Xenharmonic Wiki

3L 5s

Revision as of 13:23, 14 August 2024 by R-4981 (talk | contribs)
Todo: cleanup

Adopt zero-indexed intervals/degrees

↖ 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 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

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

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)
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
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