4L 5s (3/1-equivalent)
| ↖3L 4s⟨3/1⟩ | ↑4L 4s⟨3/1⟩ | 5L 4s⟨3/1⟩↗ |
| ←3L 5s⟨3/1⟩ | 4L 5s⟨3/1⟩ | 5L 5s⟨3/1⟩→ |
| ↙3L 6s⟨3/1⟩ | ↓4L 6s⟨3/1⟩ | 5L 6s⟨3/1⟩↘ |
4L 5s⟨3/1⟩ is a non-octave moment of symmetry scale consisting of 4 large steps and 5 small steps, repeating every interval of 3/1 (1901.955¢). This scale is made using a generator ranging from 422.657¢ to 475.489¢, or from 1426.466¢ to 1479.298¢.
Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS with equave 3/1. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament called BPS (Bohlen-Pierce-Stearns) that eliminates only the comma 245/243, so that 9/7 * 9/7 = 5/3.
This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many EDT's (and even EDOs!) besides 13-EDT.
Some low-numbered EDOs that support BPS are 19, 22, 27, 41, and 46, and some low-numbered EDTs that support it are 9, 13, 17, and 30, all of which make it possible to play BP music to some reasonable extent. These equal temperaments contain not only the Lambda BP diatonic scale, but, with the exception 9edt, also the 13-note "Lambda chromatic" MOS scale, or Lambda[13], which can be thought of as a "detempered" version of the 13-EDT Bohlen Pierce scale. This scale may be a suitable melodic substitute for the BP chromatic scale, and is basically the same as how 19-EDO and 31-EDO do not contain 12-EDO as a subset, but they do contain the meantone[12] chromatic scale.
When playing this temperament in some EDO, it may be desired to stretch/compress the tuning so that the tritave is pure, rather than the octave being pure - or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.
One can "add" the octave to BPS temperament by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This is Sensi temperament, in essence treating it as a "3.5.7.2 extension" of the original 3.5.7 BPS temperament.
Modes
| UDP | Step pattern |
|---|---|
| 8|0 | LsLsLsLss |
| 7|1 | LsLsLssLs |
| 6|2 | LsLssLsLs |
| 5|3 | LssLsLsLs |
| 4|4 | sLsLsLsLs |
| 3|5 | sLsLsLssL |
| 2|6 | sLsLssLsL |
| 1|7 | sLssLsLsL |
| 0|8 | ssLsLsLsL |
List of EDT's supporting Lambda Temperament
Below is a list of the equal-temperaments which contain a 4L+5s scale using generators between 422.7 cents and 475.5 cents.
| Steps of ED | Generator in cents | Step ratio | Comments | ||
|---|---|---|---|---|---|
| Bright | Dark | L:s | Hardness | ||
| 2\9edt | 422.657 | 1479.298 | 1:1 | 1 | Equalized 4L 5s⟨3/1⟩ |
| 15\67edt | 425.811 | 1476.144 | 8:7 | 1.143 | |
| 13\58edt | 426.3 | 1475.655 | 7:6 | 1.167 | |
| 24\107edt | 426.607 | 1475.348 | 13:11 | 1.182 | |
| 11\49edt | 426.969 | 1474.986 | 6:5 | 1.2 | |
| 31\138edt | 427.251 | 1474.704 | 17:14 | 1.214 | |
| 20\89edt | 427.406 | 1474.549 | 11:9 | 1.222 | |
| 29\129edt | 427.571 | 1474.384 | 16:13 | 1.231 | |
| 9\40edt | 427.94 | 1474.015 | 5:4 | 1.25 | |
| 34\151edt | 428.255 | 1473.7 | 19:15 | 1.267 | |
| 25\111edt | 428.368 | 1473.587 | 14:11 | 1.273 | |
| 41\182edt | 428.462 | 1473.493 | 23:18 | 1.278 | |
| 16\71edt | 428.61 | 1473.345 | 9:7 | 1.286 | |
| 39\173edt | 428.764 | 1473.191 | 22:17 | 1.294 | |
| 23\102edt | 428.872 | 1473.083 | 13:10 | 1.3 | |
| 30\133edt | 429.012 | 1472.943 | 17:13 | 1.308 | |
| 7\31edt | 429.474 | 1472.481 | 4:3 | 1.333 | Supersoft 4L 5s⟨3/1⟩ |
| 33\146edt | 429.894 | 1472.061 | 19:14 | 1.357 | |
| 26\115edt | 430.007 | 1471.948 | 15:11 | 1.364 | |
| 45\199edt | 430.09 | 1471.865 | 26:19 | 1.368 | |
| 19\84edt | 430.204 | 1471.751 | 11:8 | 1.375 | |
| 50\221edt | 430.307 | 1471.648 | 29:21 | 1.381 | |
| 31\137edt | 430.369 | 1471.586 | 18:13 | 1.385 | |
| 43\190edt | 430.442 | 1471.513 | 25:18 | 1.389 | |
| 12\53edt | 430.631 | 1471.324 | 7:5 | 1.4 | |
| 41\181edt | 430.83 | 1471.125 | 24:17 | 1.412 | |
| 29\128edt | 430.912 | 1471.043 | 17:12 | 1.417 | |
| 46\203edt | 430.985 | 1470.97 | 27:19 | 1.421 | |
| 17\75edt | 431.11 | 1470.845 | 10:7 | 1.429 | |
| 39\172edt | 431.257 | 1470.698 | 23:16 | 1.438 | |
| 22\97edt | 431.371 | 1470.584 | 13:9 | 1.444 | |
| 27\119edt | 431.536 | 1470.419 | 16:11 | 1.455 | |
| 5\22edt | 432.263 | 1469.693 | 3:2 | 1.5 | Soft 4L 5s⟨3/1⟩ |
| 28\123edt | 432.965 | 1468.99 | 17:11 | 1.545 | |
| 23\101edt | 433.118 | 1468.837 | 14:9 | 1.556 | |
| 41\180edt | 433.223 | 1468.732 | 25:16 | 1.563 | |
| 18\79edt | 433.357 | 1468.598 | 11:7 | 1.571 | |
| 49\215edt | 433.469 | 1468.486 | 30:19 | 1.579 | |
| 31\136edt | 433.534 | 1468.421 | 19:12 | 1.583 | |
| 44\193edt | 433.606 | 1468.349 | 27:17 | 1.588 | |
| 13\57edt | 433.779 | 1468.176 | 8:5 | 1.6 | |
| 47\206edt | 433.941 | 1468.014 | 29:18 | 1.611 | |
| 34\149edt | 434.003 | 1467.952 | 21:13 | 1.615 | |
| 55\241edt | 434.056 | 1467.899 | 34:21 | 1.619 | |
| 21\92edt | 434.142 | 1467.813 | 13:8 | 1.625 | |
| 50\219edt | 434.236 | 1467.719 | 31:19 | 1.632 | |
| 29\127edt | 434.305 | 1467.65 | 18:11 | 1.636 | |
| 37\162edt | 434.397 | 1467.558 | 23:14 | 1.643 | |
| 8\35edt | 434.733 | 1467.222 | 5:3 | 1.667 | Semisoft 4L 5s⟨3/1⟩ |
| 35\153edt | 435.088 | 1466.867 | 22:13 | 1.692 | |
| 27\118edt | 435.193 | 1466.762 | 17:10 | 1.7 | |
| 46\201edt | 435.273 | 1466.682 | 29:17 | 1.706 | |
| 19\83edt | 435.387 | 1466.568 | 12:7 | 1.714 | |
| 49\214edt | 435.494 | 1466.461 | 31:18 | 1.722 | |
| 30\131edt | 435.562 | 1466.393 | 19:11 | 1.727 | |
| 41\179edt | 435.643 | 1466.312 | 26:15 | 1.733 | |
| 11\48edt | 435.865 | 1466.09 | 7:4 | 1.75 | |
| 36\157edt | 436.117 | 1465.838 | 23:13 | 1.769 | |
| 25\109edt | 436.228 | 1465.727 | 16:9 | 1.778 | |
| 39\170edt | 436.331 | 1465.624 | 25:14 | 1.786 | |
| 14\61edt | 436.514 | 1465.441 | 9:5 | 1.8 | |
| 31\135edt | 436.745 | 1465.21 | 20:11 | 1.818 | |
| 17\74edt | 436.936 | 1465.019 | 11:6 | 1.833 | |
| 20\87edt | 437.231 | 1464.724 | 13:7 | 1.857 | |
| 3\13edt | 438.913 | 1463.042 | 2:1 | 2 | Basic 4L 5s⟨3/1⟩ |
| 19\82edt | 440.697 | 1461.258 | 13:6 | 2.167 | |
| 16\69edt | 441.033 | 1460.922 | 11:5 | 2.2 | |
| 29\125edt | 441.254 | 1460.701 | 20:9 | 2.222 | |
| 13\56edt | 441.525 | 1460.43 | 9:4 | 2.25 | BPS is in this region |
| 36\155edt | 441.744 | 1460.211 | 25:11 | 2.273 | |
| 23\99edt | 441.868 | 1460.087 | 16:7 | 2.286 | |
| 33\142edt | 442.004 | 1459.951 | 23:10 | 2.3 | |
| 10\43edt | 442.315 | 1459.64 | 7:3 | 2.333 | |
| 37\159edt | 442.593 | 1459.362 | 26:11 | 2.364 | |
| 27\116edt | 442.696 | 1459.259 | 19:8 | 2.375 | |
| 44\189edt | 442.783 | 1459.172 | 31:13 | 2.385 | |
| 17\73edt | 442.921 | 1459.034 | 12:5 | 2.4 | |
| 41\176edt | 443.069 | 1458.886 | 29:12 | 2.417 | |
| 24\103edt | 443.174 | 1458.781 | 17:7 | 2.429 | |
| 31\133edt | 443.313 | 1458.642 | 22:9 | 2.444 | |
| 7\30edt | 443.79 | 1458.166 | 5:2 | 2.5 | Semihard 4L 5s⟨3/1⟩ |
| 32\137edt | 444.252 | 1457.703 | 23:9 | 2.556 | |
| 25\107edt | 444.382 | 1457.573 | 18:7 | 2.571 | |
| 43\184edt | 444.479 | 1457.476 | 31:12 | 2.583 | |
| 18\77edt | 444.613 | 1457.342 | 13:5 | 2.6 | |
| 47\201edt | 444.736 | 1457.219 | 34:13 | 2.615 | |
| 29\124edt | 444.812 | 1457.143 | 21:8 | 2.625 | |
| 40\171edt | 444.902 | 1457.053 | 29:11 | 2.636 | |
| 11\47edt | 445.138 | 1456.817 | 8:3 | 2.667 | |
| 37\158edt | 445.395 | 1456.56 | 27:10 | 2.7 | |
| 26\111edt | 445.503 | 1456.452 | 19:7 | 2.714 | |
| 41\175edt | 445.601 | 1456.354 | 30:11 | 2.727 | |
| 15\64edt | 445.771 | 1456.184 | 11:4 | 2.75 | |
| 34\145edt | 445.976 | 1455.979 | 25:9 | 2.778 | |
| 19\81edt | 446.138 | 1455.817 | 14:5 | 2.8 | |
| 23\98edt | 446.377 | 1455.578 | 17:6 | 2.833 | |
| 4\17edt | 447.519 | 1454.436 | 3:1 | 3 | Hard 4L 5s⟨3/1⟩ |
| 21\89edt | 448.776 | 1453.179 | 16:5 | 3.2 | |
| 17\72edt | 449.073 | 1452.882 | 13:4 | 3.25 | |
| 30\127edt | 449.281 | 1452.674 | 23:7 | 3.286 | |
| 13\55edt | 449.553 | 1452.402 | 10:3 | 3.333 | |
| 35\148edt | 449.787 | 1452.168 | 27:8 | 3.375 | |
| 22\93edt | 449.925 | 1452.03 | 17:5 | 3.4 | |
| 31\131edt | 450.081 | 1451.874 | 24:7 | 3.429 | |
| 9\38edt | 450.463 | 1451.492 | 7:2 | 3.5 | |
| 32\135edt | 450.834 | 1451.121 | 25:7 | 3.571 | |
| 23\97edt | 450.979 | 1450.976 | 18:5 | 3.6 | |
| 37\156edt | 451.105 | 1450.85 | 29:8 | 3.625 | |
| 14\59edt | 451.311 | 1450.644 | 11:3 | 3.667 | |
| 33\139edt | 451.543 | 1450.412 | 26:7 | 3.714 | |
| 19\80edt | 451.714 | 1450.241 | 15:4 | 3.75 | |
| 24\101edt | 451.95 | 1450.005 | 19:5 | 3.8 | |
| 5\21edt | 452.846 | 1449.109 | 4:1 | 4 | Superhard 4L 5s⟨3/1⟩ |
| 21\88edt | 453.876 | 1448.079 | 17:4 | 4.25 | |
| 16\67edt | 454.198 | 1447.757 | 13:3 | 4.333 | |
| 27\113edt | 454.449 | 1447.506 | 22:5 | 4.4 | |
| 11\46edt | 454.815 | 1447.14 | 9:2 | 4.5 | |
| 28\117edt | 455.169 | 1446.786 | 23:5 | 4.6 | |
| 17\71edt | 455.398 | 1446.557 | 14:3 | 4.667 | |
| 23\96edt | 455.677 | 1446.278 | 19:4 | 4.75 | |
| 6\25edt | 456.469 | 1445.486 | 5:1 | 5 | |
| 19\79edt | 457.432 | 1444.523 | 16:3 | 5.333 | |
| 13\54edt | 457.878 | 1444.077 | 11:2 | 5.5 | |
| 20\83edt | 458.302 | 1443.653 | 17:3 | 5.667 | |
| 7\29edt | 459.093 | 1442.862 | 6:1 | 6 | |
| 15\62edt | 460.15 | 1441.805 | 13:2 | 6.5 | |
| 8\33edt | 461.08 | 1440.875 | 7:1 | 7 | |
| 9\37edt | 462.638 | 1439.317 | 8:1 | 8 | |
| 1\4edt | 475.489 | 1426.466 | 1:0 | → ∞ | Collapsed 4L 5s⟨3/1⟩ |
- Schism, by which I mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L+5s MOS.
[Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, e.g. 12edt and 27edt.]