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 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).
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 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.
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.
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)
4-Z29A.T4 → { 4, 5, 7, 11 }
4-Z15A.T8 → { 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)
4-Z29A.T7 → { 7, 8, 10, 2 }
4-Z15A.TB → { 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.
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)
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.
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.
In a previous post, we examined a b=22 block t=2-(v=12,k=6,λ=5) block design evenly distributed around the chromatic universe in both pitch-class (PC) and interval content. The design gave us 22 hexachords (132=6×22 pitches) where each PC turns up 11 times (132=11×12) and where also each of the interval-classes (1 to 6) appear exactly 5 times each. The blocks come from the 2nd block design partially reproduced here (a bleeding chunk, eliding unnecessary detail, from the previous post) as
Interval-class is defined as the shortest distance in semitones between pitch classes, modulo 12. A major sixth - an upward skip of 9 semitones - is thus equivalent to the shorter 3 semitone skip down of a minor third - octaves being ignored. The 6 semitone tritone leap, is consequently the largest interval class. Each block contains (after renumbering 1 … 12 to 0 … 11 to bring us into PC - modulo 12 - territory) 6 PCs, and thus 6×5/2=15 intervals between pairs therein. Over the 22 blocks we therefore have 22×15=330=66×5 intervals in total and the block design itself ensures that all possible intervals between 12 PCs - which is of course 66=12×11/2 - occur 5 times each.
Musically speaking, it's a very 'democratic' distribution. Not only do all pitch classes get equal representation but all interval classes too. This is an alternative way to give 'equality' to chromaticism rather than via Schoenbergian tone rows and their transformations to retrograde, inversion, and retrograde inversion; or via Hauer's tropical hexads.
By Design
Just playing a sequence of the 22 hexads - perhaps as chords, perhaps as ascending arpeggios, perhaps even as both with 11 of them in the treble and the other 11 in the bass, will result in a mathematically legitimate presentation of that equal opportunity. But it will likely not be musically interesting (e.g. see below). In any case, the blocks generated by the design aren't presented 'ordered', but due (as far as the musician is concerned) entirely to the exigencies of linear text.
We, however, can seek further structure within, structure which may pique interest. Mathematically, all we have - all we asked for - is a bunch of numbers delivered with a certain guaranteed distribution. The design's specification required nothing more. But first, here are the blocks, presented as PC Sets (in the Forte sense):
The above table is sorted by Forte's 'interval vector' classification for no particular reason other than to show that the 22 PC Sets may be aggregated into 10 distinct interval spread classes in 12 distinct Forte Prime PC Sets. We can't imagine that this is anything other than happenstance. As usual on this blog, the blue background indicates a symmetric (inverse = self) PC set and pink indicates asymmetric (with Forte A and B Forms). There's a page devoted to 12 TET hexads if more background is needed.
Non-Blocky Tetradic Links
In order to justify an order we might apply to the 22 blocks, we sought links between the blocks beyond any demanded by the block design itself. And indeed we found such a link, in the appearance of common tetrads between the hexads. Even better, we found several cycles (i.e. Hamiltonian circuits) of all 22 blocks, stepwise connected via such tetrad-sharing, in such a way that the last block connected to the first.
Each block links to three other blocks via tetrads, which means that the blocks' connectivity can be represented as a cubic graph, specifically one with 22 vertices (the blocks) and 33 edges (the linking tetrads shared between each vertex pair). And 22 of those edges form a circuit allowing the graph's presentation as a 22-gon with 11 internal edge connections.
The cubic graph exhibits an emergent symmetry having nothing to do with the block design's requirements. Nobody asked for symmetry or for common 4-sets connecting the 6-set blocks, or even anything regarding 4-sets at all.
The symmetry is doubtless due to the algorithm used to produce the design, most likely from the underlying group used to generate the design which in this case was C_{11} - Cyclic Group order 11 generated by the permutation (1,11,10,9,8,7,6,5,4,3,2) as shown in the design's output report.
The following is a possible expression of the above block order. The hex annotations under each bar label the six PCs present in the bar. If the staff could be wrapped around a circle, the four PCs 0,9,a and b in the final bar could - in principal - be tied to the same PCs in the first.
We say 'in principal' because the PCs a and b (as, respectively, a high A# and a centred B) are in the 'wrong' octave for bar #1, as are PCs 0 and 9 (as bass notes A and C - two octaves apart from their high treble appearances in bar #1). A 'true' circular join is of course possible. But as it is, it sounds like this:
Below is the graph with PC Set {0,3,8,9,10,11} at the top, labelled as vertex 0 (the arbitrarily chosen 'first' block in the circuit shown as the first column labelled (hexadecimally as) 0389AB in the image above).
Each block, labelled as vertices 0 to L clockwise from the top, is shown with its 6 constituent hexadic pitch classes (labelled in hexadecimal) in a hexagonal 'satellite' arrangement around it. Every block is connected to three others, two being nearest neighbours around the circumference and the third being somewhere across the circle.
Each of the three connections carries a common tetrad. For example vertex C - PC set {1,3,4,7,8,10} - connects on either side with vertex B = {0,3,4,6,8,10} and D = {1,2,3,4,7,11} as well as to the relatively distant vertex J = {0,1,3,5,7,8}, the common tetrads between them being respectively {3,4,8,10}, {1,3,4,7}, {1,3,7,8}
From Graph to Group
Howsoever a cyclic symmetry of C_{11} in the generation of a block design managed to percolate through to emergent common tetradicities, the 22-vertex 33-edge graph (teased out of that collection by seeking Hamiltonian circuits via those tetradic connections) has dihedral symmetry with an automorphism group isomorphic to D_{22} (or D_{11} depending upon which Dihedral Group naming convention you follow).
For the moment, we'll dispense with the hexadic attachments - the group is concerned only with the graph's symmetries but doesn't care how it got them.
As with all dihedral groups, two generators - a mirror flip and a rotation cycle - will suffice and in this case they are the permutations (1L)(2K)(3J)(4I)(5H)(6G)(7F)(8E)(9D)(AC) and (01HI98ED45L)(2GJA7FC36KB).
We're using cyclic permutation notation where each parenthesised string (in this case of graph vertex labels) cycles (i.e. moves its tail character to its head) each time the permutation is applied.
The flipper is relatively easy to see as a simultaneous swap of 10 vertex pairs over the vertical axis through 0 and B. Vertex 1 moves to L and L moves to 1, 2 moves to K and K to 2, and so on, each effectively swapping places.
Vertices 0 and B are absent from the permutation and unaffected. In other words its action permutes the sequence 0123456789ABCDEFGHIJKL into 0LKJIHGFEDCBA987654321 (stabilising 0 and B) as you'd expect of such a flipper. Hover over the image with your mouse to see the flip in action. Or click here to see the 10 components of the permutation acting in sequence. The part of the graph unaffected by the permutation is in red.
The second generator is harder to see because the twist's rotation takes place around a pair of 11-gonal paths which - due to the tetradic connectivity of the graph - is somewhat tortuously buried within the 22-gon. They're showing as two, blue and green, polygons. The cyclic permutation notation describes the simultaneous rotation through that pair of 11-gons. The first cycle moves vertex 0 to vertex 1, vertex 1 to vertex H, vertex H to vertex I, I to 9, 9 to 8 etc all the way around until the final move from vertex L to vertex 0. All vertices are moved by this operation (hence no red pieces). The group action maps the vertices 0123456789ABCDEFGHIJKL to 1HG65LKFE87234DCJI9AB0. Click here to see the two component permutations in sequence, or hover over the figure to see the complete action.
Certainly the graph may be re-presented to show the rotations more clearly - at the expense of making the Hamiltonian circuit hard to apprehend. Relabelling the vertices, as in the following graph, effectively disentangles the 11-gons. The original Hamiltonian circuit is still there, traced along the blue edges via 0123456789ABCDEFGHIJKL as before, but is no longer on the circumference.
The Hamiltonian circuit jumps back and forth between the two orbits, as in fig. 6, and is not itself an orbit of the group. The only thing the group does do for us, musically speaking, is ensure that - by its operations - it maintains a cycle of hexads (which may be changed by the operation) where adjacent hexads always have a common tetrad (which may also changed by the operation). The group has no idea it's doing this for us, all it 'knows' is that the cubic graph - of which it's an automorphism - is structured the way it is, and that it will preserve that structure.
The identity permutation - formally (0)(1)(2)(3)(4)(5)(6)(7)(8)(9)(A)(B)(C)(D)(E)(F)(G)(H)(I)(J)(K)(L) but abbreviated to () since permutations which move something to itself are conventionally omitted - and the two generator permutations (1L)(2K)(3J)(4I)(5H)(6G)(7F)(8E)(9D)(AC) and (01HI98ED45L)(2GJA7FC36KB) are only three of the twenty two elements of this group.
We can usefully notate those operations as 'e' for the identity (it's a convention), as 'f' for the flip and as 'r' for the twist/rotator permutations. In particular, r twists its way around the two eleven stage zig-zaggy polygonal paths - not the single twenty-two stage circumference. Don't be overly misled by the term 'rotation'.
You may even, if so inclined, write r as the product of two mutually exclusive, non-interfering, rotations 'p' and 'q' - respectively the permutation cycles (01HI98ED45L) - the blue 11-gon - and (2GJA7FC36KB) - the green one. Acting independently their product, r = pq = qp, is commutative. One sees that p^{2} = (0H9E4L1I8D5), that q^{2} = (2J7C6BGAF3K) and hence that r^{2} = p^{2}q^{2} = q^{2}p^{2} - and so on, for higher powers.
Reversing the strings inside the parentheses inverts the operation, explaining why 'flippers' involving solely permutation cycles of length two are self-inverting. Thus r^{-1} = p^{-1}q^{-1} = (0L54DE89IH1)(2BK63CF7AJG). So a flip followed by a rotation followed by a further flip is equivalent to the reverse rotation - true of all dihedrals.
You might also have noticed that p itself is a Hamiltonian cycle of length 11 - eleven of the blue cubic graph's edges form its 'circumference'. We also see that q is not a Hamiltonian, but that q^{4} = (276GFKJCBA3) is a Hamiltonian (as is, naturally, q^{-4}).
The following table shows all 22 of the group's elements written in terms of r and f, their orders (i.e. how many of that operation would have to be performed in sequence to be equivalent to the identity), the equivalent permutation product (in cycle notation) and the vertex action (the resulting reorder if the group were acting on a set of vertex labels).
Bear in mind that the labels on the vertices are there for our convenience only - the group operations 'see' only a bunch of (in this case 22) points abstractly linked with each other (with, in this case, 33 relations) for 'whatever reason'. There's no 'geography' but only an invariant structure which can be operated upon in 22 distinct ways whilst preserving their connectivities in exactly the same symmetric pattern.
The (D_{22}) Automorphism Group's 22 Operations
operation
order
permutation
vertex action
e
1
()
0123456789ABCDEFGHIJKL
f
2
(1L)(2K)(3J)(4I)(5H)(6G)(7F)(8E)(9D)(AC)
0LKJIHGFEDCBA987654321
fr
2
(01)(2B)(3A)(49)(5I)(6J)(7C)(8D)(GK)(HL)
10BA9IJCD43278EFKL56GH
fr^{2}
2
(0H)(37)(48)(59)(6A)(BG)(CF)(DE)(IL)(JK)
H12789A3456GFEDCB0LKJI
fr^{3}
2
(0I)(1H)(2G)(3F)(4E)(58)(67)(9L)(AK)(BJ)
IHGFE8765LKJCD43210BA9
fr^{4}
2
(09)(1I)(2J)(3C)(4D)(5E)(6F)(7K)(8L)(AB)
9IJCDEFKL0BA3456GH1278
fr^{5}
2
(08)(19)(2A)(5D)(6C)(7B)(EL)(FK)(GJ)(HI)
89A34DCB012765LKJIHGFE
rf
2
(0L)(15)(26)(3G)(4H)(9E)(AF)(BK)(CJ)(DI)
L56GH1278EFKJI9A34DCB0
r^{2}f
2
(05)(14)(23)(6B)(7A)(89)(CG)(DH)(EI)(FJ)
543210BA9876GHIJCDEFKL
r^{3}f
2
(04)(1D)(2C)(3B)(5L)(6K)(7J)(8I)(EH)(FG)
4DCB0LKJI9A321HGFE8765
r^{4}f
2
(0D)(1E)(2F)(3K)(4L)(7G)(8H)(9I)(AJ)(BC)
DEFKL56GHIJCB012789A34
r^{5}f
2
(0E)(18)(27)(36)(45)(9H)(AG)(BF)(CK)(DL)
E87654321HGFKL0BA9IJCD
r
11
(01HI98ED45L)(2GJA7FC36KB)
1HG65LKFE87234DCJI9AB0
frf
11
(0L54DE89IH1)(2BK63CF7AJG)
L0BCD43A9IJKFE8721HG65
r^{2}
11
(0H9E4L1I8D5)(2J7C6BGAF3K)
HIJKL0BCDEFG6543A98721
fr^{2}f
11
(05D8I1L4E9H)(2K3FAGB6C7J)
5LKFEDCJIHG6789AB01234
r^{3}
11
(0IE519DLH84)(2ACKG73BJF6)
I9AB01234DCJKL5678EFGH
fr^{3}f
11
(048HLD915EI)(26FJB37GKCA)
45678EFGH123A9IJKL0BCD
r^{4}
11
(094185HELID)(276GFKJCBA3)
98721HG6543AB0LKFEDCJI
fr^{4}f
11
(0DILEH58149)(23ABCJKFG67)
D43A987210BCJIHG65LKFE
r^{5}
11
(08L95I4HD1E)(2FB7KA6J3GC)
8EFGHIJKL567210BCD43A9
r^{6}
11
(0E1DH4I59L8)(2CG3J6AK7BF)
EDCJI9AB0LKFGH12345678
The Cycle Index of the group is pretty much what you'd expect of such a group, i.e. ^{1}⁄_{22}(z_{1}^{22} + 11 z_{1}^{2}z_{2}^{10} + 10 z_{11}^{2}).
Operations of order 2 are their own inverses - applied twice in succession, they 'do nothing'. This is naturally to be expected of the single flip, but we can see this also results from a flip-twist-flip-twist, or a flip-twist-twist-flip-twist-twist etc. And 11 consecutive rotations, for example, return you to whence you started.
These permutation actions by the group upon the set of labels 0 … L show that there are 22 different Hamiltonian circuits around the 22 blocks via the 33 links. But as 11 of them are just reversals (flips) of the other 11, we'd consider that there are really only 11. And as 11 is a prime number, those 11 are truly 'different' and can't be further subfactored.
Applications
There's quite a lot to play with here - its group representation shows that there are many more than simply the one way of ordering the 22 hexads (here's another one of the same ordering, but split between three instruments):
Permuting the cycle a further 10 times (or even 21 if you're happy to include reversals) will get you further 'new' sequences of tetradically linked hexads, all of them with equally occurring PC and IC frequencies. Not only that, but you may also consider splitting the sequences into separate groups of eleventh order Hamiltonian sequences. The 'democracy' of pitch class and interval class will be maintained as long as the appropriately related cycles are kept together (as per the last ten rows of the above table). That would give you 11-lengthed sequences of 12 PCs - though not all distinct. All this from just one block design. Enjoy.