Bohlen–Pierce–Stearns
Bohlen–Pierce–Stearns (BPS) is a non-octave temperament in the 3.5.7 subgroup generated by a sharp ~9/7 (or equivalently a flat ~7/3), tempering out the sensamagic comma, 245/243 so that a stack of two generators represents 5/3 in addition to 81/49. This generates a MOS scale of 4L 5s against the tritave, known as the Bohlen–Pierce Lambda scale. The "canonical" tuning for the generator is 3\13edt, representing the equal-tempered Bohlen–Pierce scale, but a range of other tunings are valid, including 4\17edt, 7\30edt, and 10\43edt.
As the generator of the Bohlen–Pierce scale, and the simplest decently accurate temperament of the 3.5.7 subgroup, this temperament fulfills a niche similar to meantone of the 2.3.5 subgroup, allowing for the tetrad 3:5:7:9 to serve as the theory's primary consonant tetrad.
For technical data, see Sensamagic clan#BPS or No-twos subgroup temperaments#BPS (currently, extensions with 2 are stored on the former page and no-twos extensions are stored on the latter).
Extensions
Several extensions of this temperament are possible to incorporate additional harmonics.
In the 11-limit, 1331/1323 is the most convenient comma that can be tempered out, which produces Mintra temperament; this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a weak extension of BPS, and a strong add-5 extension of Mintaka, which produces 5L 2s and 5L 7s MOS scales (functioning as a macro-superpyth). Simple tunings include 17edt and 39edt.
Another weak extension to add prime 17, known as Dubhe, splits the 9/7 BPS generator in half, by tempering out 2025/2023 and equating two of 17/15 to 9/7. This produces 8L 1s enneatonic and 9L 8s chromatic MOS scales. Simple tunings include 17edt and 26edt.
Strong extensions
While strong 11-limit extensions can be proposed, tempering out 77/75 in the sharper range (i.e. sharp of 3\13edt) and 1375/1323 in the flatter range, neither of these are of particular accuracy; more accurate extensions would be of considerably higher complexity. However, one could argue for the canonicity of the latter extension by being the no-twos retraction of 11-limit hedgehog temperament (which, as a member of the porcupine family, makes more sense to consider with prime 11 in mind than without it).
In the 13-limit, sharp tunings can generally map the 13th harmonic by tempering out 637/625 and identifying (25/21)2 with 13/9, which is optimal near the 30edt tuning. For flat tunings, it is more accurate to temper out 65/63 instead.
One harmonic that can be placed on the generator chain with some accuracy, compared to other primes, is the 19th. Sharp of 13edt, it is best to temper out 11907/11875 and equate (25/21)2 to 27/19, thereby having the 19th harmonic 10 generators down. But on the flat side of the spectrum, it is less complex and more accurate flat of 13edt to temper out 6561/6517, or equivalently 135/133, so that 19/9 is equated to (9/7)3, or otherwise 15/7, though this mapping of 19 is exact flat of 22edt.
Prime 2
Main article: Relationship between Bohlen-Pierce and octave-ful temperaments.
It is also possible to incorporate octaves into the BPS temperament. The logical choices for a mapping of 2 are 7 generators up (equating 2/1 to 125/63), which produces sensi temperament, and 6 generators down (equating 2/1 to 49/25), which produces hedgehog temperament.
Interval chains
These interval chains cover strong extensions of BPS. For Mintra, see Mintaka#Mintra.
In the below, tritave-reduced harmonics below 243 are indicated in bold.
| # | Cents* | Approximate Ratios | |
|---|---|---|---|
| 3.5.7 subgroup | Add-19 extension | ||
| −4 | 139.7 | 27/25, 49/45 | 21/19, 133/125 |
| −3 | 580.3 | 7/5, 243/175 | 27/19, 171/125 |
| −2 | 1020.8 | 9/5, 49/27 | 35/19, 133/75, 243/133 |
| −1 | 1461.4 | 7/3, 81/35 | 45/19, 57/25 |
| 0 | 0.0 | 1/1, 245/243 | 135/133, 175/171, 375/361 |
| 1 | 440.6 | 9/7, 35/27 | 19/15, 25/19 |
| 2 | 881.1 | 5/3, 81/49 | 57/35, 133/81, 225/133 |
| 3 | 1321.7 | 15/7, 175/81 | 19/9, 125/57 |
| 4 | 1762.2 | 25/9, 135/49 | 19/7, 375/133 |
| 5 | 300.8 | 25/21, 405/343 | 57/49, 95/81 |
| 6 | 741.4 | 75/49, 125/81 | 95/63, 361/243 |
| 7 | 1181.9 | 125/63, 675/343 | 95/49, 361/189 |
| 8 | 1622.5 | 125/49, 625/243 | 361/147, 475/189 |
| 9 | 161.1 | 375/343, 625/567 | 361/343 |
* In 3.5.7-targeted DKW tuning
Tuning spectrum
| Edt Generator |
Eigenmonzo (Unchanged-interval)* |
Generator (¢) |
Comments |
|---|---|---|---|
| 5\22 | 432.263 | ||
| 7/3 | 435.084 | 0-comma | |
| 14\61 | 436.514 | ||
| 3645/2401 | 437.449 | 1/6-comma | |
| 23\100 | 437.450 | ||
| 49/45 | 438.632 | 1/4-comma | |
| 3\13 | 438.913 | Equal-tempered Bohlen-Pierce | |
| 7/5 | 439.814 | 1/3-comma | |
| 25\108 | 440.267 | ||
| 440.340 | DR 3:5:7, close to 10/27-comma | ||
| 22\95 | 440.453 | ||
| 19\82 | 440.697 | ||
| 25/21 | 440.760 | 2/5-comma; CEE tuning | |
| 16\69 | 441.033 | ||
| 13\56 | 441.525 | ||
| 5/3 | 442.179 | 1/2-comma | |
| 10\43 | 442.315 | ||
| 17\73 | 442.921 | ||
| 7\30 | 443.790 | ||
| 175/81 | 444.544 | 2/3-comma | |
| 18\77 | 444.613 | ||
| 11\47 | 445.138 | ||
| 4\17 | 447.519 | ||
| 35/27 | 449.275 | Full comma |
* Besides the tritave
Other tunings
- DKW (3.5.7): ~3 = 1\1, ~9/7 = 440.554