The Bohlen-Pierce Site
"I think he was investigating other things"

Last modified September 8, 2009

1. Introduction

From August 19 to August 21, 1992, Andrew Goldstein of the IEEE History Center interviewed John Robinson Pierce at his home in Palo Alto, California [1].

A considerable section of this interview deals with what both Goldstein and Pierce call the "Pierce scale". At one point Pierce says (quotations from the interview with the permission of the Director of the IEEE History Center):

PIERCE: So I said to Max (Mathews), Couldn't a scale be made out of these tones (ratios 3,5,7 and 5,7,9)? So we tried to make scales. There are only two sorts of intervals in the diatonic scale: those that go up by two semi-tones and those that go up by one semi-tone. But we kept getting scales with three intervals in them, and it just didn't fit. Then, just experimenting, I found out that they fitted very well if you tried to fit them into the ratio of 3 to 1. Later on I found out that some German had proposed this 3-to-1 scale. I have acknowledged this, and I've had somebody read it to me in English. I don't have a written translation of it, and I've never read the thing [2]. I think he was investigating other things.

GOLDSTEIN: You mean someone recently or someone from far back?

PIERCE: Someone about ten years back.

GOLDSTEIN: Oh, okay. (End of quotation)

"I think he was investigating other things." The remark leads to an interesting question: what indeed drove the discoverers of the Bohlen-Pierce scale?

2. "Some German"

Going about the issue in a historic sequence, we start with the "some German", Heinz Bohlen. It is understandable that Pierce could have an only superficial impression about Bohlen's motives from somebody reading the paper in question to him once, obviously translating simultaneously. It remains a bit of a surprise though that Pierce, excellent scientist that he was, didn't show more interest in a publication that hit one of his areas of research head-on. Anyway, what could Pierce have learned from Bohlen's paper when scrutinizing it?

The paper for the first time introduced Bohlen's "13-step scale" (now Bohlen-Pierce scale) to a wider public, presenting it in both a just and an equal-tempered version. Its bulk deals with the influence that combination tones have on consonance and harmony, according to Bohlen's view. But he, in both introduction and conclusion of the paper, left little doubt about his intentions.
Quotation from the introduction: Es soll ... für die Ermittlung möglicher weiterer Skalen, die als Basis für harmonische Tonsysteme dienen könnten, ein Werkzeug gewonnen werden, das zumindest einen wesentlichen Teil der wirksamen Einflussgrößen beinhaltet. Free translation: The goal is... to create a tool that at least contains a substantial part of the influential parameters for possibly finding other scales; scales that could be the base for new harmonic tone systems.
And a quotation from the conclusion: Vorläufiges Nahziel für die weitere Arbeit ist die Untersuchung der harmonischen Eigenschaften der diatonischen Skalenvarianten: der Versuch, zumindest den Ansatz einer Art Harmonielehre zu gewinnen, um vielleicht einmal Musik hören zu können, die nicht auf den bekannten 12 Tönen beruht und doch harmonisch ist. Free translation: The preliminary short term goal for further work is an investigation of the harmonic properties of the diatonic modes; the attempt to at least gain the initiation of a kind of theory of harmony, aiming at perhaps sometime from now to listen to music which doesn't rely on the well-known 12 tones but is nevertheless harmonic.

Thus it becomes clear that harmonic music outside the 12-tone system was Bohlen's subject of investigation when he discovered what is now known as the Bohlen-Pierce scale. A discussion of cadences in connection with the diatonic modes of the scale confirms that harmony is here meant to comprise tonality though this is not specifically mentioned. In an earlier paper about the scale [3] Bohlen is more precise in this respect, as can be seen already from the paper's translated title: Attempt on the construction of a tonal system on the basis of a 13-step scale.

3. Kees van Prooijen

The next one to independently discover the 13-step scale was Kees van Prooijen. Using continued fractions analysis as a tool, he found in the course of a general mathematical investigation [4] a considerable number of possible equal-tempered scales, among them one that he called E313, the 13 step equal division of 3. The short abstract of his paper makes clear what the goal of this investigation was.
Quotation: In this article an attempt is made to construct equal-tempered scales proceeding from the harmonic constitution of musical sounds. An attempt is also made to express the suitability of such scales in objective, numerical values.

In that publication, van Prooijen did not elaborate further on this specific scale. But about 20 years later, in a Web article [5], he says something that leaves no doubt about his motivation.
Quotation: My first fascination with this scale was through the harmonic equivalence with the well known just intonation scale in normal 12-tone orientation in the octave.

4. John Robinson Pierce

According to his own account during the above mentioned interview, Pierce began working on the scale in 1982. Several papers [6, 7, 8] appearing after that year deal with the issue, but they are all more or less concerned with related psycho-acoustical theory questions and reveal little about Pierce's motivation. The above mentioned interview turns out to be a much better source in this respect.

PIERCE: ...I like striking and effective music. I think that one of the troubles with avant-garde is that they don't know what to do to be different. Boulanger says that what he likes about the Pierce scale is it gives him a chance to write tonally, which isn't avant-garde, and still be different. ...
Well, the most substantial thing I've done in connection with CCRMA, (I think it started before I actually got here) was the Pierce scale. And either this will survive or it will not. You can write attractive music in it, but a good composer can write attractive music with any sounds. ...
You could create a scale purely mathematically, but why should you expect it to have anything to do with music? But I have plausible reasons for believing that the order that has been put into the Pierce scale is an order that can be heard. ...
This first scale, the 8-tone scale in which I wrote the eight-tone canon, had dissonance and consonance. But I later realized that it didn't have harmony. You can't be in the key. Any other starting point is just the same as any starting point. ...
On the other hand, we know that changing keys can have an important and noticeable effect in music, and you can change keys in the Pierce scale. ...
Then the second time around, it has all of the qualities, in some sense, that the traditional diatonic scale has, but the notes are all different.
(End of quotation.)

There can't be any doubt that Pierce's goal was tonal harmony.

5. Conclusion

Thus we can lay Pierce's suspicion to rest: Nobody just stumbled accidentally over the scale while "investigating other things". All three discoverers of "Bohlen-Pierce" were on the same quest, looking for harmony and specifically tonality outside the well-trodden path of the Western musical system. Their means were different, with Bohlen using simple calculations to derive the elements of musical scales from both combination tones and harmonics, van Prooijen employing an elegant mathematical tool to define possible equal-tempered scales from the harmonics of a single sound, and finally Pierce experimentally seeking a suitable framework for Mathews' unusual triads. But they were certainly united in their goal: finding an alternative harmonic scale.



[1] John Pierce, Electrical Engineer, an oral history conducted in 1992 by Andrew Goldstein, IEEE History Center, Rutgers University, New Brunswick, NJ, USA
(Scroll down to headline "Computer Music and Musical Scales")

[2] H. Bohlen : 13 Tonstufen in der Duodezime. Acustica, Vol. 39 (1978) No. 2, pp.76-86. S. Hirzel Verlag, Stuttgart.

[3] H. Bohlen: Versuch über den Aufbau eines tonalen Systems auf der Basis einer 13-stufigen Skala. Manuscript, dated Dec. 12, 1972. Limited distribution list.

[4] Kees van Prooijen: A Theory of Equal-Tempered Scales. Interface, Vol. 7 (1978), pp. 45-56. Swets & Zeitlinger B.V. - Amsterdam.

[5] Kees van Prooijen: 13 tones in the 3rd harmonic

[6] M.V. Mathews, L.A. Roberts, and J.R. Pierce: Four new scales based on nonsuccessive-integer-ratio chords. J. Acoust. Soc. Amer., 75, S10(A) (1984).

[7] Max V. Mathews and John R. Pierce: Theoretical and experimental explorations of the Bohlen-Pierce scale. J. Acoust. Soc. Amer., Vol. 84, No. 4, Oct. 1988, pp. 1214-1222.

[8] Max V. Mathews and John R. Pierce: The Bohlen-Pierce Scale. Current Directions in Computer Music Research, Chapt. 13, The MIT Press, Cambridge (Massachusetts), London (England), 1989.