Bohlen-Pierce

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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 guitar. [http://chrisvaisvil.com/the-bohlen-pierce-epiphone-roadie-guitar/ Music from this guitar.
Sword BP guitars.jpg

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.


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

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