3L 5s

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Icon-Todo.png Todo: cleanup
Adopt zero-indexed intervals/degrees, add scale tree entries from previous page version
↖ 2L 4s↑ 3L 4s 4L 4s ↗
← 2L 5s3L 5s4L 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 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

Simple Tunings of 3L 5s
Scale degree Abbrev. Basic (2:1)
11edo 		Hard (3:1)
14edo 		Soft (3:2)
19edo 		Approx. ratios*
Steps ¢ Steps ¢ Steps ¢
Perfect 0-checkdegree P0chkd 0\11 0.0 0\14 0.0 0\19 0.0 1/1
Minor 1-checkdegree m1chkd 1\11 109.1 1\14 85.7 2\19 126.3 16/15, 14/13
Major 1-checkdegree M1chkd 2\11 218.2 3\14 257.1 3\19 189.5 9/8, 8/7
Minor 2-checkdegree m2chkd 2\11 218.2 2\14 171.4 4\19 252.6 9/8, 8/7
Major 2-checkdegree M2chkd 3\11 327.3 4\14 342.9 5\19 315.8 6/5, 11/9
Perfect 3-checkdegree P3chkd 4\11 436.4 5\14 428.6 7\19 442.1 14/11, 9/7
Augmented 3-checkdegree A3chkd 5\11 545.5 7\14 600.0 8\19 505.3 11/8, 18/13
Minor 4-checkdegree m4chkd 5\11 545.5 6\14 514.3 9\19 568.4 11/8, 18/13
Major 4-checkdegree M4chkd 6\11 654.5 8\14 685.7 10\19 631.6 13/9, 16/11
Diminished 5-checkdegree d5chkd 6\11 654.5 7\14 600.0 11\19 694.7 13/9, 16/11
Perfect 5-checkdegree P5chkd 7\11 763.6 9\14 771.4 12\19 757.9 14/9, 11/7
Minor 6-checkdegree m6chkd 8\11 872.7 10\14 857.1 14\19 884.2 18/11, 5/3
Major 6-checkdegree M6chkd 9\11 981.8 12\14 1028.6 15\19 947.4 7/4, 16/9
Minor 7-checkdegree m7chkd 9\11 981.8 11\14 942.9 16\19 1010.5 7/4, 16/9
Major 7-checkdegree M7chkd 10\11 1090.9 13\14 1114.3 17\19 1073.7 13/7, 15/8
Perfect 8-checkdegree P8chkd 11\11 1200.0 14\14 1200.0 19\19 1200.0 2/1

* Ratios shown are within the 50-integer limit. Automatic search may be inexact. Other interpretations are possible.

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

* Ratios shown are within the 2.3.5.7.13 subgroup. Automatic search may be inexact. Other interpretations are possible.

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
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
31\49 759.184 440.816 8:5 1.600
50\79 759.494 440.506 13:8 1.625
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
25\39 769.231 430.769 8:3 2.667
34\53 769.811 430.189 11:4 2.750
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
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
2\3 800.000 400.000 1:0 → ∞ Collapsed 3L 5s
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