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1. Chromatic BP scales

Development of the chromatic BP scale
A "Pythagorean" approach
The equally tempered BP scale
Erlich's Triple BP Scale
The 271 tone BP scale (separate page)
Continued Fractions and BP

2. Diatonic BP scales

The basic diatonic BP scale
Diatonic modes and scale variants (separate page)
Fokker lattice representation of BP (separate page)
John "Longitude" Harrison and BP
What about BP tonality? (separate page)

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 x 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:

This is the original "13 step scale" (see Literature, [1]). Actually, Heinz Bohlen arrived at that very result in 1972 using a rather sober, psycho-acoustically 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. D stands for "Major BP diesis", and d for "Minor BP diesis". Their values are:

- A "Pythagorean" approach

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.

Tetraktys

(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 "odd-number tetraktys", with the elements in the four rows now representing the numbers 1, 3, 5, and 7:

ODD-NUMBER "TETRAKTYS"

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 = R², 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 x 9:5 = 27:25
• 5:7 x 5:3 = 25:21
• 5:7 x 15:7 = 75:49
• 7:5 x 7:5 = 49:25
• 7:5 x 9:5 = 63:25
• 5:3 x 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 odd-number "tetraktys" also suggests that BP is 7-limit rather than 9-limit, despite it containing intervals like 9/7 and 9/5.

On "Pythagorean" Scale Derivation presents more thoughts on this subject.

There exists a widespread misconception about the BP scale: that it was conceived as this equally tempered scale. A look at the earliest available write-up 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.

- Erlich's Triple BP Scale

Paul Erlich discovered that a triple, equally tempered BP scale (thirty-ninth root of three) 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 quarter-tones; quarter-tones, however, with a just intonation character.

Paul Erlich recently developed a generator that permits minimum error modes of the triple BP scale.

This is the Lambda scale, and we note that it contains 10 tones, 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 versions 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. We will rather call this interval what it is technically: a dekachord. Whether or not the dekachord 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 C-major), the Lambda scale (and each of the other BP diatonic scales) 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 scale, 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 scale 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.

(For more information about BP modes see Diatonic Modes and Scale Variants).

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