3L 5s: Difference between revisions

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'''3L 5s''' refers to the structure of octave-equivalent [[MOS]] scales with generators ranging from 1\3 (one degree of [[3edo]] = 400¢) to 3\8 (three degrees of [[8edo]] = 450¢). In the case of 8edo, L and s are the same size; in the case of 3edo, s becomes so small it disappears (and all that remains are the three equal L's). The pattern is also named ''antioneirotonic'' because it is the [[oneirotonic]] (5L 3s) MOS pattern with large and small steps switched. In contrast to oneirotonic scales, which often require the usage of completely new chords to have consonant-sounding music, some checkertonic scales contain approximations to a perfect fifth (usually as a dim. chk6th or maj. chk5th), and thus can be used for traditional root-3rd-P5 harmony.
{{MOS intro}}
 
In contrast to oneirotonic (5L 3s) scales, 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.


There are two significant harmonic entropy minima with this MOS pattern. [[Sensipent family|Sensi]], in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, is the proper one. [[Meantone family #Squares|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, is improper.
There are two significant harmonic entropy minima with this MOS pattern. [[Sensipent family|Sensi]], in which the generator is a 9/7, two of them make a 5/3, and seven of them make a 3/2, is the proper one. [[Meantone family #Squares|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, is improper.

Revision as of 23:24, 31 May 2023

↖ 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, 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 ¢.

In contrast to oneirotonic (5L 3s) scales, 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.

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, is the proper one. 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, is improper.

Standing assumptions

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.

Names

The TAMNAMS name for 3L 5s is checkertonic.

Intervals

Note: In TAMNAMS, a k-step interval class in checkertonic may be called a "k-step", "k-mosstep", or "k-checkstep". 1-indexed terms such as "mos(k+1)th" are discouraged for non-diatonic mosses.

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
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