5L 4s: Difference between revisions

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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. Another attested name is '''hemifourths'''.
{{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 -->

Latest revision as of 13:56, 18 March 2025

↖ 4L 3s ↑ 5L 3s 6L 3s ↗
← 4L 4s 5L 4s 6L 4s →
↙ 4L 5s ↓ 5L 5s 6L 5s ↘ 
┌╥╥┬╥┬╥┬╥┬┐
│║║│║│║│║││
│││││││││││
└┴┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LLsLsLsLs
sLsLsLsLL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 7\9 to 4\5 (933.3 ¢ to 960.0 ¢)
Dark 1\5 to 2\9 (240.0 ¢ to 266.7 ¢)
TAMNAMS information
Name semiquartal
Prefix cthon-
Abbrev. ct
Related MOS scales
Parent 4L 1s
Sister 4L 5s
Daughters 9L 5s, 5L 9s
Neutralized 1L 8s
2-Flought 14L 4s, 5L 13s
Equal tunings
Equalized (L:s = 1:1) 7\9 (933.3 ¢)
Supersoft (L:s = 4:3) 25\32 (937.5 ¢)
Soft (L:s = 3:2) 18\23 (939.1 ¢)
Semisoft (L:s = 5:3) 29\37 (940.5 ¢)
Basic (L:s = 2:1) 11\14 (942.9 ¢)
Semihard (L:s = 5:2) 26\33 (945.5 ¢)
Hard (L:s = 3:1) 15\19 (947.4 ¢)
Superhard (L:s = 4:1) 19\24 (950.0 ¢)
Collapsed (L:s = 1:0) 4\5 (960.0 ¢)

5L 4s, named semiquartal in TAMNAMS, is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 5 large steps and 4 small steps, repeating every octave. Generators that produce this scale range from 933.3 ¢ to 960 ¢, or from 240 ¢ to 266.7 ¢. It is also equal to a degenerate form of diasem.

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.

Scale properties

This article uses TAMNAMS conventions for the names of this scale's intervals and scale degrees. The use of 1-indexed ordinal names is reserved for interval regions.

Intervals

Intervals of 5L 4s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-cthonstep Perfect 0-cthonstep P0cts 0 0.0 ¢
1-cthonstep Minor 1-cthonstep m1cts s 0.0 ¢ to 133.3 ¢
Major 1-cthonstep M1cts L 133.3 ¢ to 240.0 ¢
2-cthonstep Perfect 2-cthonstep P2cts L + s 240.0 ¢ to 266.7 ¢
Augmented 2-cthonstep A2cts 2L 266.7 ¢ to 480.0 ¢
3-cthonstep Minor 3-cthonstep m3cts L + 2s 240.0 ¢ to 400.0 ¢
Major 3-cthonstep M3cts 2L + s 400.0 ¢ to 480.0 ¢
4-cthonstep Minor 4-cthonstep m4cts 2L + 2s 480.0 ¢ to 533.3 ¢
Major 4-cthonstep M4cts 3L + s 533.3 ¢ to 720.0 ¢
5-cthonstep Minor 5-cthonstep m5cts 2L + 3s 480.0 ¢ to 666.7 ¢
Major 5-cthonstep M5cts 3L + 2s 666.7 ¢ to 720.0 ¢
6-cthonstep Minor 6-cthonstep m6cts 3L + 3s 720.0 ¢ to 800.0 ¢
Major 6-cthonstep M6cts 4L + 2s 800.0 ¢ to 960.0 ¢
7-cthonstep Diminished 7-cthonstep d7cts 3L + 4s 720.0 ¢ to 933.3 ¢
Perfect 7-cthonstep P7cts 4L + 3s 933.3 ¢ to 960.0 ¢
8-cthonstep Minor 8-cthonstep m8cts 4L + 4s 960.0 ¢ to 1066.7 ¢
Major 8-cthonstep M8cts 5L + 3s 1066.7 ¢ to 1200.0 ¢
9-cthonstep Perfect 9-cthonstep P9cts 5L + 4s 1200.0 ¢

Generator chain

Generator chain of 5L 4s
Bright gens Scale degree Abbrev.
13 Augmented 1-cthondegree A1ctd
12 Augmented 3-cthondegree A3ctd
11 Augmented 5-cthondegree A5ctd
10 Augmented 7-cthondegree A7ctd
9 Augmented 0-cthondegree A0ctd
8 Augmented 2-cthondegree A2ctd
7 Major 4-cthondegree M4ctd
6 Major 6-cthondegree M6ctd
5 Major 8-cthondegree M8ctd
4 Major 1-cthondegree M1ctd
3 Major 3-cthondegree M3ctd
2 Major 5-cthondegree M5ctd
1 Perfect 7-cthondegree P7ctd
0 Perfect 0-cthondegree
Perfect 9-cthondegree		 P0ctd
P9ctd
−1 Perfect 2-cthondegree P2ctd
−2 Minor 4-cthondegree m4ctd
−3 Minor 6-cthondegree m6ctd
−4 Minor 8-cthondegree m8ctd
−5 Minor 1-cthondegree m1ctd
−6 Minor 3-cthondegree m3ctd
−7 Minor 5-cthondegree m5ctd
−8 Diminished 7-cthondegree d7ctd
−9 Diminished 9-cthondegree d9ctd
−10 Diminished 2-cthondegree d2ctd
−11 Diminished 4-cthondegree d4ctd
−12 Diminished 6-cthondegree d6ctd
−13 Diminished 8-cthondegree d8ctd

Modes

Scale degrees of the modes of 5L 4s
UDP Cyclic
order 		Step
pattern 		Scale degree (cthondegree)
0 1 2 3 4 5 6 7 8 9
8|0 1 LLsLsLsLs Perf. Maj. Aug. Maj. Maj. Maj. Maj. Perf. Maj. Perf.
7|1 8 LsLLsLsLs Perf. Maj. Perf. Maj. Maj. Maj. Maj. Perf. Maj. Perf.
6|2 6 LsLsLLsLs Perf. Maj. Perf. Maj. Min. Maj. Maj. Perf. Maj. Perf.
5|3 4 LsLsLsLLs Perf. Maj. Perf. Maj. Min. Maj. Min. Perf. Maj. Perf.
4|4 2 LsLsLsLsL Perf. Maj. Perf. Maj. Min. Maj. Min. Perf. Min. Perf.
3|5 9 sLLsLsLsL Perf. Min. Perf. Maj. Min. Maj. Min. Perf. Min. Perf.
2|6 7 sLsLLsLsL Perf. Min. Perf. Min. Min. Maj. Min. Perf. Min. Perf.
1|7 5 sLsLsLLsL Perf. Min. Perf. Min. Min. Min. Min. Perf. Min. Perf.
0|8 3 sLsLsLsLL Perf. Min. Perf. Min. Min. Min. Min. Dim. Min. Perf.

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.

Tuning ranges

Hard-of-basic

Hard-of-basic tunings have 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 "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 (2/1 ≤ L/s ≤ 3/1) tunings.

14edo (L/s = 2/1) 47edo (L/s = 7/3) 33edo (L/s = 5/2) 52edo (L/s = 8/3) 19edo (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 (octave − 4g) 171.43 178.72 181.81 184.62 189.47
s (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 ¢), 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
  • 52edo semiquartal has close 22/19 (error +0.04 ¢)
  • 19edo semiquartal has close 6/5 (error +0.15 ¢) and 28/27 (error +0.20 ¢)

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 (2/1 ≤ L/s ≤ 3/1) tunings.

19edo 24edo 29edo
Generator (g) 4\19, 252.63 5\24, 250.00 6\29, 248.28
L (octave − 4g) 189.47 200.00 206.90
s (5g − octave) 63.16 50.00 41.38

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.

The sizes of the generator, large step and small step of 5L 4s are as follows in various soft-of-basic tunings.

23edo 32edo 37edo
Generator (g) 5\23, 260.87 7\32, 262.50 8\37, 259.46
L (octave − 4g) 156.52 150.00 162.16
s (5g − octave) 104.35 112.50 97.30

Tuning examples

An example in the Diasem Lydian mode LSLSLMLSLM with M and S equated. (score)

14edo, basic semiquartal

19edo, hard semiquartal

23edo, soft semiquartal

24edo, superhard semiquartal

33edo, semihard semiquartal

Scale tree

Scale tree and tuning spectrum of 5L 4s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
7\9 933.333 266.667 1:1 1.000 Equalized 5L 4s
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 Supersoft 5L 4s
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 1.429
18\23 939.130 260.870 3:2 1.500 Soft 5L 4s
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 Semisoft 5L 4s
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 5L 4s
Scales with tunings softer 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 Semihard 5L 4s
67\85 945.882 254.118 13:5 2.600 Golden semaphore
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 Hard 5L 4s
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 Superhard 5L 4s
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 → ∞ Collapsed 5L 4s

Gallery

An alternative diagram with branch depth = 5

A voice-leading sketch in 24edo by Jacob Barton:

Music

Frédéric Gagné
Inthar
Starshine
Sevish
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