3L 5s
| ↖2L 4s | ↑3L 4s | 4L 4s↗ |
| ←2L 5s | 3L 5s | 4L 5s→ |
| ↙2L 6s | ↓3L 6s | 4L 6s↘ |
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.
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.
- 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 | ||||||