3L 5s: Difference between revisions

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== Theory ==
== Theory ==
In contrast to oneirotonic ([[5L 3s]]), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth ([[3/2]], usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.
In contrast to oneirotonic ([[5L 3s]]), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth ([[3/2]], usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.


=== Low harmonic entropy scales ===
=== Low harmonic entropy scales ===
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! (9)
! (9)
|-
|-
| <nowiki>7|0</nowiki>
| 7{{pipe}}0
| Anti-Sarnathian
| Anti-Sarnathian
| King
| King
| LsLssLss
| LsLssLss
| J
| J
| K&
| K&amp;
| L
| L
| M&
| M&amp;
| N&
| N&amp;
| O
| O
| P&
| P&amp;
| Q
| Q
| (J)
| (J)
|-
|-
| <nowiki>6|1</nowiki>
| 6{{pipe}}1
| Anti-Hlanithian
| Anti-Hlanithian
| Queen
| Queen
| LssLsLss
| LssLsLss
| J
| J
| K&
| K&amp;
| L
| L
| M
| M
| N&
| N&amp;
| O
| O
| P&
| P&amp;
| Q
| Q
| (J)
| (J)
|-
|-
| <nowiki>5|2</nowiki>
| 5{{pipe}}2
| Anti-Kadathian
| Anti-Kadathian
| Marshall
| Marshall
| LssLssLs
| LssLssLs
| J
| J
| K&
| K&amp;
| L
| L
| M
| M
| N&
| N&amp;
| O
| O
| P
| P
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| (J)
| (J)
|-
|-
| <nowiki>4|3</nowiki>
| 4{{pipe}}3
| Anti-Mnarian
| Anti-Mnarian
| Cardinal
| Cardinal
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| L
| L
| M
| M
| N&
| N&amp;
| O
| O
| P
| P
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| (J)
| (J)
|-
|-
| <nowiki>3|4</nowiki>
| 3{{pipe}}4
| Anti-Ultharian
| Anti-Ultharian
| Rook
| Rook
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| (J)
| (J)
|-
|-
| <nowiki>2|5</nowiki>
| 2{{pipe}}5
| Anti-Celephaïsian
| Anti-Celephaïsian
| Bishop
| Bishop
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| (J)
| (J)
|-
|-
| <nowiki>1|6</nowiki>
| 1{{pipe}}6
| Anti-Illarnekian
| Anti-Illarnekian
| Knight
| Knight
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| (J)
| (J)
|-
|-
| <nowiki>0|7</nowiki>
| 0{{pipe}}7
| Anti-Dylathian
| Anti-Dylathian
| Pawn
| Pawn

Revision as of 16:30, 21 January 2025

↖ 2L 4s ↑ 3L 4s 4L 4s ↗
← 2L 5s 3L 5s 4L 5s →
↙ 2L 6s ↓ 3L 6s 4L 6s ↘ 
┌╥┬╥┬┬╥┬┬┐
│║│║││║│││
││││││││││
└┴┴┴┴┴┴┴┴┘
Scale structure
Step pattern LsLssLss
ssLssLsL
Equave 2/1 (1200.0 ¢)
Period 2/1 (1200.0 ¢)
Generator size
Bright 5\8 to 2\3 (750.0 ¢ to 800.0 ¢)
Dark 1\3 to 3\8 (400.0 ¢ to 450.0 ¢)
TAMNAMS information
Name checkertonic
Prefix check-
Abbrev. chk
Related MOS scales
Parent 3L 2s
Sister 5L 3s
Daughters 8L 3s, 3L 8s
Neutralized 6L 2s
2-Flought 11L 5s, 3L 13s
Equal tunings
Equalized (L:s = 1:1) 5\8 (750.0 ¢)
Supersoft (L:s = 4:3) 17\27 (755.6 ¢)
Soft (L:s = 3:2) 12\19 (757.9 ¢)
Semisoft (L:s = 5:3) 19\30 (760.0 ¢)
Basic (L:s = 2:1) 7\11 (763.6 ¢)
Semihard (L:s = 5:2) 16\25 (768.0 ¢)
Hard (L:s = 3:1) 9\14 (771.4 ¢)
Superhard (L:s = 4:1) 11\17 (776.5 ¢)
Collapsed (L:s = 1:0) 2\3 (800.0 ¢)

3L 5s, named checkertonic in TAMNAMS (also known as anti-oneirotonic), is a 2/1-equivalent (octave-equivalent) moment of symmetry scale containing 3 large steps and 5 small steps, repeating every octave. Generators that produce this scale range from 750 ¢ to 800 ¢, or from 400 ¢ to 450 ¢.

Name

TAMNAMS suggests the temperament-agnostic name checkertonic for this scale.

Intervals

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 of 3L 5s
Intervals Steps
subtended 		Range in cents
Generic Specific Abbrev.
0-checkstep Perfect 0-checkstep P0chks 0 0.0 ¢
1-checkstep Minor 1-checkstep m1chks s 0.0 ¢ to 150.0 ¢
Major 1-checkstep M1chks L 150.0 ¢ to 400.0 ¢
2-checkstep Minor 2-checkstep m2chks 2s 0.0 ¢ to 300.0 ¢
Major 2-checkstep M2chks L + s 300.0 ¢ to 400.0 ¢
3-checkstep Perfect 3-checkstep P3chks L + 2s 400.0 ¢ to 450.0 ¢
Augmented 3-checkstep A3chks 2L + s 450.0 ¢ to 800.0 ¢
4-checkstep Minor 4-checkstep m4chks L + 3s 400.0 ¢ to 600.0 ¢
Major 4-checkstep M4chks 2L + 2s 600.0 ¢ to 800.0 ¢
5-checkstep Diminished 5-checkstep d5chks L + 4s 400.0 ¢ to 750.0 ¢
Perfect 5-checkstep P5chks 2L + 3s 750.0 ¢ to 800.0 ¢
6-checkstep Minor 6-checkstep m6chks 2L + 4s 800.0 ¢ to 900.0 ¢
Major 6-checkstep M6chks 3L + 3s 900.0 ¢ to 1200.0 ¢
7-checkstep Minor 7-checkstep m7chks 2L + 5s 800.0 ¢ to 1050.0 ¢
Major 7-checkstep M7chks 3L + 4s 1050.0 ¢ to 1200.0 ¢
8-checkstep Perfect 8-checkstep P8chks 3L + 5s 1200.0 ¢

Notation

The TAMNAMS system is used in this article to refer to 3L 5s step size ratios and step ratio ranges.

The notation used in this article is JKLMNOPQJ = sLssLsLs (Anti-Ultharian), &/@ = up/down by chroma.

Theory

In contrast to oneirotonic (5L 3s), which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth (3/2, usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.

Low harmonic entropy scales

There are two significant harmonic entropy minima with this MOS pattern:

  • Sensi, in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, which is proper.
  • Squares, in which the generator is also a 9/7, but two of them make an 18/11 and four of them make a 4/3, which is improper.

Tuning ranges

Simple tunings

Simple Tunings of 3L 5s
Scale degree Abbrev. Basic (2:1)
11edo 		Hard (3:1)
14edo 		Soft (3:2)
19edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-checkdegree P0chkd 0\11 0.0 0\14 0.0 0\19 0.0
Minor 1-checkdegree m1chkd 1\11 109.1 1\14 85.7 2\19 126.3
Major 1-checkdegree M1chkd 2\11 218.2 3\14 257.1 3\19 189.5
Minor 2-checkdegree m2chkd 2\11 218.2 2\14 171.4 4\19 252.6
Major 2-checkdegree M2chkd 3\11 327.3 4\14 342.9 5\19 315.8
Perfect 3-checkdegree P3chkd 4\11 436.4 5\14 428.6 7\19 442.1
Augmented 3-checkdegree A3chkd 5\11 545.5 7\14 600.0 8\19 505.3
Minor 4-checkdegree m4chkd 5\11 545.5 6\14 514.3 9\19 568.4
Major 4-checkdegree M4chkd 6\11 654.5 8\14 685.7 10\19 631.6
Diminished 5-checkdegree d5chkd 6\11 654.5 7\14 600.0 11\19 694.7
Perfect 5-checkdegree P5chkd 7\11 763.6 9\14 771.4 12\19 757.9
Minor 6-checkdegree m6chkd 8\11 872.7 10\14 857.1 14\19 884.2
Major 6-checkdegree M6chkd 9\11 981.8 12\14 1028.6 15\19 947.4
Minor 7-checkdegree m7chkd 9\11 981.8 11\14 942.9 16\19 1010.5
Major 7-checkdegree M7chkd 10\11 1090.9 13\14 1114.3 17\19 1073.7
Perfect 8-checkdegree P8chkd 11\11 1200.0 14\14 1200.0 19\19 1200.0

Parasoft tunings

Parasoft tunings (step ratios 4:3 to 3:2) are associated with sensi tempermament.


Parasoft Tunings of 3L 5s
Scale degree Abbrev. Supersoft (4:3)
27edo 		7:5
46edo 		Soft (3:2)
19edo
Steps ¢ Steps ¢ Steps ¢
Perfect 0-checkdegree P0chkd 0\27 0.0 0\46 0.0 0\19 0.0
Minor 1-checkdegree m1chkd 3\27 133.3 5\46 130.4 2\19 126.3
Major 1-checkdegree M1chkd 4\27 177.8 7\46 182.6 3\19 189.5
Minor 2-checkdegree m2chkd 6\27 266.7 10\46 260.9 4\19 252.6
Major 2-checkdegree M2chkd 7\27 311.1 12\46 313.0 5\19 315.8
Perfect 3-checkdegree P3chkd 10\27 444.4 17\46 443.5 7\19 442.1
Augmented 3-checkdegree A3chkd 11\27 488.9 19\46 495.7 8\19 505.3
Minor 4-checkdegree m4chkd 13\27 577.8 22\46 573.9 9\19 568.4
Major 4-checkdegree M4chkd 14\27 622.2 24\46 626.1 10\19 631.6
Diminished 5-checkdegree d5chkd 16\27 711.1 27\46 704.3 11\19 694.7
Perfect 5-checkdegree P5chkd 17\27 755.6 29\46 756.5 12\19 757.9
Minor 6-checkdegree m6chkd 20\27 888.9 34\46 887.0 14\19 884.2
Major 6-checkdegree M6chkd 21\27 933.3 36\46 939.1 15\19 947.4
Minor 7-checkdegree m7chkd 23\27 1022.2 39\46 1017.4 16\19 1010.5
Major 7-checkdegree M7chkd 24\27 1066.7 41\46 1069.6 17\19 1073.7
Perfect 8-checkdegree P8chkd 27\27 1200.0 46\46 1200.0 19\19 1200.0

Modes

Scale degrees of the modes of 3L 5s
UDP Cyclic
order 		Step
pattern 		Scale degree (checkdegree)
0 1 2 3 4 5 6 7 8
7|0 1 LsLssLss Perf. Maj. Maj. Aug. Maj. Perf. Maj. Maj. Perf.
6|1 6 LssLsLss Perf. Maj. Maj. Perf. Maj. Perf. Maj. Maj. Perf.
5|2 3 LssLssLs Perf. Maj. Maj. Perf. Maj. Perf. Min. Maj. Perf.
4|3 8 sLsLssLs Perf. Min. Maj. Perf. Maj. Perf. Min. Maj. Perf.
3|4 5 sLssLsLs Perf. Min. Maj. Perf. Min. Perf. Min. Maj. Perf.
2|5 2 sLssLssL Perf. Min. Maj. Perf. Min. Perf. Min. Min. Perf.
1|6 7 ssLsLssL Perf. Min. Min. Perf. Min. Perf. Min. Min. Perf.
0|7 4 ssLssLsL Perf. Min. Min. Perf. Min. Dim. Min. Min. Perf.

Proposed names

The modes of checkertonic can be named after its sister MOS 5L 3s (oneirotonic). R-4981 has also proposed names based on grand chess pieces.


Modes of 3L 5s
UDP Cyclic
order		 Step
pattern
7|0 1 LsLssLss
6|1 6 LssLsLss
5|2 3 LssLssLs
4|3 8 sLsLssLs
3|4 5 sLssLsLs
2|5 2 sLssLssL
1|6 7 ssLsLssL
0|7 4 ssLssLsL

The order of modes on the white keys JKLMNOPQ are:

  • J Anti-Ultharian, Rook
  • K Anti-Hlanithian, Queen
  • L Anti-Illarnekian, Knight
  • M Anti-Mnarian, Cardinal
  • N Anti-Sarnathian, King
  • O Anti-Celephaïsian, Bishop
  • P Anti-Kadathian, Marshall
  • Q Anti-Dylathian, Pawn
Scale degrees (on J, sLssLsLs = JKLMNOPQ)
UDP Anti-modes of 5L 3s Chess-based names Step pattern 1 2 3 4 5 6 7 8 (9)
7|0 Anti-Sarnathian King LsLssLss J K& L M& N& O P& Q (J)
6|1 Anti-Hlanithian Queen LssLsLss J K& L M N& O P& Q (J)
5|2 Anti-Kadathian Marshall LssLssLs J K& L M N& O P Q (J)
4|3 Anti-Mnarian Cardinal sLsLssLs J K L M N& O P Q (J)
3|4 Anti-Ultharian Rook sLssLsLs J K L M N O P Q (J)
2|5 Anti-Celephaïsian Bishop sLssLssL J K L M N O P Q@ (J)
1|6 Anti-Illarnekian Knight ssLsLssL J K L@ M N O P Q@ (J)
0|7 Anti-Dylathian Pawn ssLssLsL J K L@ M N O@ P Q@ (J)

Temperaments

The major temperaments in this area are:

  • Sensi (Parasoft checkertonic)
  • Squares (Parahard checkertonic)

Music

Scale tree

Generator ranges:

  • Chroma-positive generator: 750 cents (5\8) to 800 cents (2\3)
  • Chroma-negative generator: 400 cents (1\3) to 450 cents (3\8)

Template:Scale tree

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