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

hekts

L s notes
1/4 475.489

325

0
8/33 461.08

315.1515

403.445

275.758

57.635

39.394

7/29 459.093

313.793

393.508

268.9655

65.585

44.828

13/54 457.878

312.963

387.435

264.815

70.428

48.148

6/25 456.469

312

380.391

260

76.078

52

17/71 455.398

311.267

375.033

256.338

80.364

54.9295

11/46 454.815

310.8695

372.122

254.348

82.694

56.522

16/67 454.198

310.448

369.036

252.239

85.162

58.209

5/21 452.846

309.524

362.277

247.619

90.569

61.905

19/80 451.714

308.75

356.617

243.75

95.098

65

14/59 451.311

308.4745

354.602

242.373

96.71

66.102

23/97 450.979

308.247

352.94

241.234

98.039

67.01

9/38 450.463

307.895

350.36

239.474

100.103

68.421

22/93 449.925

307.527

347.669

237.634

102.256

69.8925

13/55 449.553

307.273

345.81

236.364

103.743

70.909

17/72 449.073

306.944

343.4085

234.722

105.664

72.222

448.421

306.499

340.148

232.494

108.2725

74.005

4/17 447.518

305.882

335.639

229.412

111.88

76.471

Canonical BP scales are between here...
19/81 446.137

304.938

328.733

224.691

117.405

80.247

15/64 445.771

304.6875

326.8985

223.4375

118.872

81.25

445.533

304.525

325.711

222.626

119.822

81.899

26/111 445.503

304.5045

325.559

222.5225

119.943

81.982

11/47 445.138

304.255

323.737

221.277

121.401

82.989

29/124 444.812

304.032

322.105

220.161

122.705

83.871

Golden BP is near here
18/77 444.613

303.896

321.109

219.4805

123.5035

84.416

25/107 444.382

303.738

319.955

218.692

120.427

85.047

7/30 443.7895

303.333

316.9925

216.667

126.797

86.667

24/103 443.174

302.913

313.915

214.563

129.26

88.3495

17/73 442.921

302.74

312.65

213.699

130.271

89.041

27/116 442.696

302.586

311.527

212.931

131.169

89.65

10/43 442.315

302.326

309.621

211.628

132.6945

90.698

23/99 441.868

302.02

307.387

210.101

134.482

91.919

13/56 441.525

301.786

305.671

208.929

135.854

92.857

16/69 441.033

301.449

303.21

207.246

137.823

94.203

3/13 438.913

300

292.6085

200

146.304

100

...and here

Boundary of propriety for Lambda scale

17/74 436.935

298.649

282.723

193.243

154.2125

105.405

14/61 436.514

298.361

280.616

191.803

155.897

106.557

25/109 436.228

298.165

279.186

190.826

157.042

207.339

11/48 435.865

297.917

277.368

189.583

158.496

108.333

30/131 435.562

297.71

275.856

188.55

159.706

109.16

19/83 435.387

297.59

274.981

187.952

160.406

109.639

27/118 435.193

297.458

274.0105

187.288

161.183

110.1695

8/35 434.733

297.143

271.707

185.714

163.025

111.429

29/127 434.305

296.85

269.568

184.252

164.736

112.598

21/92 434.142

596.739

268.7545

183.696

165.387

113.0435

34/149 434.003

296.644

268.061

183.2215

165.942

113.423

Golden Lambda scale is near here

18\7*30\11=7

13/57 433.779

296.491

266.941

182.456

166.838

114.035

18\7*30\11=7
31/136 433.534

296.3235

265.714

181.618

167.8195

114.706

18/79 433.356

296.2025

264.829

181.013

168.528

115.189

23/101 433.1185

296.04

263.637

180.198

169.484

115.842

5/22 432.2625

295.4545

259.3575

177.273

172.905

118.182

22/97 431.371

294.845

254.901

174.227

176.47

120.619

17/75 431.11

294.667

253.594

173.333

177.516

121.333

29/128 430.912

294.531

252.603

172.626

178.308

121.875

12/53 430.631

294.334

251.202

171.698

179.43

122.6415

31/137 430.369

294.161

249.892

170.803

180.4775

123.358

19/84 430.204

294.048

249.066

170.238

181.139

123.8095

26/115 430.007

293.913

248.081

169.565

181.927

124.348

7/31 429.474

293.548

245.4135

167.742

184.06

125.8065

23/102 428.872

293.137

242.406

165.686

186.466

127.451

16/71 428.6095

292.958

241.093

164.789

187.517

128.169

25/111 428.368

292.793

239.886

163.964

188.482

128.829

9/40 427.94

292.5

237.744

162.5

190.1955

130

20/89 427.406

292.135

235.073

160.674

192.3325

131.461

11/49 426.9695

291.837

232.892

159.184

194.077

132.653

13/58 426.3

291.379

229.546

156.897

196.754

134.483

2/9 422.657

288.889

211.328

144.444

Separatrix of Lambda and Anti-Lambda scales
13/59 419.075

286.441

225.656

154.237

193.419

132.203

11/50 418.43

286

228.235

156

190.1955

130

20/91 418.01

285.714

229.907

157.143

188.105

128.571

9/41 417.502

285.365

231.946

158.537

185.557

126.829

25/114 417.095

285.088

233.573

159.649

183.522

125.439

16/73 416.867

284.9315

234.488

160.273

182.379

124.6575

23/105 416.619

284.762

235.48

160.952

181.139

123.8095

7/32 416.053

284.375

237.744

162.5

178.308

121.875

26/119 415.553

284.034

239.742

163.8655

175.811

120.168

19/87 415.3695

283.908

240.477

164.368

174.892

119.54

31/142 415.2155

283.803

241.093

164.789

174.123

119.014

12/55 414.972

283.636

242.067

165.4545

172.905

118.182

29/133 414.712

283.459

243.107

166.165

171.605

117.293

17/78 414.528

283.333

243.84

166.667

170.688

116.667

22/101 414.287

283.168

244.806

167.327

169.481

115.842

5/23 413.4685

282.609

248.081

169.565

165.387

113.0435

23/106 412.688

282.0755

251.202

171.698

161.487

110.377

18/83 412.472

281.928

252.066

172.289

160.406

109.639

31/143 412.312

281.818

252.707

172.727

159.605

109.091

13/60 412.09

281.667

253.594

173.333

158.496

108.333

34/157 411.888

281.529

254.402

173.885

157.487

107.643

Golden Anti-Lambda scale is near here
21/97 411.7635

281.443

254.901

174.227

156.862

107.2165

29/134 411.617

281.343

255.4865

174.627

156.131

103.716

8/37 411.2335

281.081.

257.021

175.676

154.2125

105.405

27/125 410.822

280.8

258.666

176.8

152.156

104

19/88 410.649

280.682

259.3575

177.273

151.292

103.409

30/139 410.494

280.5755

259.9795

177.698

150.514

102.878

11/51 410.2255

280.392

261.053

178.431

149.173

101.961

25/116 409.904

280.172

262.339

179.31

147.5655

100.862

14/65 409.652

280

263.348

180

146.304

100

17/79 409.2815

279.747

264.819

181.013

144.452

98.734

3/14 407.562

278.571

271.708

185.714

135.854

92.857

Boundary of propriety for Anti-Lambda scale
16/75 405.75

277.333

278.953

190.667

126.797

86.667

13/61 405.335

277.049

280.616

191.803

124.718

85.245

23/108 405.046

276.852

281.771

192.593

123.275

84.259

10/47 404.671

276.596

283.27

193.617

121.401

82.979

27/127 404.353

276.378

284.544

194.488

119.808

81.89

17/80 404.165

276.25

285.293

195

118.873

81.25

24/113 403.955

276.106

286.135

195.575

117.82

80.531

7/33 403.445

275.758

288.175

196.97

115.27

78.788

25/118 402.957

275.424

290.129

198.305

112.828

77.119

18/85 402.767

275.294

290.887

198.8235

111.88

76.471

29/137 402.604

275.1825

291.5405

199.27

111.063

75.912

11/52 402.337

275

292.6085

200

109.728

75

26/123 402.039

274.797

293.79

200.813

108.241

73.984

402.015

274.78

293.896

200.88

108.118

73.9

15/71 401.8215

274.648

294.669

201.4085

107.152

73.239

19/90 401.524

274.444

295.829

202.222

105.664

72.222

4\19 400.412

273.684

300.309

205.263

100.103

68.421

399.692

273.193

303.1855

207.2295

96.507

65.963

17/81 399.176

272.8395

305.252

208.642

93.924

64.1975

13/62 398.797

272.551

306.767

209.677

92.03

62.903

22/105 398.505

272.381

307.936

210.476

90.569

61.905

9/43 398.084

272.093

309.621

211.628

88.463

60.465

23/110 397.6815

271.818

311.229

212.727

86.4525

59.091

14/67 397.423

271.641

312.261

213.433

85.162

58.209

19/91 397.1115

271.429

313.509

214.286

83.602

57.143

5/24 396.241

270.833

316.9925

216.667

79.248

54.167

16/77 395.211

270.13

321.109

219.4805

74.102

50.649

11/53 394.745

269.811

322.9735

220.755

71.772

49.057

17/82 394.308

269.8512

324.724

221.951

69.584

47.561

6/29 393.508

268.9655

327.923

224.138

65.585

44.8275

13/63 392.467

265.254

332.087

226.984

60.3795

41.27

7/34 391.579

267.647

335.639

229.412

55.94

38.235

8/39 390.145

266.667

341.3765

233.333

48.768

33.333

1/5 380.391

260

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