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.

External image: http://www.ronsword.com/images/Sword_BP_guitars.jpg

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External image: http://ronsword.com/bpguitar.JPG

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Ron Sword and his 9-string 13-tone BP Touchstick/Guitar crossover instrument (aka: "the Bohlen-Box")

Triple Bohlen-Pierce[edit]

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.

External image: http://www.ronsword.com/images/3BP3.JPG

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(triple bohlen-pierce 39√3 Classical Guitar by Ron Sword)

links to ronsword.com broken Spt3125 (talk) 03:24, 22 September 2018 (UTC)

Theory[edit]

Bohlen Pierce website

Wikipedia Bohlen-Pierce scale

Bohlen-Pierce Scale Research by Elaine Walker

Sword, Ronald. "Bohlen Pierce Scales for Guitar" IAAA Press, UK-USA. First Ed: May 2009.

Intervals of BP

EDTs compatible with the BP nonatonic scale[edit]

The Lambda MOS family of 4L+5s is well known for accurately representing the 3.5.7 subgroup when the generator is near the boundary of propriety, such as in the case of the Bohlen Pierce scale. 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, eg. 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