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.

Extensions

Several extensions of this temperament are possible to incorporate additional harmonics.

While 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. The convenient comma here is 1331/1323 that can be tempered out, but this splits the 9/7 generator (plus a tritave) in three and therefore functions instead as a 5-limit extension of Mintaka.

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.

Interval chains

In the below, tritave-reduced harmonics below 243 are indicated in bold.

Basic 3.5.7 temperament

#	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, 875/729
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

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
	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