The Bohlen-Pierce Site
First notes on BP

Last modified: September 5, 2008


The first available written account of what much later will be dubbed the "Bohlen-Pierce scale", or BP, are 24 numbered pages of untitled, pencil-written notes in German. They are not dated, but the context indicates that they precede Heinz Bohlen's manuscript "Die Bildungsgesetze des 12-stufigen Tonsystems und ihre Anwendung auf einen Sonderfall" from July 1972 by a time interval sufficient to read and at least partly digest Hermann Grabner's "Allgemeine Musiklehre". Thus it seems justified to date them roughly into spring 1972. The notes don't carry any references to their author either, but they are clearly in Bohlen's handwriting. Their general appearance is that of typical "engineering notes", jotted down to archive thoughts and investigation results rather than to publish them.

The original manuscript is now archived at Huygens-Fokker Foundation (Stichting Huygens-Fokker), Amsterdam.

In the beginning, a clear logical sequence of the issues discussed is not discernible, just a collection of loosely related statements. That changes later. The first page is a typical example of the bone-dry style in which the notes are written.

Nothing betrays the excitement of the author nor the surprising outcome of the investigation. There is no introduction, no hint at what he is venturing for. The following is a free translation:

Basic mathematical function of a scale

The impression of a pleasant natural sound is conveyed by all sequences of pure tones (sine waves) which are based on the principle of simple exponential functions:

fn / f0 = Kn/s

where f0 is the frequency of the fundamental tone of the sequence, fn is the frequency of the nth tone, n is its order, s the total number of tone steps in the sequence, and K the base.

Adhering to this function simultaneously creates a suitable (tempered) tuning mode for keyboard instruments.

Example: K = 2 (octave system) [and] s =12 [results in] fn / f0 = 2n/12, the tempered tuning of the 12-step tone system.

2nd example: Krenek, Pentecost Oratory "Spiritus Intelligentiae, Sanctus"
fn/ f0 = 2n/13 with 330 Hz as "central tone".

3rd example: The Arabic scale "Saba" would, in a tempered manner, approach fn / f0 = 2.4n/30.

The expression "pleasant natural sound" refers to melodic rather than harmonic properties, obviously, but the author, not addressing an audience here, doesn't bother to explain. He doesn't seem to be too much blinkered by excessive knowledge of musical science, either. However, it is questionable whether he would have set out for his investigation if he would have been.
Anyway, he continues. Next he turns to non-linear effects, discussing harmonics (page 2) and combination tones (page 3). As an electrical engineer, he is in safe waters here. On page 4 we find something that is significant for the author's view point, developed from his investigations and own experiments with sound generators:

Recognizability of intervals

Beyond about 1000 Hz, the capability of the human ear (even the trained one) to clearly recognize intervals begins to falter rapidly. The ability to correlate harmonics with a fundamental ("harmonic overtone series") is therefore limited. This is naturally also valid for combination tones in the area of high frequencies, not so, however, for those at low frequencies (difference tones). The clearest recognizability is related to combination tones in close proximity of the original sound.

Nevertheless, on pages 5 and 6, the author embarks on the derivation of a scale from the harmonics, following the simple request that there shall exist identical harmonics of the fundamental tone f0 and any scale tone fx (Sound with consonant harmonics):

q fx = p f0

where p >(or =) 1, and 0 < q <(or =) p.

He arrives, after some manipulation, requiring fill-in in accordance to given examples, at the just major scale in its Ptolemaic version: 1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1, criticizing:

- Demand for consonant harmonics not met for second and seventh
- Process had to be cut short at 5th harmonic. The 6th harmonic would have yielded the minor third, the 7th would have lead to several unsuitable intervals.

The author continues on page 7 by deriving the Aeolic minor scale (1/1, 9/8, 6/5, 4/3, 3/2, 8/5, 16/9, 2/1) from an inversion of the method described. He complains that this is an artifact, only justified in retrospect by the chromatic scale. Pages 8 and 9 are dedicated to a scale derivation from the combination tones:

Request: Consonance between combination tones, especially at close vicinity to the original interval f0, fx:

p f0 - q fx = fx or p fx - q f0 = f0

leading to fx / f0 = p / (q + 1) and fx / f0 = (q+1) / p

where p, q > (or =) 1

The result is the just chromatic scale. On page 10 the author compares this scale with its equally tempered version, emphasizing that the observed defect of less than 1 % is simply good luck. Summarizing he sketches on pages 11 and 12 what he considers

Design laws for a scale

A scale has to be simultaneously in accordance with the two following, entirely independent laws (maximum deviation between intervals 1 %):

1. (Principle of consonance) The scale is to be built such that the majority of its intervals is in consonance with the combination tones generated. Mathematically:

fx / f0 = p / (q +1) and fx / f0 = (q + 1) / p respectively,

where p, q are whole numbers larger or equal to 1.

The scale thus generated is chromatic. A formal derivation from the overtones of single tones (physically less justified) and the inversion of this process lead to diatonic scales which again in combination produce the chromatic scale.

2. (Principle of equidistance) To meet the request for a naturally pleasing sound, the scale has to be built in accordance with an exponential function:

fn / f0 = Kn/s

An exact agreement with this request simultaneously permits equal temperament.

These rather austere demands would certainly not go undisputed with many a theorist. It is also interesting that Bohlen sees equal temperament as important not just considering practicability (keyboard instruments) but rather more under melodic aspects (which he calls naturally pleasant sound). Of even greater concern to him, however, is obviously his law no. 1, consonance. We will see in the following where this stubborn dedication to two seemingly diverging principles leads.

On page 13 (below) the author heads for his surprising next issue, again without any transition:

Derivation of a scale from the combination tones, assuming only odd-number harmonics


fx / f0 = p / (q + 1) and fx / f0 = (q +1) / p respectively,

where p, q are whole numbers larger or equal to 1,
p are odd numbers, p + q are odd numbers (i.e. q are straight numbers), q < p

On page 15 (below), the author produces the result, a non-octave 13-step scale filling the frame of the twelfth: 1/1, 27/25, 25/21, 9/7, 7/5, 75/49, 5/3, 9/5, 49/25, 15/7, 7/3, 63/25, 25/9, 3/1.

It was certainly fascinating, but the notes do not betray any emotion. Instead, the author immediately pursues the generation of related diatonic scales. On pages 16 to 21, he describes four of them:

"Dur I": 1/1, 27/25, 9/7, 7/5, 5/3, 9/5, 49/25, 7/3, 63/25, 3/1
"Dur II": 1/1, 25/21, 9/7, 7/5, 5/3, 9/5, 15/7, 7/3, 63/25, 3/1
"Moll I": 1/1, 25/21, 9/7, 75/49, 5/3, 9/5, 15/7, 7/3, 25/9, 3/1
"Moll II": 1/1, 27/25, 9/7, 7/5, 5/3, 9/5, 15/7, 7/3, 25/9, 3/1

Of special interest is "Moll II" on page 21 (see below).

This is the scale that was to be independently discovered and named "P3579" by John Robinson Pierce about 10 years later and today carries his name. Bohlen's comment on the scale on page 21 translates to:

Scale with the most simple numerical relations, therefore possible base scale, on the other hand poor regarding tension.

So far, the author is only concerned with the generation of his "13-step scale" in accordance with his first "law", the consonance principle. That the scale meets the demand of his second "law", the equidistance principle, is only matter-of-factly documented on two of the first pages of the following stack of notes. They carry the page numbers 24 and 24b (the numeration of pages continues, with many pages now extended using alphabetical indices). Page 24 contains a table listing the chromatic version as well as Bohlen's "Gamma" and "Delta" scale in both just and equally tempered tuning.

"Gamma" is a modified "Moll I" (63/25 has been replaced by "lead tone" 25/9) and "Delta" is identical with "Dur I". A diagram on page 24b (see below) shows that the deviation between just tuning and equal temperament remains well below 1 %.