3L 5s: Difference between revisions

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Line 35: Line 35:
! #Gens up
! #Gens up
|-
|-
| min. sen2nd
| min. chk2nd
| 1\11, 109.1
| 1\11, 109.1
| 1\14, 85.7
| 1\14, 85.7
Line 42: Line 42:
| +3
| +3
|-
|-
| maj. sen2nd
| maj. chk2nd
| 2\11, 218.2
| 2\11, 218.2
| 3\14, 257.1
| 3\14, 257.1
Line 49: Line 49:
| -5
| -5
|-
|-
| min. sen3rd
| min. chk3rd
| 2\11, 218.2
| 2\11, 218.2
| 2\14, 171.4
| 2\14, 171.4
Line 56: Line 56:
| +6
| +6
|-
|-
| maj. sen3rd
| maj. chk3rd
| 3\11, 327.3
| 3\11, 327.3
| 4\14, 342.9
| 4\14, 342.9
Line 63: Line 63:
| -2
| -2
|-
|-
| perf. sen4th
| perf. chk4th
| 4\11, 436.4
| 4\11, 436.4
| 5\14, 428.6
| 5\14, 428.6
Line 70: Line 70:
| +1
| +1
|-
|-
| aug. sen4th
| aug. chk4th
| 5\11, 545.5
| 5\11, 545.5
| 7\14, 600.0
| 7\14, 600.0
Line 77: Line 77:
| -7
| -7
|-
|-
| min. sen5th
| min. chk5th
| 5\11, 545.5
| 5\11, 545.5
| 6\14, 514.3
| 6\14, 514.3
Line 84: Line 84:
| +4
| +4
|-
|-
| maj. sen5th
| maj. chk5th
| 6\11, 656.6
| 6\11, 656.6
| 8\14, 685.7
| 8\14, 685.7
Line 91: Line 91:
| -4
| -4
|-
|-
| dim. sen6th
| dim. chk6th
| 6\11, 656.6
| 6\11, 656.6
| 7\14, 600.0
| 7\14, 600.0
Line 98: Line 98:
| +7
| +7
|-
|-
| perf. sen6th
| perf. chk6th
| 7\11, 763.6
| 7\11, 763.6
| 8\14, 771.4
| 8\14, 771.4
Line 105: Line 105:
| -1
| -1
|-
|-
| min. sen7th
| min. chk7th
| 8\11, 872.7
| 8\11, 872.7
| 10\14, 857.1
| 10\14, 857.1
Line 112: Line 112:
| +2
| +2
|-
|-
| maj. sen7th
| maj. chk7th
| 9\11, 981.8
| 9\11, 981.8
| 12\14, 1028.6
| 12\14, 1028.6
Line 119: Line 119:
| -6
| -6
|-
|-
| min. sen8th
| min. chk8th
| 9\11, 981.8
| 9\11, 981.8
| 11\14, 942.9
| 11\14, 942.9
Line 126: Line 126:
| +5
| +5
|-
|-
| maj. sen8th
| maj. chk8th
| 10\11, 1090.9
| 10\11, 1090.9
| 13\14, 1114.3
| 13\14, 1114.3
Line 157: Line 157:
| 0
| 0
|-
|-
| min. sen2nd
| min. chk2nd
| 2\19, 126.3
| 2\19, 126.3
| 3\27, 133.3
| 3\27, 133.3
Line 165: Line 165:
| +3
| +3
|-
|-
| maj. sen2nd
| maj. chk2nd
| 3\19, 189.5
| 3\19, 189.5
| 4\27, 177.8
| 4\27, 177.8
Line 173: Line 173:
| -5
| -5
|-
|-
| min. sen3rd
| min. chk3rd
| 4\19, 252.6
| 4\19, 252.6
| 6\27, 266.7
| 6\27, 266.7
Line 181: Line 181:
| +6
| +6
|-
|-
| maj. sen3rd
| maj. chk3rd
| 5\19, 315.8
| 5\19, 315.8
| 7\27, 311.1
| 7\27, 311.1
Line 189: Line 189:
| -2
| -2
|-
|-
| perf. sen4th
| perf. chk4th
| 7\19, 442.1
| 7\19, 442.1
| 10\27, 444.4
| 10\27, 444.4
Line 197: Line 197:
| +1
| +1
|-
|-
| aug. sen4th
| aug. chk4th
| 8\19, 505.3
| 8\19, 505.3
| 11\27, 488.9
| 11\27, 488.9
Line 205: Line 205:
| -7
| -7
|-
|-
| min. sen5th
| min. chk5th
| 9\19, 568.4
| 9\19, 568.4
| 13\27, 577.8
| 13\27, 577.8
Line 213: Line 213:
| +4
| +4
|-
|-
| maj. sen5th
| maj. chk5th
| 10\19, 631.6
| 10\19, 631.6
| 14\27, 622.2
| 14\27, 622.2
Line 221: Line 221:
| -4
| -4
|-
|-
| dim. sen6th
| dim. chk6th
| 11\19, 694.7
| 11\19, 694.7
| 16\27, 711.1
| 16\27, 711.1
Line 229: Line 229:
| +7
| +7
|-
|-
| perf. sen6th
| perf. chk6th
| 12\19, 757.9
| 12\19, 757.9
| 17\27, 755.6
| 17\27, 755.6
Line 237: Line 237:
| -1
| -1
|-
|-
| min. sen7th
| min. chk7th
| 14\19, 884.2
| 14\19, 884.2
| 20\27, 888.9
| 20\27, 888.9
Line 245: Line 245:
| +2
| +2
|-
|-
| maj. sen7th
| maj. chk7th
| 15\19, 947.4
| 15\19, 947.4
| 21\27, 933.3
| 21\27, 933.3
Line 253: Line 253:
| -6
| -6
|-
|-
| min. sen8th
| min. chk8th
| 16\19, 1010.5
| 16\19, 1010.5
| 23\27, 1022.2
| 23\27, 1022.2
Line 261: Line 261:
| +5
| +5
|-
|-
| maj. sen8th
| maj. chk8th
| 17\19, 1073.7
| 17\19, 1073.7
| 24\27, 1066.7
| 24\27, 1066.7
Line 270: Line 270:
|}
|}


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


== Modes ==
== Modes ==

Revision as of 08:14, 15 August 2022

↖ 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 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.

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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