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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 => Music from this guitar
external image 052-crop.jpg
external image Sword_BP_guitars.jpg

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

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.

external image 3BP3.JPG(triple bohlen-pierce 39√3 Classical Guitar by Ron Sword)

Theory

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

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


Physical instruments tuned to the BP scale

Bohlen Pierce guitar
Clarinets
Metallophone
Electronic Organ
Stredici
Kalimba (Mbira)
Pedal Steel Guitar

Compositions

A Mean Little Voice by Stephen Weigel
Ask For It by Chris Vaisvil
Links to available music written in BP at above website.
Bohl-en Roll by Carlo Serafini (blog entry)
Bohlen-Pierce electric guitar improvisation by Jean-Pierre Poulin
Bohlen-Pierce "Stretched Chroma" Acoustic Improvisation by Ron Sword
Reminiscences by Steven Yi
Roll'n'Peace by Jean-Pierre Poulin
Comets Over Flatland 1 by Randy Winchester
Comets Over Flatland 2 by Randy Winchester
Comets Over Flatland 3 by Randy Winchester
Comets Over Flatland 4 by Randy Winchester
Bohlen-Pierce Island audio by Chris Vaisvil
Mesonic Atom by Chris Vaisvil
Bending the Rules by Chris Vaisvil
Bohlen-Pierce Canon by Kjell Hansen.
Bohlen's Pierced Waltz by Chris Vaisvil
The Complex Plane by Chris Vaisvil

See also