Bohlen-Pierce-Stearns
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.
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 |
Other tunings
- DKW (3.5.7): ~3 = 1\1, ~9/7 = 440.554