The Bohlen-Pierce (BP) scale is a nonoctave scale, a 13-part equal division of the perfect-twelfth (3/1) or Tritave (13edt). Each step is about 146 ¢, making it a macrotonal scale. It is closely related to the rank two temperament bohpier. Bohlen-Pierce is normally thought of (if not in these terms, then in fact) as a temperament defined on the 3.5.7 subgroup. However, it (or at least 3.5.7-limit 13edt) can be extended to the 3.5.7.11/4 subgroup. This extension is controversial because of the presence of 2 in the denominator of 11/4, but the interval is present in the sense that 3^(12\13) provides an approximation to it. Chords of Bohlen-Pierce, from this extended perspective, may be found listed on the page chords of bohpier. Bohlen-Pierce was discovered independently by Heinz Bohlen, John Pierce, Kees van Prooijen, and perhaps others, usually noticed for its good approximation of odd-number just ratios 3:5, 5:7, 3:7, etc.; but not necessarily 4:11, 5:6, 6:7, etc.

Chris Vaisvil's BP electric guitar. Music from this guitar.

The Rank-2 "Lambda" Temperament and the BP nonatonic scale

Suggested for use as a "diatonic scale" when playing Bohlen-Pierce is the 9-note Lambda scale, which is the 4L5s MOS. This can be thought of as an MOS generated by a 3.5.7 rank-2 temperament that eliminates only the comma 245/243, so that 9/7 * 9/7 = 5/3.

This is a very good temperament on the 3.5.7 subgroup, and additionally is supported by many EDT's (and even EDOs!) besides 13-EDT.

Some low-numbered EDOs that support Lambda are 19, 22, 27, 41, and 46, all of which make it possible to play BP music to some reasonable extent. These EDOs contain not only the Lambda BP diatonic scale, but also the 13-note "Lambda chromatic" MOS scale, or Lambda[13], which can be thought of as a "detempered" version of the 13-EDT Bohlen Pierce scale. This scale may be a suitable melodic substitute for the BP chromatic scale, and is basically the same as how 19-EDO and 31-EDO do not contain 12-EDO as a subset, but they do contain the meantone[12] chromatic scale.

When playing this temperament in some EDO, it may be desired to stretch/compress the tuning so that the tritave is pure, rather than the octave being pure - or in general, to minimize the error on the 3.5.7 subgroup while ignoring the error on 2/1.

One can "add" the octave to Lambda temperament by simply creating a new mapping for 2/1. A simple way to do so is to map the 2/1 to +7 of the ~9/7 generators, minus a single tritave. This is Sensi temperament, in essence treating it as a "3.5.7.2 extension" of the original 3.5.7 Lambda temperament.

List of EDT's supporting Lambda Temperament

Below is a list of the equal-temperaments which contain a 4L+5s scale using generators between 422.7 cents and 475.5 cents.

L=1 s=0 4 edt

L=1 s=1 9 edt (5flat40 7sharp18)

L=2 s=1 13 (5flat7 7flat3)

L=3 s=1 17 (5sharp10 7flat12)

L=3 s=2 22 (~14edo)

L=4 s=1 21

L=4 s=3 31

L=5 s=1 25

L=5 s=2 30 (~19edo) (5sharp3 7flat8)

L=5 s=3 35 (~22edo) (5flat14 7sharp0)

L=5 s=4 40

L=6 s=1 29

L=6 s=5 49 (~31EDO) (5sharp8 7sharp8) (Schism*)

L=7 s=1 33

L=7 s=2 38 (~24edo)

L=7 s=3 43 (~27edo) (5sharp0 7flat6)

L=7 s=4 48 (5flat13 7flat0)

L=7 s=5 53

L=7 s=6 58 5sharp1 7sharp10 (Schism*)

Schism, by which I mean, the most accurate value for 5/3 and-or 7/3 is found outside the 4L+5s MOS.

[Also, the way I see it, as 4edt and 9edt are comparable to 5edo and 7edo, then the "counterparts" of Blackwood and Whitewood would be found in multiples therein and would be octatonic and octadecatonic, e.g. 12edt and 27edt.]

Generator							cents	L	s	notes
1/4							475.49		0
						8/33	461.08	403.445	57.635
					7/29		459.09	393.505	65.585
						13/54	457.88	387.435	70.455
				6/25			456.47	380.39	76.08
						17/71	455.4	375.03	80.37
					11/46		454.815	372.12	82.695
						16/67	454.2	369.03	85.17
			5/21				452.85	362.28	90.57
						19/80	451.71	356.62	94.91
					14/59		451.31	354.6	96.7
						23/97	450.98	352.94	98.04
				9/38			450.46	350.36	100.1
						22/93	449.925	347.67	102.255
					13/55		449.55	345.81	103.74
						17/72	449.07	343.41	105.66
							448.42	340.15	108.27
		4/17					447.52	335.64	111.88	Canonical BP scales are between here...
						19/81	446.14	328.73	117.41
					15/64		445.77	327.1	118.87
							445.53	325.71	119.82
						26/111	445.5	325.56	119.94
				11/47			445.14	323.74	121.4
						29/124	444.81	322.105	122.705	Golden BP is near here
					18/77		444.61	321.1	123.51
						25/107	444.38	319.955	120.425
			7/30				443.79	316.99	126.8
						24/103	443.17	313.915	129.265
					17/73		442.92	312.65	130.27
						27/116	442.7	311.53	131.17
				10/43			442.315	309.62	132.695
						23/99	441.87	307.39	134.48
					13/56		441.525	305.67	135.845
						16/69	441.03	303.21	137.82
	3/13						438.91	292.61	146.3	...and here Boundary of propriety for Lambda scale
						17/74	436.935	282.72	154.215
					14/61		436.515	280.61	155.905
						25/109	436.23	279.19	157.04
				11/48			435.845	277.37	158.495
						30/131	435.56	275.86	159.7
					19/83		435.39	274.98	160.41
						27/118	435.19	274.01	161.18
			8/35				434.73	271.71	163.02
						29/127	434.305	269.57	164.735
					21/92		434.14	268.755	165.385
						34/149	434	268.06	165.94	Golden Lambda scale is near here 18\7*30\11=7
				13/57			433.78	266.94	166.84	18\7*30\11=7
						31/136	433.53	265.71	167.62
					18/79		433.36	264.83	168.53
						23/101	433.11	263.64	169.47
		5/22					432.26	259.36	172.905
						22/97	431.37	254.9	176.47
					17/75		431.11	253.59	177.52
						29/128	430.91	252.6	178.31
				12/53			430.63	251.2	179.43
						31/137	430.37	249.89	180.48
					19/84		430.2	249.065	181.135
						26/115	430.01	248.08	181.93
			7/31				429.47	245.41	184.06
						23/102	428.87	242.41	186.46
					16/71		428.61	241.09	187.59
						25/111	428.37	239.89	188.48
				9/40			427.94	237.74	190.2
						20/89	427.41	235.07	192.34
					11/49		426.97	232.89	194.08
						13/58	426.3	229.55	196.75
2/9							422.66	211.33		Separatrix of Lambda and Anti-Lambda scales
						13/59	419.075	225.66	193.41
					11/50		418.43	228.235	190.2
						20/91	418.015	229.91	188.105
				9/41			417.5	231.95	185.56
						25/114	417.095	233.57	183.52
					16/73		416.87	234.49	182.38
						23/105	416.62	235.48	181.14
			7/32				416.05	237.74	178.31
						26/119	415.55	239.74	175.81
					19/87		415.37	240.48	174.89
						31/143	415.215	241.09	174.12
				12/55			414.97	242.07	172.905
						29/133	414.71	243.11	171.605
					17/78		414.53	243.84	170.69
						22/101	414.29	244.81	169.48
		5/23					413.47	248.08	165.39
						23/106	412.7	251.2	161.49
					18/83		412.47	252.06	160.41
						31/143	412.31	252.71	159.605
				13/60			412.09	253.59	158.5
						34/157	411.89	254.4	157.49	Golden Anti-Lambda scale is near here
					21/97		411.76	254.9	156.86
						29/134	411.625	255.49	156.13
			8/37				411.23	257.02	154.21
						27/125	410.82	258.67	152.16
					19/88		410.65	259.36	151.29
						30/139	410.49	259.98	150.51
				11/51			410.23	261.05	149.17
						25/116	409.9	262.34	147.565
					14/65		409.75	263.35	146.3
						17/79	409.28	264.83	144.45
	3/14						407.56	271.71	135.85	Boundary of propriety for Anti-Lambda scale
						16/75	405.75	278.95	126.8
					13/61		405.345	280.62	124.72
						23/108	405.05	281.77	123.275
				10/47			404.7	283.29	121.4
						27/127	404.35	284.54	119.81
					17/80		404.165	285.29	118.87
						24/113	403.955	286.135	117.82
			7/33				403.445	288.175	115.27
						25/118	402.955	290.13	112.83
					18/85		402.77	290.89	111.88
						29/137	402.6	291.54	111.06
				11/52			402.35	292.61	109.73
						26/123	402.05	293.8	108.24
							402.01	293.9	108.11
					15/71		401.83	294.67	107.15
						19/90	401.52	295.86	105.66
		4\19					400.41	300.31	100.1
							399.69	303.185	96.51
						17/81	399.18	305.25	93.92
					13/62		398.8	306.77	92.03
						22/105	398.515	307.94	90.57
				9/43			398.08	309.62	88.46
						23/110	397.68	311.23	86.45
					14/67		397.42	312.26	85.16
						19/91	397.11	313.51	83.6
			5/24				396.24	316.99	79.25
						16/77	395.21	321.11	74.1
					11/53		394.745	322.93	71.77
						17/82	394.31	324.72	69.58
				6/29			393.505	327.92	65.585
						13/63	392.47	332.09	60.38
					7/34		391.58	335.64	55.94
						8/39	390.145	341.38	48.77
1/5							380.39		0

Triple Bohlen-Pierce

Proposed by Paul Erlich, is the Triple Bohlen-Pierce Scale, or 39th root of 3. It approximates additional odd harmonics and can be used in a variety of ways, for both just intonation chords and harmonies, as standard Bohlen-Pierce scale interlocking three times with calm sounding quarter-tones, and for various JI modulations.