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 ==
Line 16: Line 13:
=== Modes ===
=== Modes ===
{{MOS mode degrees}}
{{MOS mode degrees}}


Inthar and cellularAutomaton have proposed mode names based on scientific names of various corvids. The names as of 5/2/23{{Clarify}} are as follows.
Inthar and cellularAutomaton have proposed mode names based on scientific names of various corvids. The names as of 5/2/23{{Clarify}} are as follows.
Line 56: Line 52:
! [[19edo]] ({{nowrap|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
Line 79: Line 75:


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{{pipe}}7 and 0{{pipe}}8
* 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¢)
* 52edo semiquartal has close 22/19 (error +0.04¢)
* 19edo semiquartal has close 6/5 (error +0.15¢) and 28/27 (error +0.20¢)
* 19edo semiquartal has close 6/5 (error +0.15¢) and 28/27 (error +0.20¢)
Line 95: Line 91:
! [[29edo]]
! [[29edo]]
|-
|-
| generator (g)
| Generator (g)
| 4\19, 252.63
| 4\19, 252.63
| 5\24, 250.00
| 5\24, 250.00
Line 122: Line 118:
! [[37edo]]
! [[37edo]]
|-
|-
| generator (g)
| Generator (g)
| 5\23, 260.87
| 5\23, 260.87
| 7\32, 262.50
| 7\32, 262.50

Revision as of 15:15, 30 September 2024

↖ 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 ¢

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.

Inthar and cellularAutomaton have proposed mode names based on scientific names of various corvids. The names as of 5/2/23[clarification needed] are as follows.

Modes of 5L 4s
UDP Cyclic
order		 Step
pattern
8|0 1 LLsLsLsLs
7|1 8 LsLLsLsLs
6|2 6 LsLsLLsLs
5|3 4 LsLsLsLLs
4|4 2 LsLsLsLsL
3|5 9 sLLsLsLsL
2|6 7 sLsLLsLsL
1|7 5 sLsLsLLsL
0|8 3 sLsLsLsLL

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 "ultramajor" chords and "inframinor" triads could be viewed as approximating 10:13:15 and 26:30:39. 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

Template:Scale tree

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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