1. Fundamental BP scales
Harmonic
scale essentials (separate
page)
Development
of the just chromatic scale
Continued Fractions
and BP (separate page)
A "Pythagorean"
approach
The equally tempered
scale
The reference BP
diatonic mode
Fokker lattice representation
of BP (separate page)
What about
BP tonality? (separate
page)
2. BP modes (on separate pages)
Basic
BP diatonic modes
Kees van
Prooijen's 7step modes
3. BP tunings (on separate pages)
Dave
Benson's BP Pythagorean tuning
Dan Stearns'
BP "meantone" rotations
Dave Keenan's
minimum error approach
Paul Erlich's
TOP paradigm approach
John "Longitude"
Harrison and BP
4. More than 13 steps
per tritave
Erlich's Triple
BP Scale
Minimum
error generator for the triple BP scale (separate page)
The 271 tone
BP scale (separate page)
One
basic idea leading to BP is simple. It is generally accepted that
the major triad 6/5/4 possesses almost ideal harmonic properties,
and that the interval with the highest consonance is the octave
(2/1). Now let's assume that the traditional major scale of the
Western musical system developed from the desire to accommodate
the major triad (be it as an entity or in parts), in as many varieties
as possible, within the framework of the octave. If we make this
assumption, then human curiosity may demand that we try this process
again, this time under modified conditions. There is indeed another
pair of chords that we can use. 7/5/3 is another triad of high
harmonic value, though entirely incompatible with the traditional
scale; and the twelfth (3/1) offers itself as an alternate frame
interval, with a consonance grade second only to the octave.
Let's do the experiment right here. Four values of the assumed
scale are present from the very beginning: 1/1 as the base tone,
then 5/3 and 7/3 to form the 7/5/3 triad with the base tone, and
finally 3/1 as the top tone (first phase). Now we want to play
the triad so that it ends with the top tone (we could also say
that we want to suspend it from the top tone). This then leads
to two new tones: 9/7 and 15/7 (second phase). But without recognizing
this immediately, we have already created another triad: taking
5/3 as a starting point, 7/3 and 3/1 form a triad 9/7/5 with it.
With the base tone of the scale as support, this generates 7/5
and 9/5 as new members of our scale (third phase). The following
table shows what we have achieved:





























This
is the basic harmonic framework of the BP scale. Calculating the
distances between the tones that we found reveals something interesting:
we have three different categories. There is a small one (either
1.0800 or 1.0889), a medium one (1.1905), and a large one (1.2857).
It soon becomes obvious that the medium one is about twice the
small ones (second route of 1.1905 equals 1.0911), and that the
large one is the sum of the medium one and a small one (1.0800
× 1.1905 = 1.2857). Thus the relation small:medium:large
is about 1:2:3. This is simply fortuitous, but it rings a bell!
If we would insert a tone at a small or a medium distance into
each of the large distances, it would leave us with a scale containing
only two categories of distances ("small" and "medium"),
which could be considered semitones and whole tones. Filling each
of the whole tone gaps in turn with semitones would produce a
chromatic scale.
We start with the large gap at the beginning of the scale. Replacing
it by the already mentioned combination of whole tone and semitone,
and inserting in turn the same semitone into the whole tone, adds
two new members to the scale: 27/25 and 25/21. In order to avoid
rather awkward fractions, we discontinue the use of 27/25 in the
replacement of the whole tone 25/21 between 5/3 and 7/5. Suspending
instead the other known semitone, 49/45, from 5/3 leads to 75/49
as the next member of our scale. Respecting the "symmetry
axis" between 5/3 and 9/5 that rules the harmonic framework
in the previous table, we simply maintain it while filling in
the rest of the gaps. Thus we arrive at a scale with thirteen
steps:

(relation to base) 
to previous step 










































This is the original "13 step scale" (see Literature, [1]). Actually, Heinz Bohlen arrived at that very result in 1972 using a rather sober, psychoacoustically based mathematical approach, detailed in his first manuscript on the issue.
Note that the number of differing semitones has doubled; there are now four of them. Investigating them a little further shows that they are related to each other in a distinct way:








We might say that the different semitones form four corners of a diamond, the distances between them being either D or d. This picture is an attempt to illustrate this:
D stands for "Major BP diesis", and d for "Minor BP diesis". Their values are:









An entirely different way to derive the BP scale is a purely mathematical approach, similar to the method that Pythagoras and his disciples have supposedly used when creating the Pythagorean scale. The basic Pythagorean intervals (4:3, 3:2, and 2:1) are related to the symbol of the tetraktys, shown in the following figure.
(That stars are chosen to mark the elements of the tetraktys has no deeper meaning: it's just convenient when using PowerPoint to generate these pictures.) If we take away the last row of this figure, like shown below,
and superimpose the remnant to the original symbol, we obtain a kind of "oddnumber tetraktys", with the elements in the four rows now representing the numbers 1, 3, 5, and 7:
A configuration of this kind has some neat mathematical properties. If we call R the number of rows and n the total number of elements, we find that their relation is simply n = R^{2}, and the number r of elements in any given row is r = 2 R  1. But let us evaluate the musical potential of the numbers represented by the different rows in this new symbol. We can do this in three subsequent steps.
Step 1: Original intervals
•
3:1 (the "twelfth") is occupying the place of
the octave.
• Original intervals fitting into this frame are 5:3,
7:3, and 7:5.
Step 2: Inversion of original intervals
•
1:3, transferred into the 3:1 frame by multiplying it with 3,
yields 1:1.
• Similarly, 3:5 leads to 9:5, 3:7 to 9:7,
and 5:7 to 15:7.
Step 3: Combination of intervals resulting from steps 1 and 2
•
3:5 × 9:5 = 27:25
• 5:7 × 5:3 = 25:21
• 5:7 × 15:7 = 75:49
• 7:5 × 7:5 = 49:25
• 7:5 × 9:5 = 63:25
• 5:3 × 5:3 = 25:9
• All other combinations lead to already known intervals,
or fall outside the 3:1 frame.
Arranging all intervals found this way in ascending order reveals the complete chromatic BP scale:
This is a surprising result because it is not at all trivial. If we would apply the same method to the Pythagorean tetraktys, it would yield only 5 out of 13 (if we include unison) intervals of the chromatic Western scale: 1/1, 4/3, 3/2, 16/9, 2/1. The viability of a derivation of the complete BP scale from the oddnumber "tetraktys" also supports the view that BP is 7limit.
On "Pythagorean" Scale Derivation presents more thoughts on this subject.
The
number of four different semitone sizes in the chromatic scale
makes one wonder: assume one wanted to tune a fretted or keyboard
instrument to BP, would there be a tempered scale available to
tune that instrument in an adequate manner? The surprising answer
is that this can be achieved with the smallest possible number
of steps. The just chromatic scale has 13 steps, and again 13
steps are sufficient to create an equally tempered scale. If we
call n the step number, then the frequency ratios for each step
of this scale to its fundamental step abide by the formula
This scale deviates less from the BP scales in just tuning than the equally tempered western scale (f_{}n/f_{}0 = 2^{}n/12) from its just relatives. This is remarkable because it is simply accidental. (If in 1972 Heinz Bohlen would have known anything about Continued Fractions, this result wouldn't have surprised him as much as it actually did.) A half tone step in this scale equals the thirteenth route of three (or 146.3 cent). The following table shows the cent values for the BP chromatic scale in equal temperament compared to that in just tuning:


[cent] 

























































There exists a widespread misconception about the BP scale: that it was conceived as this equally tempered scale. A look at the earliest available writeup on the scale ( Literature, [1]), however, makes clear that this is not the case. The scale was designed in just tuning; its suitability to be turned into an equally tempered scale with again 13 steps was so obvious, however, that it was detected within days. The minimal defect was considered to be a supporting fact, because it proved that the scale abided by both of Bohlen's independent conditions for a viable scale: majority of intervals in consonance with their combination tones, and approximate equidistance of the steps.
This defect could have been smaller still if Bohlen had chosen different just nonbasic intervals, as for instance 25/23 (144 cents) as step 1, 13/11 (289 cents) as step 2, 33/13 (1613 cents) as step 11, and 69/25 (1758 cents) as step 12. But he decided against this manipulation for two reasons. It would have introduced 11, 13 and 23limit intervals into a basically 7limit scale, and it would have increased the number of just semitones from 4 to 6.
Paul Erlich discovered that a triple, equally tempered BP scale (f_{}n/f_{}0 = 3^{}n/39) accommodates intervals consisting of odd numbers up to 15 with astounding accuracy. The scale has a number of voids which could easily be filled by given example. Its mathematical properties are in any case surprising enough, as shown in the following table (new scale members are highlighted):



















































































































The additional scale members have an even smaller defect than the original ones. The scale is certainly worthwhile to be investigated further, as a quasi microtonal version of BP. Also remarkable is the closeness of its smallest steps to quartertones; quartertones, however, with a just intonation character.
Paul Erlich recently developed a generator that permits minimum error modes of the triple BP scale.
9 out of 13
Returning
to the basic harmonic framework of the BP scale,





























we
recognize that we need to insert a single tone into each of the
two large (9/7 = 1.2857) gaps only in order to arrive at a diatonic
scale, containing semitone and whole tone steps. If we decide
to start such a scale with a hearty whole tone step (25/21 = 1.1905),
and to end it with a lead tone step (27/25 = 1.0800), then it
will contain 9 steps out of the original 13, and it will look
like this:












































This
is the Lambda mode, and we note that its 10 tones are divided
by 4 whole tone steps, all 25/21, and 5 semitone steps, alternating
between 27/25 and 49/45. We will see later that all diatonic modes
of the BP scale share these properties.
For reasons of convenience, the tones are named like shown in
column 3. These tone names have been proposed by Manuel
Op de Coul in 2000 for mnemonic reasons. (Bohlen had named the
tones, less practical, k [later i], l,m,n,o,r,s,t,u.) We see that
there is also something like a program in the choice of names:
they repeat at a ratio of 3, claiming that three times the frequency
has the same meaning for this scale like the octave (two times
the frequency) has for the traditional western scale. It should
be mentioned that in the seventies Bohlen has called the interval
3/1 a "dekade", because it contains 10 tones. This was
in analogy to the octave, as containing 8 tones. Later Pierce
created the expression "tritave". Both names make the
above claim. Whether or not the tritave is an analogy to the octave
shall be discussed at another occasion.
More analogies to a traditional diatonic scale are easily (perhaps
too easily) detected. For instance: Like a traditional scale can
be considered as consisting of two tetrachords separated by a
whole tone (c,d,e,f and g,a,b,c' in the case of Cmajor), the
Lambda mode can be described as consisting of two pentachords
(C,D,E,F,G and H,J,A,B,C'), separated by a small semitone. In
the Lambda mode, the two pentachords are not identical. Each pentachord
has 5/3 as a frame interval, and each one is formed by a series
of two whole tones, one small semitone and one minor semitone.
In more general terms, by defining a semitone step as "1"
and a whole tone step as "2", we can write the Lambda
mode in a kind of shorthand code as
with
the first group of numbers describing the first pentachord, the
last group describing the second pentachord, and the single semitone
step in the center indicating their separation.