The Bohlen-Pierce Site
BP Notation and Instruments

Last updated: June 29, 2011

- 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:

 BP (JI) Traditional (ET) Pitch [Hz] A1 E 82.4 A b + 2 cent 247.2

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:

C = R13/3n.

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.)

 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

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 minimum error scale development, using a generator of 439.82 cents, or Paul Erlich's TOP paradigm approach appear to be elegant alternatives.

The question of partials and timbre

When in 1972 Heinz Bohlen developed the 13-step scale that is now called BP, he was driven by curiosity to find out how a scale would look like that was solely based on sounds whose partials consisted of odd harmonics only. Thus he chose square waves as the waveform for the tones of his home made electronic organ since (ideal) square waves consist of only odd harmonics. The first scientist to comment on the new scale, Jürgen Meyer in 1973, considered the need for instruments featuring exclusively odd harmonics crucial and saw this demand as a serious impediment for the acceptance of BP. So did later John Pierce, and he was seconded in his opinion by William Sethares' book "Tuning, Timbre, Spectrum, Scales". The same view led to Georg Hajdu's Bohlen-Pierce clarinet project which caused the development of BP clarinets by Stephen Fox. Clarinets feature mainly odd harmonics.

It is quite obvious that avoiding even harmonics makes BP compositions sound smooth, and most composers go to great lengths to take care of this issue. However, Charles Carpenter's compositions, for example, show little consideration for this request and let the even harmonics freely add spice to the performance. When Heinz Bohlen, in 1997, refretted an acoustical guitar to BP, he found that the instrument didn't sound strange, despite the now unavoidable even harmonics. BP metallophones, also built by Stephen Fox, show a similar result, although their partials are anything else than odd harmonics. And singers like Elaine Walker and Maureen Chowning have demonstrated that the human voice, despite being by far not devoid of even harmonics, is an instrument absolutely fit for BP.

Thus it seems that the answer to the question whether or not to permit other partials than odd harmonics in BP music, depends solely on the intentions of the composer.

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
Heinz Bohlen 1972/73

According to some still existing invoices, Bohlen bought the bulk of parts for this electronic organ in late 1972. Construction started immediately. Case, keyboard frame and a variety of electrical parts were standard elements of the Böhm BnT, a do-it-yourself organ supplied by Dr. Böhm & Co., at that time located in Minden, Germany. The order also 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 C1 (32.04 Hz) to G2 (1,436.4 Hz). Thus F1 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. The reverberation output amplifier has been supplied by Hammond.

BP electronic organ, detail
Heinz Bohlen 1972/73

Since Bohlen had been basing the original derivation of his 13-step scale (BP) on the hypothetical assumption of tones featuring odd harmonics only, he chose square-waves as the waveform for the sound generators (ideal square-waves are composed of only odd harmonics). Four TCA 430 (a gift from ITT) are the basic elements of 13 such 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.

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

l = L (1 - 3-n/13).

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 :

s = l - δ = l(1 - δ/l).

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

δ/l = (0.6 + 2.4 e-l/86 mm) %.

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

 Fret no. Theoretical l [mm] Practical s [mm] 0 0 0 1 52.8 51.8 2 101.4 100.1 3 146.0 144.5 4 187.0 185.4 5 224.7 223.0 6 259.3 257.4 7 291.1 289.1 8 320.4 318.3 9 347.3 345.1 10 372.0 369.7 11 394.6 392.1 12 415.5 412.9 13 434.7 432.0 14 452.3 449.5

String tuning

Bohlen has refretted a standard acoustical guitar (Hohner HW 300 G) this way. He used 10 frets. The tuning he chose is as follows:

 String I (lowest) II III IV V VI (highest) BP tone A1 C E G A C1

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

 String I II III IV V VI Tune full string 4th fret 1st fret 1st fret full string full string to E (low) d + 18ct d + 18ct g - 47ct b + 2ct 3rd fret of V

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

 String I II III IV V VI Tune full string to E (low) 3rd fret of I 3rd fret of II 3rd fret of III 4th fret of IV 3rd fret of V

BP acoustical guitar, Hohner
Refretted by Heinz Bohlen 1997

Jean-Pierre Poulin uses this procedure for his electrical BP guitars, too:

BP electrical guitar by Jean-Pierre Poulin

BP acoustical guitars by Ron Sword (BP, BP cut-out, triple BP)

BP clarinets

Georg Hajdu was possibly the first one to realize that clarinets, because of producing 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 pictures below show one of the first clarinets. It is obvious that, due to the clarinet's affinity to BP, this instrument requires considerably less keys than a traditional clarinet. Meanwhile, a tenor clarinet, also manufactured by Stephen Fox, has become available, too.

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

BP soprano clarinet by Stephen Fox, 2007, details
(Picture courtesy of Georg Hajdu, Hamburg)

There is a related article (in German) on the website of Hochschule für Musik und Theater, Hamburg.
Nora-Louise Müller discusses her
BP clarinet research on her website.

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 Bb (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)

A BP panflute

The flutist Arturo Grolimund developed the idea that an ideal instrument for playing BP should be a panflute since, like the clarinette, panflute pipes are closed at one end and thus produce primarily odd harmonics. He was able to convince panflute maker, teacher and virtuoso Ulrich Herkenhoff in Munich to build such an instrument for him. Arturo Grolimund's panflute features 27 pipes, all made from bamboo and covering a pitch range of two tritaves from 265 to 2385 Hz. The "foot" is crafted from mahogany.

BP panflute by Ulrich Herkenhoff, 2009
(Picture courtesy of Ulrich Herkenhoff, Munich)

Stredici

A totally unusual BP instrument has been named "Stredici" and has been built under the direction of David Lieberman by students of his architecture class. It is a wooden structure, 16 feet (~ 488 cm) long, and it carries 27 strings. The strings are ment to be plucked as well as bowed or even beaten. This video presents the stredici in action, introduceded by the tranSpectra group and played by Neal Evans on the occasion of the 2009 Open Ears festival.

Stredici by David Lieberman and his architecture class, 2009
(Picture courtesy of David Lieberman, Toronto)