3L 5s

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↖2L 4s ↑3L 4s 4L 4s↗
←2L 5s3L 5s 4L 5s→
↙2L 6s ↓3L 6s 4L 6s↘
Scale structure
Brightest mode LsLssLss
Equave (cents) 2/1 (1200.0¢)
Period (cents) 2/1 (1200.0¢)
Generator ranges
Bright 5\8 (750.0¢) to 2\3 (800.0¢)
Dark 1\3 (400.0¢) to 3\8 (450.0¢)
TAMNAMS information
Name checkertonic
Prefix check-
Abbrev. chk
Related scales
Parent 3L 2s
Sister 5L 3s
Daughters 8L 3s, 3L 8s
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 assumes TAMNAMS for naming step ratios, intevrvals, and scale degrees.

Names for this scale's intervals (mossteps) and scale degrees (mosdegrees) are based on the number of large and small steps from the root, starting at 0 (0-mosstep and 0-mosdegree) for the unison, per TAMNAMS. Ordinal names, such as mos-1st for the unison, are discouraged for non-diatonic MOS scales.

Intervals of 3L 5s
Intervals (with relation to root) Size Abbrev.
Generic Specific L's and s's Range in cents
0-checkstep (root) Perfect 0-checkstep 0 0.0¢ P0ms
1-checkstep Minor 1-checkstep s 0.0¢ to 150.0¢ m1ms
Major 1-checkstep L 150.0¢ to 400.0¢ M1ms
2-checkstep Minor 2-checkstep 2s 0.0¢ to 300.0¢ m2ms
Major 2-checkstep L + s 300.0¢ to 400.0¢ M2ms
3-checkstep Perfect 3-checkstep L + 2s 400.0¢ to 450.0¢ P3ms
Augmented 3-checkstep 2L + s 450.0¢ to 800.0¢ A3ms
4-checkstep Minor 4-checkstep L + 3s 400.0¢ to 600.0¢ m4ms
Major 4-checkstep 2L + 2s 600.0¢ to 800.0¢ M4ms
5-checkstep Diminished 5-checkstep L + 4s 400.0¢ to 750.0¢ d5ms
Perfect 5-checkstep 2L + 3s 750.0¢ to 800.0¢ P5ms
6-checkstep Minor 6-checkstep 2L + 4s 800.0¢ to 900.0¢ m6ms
Major 6-checkstep 3L + 3s 900.0¢ to 1200.0¢ M6ms
7-checkstep Minor 7-checkstep 2L + 5s 800.0¢ to 1050.0¢ m7ms
Major 7-checkstep 3L + 4s 1050.0¢ to 1200.0¢ M7ms
8-checkstep (octave) Perfect 8-checkstep 3L + 5s 1200.0¢ P8ms

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

Degree Size in 11edo (basic) Size in 14edo (hard) Size in 19edo (soft) Note name on J #Gens up
min. chk2nd 1\11, 109.1 1\14, 85.7 2\19, 126.3 K +3
maj. chk2nd 2\11, 218.2 3\14, 257.1 3\19, 189.5 K& -5
min. chk3rd 2\11, 218.2 2\14, 171.4 4\19, 252.6 L@ +6
maj. chk3rd 3\11, 327.3 4\14, 342.9 5\19, 315.8 L -2
perf. chk4th 4\11, 436.4 5\14, 428.6 7\19, 442.1 M +1
aug. chk4th 5\11, 545.5 7\14, 600.0 8\19, 505.3 M& -7
min. chk5th 5\11, 545.5 6\14, 514.3 9\19, 568.4 N +4
maj. chk5th 6\11, 656.6 8\14, 685.7 10\19, 631.6 N& -4
dim. chk6th 6\11, 656.6 7\14, 600.0 11\19, 694.7 O@ +7
perf. chk6th 7\11, 763.6 8\14, 771.4 12\19, 757.9 O -1
min. chk7th 8\11, 872.7 10\14, 857.1 14\19, 884.2 P +2
maj. chk7th 9\11, 981.8 12\14, 1028.6 15\19, 947.4 P& -6
min. chk8th 9\11, 981.8 11\14, 942.9 16\19, 1010.5 Q@ +5
maj. chk8th 10\11, 1090.9 13\14, 1114.3 17\19, 1073.7 Q -3

Parasoft

Parasoft checkertonic is the narrow region between 7\19 (442.1¢) and 10\27 (444.4¢).

Sortable table of major and minor intervals in parasoft checkertonic tunings:

Degree Size in 19edo (soft) Size in 27edo (supersoft) Size in 46edo Note name on J Approximate ratios #Gens up
unison 0\19, 0.00 0\27, 0.00 0\46, 0.00 J 1/1 0
min. chk2nd 2\19, 126.3 3\27, 133.3 5\46, 130.4 K 14/13 +3
maj. chk2nd 3\19, 189.5 4\27, 177.8 7\46, 182.6 K& 10/9 -5
min. chk3rd 4\19, 252.6 6\27, 266.7 10\46, 260.9 L@ 7/6 +6
maj. chk3rd 5\19, 315.8 7\27, 311.1 12\46, 313.0 L 6/5 -2
perf. chk4th 7\19, 442.1 10\27, 444.4 17\46, 443.5 M 9/7, 13/10 +1
aug. chk4th 8\19, 505.3 11\27, 488.9 19\46, 495.7 M& 4/3 -7
min. chk5th 9\19, 568.4 13\27, 577.8 22\46, 573.9 N 7/5, 18/13 +4
maj. chk5th 10\19, 631.6 14\27, 622.2 24\46, 626.1 N& 10/7, 13/9 -4
dim. chk6th 11\19, 694.7 16\27, 711.1 27\46, 704.3 O@ 3/2 +7
perf. chk6th 12\19, 757.9 17\27, 755.6 20\46, 756.5 O 14/9, 20/13 -1
min. chk7th 14\19, 884.2 20\27, 888.9 34\46, 887.0 P 5/3 +2
maj. chk7th 15\19, 947.4 21\27, 933.3 36\46, 939.1 P& 12/7 -6
min. chk8th 16\19, 1010.5 23\27, 1022.2 39\46, 1017.4 Q@ 9/5 +5
maj. chk8th 17\19, 1073.7 24\27, 1066.7 41\46, 1069.6 Q 13/7 -3

Tunings in this region have a regular temperament interpretation called sensi.

Modes

Checkertonic modes can be named by prefixing anti- to their counterpart modes in the MOS sister oneirotonic.

  1. Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss
  2. Anti-Hlanithian (lə-NITH-iən): LssLsLss
  3. Anti-Kadathian (kə-DA(H)TH-iən): LssLssLs
  4. Anti-Mnarian (mə-NA(I)R-iən): sLsLssLs
  5. Anti-Ultharian (ul-THA(I)R-iən): sLssLsLs
  6. Anti-Celephaïsian (kel-ə-FAY-zhən): sLssLssL
  7. Anti-Illarnekian (ill-ar-NEK-iən): ssLsLssL
  8. Anti-Dylathian (də-LA(H)TH-iən): ssLssLsL

The modes on the white keys JKLMNOPQJ are:

  • J Anti-Ultharian
  • K Anti-Hlanithian
  • L Anti-Illarnekian
  • M Anti-Mnarian
  • N Anti-Sarnathian
  • O Anti-Celephaïsian
  • P Anti-Kadathian
  • Q Anti-Dylathian
Table of modes (based on J, from brightest to darkest)
Mode 1 2 3 4 5 6 7 8 (9)
Anti-Sarnathian J K& L M& N& O P& Q (J)
Anti-Hlanithian J K& L M N& O P& Q (J)
Anti-Kadathian J K& L M N& O P Q (J)
Anti-Mnarian J K L M N& O P Q (J)
Anti-Ultharian J K L M N O P Q (J)
Anti-Celephaïsian J K L M N O P Q@ (J)
Anti-Illarnekian J K L@ M N O P Q@ (J)
Anti-Dylathian J K L@ M N O@ P Q@ (J)

Temperaments

The major temperaments in this area are:

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

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)
Generator Cents L s L/s Comments
5\8 750.000 1 1 1.000
27\43 753.488 6 5 1.200
22\35 754.286 5 4 1.250
39\62 754.839 9 7 1.286
17\27 755.556 4 3 1.333
46\73 756.164 11 8 1.375
29\46 756.522 7 5 1.400 Sensi is in this region
41\65 756.923 10 7 1.429
12\19 757.895 3 2 1.500
43\68 758.824 11 7 1.571 Clyde
31\49 759.184 8 5 1.600
50\79 759.494 13 8 1.625 Golden checkertonic/sentry (759.4078¢)
19\30 760.000 5 3 1.667
45\71 760.563 12 7 1.714
26\41 760.976 7 4 1.750
33\52 761.538 9 5 1.800
7\11 763.636 2 1 2.000 Basic checkertonic
(Generators smaller than this are proper)
30\47 765.957 9 4 2.250
23\36 766.667 7 3 2.333
39\61 767.213 12 5 2.400
16\25 768.000 5 2 2.500
41\64 768.750 13 5 2.600 Unnamed golden tuning (768.8815¢)
25\39 769.231 8 3 2.667
34\53 769.811 11 4 2.750 Hamity
9\14 771.429 3 1 3.000
29\45 773.333 10 3 3.333
20\31 774.194 7 2 3.500 Squares is in this region
31\48 775.000 11 3 3.667
11\17 776.471 4 1 4.000
24\37 778.378 9 2 4.500
13\20 780.000 5 1 5.000
15\23 782.609 6 1 6.000 Roman↓, Hocus
2\3 800.000 1 0 → inf