Notation
Pitch correlation
Just intonation, equal temperament, or...?
Tuning synthesizers to BP
The first BP instrument
BP guitars
BP clarinets
A BP metallophone
Link: Stephen Fox' BP clarinet project
Link: Jean Pierre Poulin's electrical BP guitar

- Pitch correlation

Principally, it doesn't matter whether there is any relation in pitch between BP and, for instance, the traditional Western scale. For practical reasons, like instrument tuning, however, it seems to be preferable to establish something like a convention in this respect. This will take some time.

The following correlation has been useful when tuning a guitar to BP:

In this case BP A₁ is at the same time the lowest tone in compositions for the human voice, and thus it happens that BP A₂ is the lowest tone on the piano.

Stephen Fox chose A = 440 Hz for his BP clarinets and may well have set a trend by doing so.

- Just intonation, equal temperament, or...?

Like the traditional Western scale, the Bohlen-Pierce scale suffers from a comma problem. It is even worse in this case. A cycle of any 13 of the BP intervals leads back to the original tone (off-set by multiples of 3 ignored), but none of the just intervals in BP can be used to tune an instrument this way without running into a serious "wolf" condition. The following short calculation will demonstrate this.

If R is the ratio of a BP interval, and n is its step number, then the value C of the resulting comma is:

The values for commas resulting from the main BP intervals are shown in this table. ( For comparison: the comma resulting from a cycle of just fifths in the traditional Western scale is C = (3/2)¹²/2⁷ = 1.0136 or 23.4 cent.)

We see that any possible BP comma is much worse than the comma of the traditional Western scale. Thus, in cases where just intonation is not possible, equal temperament seemed to be the most recommendable tuning approach for BP for a while. This is supported by the fact that the defect of the equal temperament BP scale is considerably lower than that of the ET Western scale. However, Dave Keenan's recent minimum error scale development, using a generator of 439.82 cents, appears to be a much more elegant solution to the problem.

Tuning synthesizers or keyboards to BP

Advice on how to tune synthesizers, keyboards and sound cards to BP can be obtained from Robert Walker's software Fractal Tune Smithy and from Manuel Op de Coul's code Scala.

Scala (the more demanding tool, and a freeware) can upload tuning tables to families of synthesizers permitting this process, while Fractal Tune Smithy (easier to handle, shareware) supports real-time tuning for keyboards that feature General MIDI and possess suitable MIDI Out and MIDI In interfaces to a computer.

More expensive, but very convenient for synthesizers and keyboards with MIDI In and MIDI out sockets, and not requiring the use of a computer, is TBX1, a real-time tuning hardware solution by Aaron Hunt.

Help when tuning specifically Kurzweil K2000 and K2500 is supplied by a website of John Loffink.

The first BP instrument

BP electronic organ, 1972/73

According to some still existing invoices, Heinz Bohlen bought the bulk of parts for this electronic organ in late 1972. Construction started immediately. Case, keyboard and most electrical parts were supplied by Dr. Böhm & Co., at that time located in Minden, Germany. The order contained 11 white C-keys, 11 white D-keys, 10 white E-keys and 14 black keys. Understandably, that created some confusion in this fairly conservative company.

The 46 keys cover a range of about 3.5 dekachords (tritaves) from C (32.04 Hz) to G" (1,436.4 Hz). Thus F' happens to be at 440 Hz. The white keys represent the Gamma scale (1212 1 1221). The seven yellow stripes visible on the keyboard are just covers for the unused slots in the C-D-E groups. The organ has 28 stops. Four TCA 430 (a gift from ITT) are the basic elements of 13 square-wave generators, operating from 2,595.3 Hz to 7,154.9 Hz, and fifty-two FCJ 121 (supplier: VALVO) form the 3:1 divider circuits. These divider circuits created some headaches; the problem was solved by Bernd Seidel, a friend and engineer. The reverberation output amplifier has been supplied by Hammond.

Despite its rather crude sound generation system this simple organ has contributed much to the understanding of the harmonic potential of the BP scale, in a time when synthesizers were still highly expensive luxury items. It has been played a lot in its early days, and by quite many persons; sometimes with serious scientific intentions, but more often just for fun. It is still around, but in need of repair.

BP guitars

Being restricted to synthesizers in the first time was a major obstacle for the introduction of BP. Synthesizers are marvelous instruments; for getting acquainted with novel musical material, however, it would be preferable to de-emphasize the strangeness of the experience by using traditional instruments. Refretting a guitar is one of the possible ways to achieve this.

Theoretical fret positions

If we call the length of a full string between nut and bridge L, and the number of frets we want to position (starting from the nut) n, then the theoretical distance l between fret and nut can be calculated as

Practical fret positions

In real life, however, the fret positions require some consideration of the increase in string tension when the string is clamped down behind the fret. To compensate this effect, the fret is moved back a little in the direction of the nut to the position s :

Investigation of practical guitars reveals the following as a useful approximation for "delta":

Thus practical fret positions for a BP guitar with 652 mm strings turn out to be:

String tuning

The site's caretaker has refretted a standard acoustical guitar (Hohner HW 300 G) this way. The tuning he chose is as follows:

Using an electronic tuning aid, this can be achieved by tuning the strings in the following way:

In the absence of an electronic tuning aid, the procedure is as follows:

BP acoustical guitar, 1997

Jeanpierre Poulin uses this procedure for his electrical BP guitars, too.

BP clarinets

Georg Hajdu was possibly the first to realize that clarinets, because of exhibiting sounds containing only odd harmonics and overblowing at the twelfth (the "tritave"), would be extremely suitable instruments for the Bohlen-Pierce scale. At his instigation Stephen Fox in Toronto has developed and built clarinets that are tuned to BP. A small body of compositions has already been written for them, and they are presently introduced to the public through workshops and concerts. The picture below shows one of the first clarinets. It is obvious that, due to the clarinet's affinity to BP, this instrument has needed considerably less keys than a traditional clarinet.

BP soprano clarinet by Stephen Fox, 2007
(Picture courtesy of Stephen Fox, Toronto)

Please see also a related article (in German) on the website of Hochschule für Musik und Theater, Hamburg.

A BP metallophone

At his own instigation Stephen Fox also built a BP metallophone. It is presently owned by this site's caretaker. Its pitch range reaches from 265 Hz to 795 Hz, that is a full tritave based on B^b (when choosing the Lambda scale as a reference). A remarkable property of this instrument is that, probably due to a metallophone's specific timbre, BP chords and intervals loose much of their "strangeness".

BP metallophone by Stephen Fox, 2007
(Picture courtesy of Stephen Fox, Toronto)

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