The Bohlen-Pierce Site
BP Interval Properties

Last updated: January 24, 2010

 

On this page:

Tritave
BP intervals
Enharmonics
Pentachords



- Tritave

The compass interval of the Bohlen-Pierce scale is 3:1. The question how to name it arose early on. In the traditional Western scale, this ratio represents the perfect twelfth. But since in none of the BP scales, be it diatonic or chromatic, 3:1 would mark the twelfth step, nor the twelfth note, this name didn't seem to be applicable in connection with BP.

So in analogy to the traditional Western scale, where the compass interval 2:1 is called "octave", as representing the eighth tone in a diatonic scale, Heinz Bohlen decided in 1972 to name 3:1 "Dekade" or "decade", respectively, because BP diatonic scales possess 10 tones. He was not really happy with that expression, though, being aware that the correct analogy to "octave" should have been "decime". But "Dezime" was already occupied, because it stands for the tenth (5:2) in German music language.

Nevertheless, he also had a hard time getting used to "tritave", an artificial word that John Pierce invented when he came across BP. But despite Bohlen's aversion against verbal contraptions, he had to admit that it served well the claim that 3:1 has the same pivotal importance for BP, which the octave has for the traditional Western scale. Thus "tritave" has become the generally accepted expression for 3:1.

Funny enough, John Pierce's creation has meanwhile invaded his (and also Heinz Bohlen's) original field of expertise, microwave technology. A recent paper on a 6-18 GHz solid-state receiver uses the expression "tritave" for just this frequency relation.



- BP intervals

Most of the intervals appearing in the Bohlen-Pierce scale are not at all novel ones. But only two of them are also used in conventional just Western tuning:
· 5/3, the just major sixth,
· 3/1, the perfect twelfth.

Two others resemble old acquaintances out of the equal-tempered western scale (12tET):
· 25/21 (302 cents) is almost identical with the equal-tempered minor third (300 cents), and is therefore called "quasi-tempered minor third";
· 63/25 is practically identical with the equal-tempered major tenth (1600 cents both) and consequently named "quasi-tempered major tenth".

Others are known as consonant and not entirely "unheard of", but are incompatible with conventional western tuning:
· 9/7, the supermajor or septimal major third, may occur in bagpipe tuning between low G (7/8) and B (9/8) [*],
· 7/5, the septimal diminished or subminor fifth (Huygen's tritone), also sometimes appearing in bagpipe tuning between C (5/4) and high G (7/4) [*],
· 9/5, the just or acute (or natural) minor seventh, is sometimes used as high G in bagpipe tuning [*],
· 15/7, the septimal minor ninth,
· 7/3, the minimal or septimal minor tenth. And
· 27/25, though certainly not consonant, is known as great (or large) limma, and can appear in bagpipe tuning between F (5/3) and high G (9/5) as well as between C (5/4) and D (27/20) [*].

[*] Ewan Macpherson: The Pitch and Scale of the Great Highland Bagpipe.
Reference is low A (1/1).


When used in context with the BP scale, these names are meaningless, however. Thus, as long as we discuss these intervals in connection with BP, we give them (and their enharmonics) simple names as follows:

 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, or tritave

The BP scale is what theorists call "7-limit". Each single interval (and all its enharmonics) can be mathematically expressed as

3a 5b 7c,

the exponents a, b, c being integers of the form -5,...-2,-1,0,+1,+2,...+6. The enharmonics table in the following chapter shows the exponents for all BP enharmonic intervals.

(For comparison: the traditional Western scale is "5-limit": 2a 3b 5c, while the "triple BP scale" is "13-limit": 3a 5b 7c 11d 13e.)




-
Enharmonics

If we try other chromatic scale members than 1/1 as base tones for the intervals listed above, we encounter various enharmonic tones, i.e. tones that are close to those already known, however not exactly identical. Not surprisingly they show the same relationship among each other that we detected when investigating the different semitones: they form a chain, consisting of diamonds similar to the one we know already (and including this one). The following table tries to illustrate that chain (interval spans in brackets are theoretical only):

 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

Following suggestions of Manuel Op de Coul we can address the intervals contained in a diamond as an interval class. Adding interval names we arrive at the table below (Ni stands for the number of co-incidences, i.e. how many times each enharmonic appears in the circle of possible scales):

 Interval Class

Ni 

 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

189/175

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
Al-Hwarizmi's lute middle finger

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

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"


Pentachords

In the same way the traditional Western scale can be divided into two tetrachords, any diatonic BP scale can be considered as to consist of two pentachords, as for instance

C D E F G and H J A B C.

In all basic modes, including van Prooijen's 7-step modes, both pentachords consist of six semitone steps, with the highest and lowest tone forming the ratio 5/3 (just major sixth, BP sixth), and they are separated by the interval 27/25 (great limma, BP small semitone).

All four Walker modes possess two different pentachords, one again consisting of six semitone steps with a total span of 5/3, the other one having only five semitone steps and a span of 75/49 (BP fifth). In these modes the pentachords are separated by 147/125, a BP enharmonic whole tone.

Benson's Pythagorean BP scale consists of two equal pentachords 81/49 (enharmonic BP sixth), separated by 2401/2187, an interval exactly half ways between a BP major semitone and a BP great semitone.

Regarding pentachords, Stearn's BP Meantone Rotations behave in accordance with what has been said about the basic BP modes and the Walker modes above, just replacing the pentachords 5/3 and 75/49 with 868 and 766 cents, respectively, and the separating tones 27/25 and 147/125 with 166 and 268 cents, respectively.

Similarly, in Keenan's Minimum Error modes the larger pentachord is represented by 879.6 cents, the smaller one by 725.2 cents, and the two separating intervals by 142.8 cents and 297.2 cents, respectively.



Back to the top of this page