Block Around the Clock Group

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

… autGroup:=Group([(1,11,10,9,8,7,6,5,4,3,2)]), blockNumbers:=[22], blockSizes:=[6],
blocks:=[[1,2,3,4,7,12], [1,2,3,6,11,12], [1,2,4,6,8,9], [1,2,5,6,8,10], [1,2,5,10,11,12], [1,3,4,7,8,10], [1,3,5,6,9,10], [1,3,5,7,8,11], [1,4,5,7,9,11], [1,4,9,10,11,12], [1,6,7,8,9,12], [2,3,4,5,8,12], [2,3,5,7,9,10], [2,3,6,7,9,11], [2,4,5,8,9,11], [2,4,6,7,10,11], [2,7,8,9,10,12], [3,4,5,6,9,12], [3,4,6,8,10,11], [3,8,9,10,11,12], [4,5,6,7,10,12], [5,6,7,8,11,12]],… lambdas:=[5], t:=2), v:=12)

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):

Blocks as Pitch Class Sets from the design
block≡ PC SetiStringFort(e)ifications

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.

22 hexad blocks linked by tetrads

An Ordered Block Design

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 C11 - 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.

Hexachord sequence tied across bars by common tetrads.

Musical Implementation of Block Design

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).

Graph of 22 Hexadic Blocks with 33 Tetradic Commonalities
0 B A 9 8 3 0 1 B A 9 8 7 2 2 A 9 7 5 3 2 3 9 7 6 3 2 0 4 B 6 3 2 1 0 5 B A 5 2 1 0 6 A 8 6 5 2 1 7 A 9 6 5 3 1 8 B 9 6 5 4 3 9 B A 7 6 5 4 A A 7 6 4 2 0 B A 8 6 4 3 0 C A 8 7 4 3 1 D B 7 4 3 2 1 E B 8 5 4 3 2 F 9 8 5 4 2 0 G 9 8 6 4 2 1 H B 9 8 7 6 1 I B 8 7 6 5 0 J 8 7 5 3 1 0 K 9 7 5 4 1 0 L B A 9 4 1 0

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 C11 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 D22 (or D11 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.

The flip permutation (leaving tetradic connections unaltered)
0 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L

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 twist permutation (leaving tetradic connections unaltered)
0 1 2 3 4 5 6 7 8 9 A B C D E F G H I J K L

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 twist (rotation) permutation on a relabeled graph
0 1 H I 9 8 7 A J G 2 B K 6 3 C F E D 4 5 L

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 p2 = (0H9E4L1I8D5), that q2 = (2J7C6BGAF3K) and hence that r2 = p2q2 = q2p2 - 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-1q-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 q4 = (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 (D22) Automorphism Group's 22 Operations
operation order permutation vertex action

The Cycle Index of the group is pretty much what you'd expect of such a group, i.e. 122(z122 + 11 z12z210 + 10 z112).

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.


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):

Rendering hexads with common tetrads across bars with a trio

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.