BPS
Bohlen-Pierce-Stearns (BPS) is a 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, which 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.
While strong 11-limit extensions can be proposed, tempering out 77/75 in the flat range and 1375/1323 in the sharp 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).
Sharp tunings generally possess a more convenient 13th harmonic than 11th, by tempering out 637/625 and identifying (25/21)2 with 13/9, which is optimal near the 30edt tuning. It is then very easy to insert in 19 by tempering 247/245, and identifying 13/9 with 27/19, therefore placing the 19th harmonic 10 generators down; this extension is optimal near the 56edt tuning.
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 |
|---|---|---|
| -4 | 139.7 | 27/25, 49/45 |
| -3 | 580.3 | 7/5, 243/175 |
| -2 | 1020.8 | 9/5, 49/27 |
| -1 | 1461.4 | 7/3, 81/35 |
| 0 | 0.0 | 1/1 |
| 1 | 440.6 | 9/7, 35/27 |
| 2 | 881.1 | 5/3, 81/49 |
| 3 | 1321.7 | 15/7, 175/81 |
| 4 | 1762.2 | 25/9, 135/49 |
| 5 | 300.8 | 25/21, 405/343 |
| 6 | 741.4 | 75/49, 125/81 |
| 7 | 1181.9 | 125/63, 675/343 |
| 8 | 1622.5 | 125/49, 625/243 |
| 9 | 161.1 | 375/343, 625/567 |
| # | Cents* | Approximate Ratios |
|---|---|---|
| -4 | 131.4 | 27/25, 49/45, 125/117 |
| -3 | 574.0 | 7/5, 243/175, 125/91 |
| -2 | 1016.7 | 9/5, 49/27 |
| -1 | 1459.3 | 7/3, 81/35 |
| 0 | 0.0 | 1/1 |
| 1 | 442.6 | 9/7, 35/27 |
| 2 | 885.3 | 5/3, 81/49 |
| 3 | 1327.9 | 15/7, 175/81, 273/125 |
| 4 | 1770.5 | 25/9, 135/49, 351/125 |
| 5 | 311.2 | 91/75, 25/21, 343/285, 405/343 |
| 6 | 753.8 | 39/25, 75/49, 147/95, 125/81 |
| 7 | 1196.4 | 91/45, 189/95, 125/63 |
| 8 | 1639.1 | 13/5, 637/243, 243/95, 125/49 |
| 9 | 179.7 | 39/35, 91/81, 21/19, 375/343 |
| 10 | 622.4 | 13/9, 27/19, 625/441 |
| 11 | 1065.0 | 13/7, 455/243, 243/133, 35/19 |
| 12 | 1507.6 | 65/27, 117/49, 45/19 |
| 13 | 48.3 | 65/63, 135/133, 175/171 |
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 | |
| 16\69 | 441.033 | ||
| 19/9 | 441.226 | ||
| 13\56 | 441.525 | ||
| 5/3 | 442.179 | 1/2-comma | |
| 10\43 | 442.315 | ||
| 17\73 | 442.921 | ||
| 7\30 | 443.790 | ||
| 13/9 | 444.053 | ||
| 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 |
Other tunings
- DKW (3.5.7): ~3 = 1\1, ~9/7 = 440.554