The Bohlen-Pierce Site
BP Scale Structures

Last updated: May 31, 2009

Contents of this chapter:

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 7-step modes

3. BP tunings (on separate pages)

Dave Benson's BP Pythagorean tuning
Dan Stearns' BP "meantone" rotations
Dave Keenan's minimum error 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)

Development of the chromatic Bohlen-Pierce (BP) scale

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:

 Result of first phase Result of second phase Result of third phase Relation to previous tone 1/1 1/1 - 9/7 9/7 = 1.2857 7/5 49/45 = 1.0889 5/3 25/21 = 1.1905 9/5 27/25 = 1.0800 15/7 25/21 = 1.1905 7/3 49/45 = 1.0889 3/1 3/1 9/7 = 1.2857

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:

 Step no. Tone (relation to base) Relation to previous step 0 1/1 - 1 27/25 27/25 2 25/21 625/567 3 9/7 27/25 4 7/5 49/45 5 75/49 375/343 6 5/3 49/45 7 9/5 27/25 8 49/25 49/45 9 15/7 375/343 10 7/3 49/45 11 63/25 27/25 12 25/9 625/567 13 3/1 27/25

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:

 Small semitone 27/25 + D = Major semitone 375/343 + d = Large semitone 625/567 + d = Minor semitone 49/45 + D =

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:

 Name Ratio Cent Major BP diesis (D) 3125/3087 = 1.0123 21.2 Minor BP diesis (d) 245/243 = 1.0082 14.2

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 × 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:

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.

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 supports the view that BP is 7-limit.

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

The equally tempered BP scale

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

fn/f0 = 3n/13.

This scale deviates less from the BP scales in just tuning than the equally tempered western scale (fn/f0 = 2n/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:

 Step no. Equal temperament [cent] Just tuning [cent] Deviation (defect) [cent] 0 0 0 0 1 146 133 + 13 2 293 302 - 9 3 439 435 + 4 4 585 583 + 2 5 732 737 - 5 6 878 884 - 6 7 1024 1018 + 6 8 1170 1165 + 5 9 1317 1319 - 2 10 1463 1467 - 4 11 1609 1600 + 9 12 1756 1769 + 13 13 1902 1902 0

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.

This defect could have been smaller still if Bohlen had chosen different just non-basic 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 23-limit intervals into a basically 7-limit scale, and it would have increased the number of just semitones from 4 to 6.

Erlich's Triple BP Scale

Paul Erlich discovered that a triple, equally tempered BP scale (fn/f0 = 3n/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):

 Step no. Ratio JT [cent] ET [cent] Defect [cent] 3 27/25 133.2 146.3 13.1 5 15/13 247.7 243.8 - 3.9 6 13/11 289.2 292.6 3.4 6 25/21 301.8 292.6 - 9.2 7 11/9 347.4 341.4 - 6.0 9 9/7 435.1 438.9 3.8 11 15/11 536.9 536.5 - 0.4 12 7/5 582.5 585.2 2.7 13 13/9 636.6 634.0 - 2.6 15 75/49 736.9 731.5 5.4 16 11/7 782.5 780.3 - 2.2 18 5/3 884.4 877.8 - 6.6 21 9/5 1017.6 1024.2 6.6 22 13/7 1071.7 1072.9 1.2 24 49/25 1165.0 1170.5 5.5 27 15/7 1319.4 1316.8 - 2.6 28 11/5 1365.0 1365.5 0.5 30 7/3 1466.9 1463.1 - 3.8 33 63/25 1600.1 1609.4 9.3 34 13/5 1654.2 1658.2 4.0 36 25/9 1768.7 1755.7 - 13.0 39 3/1 1902.0 1902.0 0

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.

The reference BP diatonic mode

9 out of 13

Returning to the basic harmonic framework of the BP scale,

 Result of first phase Result of second phase Result of third phase Relation to previous tone 1/1 1/1 - 9/7 9/7 = 1.2857 7/5 49/45 = 1.0889 5/3 25/21 = 1.1905 9/5 27/25 = 1.0800 15/7 25/21 = 1.1905 7/3 49/45 = 1.0889 3/1 3/1 9/7 = 1.2857

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:

 Step no. Relation to base tone Name Distance to previous tone 0 1/1 C - 1 25/21 D 25/21 2 9/7 E 27/25 3 7/5 F 49/45 4 5/3 G 25/21 5 9/5 H 27/25 6 15/7 J 25/21 7 7/3 A 49/45 8 25/9 B 25/21 9 3/1 C' 27/25

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 C-major), 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

2112 1 2121,

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.