')
|-
| |LsLsLsLLs
| style="text-align:center;" |<nowiki>5|3</nowiki> || Podocian || Ground jay (genus ''Podoces'')
|-
| |LsLsLsLsL
| style="text-align:center;" |<nowiki>4|4</nowiki> || Nucifragan || Nutcracker (genus ''Nucifraga'')
|-
| |sLLsLsLsL
| style="text-align:center;" |<nowiki>3|5</nowiki> || Coracian || Common raven (''Corvus corax'')
|-
| |sLsLLsLsL
| style="text-align:center;" |<nowiki>2|6</nowiki> || Frugilegian || Rook (''Corvus frugilegus'')
|-
| |sLsLsLLsL
| style="text-align:center;" |<nowiki>1|7</nowiki> || Temnurian || Ratchet-tailed treepie (genus ''Temnurus'')
|-
| |sLsLsLsLL
| style="text-align:center;" |<nowiki>0|8</nowiki> || Pyrrhian || Chough (genus ''Pyrrhocorax'')
|}
Note that the darkest two modes have no diatonic or [[armotonic]] fifth on the root in nonextreme semiquartal tunings.


== Music ==
== Music ==
* [https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] by Starshine, in [[19edo]]
*[https://soundcloud.com/starshine99/rins-ufo-ride ''Rin's UFO Ride''] by Starshine, in [[19edo]]
* [[:File:Dream EP 14edo Sketch.mp3|''Dream EP 14edo Sketch'']] by [[Inthar]], a short swing ditty in [[14edo]], in the 212121221 mode
*[[: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: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]]
*[[: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}}
*[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 -->
 
== Scale tree ==
[[File:Hemifourths.png|thumb|An alternative diagram with branch depth = 5]]
 
{| class="wikitable center-all"
! colspan="6" rowspan="2" | 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:9-tone scales]]
[[Category:Semiquartal| ]] <!-- main article -->
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