The Bohlen-Pierce Scale

Elaine Walker's research paper | Some background on the Bohlen-Pierce Scale


I wrote this research paper for an Advanced Acoustics class at New York University in the Spring of 2001 while obtaining my masters degree in Music Technology. It included listening tests with nine subjects who collectively had a wide variety of musical training. Some were beginners and some were long-time professional musicians.

The main Bohlen-Pierce Scale website, run by Heinz Bohlen:

Other research: Tonality, Harmonic Progressions, and Voice Leading with the Bohlen-Pierce Scale - Dustin Schallert (2012)


Elaine Walker:
Love Song (2011, ZIA Drum N Space)
Greater Good (2011, ZIA Drum N Space)
Love Song (2008, performance by 17 Tone Piano Project. Thanks guys!)
Stick Men (1992, ZIA v1.5)

Other music:
I Know of No Geometry (Dr. Richard Boulanger)
Solemn Song for Evening (Dr. Richard Boulanger, live performance at the Bohlen-Pierce Symposium 2010)
Bohlen-Pierce Minus 1 (Carlo Serafini)


Bohlen-Pierce Scale Symposium, Boston 2010: Videos of Performances and Lectures
My lecture at the Bohlen-Pierce Scale Symposium, Boston 2010: 5 Videos
AXIS keyboard for the Bohlen-Pierce Scale: 11 Videos


Some modes and chords are mapped out on this site, which you may find useful.


The videos above feature these three types of Bohlen-Pierce keyboards! These keyboards are all discussed here.
BP sonome in DurII mode

This is Heinz Bohlen's original homemade electronic organ from 1972, in Gamma Mode. To the right, Heinz is tinkering with the transcription of a Christmas carol into his "13-Step Scale" (1973).

In 2000 when I was doing research in BP at NYU, the past research I had found all referred to Lambda as the main mode of choice, and not Gamma, the mode Heinz Bohlen's keyboard was using (or Dur II, which has the same black/white key pattern. At the time I did not know which mode Heinz's keyboard was referring to).

So at that time (2000) I drew out a Lambda keyboard and I liked the simplicity of it.

Ten years later I finally built a Lambda mode keyboard, and found it was a nightmare to play. Using black keys as white keys where there are 3 in a row made it even worse, because it was harder to see the 4-black-key groupings. As well, I first realized the issue of shaping the white keys. I had ordered extra C's and F's and shaved them narrower. By the way, I built this from Korg Poly-800, which I do NOT recommend for several reasons. I just happened to have one lying around.

Below is a diagram of the same keyboard, to make it more clear how it was made. G# was painted white to become a Bohlen-Pierce E. They grey shaded keys - H, J, A, B - were shaved thinner in a symmetrical manner so that they black keys - H#, A# - are centered. The pattern - C, C#, D, E, F, F#, G - are the same keys and same widths as a normal 12 tone pattern - F, F#, G, G#, A, A#, B.

A Bohlen-Pierce Scale keyboard in Dur II Mode (or Gamma Mode, which looks the same) can be made with a similar method, but requires two white-painted black keys per tritave - E and A - instead of just E. I think that Dur II is easier to see and play since the pattern is more varied. The extra white-painted black keys results in narrower tritaves which are easier to reach.

The keyboards above can be made by modifying most store bought keyboards and ordering extra keys, but the downside is that they have unequal widths of white keys. In my opinion, this is OK for composing, but less than ideal for performance.

Another option - a Bohlen-Pierce Scale keyboard with equal spacing on top and equal spacing on bottom - does not work out very well either. In the picture below, the Cs are centered, but it could potentially be cut so that other keys are centered instead. No matter how they are aligned, some keys - like B and D - are inevitably shifted over one way or the other and not aligned with the top portion of the key. This is be less than ideal for performance as well.

This is the same keyboard without the lines:

One way to get around the hassle of shaving extra white keys narrower or building a new keyboard from scratch altogether, such as the adjustable keyboard, is to simply saw off the wide part of the narrow keys. The black and white keys can then be freely interchanged. This isn’t ideal either since pianists may miss the wide part of the white keys for doing glissandos, and generally are used to having the black keys shorter than the white keys. I do find it pleasant to perform on, however, in the form of a Vertical Keyboard, a patent pending invention that is currently for sale (see Vertical
BP-Tar (Lambda Mode)

BP-Tar (Dur II Mode)


Another style of keyboard that is great for the Bohlen-Pierce Scale is the sonome (“sonome” is the generic name for this type of hexagon keyboard, like "piano"). Many useful chords in the Bohlen-Pierce Scale that are wide and hard to reach on a traditional style piano keyboard, are closer together and easy to reach on a sonome.

Bohlen-Pierce Scale, Lambda Mode:

BP sonome in Lambda mode

Lambda Mode (pictured above) is the mode of choice for most Bohlen-Pierce Scale fans. However, it’s interesting that Dur II mode (created by simply moving the last accidental down one half step) is easier to see and play on both a traditional style piano keyboard and a sonome.

Bohlen-Pierce Scale, Dur II Mode:

BP sonome in DurII mode
Watch videos about rearranging the hexagon keys of a sonome for the Bohlen-Pierce Scale here.

Some modes and chords are mapped out on this site, which you may find useful.

The Two Inventors of the Bohlen-Pierce Scale

Heinz Bohlen at Stanford, 1935–
(Picture by Elaine Walker, 2001)

John Robinson Pierce, 1910–2002
(Picture courtesy of CCRMA, Stanford, CA)

The story of the Bohlen-Pierce Scale and it’s inventors is a bittersweet one. Sadly, the two microwave engineers who invented the Scale independently six years apart, never met even though they both lived in the San Francisco area for a number of years. When John Pierce passed away in 2002, Heinz's dream of meeting him and discussing their co-invented scale in person, died with him. Heinz was also a fan of John Pierce because of his fame in the microwave engineering world. John Pierce is credited for inventing the idea of the communication satellite. In turn, John Pierce may have been envious that Heinz Bohlen invented the BP Scale six years before he did.

It all started in the early 1970s when Heinz Bohlen became curious as to why musicians always use the same 12 note per octave tuning. After getting some unsatisfactory answers from musicians, he took it upon himself to research music and tonality. It was finally the understanding of combination tones that lead him to believe he could use the same method that lead to the 12 tone equal temperament, to devise another tuning from a different framework. He understood that the 12 tone scale was based around the a major triad, the inversion, and then filling in the gaps. So, he started with a new triad that was very harmonically pure, but not contained within the normal 12 tone framework. He made the stunning discovery that a scale of thirteen almost equal steps within the framework of an octave and a fifth (which John Pierce later dubbed as a tritave) contained this very pure triad. He recognized that this tuning shared a duality with the traditional Western 12 tone tuning, it had harmonic value, and that it would be a valid compositional tool.

Amazingly, during the same time frame, another man named Kees van Prooijen had independently invented the BP Scale during his research in generic equal temperaments. But even more amazingly, a second microwave engineer named John R. Pierce made the same discovery, six years later. He arrived at the tuning using a different method, but for the same reason - Curiosity. Leave it to microwave engineers to invent a tuning with no octaves! When John Pierce realized it had already been invented by Heinz Bohlen, the tuning was dubbed the “Bohlen-Pierce Scale. He was apparently unaware of Kees van Proojijen. Coincidentally, John Pierce wrote “The Science of Musical Sound”, which was my favorite childhood coffee table book. When my professor at Berklee College of Music, Dr. Boulanger, introduced me to the BP Scale in 1991, I knew the name John R. Pierce sounded familiar. That XMas I found the book in my parents living room and remembered how I loved the book when I was younger.

I had the pleasure of meeting Heinz Bohlen in 2001 in San Jose where he lived when I was on a trip to visit Stanford. We both ventured to Stanford looking for Max Mathews (the “Father of computer music”), but he was not there that week. We had hopes of having a grand reunion of Max Mathews, John Pierce, Heinz Bohlen, and myself. Dr. Richard Boulanger has also worked extensively with Max Mathews and John Pierce on Bohlen-Pierce Scale music and research. He informed me that John Pierce was in the hospital with Alzheimer's and was not taking visitors. Alas, the reunion never happened, but I found Heinz Bohlen to be delightful company over sushi. He took a great interest in my Chaos Melody Theory research and we talked about chaos theory more than the BP Scale! Heinz and I have kept in touch and remained friends ever since. - EW

Some background on the Bohlen-Pierce Scale and its inventors

Heinz and Elaine at Stanford, 2001

The Science of Musical Sound was my favorite childhood "coffee table" book. It's a coincidence. No joke!