The interval relation of the "wide" BohlenPierce triad is 7/5/3. When creating a twodimensional lattice with progressions of its components 5/3 (5limit) and 7/5 (7limit) on the axes, the 13 steps of the chromatic just BP scale appear in the neat shape of a diamond (see figure below).
The
figure is produced just as other lattices in general use except
that in this case the octave has been replaced by its "tritave"
(perfect twelfth) counterpart. (For information about the world
of lattices, unison vectors and periodicity blocks
see Paul Erlich's "A
gentle introduction to Fokker periodicity blocks".)
However,
the symmetric shape of the diamond is misleading as far as the
related unison vectors are concerned. These turn out to be, as
reported in a note of Paul Erlich of October 15, 1999,
responding to Kees van Prooijen:
[ 2 3 ] or 245/243 (the Minor BP Diesis), corresponding to 14.19
cents ; and
[ 3 2 ] or 3125/3087 (the Major BP Diesis), corresponding to
21.20 cents.
The determinant of the matrix (sorry, a matrix is not easily reproducible in html) resulting from these vectors is indeed 13.
The following figure depicts how the vectors (represented by arrows) confine the 13 notes shown above (here represented by X's for greater clarity). Like in the diagram above, the horizontal increments from square to square are 7/5, the vertical ones 5/3.
It becomes obvious that the chromatic BP scale is a periodicity block.
In the same note, Paul Erlich points out that "Bohlen's diatonic JI scale" (BP Lambda is the mode addressed here) is a periodicity block with 245/243 (Minor BP Diesis, 14.19 cents) and 625/567 (Great BP Semitone, 168.61 cents) as unison vectors. In the lattice chosen above these vectors appear as [ 2 3 ] and [ 1 3 ], resulting in a determinant of 9, which is indeed the number of steps in a diatonic BP mode. The following diagram demonstrates that by carefully positioning the vectors one can actually achieve the required enclosure for BP Lambda, here again marked by X's (horizontal increments 7/5, horizontal ones 5/3, as before):
(Shifting the parallelogram a little lower results in a slight improvement of the demonstration). BP Harmonic, the inversion of the Lambda mode, can be enclosed by the same unison vectors, by the way. In this lattice system, inversions reveal the same pattern as the originals, just flipped around both the vertical and the horizontal axis.
It is probably not easily evident from this precariously positioned parallelogram that Lambda is a periodicity block. This can be made clearer by a slight variation of the method, however. The following larger cut of the same lattice shows Lambda again in the center, this time marked by 0's. It is surrounded by identical patterns, each marked by a different letter. It becomes obvious that these patterns tile the whole infinite plane, created by horizontal 7/5 and vertical 5/3 increments, without overlaps or voids.
This procedure should not be inverted, naturally. Not every pattern that tiles the plane without voids or overlaps can be considered a periodicity block (or can it?). And thus the hunt is on to find suitable unison vectors for other existing diatonic BP modes.
The field for this hunt, however, soon turns out to be fairly limited. The following diagram shows an extended part of the BP lattice, compared to the first diagram above (the larger numbers require smaller fonts and may be difficult to read, but the essential ones are repeated below):


2187 



11907 
1701 
243 
729 
729 


9261 
1323 
567 
81 
81 
243 
243 


7203 
3087 
441 
63 

27 
27 
135 
2025 


16807 
2401 
343 
49 



45 
225 
1125 
5625 

16807 
2401 
343 





125 
1875 
9375 

16807 
2401 
343 
49 



125 
625 
3125 
15625 

2401 
343 
49 
35 
25 
125 
625 
3125 
15625 


343 
245 
175 
125 
625 
3125 
15625 


1225 
875 
625 
3125 
15625 


4375 
3125 
15625 












The center is occupied by the diamond (highlighted) representing the 13 steps of the chromatic scale; each diatonic mode is contained within that diamond. It is obvious that suitable unison vectors for these scales must not be much longer than the borderlines of the central diamond. The numbers surrounding the central diamond extend the lattice to a distance from the center that exceeds this length. Searching for small intervals, we find only eight of them (marked bold) in the thus defined area, including those that have been mentioned above. These eight intervals are listed below. We recognize the four BP semitones (compare "Development of the chromatic BP scale") and the four BP diesises (compare "The 271 tone BP scale"):




































Two
of the semitones are eliminated by the condition that the determinants
of each pair that these vectors can form are either 9 (for any
diatonic scale) or 13 (for the chromatic scale).
27/25 [ 0 2 ] and 49/45 [ 2 1 ] cannot produce these numbers
in combination with any other small interval.
Thus
only six remain, offering five times 13step combinations :
[ 2 3 ] & [ 3 2 ] (already used for the chromatic scale),
[ 2 3 ] & [ 5 1 ], [ 2 3 ] & [ 1 5 ],
[ 5 1 ] & [ 3 2 ], [ 3 2 ] & [ 1 5 ],
but only three times 9step combinations:
[ 2 3 ] & [ 1 3 ] (already used for the Lambda mode), [ 2
3 ] & [ 3 0 ], [ 3 0 ] & [ 1 3 ].
The 9step combinations are of major interest here. Contemplating the one that already yielded both the Lambda and the Harmonic mode ( [2 3 ] & [ 1 3 ] ), we might be interested what happens when we shift the vector parallelogram a little.
Shifting it by less than one cell's width results in modes where the distance between two of the tones frequently becomes an unpractical diesis (namely [ 2 3 ] ); shifting it by one cell, however, leads to interesting new versions of diatonic modes.
A downward shift of the Lambda block by one cell, for instance, produces the Harmonic mode (step sequence 1212 1 2112) with a twist: 7/3 is replaced by its enharmonic sister 81/35. Shifting the same block to the right by one cell results in Dur I (1212 1 1212) with two enharmonic substitutions.
An upward shift by one cell, however, yields (with three enharmonic substitutions this time) a member of a totally novel class of diatonic BP modes: 2121 2 1121. The two pentachords are separated by a whole tone rather than by a semitone as in the known class. A similar effect happens as the result of a onecell shift of the Lambda block to the left: 2121 2 1212 with again three enharmonic replacements. This opens a whatever's box of potentially 36 more diatonic BP modes.
An application for the pair [ 3 0 ] & [ 1 3 ] is easily found: it fits the regular shape of Dur II quite well. The following diagram shows this:
Dur II is a symmetric mode; thus no second mode can be defined by the same combination of unison vectors without enharmonic substitutions. But since Dur II has the step sequence 2112 1 2112, we can already anticipate that shifting it results in, amongst others, an enharmonically changed version of the Gamma mode (step sequence 1212 1 1221).
The caretaker wants to thank Paul Erlich for valuable hints and corrections.