BPS
BPS (for Bohlen–Pierce–Stearns) 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⟨3/1⟩, 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.
| BPS |
3.5.7 49-throdd-limit: 9.46 ¢
3.5.7 49-throdd-limit: 9 notes
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 3:5:7:9 tetrad to serve as BPS' primary consonance, similar to how the 4:5:6 triad serves as meantone's primary consonance.
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⟨3/1⟩ and 5L 7s⟨3/1⟩ 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⟨3/1⟩ enneatonic and 9L 8s⟨3/1⟩ 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
It is also possible to incorporate octaves into BPS. The logical choices for a mapping of 2 are 7 generators up (equating 2/1 to 125/63), which produces sensi, and 6 generators down (equating 2/1 to 49/25), which produces hedgehog.
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
Tunings
Norm-based tunings
| Euclidean | |||
|---|---|---|---|
| Constrained | Constrained & skewed | Destretched | |
| Tenney | CTE: ~9/7 = 441.1431 ¢ | CWE: ~9/7 = 440.6646 ¢ | POTE: ~9/7 = 440.4881 ¢ |
Other tunings
- DKW (3.5.7): ~3 = 1901.955 ¢, ~9/7 = 440.554 ¢
Tuning spectrum
| Edt generator |
Unchanged interval (eigenmonzo)* |
Generator (¢) | Comments |
|---|---|---|---|
| 5\22 | 432.263 | ||
| 8\35 | 434.733 | ||
| 9/7 | 435.084 | Untempered | |
| 11\48 | 435.865 | ||
| 14\61 | 436.514 | ||
| 438.038 | DR 5:7:9, close to 5/24-comma | ||
| 135/49 | 438.632 | 1/4-comma | |
| 3\13 | 438.913 | Equal-tempered Bohlen–Pierce | |
| 15/7 | 439.814 | 1/3-comma | |
| 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 | ||
| 2/1 | 443.136 | Sensi mapping of 2/1 to ~125/63 | |
| 7\30 | 443.790 | ||
| 11\47 | 445.138 | ||
| 4\17 | 447.519 | ||
| 35/27 | 449.275 | Full comma |
* Besides the tritave