Bohlen-Pierce-Stearns

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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.

Basic BPS (extension-agnostic)
# 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
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