3L 5s
↖2L 4s | ↑3L 4s | 4L 4s↗ |
←2L 5s | 3L 5s | 4L 5s→ |
↙2L 6s | ↓3L 6s | 4L 6s↘ |
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 (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.
- Anti-Sarnathian (sar-NA(H)TH-iən): LsLssLss
- Anti-Hlanithian (lə-NITH-iən): LssLsLss
- Anti-Kadathian (kə-DA(H)TH-iən): LssLssLs
- Anti-Mnarian (mə-NA(I)R-iən): sLsLssLs
- Anti-Ultharian (ul-THA(I)R-iən): sLssLsLs
- Anti-Celephaïsian (kel-ə-FAY-zhən): sLssLssL
- Anti-Illarnekian (ill-ar-NEK-iən): ssLsLssL
- 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
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:
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 |