3L 5s

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↖ 2L 4s ↑ 3L 4s 4L 4s ↗
← 2L 5s 3L 5s 4L 5s →
↙ 2L 6s ↓ 3L 6s 4L 6s ↘ 
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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 diatonic interval categories.
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

Lua error in Module:MOS_tunings at line 299: attempt to call field 'search_by_args_within_equave' (a nil value).

Parasoft tunings

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

Lua error in Module:MOS_tunings at line 304: attempt to call field 'parse_search_args' (a nil value).

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		 Anti-modes of 5L 3s Grand chess names[proposed]
7|0 1 LsLssLss Anti-Sarnathian (sar-NA(H)TH-iən) King
6|1 6 LssLsLss Anti-Hlanithian (lə-NITH-iən) Queen
5|2 3 LssLssLs Anti-Kadathian (kə-DA(H)TH-iən) Marshall
4|3 8 sLsLssLs Anti-Mnarian (mə-NA(I)R-iən) Cardinal
3|4 5 sLssLsLs Anti-Ultharian (ul-THA(I)R-iən) Rook
2|5 2 sLssLssL Anti-Celephaïsian (kel-ə-FAY-zhən) Bishop
1|6 7 ssLsLssL Anti-Illarnekian (ill-ar-NEK-iən) Knight
0|7 4 ssLssLsL Anti-Dylathian (də-LA(H)TH-iən) Pawn

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)


Scale Tree and Tuning Spectrum of 3L 5s
Generator(edo) Cents Step ratio Comments
Bright Dark L:s Hardness
5\8 750.000 450.000 1:1 1.000 Equalized 3L 5s
27\43 753.488 446.512 6:5 1.200
22\35 754.286 445.714 5:4 1.250
39\62 754.839 445.161 9:7 1.286
17\27 755.556 444.444 4:3 1.333 Supersoft 3L 5s
46\73 756.164 443.836 11:8 1.375
29\46 756.522 443.478 7:5 1.400 Sensi (optimal around here)
41\65 756.923 443.077 10:7 1.429
12\19 757.895 442.105 3:2 1.500 Soft 3L 5s
43\68 758.824 441.176 11:7 1.571 Clyde
31\49 759.184 440.816 8:5 1.600
50\79 759.494 440.506 13:8 1.625 Golden sentry (759.4078¢)
19\30 760.000 440.000 5:3 1.667 Semisoft 3L 5s
45\71 760.563 439.437 12:7 1.714
26\41 760.976 439.024 7:4 1.750
33\52 761.538 438.462 9:5 1.800
7\11 763.636 436.364 2:1 2.000 Basic 3L 5s
Scales with tunings softer than this are proper
30\47 765.957 434.043 9:4 2.250
23\36 766.667 433.333 7:3 2.333
39\61 767.213 432.787 12:5 2.400
16\25 768.000 432.000 5:2 2.500 Semihard 3L 5s
41\64 768.750 431.250 13:5 2.600 Unnamed golden tuning (768.8815¢)
25\39 769.231 430.769 8:3 2.667
34\53 769.811 430.189 11:4 2.750 Hamity
9\14 771.429 428.571 3:1 3.000 Hard 3L 5s
29\45 773.333 426.667 10:3 3.333
20\31 774.194 425.806 7:2 3.500 Squares (optimal around here)
31\48 775.000 425.000 11:3 3.667
11\17 776.471 423.529 4:1 4.000 Superhard 3L 5s
24\37 778.378 421.622 9:2 4.500
13\20 780.000 420.000 5:1 5.000
15\23 782.609 417.391 6:1 6.000 Roman↓, hocus↓
2\3 800.000 400.000 1:0 → ∞ Collapsed 3L 5s
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