On this page:

Basic BP diatonic modes

Kees van Prooijen's 7-step modes

Dave Benson's BP-Pythagorean tuning

Dan Stearns' BP "meantone" rotations

Dave Keenan's minimum error approach

Paul Erlich's TOP paradigm approach

Minimum error generator for the triple BP scale

John "Longitude" Harrison and BP

Basic BP diatonic modes

The first diatonic BP modes that Heinz Bohlen evaluated and propagated were much influenced by the leading modes of the Western system. Already their names betray that: Dur I, Dur II, Moll I and Moll II (Dur = Major, Moll = Minor in German music terminology). Dur II and Moll II (which later became John Pierce's preferred scale) are actually of high tonal value, but at first Bohlen ignored that and rather settled for Moll I, which he renamed Delta, and for Gamma, an offspring of Dur I through the introduction of a lead tone. Later, when tonality aspects became more and more the driving force, Bohlen introduced Harmonic and Lambda. They rival Dur II and Moll II with regard to tonal capability but offer more melodic tension. Lambda is presently the diatonic reference mode for the BP system.

All diatonic modes mentioned so far abide by one unwritten law: they consist of two pentachords each spanning 6 halftones, separated by another halftone. Elaine Walker ignored this veto by proposing four new modes with unequal pentachords (spanning 6 and 5 halftones) that are consequently separated by a whole tone. Walker's bold act of musical disobedience completed one family of possible basic BP modes. There are more families, however, as will be shown below.

This picture shows the Lambda family of modes. The colored segments of the outer ring represent the 9 tones of each diatonic mode. Each mode starts at the segment that bears its name. Moving the outer ring around the inner one enables each mode to be based on any step of the chromatic BP scale. Note: This is not a key circle, but three steps in any direction will alter the key signature by one accidental (clockwise by a sharp, anti-clockwise by a flat). The position shown is that with no key signature at all.

There are five modes in this family that contain only pentachords consisting of six halftone steps:

The four modes of Elaine Walker have mixed pentachords:

The Lambda family is defined by the fact that there are never 2 whole note steps in a row, which is certainly useful for melodic developments. If that request is ignored, four other families of basic BP modes become available. The first one is the Gamma family:

Only Dur II and Gamma have been superficially explored, the rest (here indicated as Xn) is quasi unknown territory. X5 is identical with the inverted Gamma mode listed by Manuel Op de Coul. Again we find five versions with equal pentachords (Gamma, Dur II, X1, X4 and X5), and four versions with unequal pentachords (X2, X3, X6 and X7). The members of this family are of mixed value regarding their usefulness for tonal music; Dur II, as already mentioned, has high value in this respect, while X6 seems to be almost useless, at least from a theoretical point of view.

A closer look at the picture reveals that three other versions of the Gamma family are possible, with again nine members each, which means that the total number of possible "basic" BP modes is 45. That is only of academical interest. It goes without saying that 2 or 3 good ones would be all that is needed, and it is highly likely that the Lambda family can provide those.

For acoustic examples of some of the diatonic mode scales listed above see Elaine Walker's BP page.

Kees van Prooijen's 7-step modes

When developing his version of diatonic BP scales, Kees van Prooijen stayed even closer to the traditional system than Bohlen had done in the beginning. His fundamental modes are 7-step scales, consequently named Major and Minor. They are derived through locating the tones of the traditional scales in a lattice based on the parameters 3/2 and 5/4, then replacing the parameters by 5/3 and 7/3. Thus the Major mode consists of

while the Minor mode takes the form

Further modes are generated through modulations to the dominant or the sub-dominant, respectively.

For details please see van Prooijen's "13 tones in the 3rd Harmonic".

In the chapter "More Scales and Temperaments" of his lecture "Mathematics and Music" Dave Benson employs a method to develop the Lambda scale that is strictly related to the Pythagorean scope. He uses the ratio 7/3 as a base to develop the scale in 3 upward and 6 downward steps from the base tone, thus arriving at the scale marked bold below (the just ratios are listed for comparison):

This is still the Lambda mode, but the deviations in several of the steps are significant and certainly leading to a different sound impression of this tuning.

By the way, Benson's procedure is naturally not limited to producing a new version of the Lambda mode only: 5 steps upward and 4 down would have generated Moll II (or Pierce), 6 upward and 3 down would have lead to Dur I, and so on.

Dan Stearns' BP "meantone" rotations

Employing a generator of 434 cents, composed of 9/7 (strictly BP) reduced by 1/6th of the 118098/117649 comma
(2x310/76, not strictly BP), Dan Stearns arrived at a Lambda "meantone" tuning (Dave Keenan and Paul Erlich
prefer to call it a linear temperament) that appears quite innocent at the first look:

All notes are placed within one of the BP diamonds, thus everything appears quite BP-like until one discovers that there are just octaves (for instance between D and A) and just fifths (for instance between F and J) involved! This was Stearns' intention, obviously. As far as this deviates from the original BP concept, it nevertheless seems to be a variety worth investigating. Here is the complete set of Stearns' BP "meantone" rotations through the Lambda family:

Dave Keenan's minimum error approach

Following a suggestion by Paul Erlich, Dave Keenan developed a generator that seeks to minimize the maximum error of the consonant BP intervals 9/7, 7/5, 5/3, 9/5 and 7/3. This generator is 439.82 cents, close to 9/7 (435.08 cents). The diatonic modes based on this generator feature maximum errors of 4.8 cents from BP just intonation. The gamut of Lambda family modes, recreated with this generator, is shown below:

Keenan's approach makes an excellent instrument tuning for BP, because it permits easy modulation while maintaining a large number of consonant chords.

Paul Erlich's TOP paradigm approach

From a letter of Paul Erlich, dated May 19, 2004:

"...The Tenney HD (Harmonic Distance) function of a ratio n/d is log(n·d). The TOP paradigm minimizes the maximum Tenney-weighted error over all ratios, no matter how complex (though of course only the simplest ones matter in reality, and the condition can be equivalently stated with respect to some more limited set or even just the primes alone). In the case where only a single comma vanishes, I discovered that this optimum tuning is easy to calculate...

...BP diatonic scales, with their two possible sizes for each generic interval, arise from Just Intonation naturally if the ratio 245:243 (minor BP diesis, 14.191 cents in JI) vanishes. (In fact, the unusually great suitability of the BP diatonic scale for {3,5,7} harmony is entirely equivalent to the unusually small separation between 243 and 245 in the list of numbers 3p·5q·7r.) The most efficient way of doing this would widen the representation of prime 3 (the only prime factor of 243) while narrowing the representations of primes 5 and 7 (the prime factors of 245)."

The factor required for widening or narrowing the logarithmic representation of the primes in this case is the logarithm of the ratio divided by its Tenney HD function:

Specifically, prime 3 gets tempered to

prime 5 gets tempered to

and prime 7 gets tempered to

"It's easily seen that this results in the BP diatonic scale having only two step sizes.
In JI, the two sizes of small step are 27/25 and 49/45. 27/25 is 33·5^-2, and 49/45 is 3^-2·5^-1·72. Their tempered representations are thus

These are identical, so the number of step sizes in the BP diatonic scale is reduced from 3 to 2.
The large step, beginning as 25/21 or 3^-1·52·7^-1 in JI, becomes

Paul Erlich's idea provides another excellent instrument tuning option, with very good approaches of the main BP consonances, as the comparison with the JI tuning of the Lambda mode in the following table shows:

Minimum error generator for the triple BP scale
(from a letter by Paul Erlich, dated September 27, 2001)

Also instigated by Paul Erlich, Manuel Op de Coul performed a similar exercise for Erlich's triple BP scale, which yielded a generator of 780.352 cents. This results in a 39-tone tuning with 22 small and 17 large steps within the 3/1 boundary. The errors are as follows:

Aside from the 39-tone scale, this generator produces proper MOS( two-step size) scales with 5 and 17 tones per 3/1, and improper ones with 7, 12 and 22 tones. These would be worth trying out if one wishes to get these ratios of 11 and 13 into music.