The Polygony And The Octasy

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.


We have, in this dodecaphonic universe, four all-interval tetrads. Some may better know these as, respectively, PC Sets 4-Z29A, 4-Z15A, 4-Z29B and 4-Z15B 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).

0 1 2 3 4 5 6 7 8 9 A B 4-Z29A 4-Z15A 4-Z29B 4-Z15B
The Tetrarchy

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

A pair of non-interfering tetrads

Due to the non-colliding pitch class limitation, it turns out that - of the 66 possible ways to combine two of them - there are only 14 non-colliders. 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 8-28, 8-25, 8-26, 8-9, 8-17, 8-10, 8-3.

8-28 8-25 8-26 8-9 8-17 8-10 8-3
The Octarchy

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 4-Z29A rendered as the set of pitches C, C♯, E♭, G and the set 4-Z15A (prime-form 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).

Forte-wise, this might be expressed as something like 8-26 = 4-Z29A + 4-Z15A.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 prime-form 4-29A 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.

The Octacy

So a second example of such a ‘Fortean operational notation’ is demonstrated by 8-25's ‘11 o'clock’ orientation showing the image 4-Z29A + 4-Z15A.T5 = 8-25.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 8-25 to prime form's ‘midnight’) would require an application of .T1, and, bearing in mind that .TB.T1 ≡ .T0 is effectively a no-op, the equality could be reversed and rewritten as 8-25 = 4-Z29A.T1 + 4-Z15A.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 8-26) gives us (in two modes of the same sequence) both the Bebop Major and Bebop Harmonic Minor.

Forte 8-26.T4 as Major Bebop C, D, E, F, G, A♭, A, B (PC 0 = C)
C C♯ D E♭ E F F♯ G A♭ A B♭ B 4-Z29A.T4
→ { 4, 5, 7, 11 }
→ { 8, 9, 0, 2 }
{ E, F, G, B } ∪ { A♭, A, C, D }
Forte 8-26.T7 as Harmonic Minor Bebop C, D, E♭, F, G, A♭, B♭, B (PC 0 = C)
→ { 7, 8, 10, 2 }
→ { 11, 0, 3, 5 }
{ G, A♭, B♭, D } ∪ { B, C, E♭, F }
Major and Minor Beboppery from All Interval Sets

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 Forte-speak as 8-23. It's not one of our all-interval 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 8-23 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.

G A♭ A B♭ B C C♯ D E♭ E F G♭ D E♭ E F G♭ G G♯ A B♭ B C D♭ 8-23 8-23.T9 (with PC 0 = G)
G, A, B, C, D, E, F, G♭
Bebop Dominant in G
8-23.T2 (with PC 0 = D)
D, E, F, G♭, G, A, B, C
Bebop Dorian in D
Dominant and Dorian Beboppery modes of 8-23

Dominant Flat Nine

Another Bebop scale related to an all interval set is the Bebop Dominant Flat Nine. As this scale is not self-inverting, 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 8-Z15B - or operationally 8-Z15B.T9.

This set's ‘unused’ pitch classes, viz. { 2, 4, 10, 11 }, can be operationally written as 4-Z15A.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, 8-Z15B + 4-Z15A.TA = 12-1 (where 12-1 is Fortean for the complete chromatic scale).

The transposition of the above by 4 semitones - to modally shift 8-Z15B 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)
8-Z15B 4-Z15A.TA
{ 10, 11, 2, 4 }
≡ -8-Z15B
8-Z15B.T4 ≡
{ 0, 1, 4, 5, 7, 9, 10, 11 }
Bebop Δ♭9 + 4-Z15B.T2 = Chromatic

Forte: 8-Z15B.T4 + 4-Z15A.T2 = 12-1
Flat Nine Beboppery as an anti All-Interval Set

By the way, it's no coincidence that both of these sets share the same number 15 (in 8-Z15B and 4-Z15A) - 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 non-invertible PC sets categorised by Forte as 8-22A and 8-27A.

The sets (as scales) are known as the Altered Bebop Dorian and the Bebop Melodic Minor.

A B♭ B C D♭ D E♭ E F G♭ G A♭ 8-22A 8-27A 8-22A.TA (PC 0 = D)
D, E, F, G, A, B, C, D♭
Bebop Altered Dorian in D
8-27A.T7 (PC 0 = A)
A, B, C, D, E, F, G♭, A♭
Bebop Melodic Minor in A
Bebopperies unrelated to All-Interval Sets

Of course when we say that these octatonic sets are unrelated to all-interval sets, this does not mean that one cannot extract an all-interval subset from them. For example a 4-Z29A 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 all-interval set (inversions included).

It's also possible to pick out a 4-Z15A 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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