The Bohlen-Pierce Site
BP Names and Definitions


Last modified: August 24, 2013

 

This page deals with names and definitions of scales, intervals etc. as related to the Bohlen-Pierce scale. It does not provide explanations; they can be found on other, subject-related pages at the BP site.

Contents:
Basic just intonation BP interval names
Complete BP interval list
Tone names in diatonic scales
Notation
List of diatonic modes
Commas
List of interval elements and their structures
BP triads
Pitch correlation between BP and the traditional Western scale

Basic just intonation BP interval names:

 Span of interval (ratio)

 Name
 27/25  BP first
 25/21  BP second
 9/7  BP third
 7/5  BP fourth
 75/49  BP fifth
 5/3  BP sixth
 9/5  BP seventh
 49/25  BP eighth
 15/7  BP ninth
 7/3  BP tenth
 63/25  BP eleventh
 25/9  BP twelfth
 3/1  BP thirteenth

This scale containes four different semitones:

- Small semitone 27/25
- Minor semitone 49/45
- Major semitone 375/343
- Great semitone 625/567

The differences between these semitones are defined as dieses. A simple way to imagine the relationship between semitones and dieses is the semitone diamond (not shown "to scale" in the following picture):

 Name

Ratio

Cent

 Small BP diesis (D - d)

16875/16807 = 1.0040

6.990

 Minor BP diesis (d)

245/243 = 1.0082

14.191

 Major BP diesis (D)

 3125/3087 = 1.0123

 21.181

 Great BP diesis (D + d)

 15625/15309 = 1.0206

35.371

Developing the JI scale from other base tones than 1/1 leads to enharmonics, arranged in a "chain of semitone diamonds":

 Diamond #

 Interval name

 Interval span [ratio]

 Exp's a,b,c

 Interval span [cent]

 -

 (reference)

 1/1

 0,0,0

 0

 I

 BP first

 27/25

 3,-2,0

 133

 49/45

 -2,-1,2

 147

 375/343

 1,3,-3

 154

 625/567

 -4,4,-1

 169

 II

BP second

729/625

 6,-4,0

266

147/125

 1,-3,2

281

(405/343)

 4,1,-3

288

25/21

 -1,2,-1

302

III

BP third

(3969/3125)

 4,-5,2

414

2401/1875

 -1,-4,4

428

9/7

 2,0,-1

 435

35/27

 -3,1,1

449

IV

BP fourth

 243/175

 5,-2,-1

568

7/5

 0,-1,1

583

(3375/2401)

 3,3,-4

590

625/441

 -2,4,-2

604

V

BP fifth

189/125

 3,-3,1

716

343/225

 -2,-2,3

730

75/49

 1,2,-2

737

125/81

 -4,3,0

751

VI

BP sixth

(5103/3125)

6,-5,1

849

1029/625

 1,-4,3

863

81/49

 4,0,-2

870

5/3

 -1,1,0

884

VII

BP seventh

9/5

 2,-1,0

1018

49/27

 -3,0,2

1032

625/343

 0,4,-3

1039

(3125/1701)

 -5,5,-1

1053

VIII

BP eighth

243/125

 5,-3,0

1151

49/25

 0,-2,2

1165

675/343

 3,2,-3

1172

125/63

 -2,3,-1

1186

IX

 BP ninth

 1323/625

 3,-4,2

1298

(2401/1125)

 -2,-3,4

1312

15/7

 1,1,-1

1319

175/81

 -4,2,1

1334

X

BP tenth

81/35

 4,-1,-1

1453

7/3

 -1,0,1

1467

5625/2401

 2,4,-4

1474

(3125/1323)

 -3,5,-2

1488

XI

BP eleventh

63/25

 2,-2,1

1600

(343/135)

 -3,-1,3

1614

125/49

 0,3,-2

1621

625/243

 -5,4,0

1635

XII

BP twelfth

1701/625

 5,-4,1

1733

343/125

 0,-3,3

1748

135/49

 3,1,-2

1755

25/9

 -2,2,0

1769

-

BP thirteenth

 3/1

 1,0,0

1902

The distances between diamonds are called "links".

Complete BP interval list based on a suggestion by Manuel Op de Coul:

(Nc = number of coincidences)

 Interval Class

Nc 

 Ratio

 Cents

 Name

 0

 -

 1/1

 0
 unison

1

 5

 27/25

133.238
 great limma,
BP small semitone

 4

49/45

147.428
 BP minor semitone

 2

375/343

154.418
 BP major semitone

2

625/567

168.609
 BP great semitone

2

1

729/625

266.475
  

4

147/125

280.666
  

0

405/343

287.656
  

8

25/21

301.847
 BP second,
quasi-tempered minor third

3

0

3969/3125

413.903
  

1

2401/1875

428.094
  

8

9/7

435.084
 septimal major third,
BP third

4

35/27

449.275
 9/4-tone, septimal semi-diminished fourth

4

2

243/175

568.322
  

8

7/5

582.512
 septimal tritone,
BP fourth

0

3375/2401

589.503
  

3

625/441

603.693
  

5

 4

189/125

715.750
  

2

343/225

729.940
  

5

75/49

736.931
 BP fifth

2

125/81

751.121
  

6

0

5103/3125

848.987
  

2

1029/625

863.178
  

3

81/49

870.168
  

8

5/3

884.359
 major sixth,
BP sixth

7

8

9/5

1017.596
 just minor seventh,
BP seventh

3

49/27

1031.787
  

2

625/343

1038.777
  

0

3125/1701

1052.968
  

8

2

243/125

1150.834
 octave minus maximal diesis

5

49/25

1165.024
 BP eighth

2

675/343

1172.015
  

4

125/63

1186.205
  

9

3

1323/625

1298.262
  

0

2401/1125

1312.452
  

8

15/7

1319.443
 septimal minor ninth,
BP ninth

2

175/81

1333.633
  

10

4

81/35

1452.680
  

8

7/3

1466.871
 minimal tenth,
BP tenth

1

5625/2401

1473.861
  

0

3125/1323

1488.052
  

11

8

63/25

1600.108
 BP eleventh,
quasi-equal major tenth

0

343/135

1614.299
  

4

125/49

1621.289
  

1

625/243

1635.480
  

12

2 

1701/625

 1733.346
  

2

343/125

1747.537
  

 4

135/49

1754.527
  

5

25/9

1768.717
 classic augmented eleventh, BP twelfth

 0

-

3/1

1901.955
 just twelfth,
BP thirteenth, "tritave"

Tone names in diatonic scales
(the reference scale here is the Lambda mode):

Tone no.

Relation to base tone

Name
(*)

Distance to previous tone

1

1/1

C

-

2

25/21

D

25/21

3

9/7

E

27/25

4

7/5

F

49/45

5

5/3

G

25/21

6

9/5

H

27/25

7

15/7

J

25/21

8

7/3

A

49/45

9

25/9

B

25/21

10

3/1

C'

27/25

(*) Named in accordance with a suggestion by Manuel op de Coul, aiming at making memorizing easier. The names previously used by Heinz Bohlen were: i, l, m, n, o, r, s, t, u, i'.

Note

Because of the site having entered a maintenance only status, several recent systems of notation and tone names, created by Stephen Fox, Georg Hajdu and Nora-Louise Müller, are not listed here. There may be others, too.

Notation

Reference scale (c-lambda):

Key signatures for other lambda scales:

Note

Because of the site having entered a maintenance only status, several recent systems of notation and tone names, created by Stephen Fox, Georg Hajdu and Nora-Louise Müller, are not listed here. There may be others, too.

List of at least somewhat evaluated diatonic modes
(1 indicates a semitone step, 2 a whole tone step):

1212 1 1212 "Dur I" (Bohlen, 1972)
1212 1 1221 "Gamma" (Bohlen, 1972)
2112 1 2112 "Dur II" (Bohlen, 1972)
2121 1 2121 "Moll I", later "Delta" (Bohlen, 1972)
1212 1 2121 "Moll II", later "Pierce" (Bohlen, 1972, and Pierce, independently in 1984)
1212 1 2112 "Harmonic scale" (Bohlen, 1997)
2112 1 2121 "Lambda" (Bohlen, 1997).

Diatonic modes in form of a table:

Evaluated BP modes

 Mode
=>

 Dur I

 Dur II

 Moll I
(Delta)

 Moll II
(Pierce)

 Gamma

 Harmonic

 Lambda

 1/1

 X

X

X

X

X

X

X

 27/25

X
    

X

X

X
  

 25/21
  

X

X
      

X

 9/7

X

X

X

X

X

X

X

 7/5

X

X
  

X

X

X

X

 75/49
    

X
        

 5/3

X

X

X

X

X

X

X

 9/5

X

X

X

X

X

X

X

 49/25

X
      

X
    

 15/7
  

X

X

X
  

X

X

 7/3

X

X

X

X

X

X

X

 63/25

X

X
      

X
  

 25/9
    

X

X

X

X

 3/1

X

X

X

X

X

X

X

Commas, depending on the interval ratio chosen to "complete" the cycle:

 Step no. n

 Ratio R

Comma C

C [cent]

 1

27/25

0.90645

 -170

 2

25/21

1.0718

 120

3

9/7

0.97166

 -49.8

4

7/5

0.97989

 -35.2

5

75/49

1.0415

70.2

6

5/3

1.0503

84.9

7

9/5

0.95212

- 84.9

8

49/25

0.96019

 -70.3

 9

15/7

1.0205

35.1

10

7/3

1.0292

49.8

11

63/25

0.93297

-120

12

25/9

 1.1031

170

List of interval elements and their structures

There are always three independent basic building blocks that can be used to describe all other intervals: two dieses and one semitone. Those chosen for the list below are the small diesis (a = D - d), the minor diesis (d), and the small semitone (ST).

 Name or symbol

 Numerical parameters m,n,o
(3m 5n 7o)

 Structure

Cents

 small diesis a

3, 4, - 5

a

6.990

minor diesis d

- 5, 1, 2

d

14.191

major diesis D

- 2, 5, - 3

a + d

21.181

great diesis

- 7, 6, - 1

a + 2d

35.371

7/5-comma

- 4, - 13, 13

3a + d

35.161

7/3-comma

23, 0, - 13

4d - a

49.772

49/25-comma

8, 26, - 26

6a + 2d

70.323

5/3-comma

- 19, 13, 0

2a + 5d

84.933

25/21-comma

- 15, 26, - 13

5a + 6d

120.095

27/25-comma

38, - 26, 0

4a +10d

169.867

small semitone

3, - 2, 0

ST

133.238

minor semitone

 - 2, - 1, 2

ST + d

147.428

major semitone

1, 3, - 3

ST + a + d

154.418

 great semitone

- 4, + 4, -1

ST + a + 2d

168.609

small link

10, - 8, 1

ST - a -2d

97.866

minor link

5, - 7, 3

ST - a - d

112.057

major link

8, - 3, - 2

 ST - d

119.047

great link

3, - 2, 0

ST

133.238

BP Triads

Wide Triad: 3:5:7, for example CGA

Narrow Triad: 5:7:9, for example CFH

Bohlen had originally named 3:5:7 the "major triad" and 5:7:9 the "minor triad". In several discussions, especially with Paul Erlich, it turned out, however, that these names unavoidably triggered the expectation that there should exist a BP tonality fully parallel to that of the traditional Western scale. To prevent people from jumping to conclusions in this respect, "wide" and "narrow" triad have been introduced as working titles.

Pitch correlation between BP and the traditional Western scale:

 BP (JI)

Traditional (ET)

Pitch [Hz]

A1

E

 82.4

A

b + 2 cent

 247.2