We return to the matter of all interval sets, as described in general in a previous post, but in particular of those in the 12 tone universe inhabited by the usual musics.
Tetrad
We have, in this dodecaphonic universe, four allinterval tetrads. Some may better know these as, respectively, PC Sets 4Z29A, 4Z15A, 4Z29B and 4Z15B in the Fortean bestiary. There are only two distinct ‘shapes’, however, as each set can be paired with its mirror image  its musical inversion (the A and B forms).
Octad = Tetrad + Tetrad
Each of these tetrads may add a transposition of one of the others to form an octad, provided that no pitch class takes up a space occupied by the other. For example we may add 4 semitones to the second (which  so transposed  no longer collides with the first) and add this to the first to produce
Due to the noncolliding pitch class limitation, it turns out that  of the 66 possible ways to combine two of them  there are only 14 noncolliders. Even then, we find that two turn out to be the well known ‘octatonic scale’ shape (covering both minor and major versions of those Jazz/Stravinsky/Messiaen/etc scales) built from four consecutive semitone+wholetone steps. These shapes result by combining one tetrad shape with the inversion of the other.
So finally, due to similar symmetries, we end up with congruences (by which we mean only modal equivalences, where the shapes are rotatable into each other) in only 7 distinct octads, all of them symmetric (i.e. inversionally identical).
These are the Fortean PC Sets known as 828, 825, 826, 89, 817, 810, 83.
If we were to take (as seems usual but it's really not compulsory) that pitch class 0 represents the note C, then an application of this particular ‘tetradic addition’ would be the addition of 4Z29A rendered as the set of pitches C, C♯, E♭, G and the set 4Z15A (primeform rendered as C, C♯, E, F♯) transposed (by the aforementioned four semitones) up to E, F, A♭, B♭. This produces the eight pitches (in order) C, C♯, E♭, E, F, G, A♭, B♭ (corresponding to pitch classes 0, 1, 3, 4, 5, 7, 8, 10).
Fortewise, this might be expressed as something like 826 = 4Z29A + 4Z15A.T4, where the .T4 operator applied to a PC set indicates its transposition (up) by four semitones.
The following image shows the seven distinct constructions in an arrangement where the primeform 429A is fixed at the top of a ‘pitch class clock’. Addends and sums are oriented appropriately with respect to it. Each tetrad pair (in pink, indicating their inherent inversional asymmetry) is in one clock and its summed octad (blue, indicating inversional symmetry) is in its own clock to its right.
It's OK to fix one tetrad (we've chosen the first) at the top in this way because any other possible tetrad pairing will be rotationally or inversionally identical to one of these shapes.
Labels are Fortean PC Set names located at PC element 0 positions. Consequently some may almost be upside down.
So a second example of such a ‘Fortean operational notation’ is demonstrated by 825's ‘11 o'clock’ orientation showing the image 4Z29A + 4Z15A.T5 = 825.TB where .TB is a transposition 11 semitones up (clockwise rotation by ‘one hour’), or 1 semitone down (an ‘hour’ anticlockwise). Shifting this expression clockwise 1 semitone (to ‘right’ the 825 to prime form's ‘midnight’) would require an application of .T1, and, bearing in mind that .TB.T1 ≡ .T0 is effectively a noop, the equality could be reversed and rewritten as 825 = 4Z29A.T1 + 4Z15A.T6 (as the operational composition .T5.T1 is, of course, .T6).
All That Bebop
There are several bebop scales, all of them  by design  octatonic.
Major and Minor
One of the prime forms above (Forte 826) gives us (in two modes of the same sequence) both the Bebop Major and Bebop Harmonic Minor.Forte 826.T4 as Major Bebop C, D, E, F, G, A♭, A, B (PC 0 = C)  
4Z29A.T4 → { 4, 5, 7, 11 } 
4Z15A.T8 → { 8, 9, 0, 2 } 

{ E, F, G, B } ∪ { A♭, A, C, D }  
Forte 826.T7 as Harmonic Minor Bebop C, D, E♭, F, G, A♭, B♭, B (PC 0 = C)  
4Z29A.T7 → { 7, 8, 10, 2 } 
4Z15A.TB → { 11, 0, 3, 5 } 

{ G, A♭, B♭, D } ∪ { B, C, E♭, F } 
Operationallywise, one might also say that BebopMajor.T3 = BebopHarmonicMinor (or, alternatively, BebopHarmonicMinor.T9 = BebopMajor), were one so seduced by operational notations.
Dominant and Dorian
The Bebop Dominant and Bebop Dorian scales are, like the preceding Major and Minor Harmonic, modal variations of the same PC Set, known in Fortespeak as 823. It's not one of our allinterval tetradic composites, but is nevertheless a symmetric set  its inversion is the same set. The figure below exhibits the rotations needed to recover the scales from the prime form  the Fortean 823 label appearing as usual at its 0 pitch class vertex.
And, just to draw attention to the fact that musical applications (instantiations) of pitch class sets do not require that pitch class zero be eternally attached to the note C, this time we'll exemplify the Dominant scale in G and the Dorian in D  they should go nicely with the above bebop major.
823.T9 (with PC 0 = G) G, A, B, C, D, E, F, G♭ Bebop Dominant in G 

823.T2 (with PC 0 = D) D, E, F, G♭, G, A, B, C Bebop Dorian in D 
Dominant Flat Nine
Another Bebop scale related to an all interval set is the Bebop Dominant Flat Nine. As this scale is not selfinverting, it can't be one of the combined tetrads. Nonetheless, as we shall soon see, it is yet related to one.
In the scale of C, it would be C, D♭ E, F, G, A, B♭ B  a mode of the PC Set { 0, 1, 3, 5, 6, 7, 8, 9 }  aka Forte 8Z15B  or operationally 8Z15B.T9.
This set's ‘unused’ pitch classes, viz. { 2, 4, 10, 11 }, can be operationally written as 4Z15A.TA. This is because { 0 + 10, 1 + 10, 4 + 10, 6 + 10 } = { 10, 11, 14, 16 }, the same as (on our 12 hour clock) { 10, 11, 2, 4 } and the irrelevancy of set element presentation order finishes it off (as { 0, 1, 4, 6 } + 10). This means that the bebop dominant flat nine is (a transposition of) the PC set complementary to our second all interval set. Or, more formally, 8Z15B + 4Z15A.TA = 121 (where 121 is Fortean for the complete chromatic scale).
The transposition of the above by 4 semitones  to modally shift 8Z15B into the actual bebop dominant flat nine scale  leaves this expression essentially unaltered, due to our modulo polynomial arithmetic. (But we might be tempted to say BebopDominantFlatNine is anti SecondAllIntervalTetrad.T2).
Bebop Dominant Flat Nine instantiated in C, D♭, E, F, G, A, B♭, B (PC 0 = C)  
4Z15A.TA { 10, 11, 2, 4 } ≡ 8Z15B 
8Z15B.T4 ≡ { 0, 1, 4, 5, 7, 9, 10, 11 } 

Bebop Δ♭9 + 4Z15B.T2 = Chromatic
Forte: 8Z15B.T4 + 4Z15A.T2 = 121 
By the way, it's no coincidence that both of these sets share the same number 15 (in 8Z15B and 4Z15A)  Forte numbered his sets fully aware of complementarities.
Another pair of Bebops
A final pair of bebop scales in this collection are found as modes of the noninvertible PC sets categorised by Forte as 822A and 827A.
The sets (as scales) are known as the Altered Bebop Dorian and the Bebop Melodic Minor.
822A.TA (PC 0 = D) D, E, F, G, A, B, C, D♭ Bebop Altered Dorian in D 

827A.T7 (PC 0 = A) A, B, C, D, E, F, G♭, A♭ Bebop Melodic Minor in A 
Of course when we say that these octatonic sets are unrelated to allinterval sets, this does not mean that one cannot extract an allinterval subset from them. For example a 4Z29A may be extracted from either of these scales, viz. (E, F, G, B) from the altered dorian and (B, C, D, G♭) from the melodic minor (both following the pattern {0, 1, 3, 7} from E and B respectively). It's simply that the four pitches remaining in each scale  respectively (A, C, D♭, D) and (E, F, A♭, A)  are not congruent with any allinterval set (inversions included).
It's also possible to pick out a 4Z15A as (D♭, D, F, G)  from the altered dorian. We leave it as an exercise for the student to spot any other possible extractions.
In any event, certainly neither octatonic set's complement is congruent to such a tetrad.
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