'No wonder kids grow up crazy. A cat's cradle is nothing but
a bunch of X's between somebody's hands, and little kids look
and look and look at all those X's ...'
'And?'
'No damn cat, and no damn cradle.'
– Kurt Vonnegut, Cat's Cradle
– Kurt Vonnegut, Cat's Cradle
To explain this partition puzzle (I know how to find the cats, but can I find cradles for all of them?), it's helpful to begin with a motivating example or two. First take a look at Table 1.
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| Table 1 |
This list of z-related[1] pairs of tetrads was extracted from the list I posted recently of the 1,591 tetrads from K4 through K24. Musicians can relate to the complete tetrad list in practical terms as either the complete tetrachord content in all harmonic spaces from the single tetrachord in the equal 4-division of the octave (4tET) through the 256 tetrachords in quartertone space (24tET), or all 4-onset rhythms possible in a single measure of 4 beats through a measure of 24 beats (disallowing rhythmic tuplets). I tend to think of it as an uninterpreted "event space" leaving it open to other applications both inside and outside the music domain.
Next take a look at this fairly simple diagram (Example 1) which represents the entry at the top of Table 1 as two tetragons inscribed in K8-space[2]:
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| Example 1 |
This is the first appearance of z-related sets in any K-space[3]. The box to the lower left indicates two "descriptions" – shape and content – of the figures inscribed in the K8-space. On the left are the interval strings for the two figures, (1214⤸ and (1331⤸ which describe the unique shapes of the inscribed polygons. All intervals in a string are measured clockwise. On the right in the box is the interval-class vector [2121] common to both strings. An interval class (ic) is defined as the shortest distance between two nodes, whether that is measured clockwise or counterclockwise. And the interval-class vector of any inscribed polygon describes the total ic content of the polygon. Examples 2a & 2b demonstrate the ic-vector equivalence of the two figures.
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| Example 2a: (1214⤸ Example 2b: (1331⤸ (ic content in both figures is [2121]: two ic1's,one ic2, two ic3's & one ic4) |
Example 1 is also the first appearance of z-related sets that can be explained by the Babbitt "hexachord theorem." Of course, the theorem does not only apply in K12-space where it was first observed. It should be thought of as a set-complement theorem that applies in any K2n-space (i.e., any K-space with an even number of nodes). An informal generalized statement of the theorem might be:
Babbitt Complement Theorem. When you choose any n nodes from any K2n-space that define a structure S, their complement – the n remaining nodes – will define a structure T that has the same interval-class vector as S.
The two figures described by the two complementary sets of n nodes in K2n-space might be transformationally related by rotation (musical transposition) or reflection (musical inversion)[5]; but if they are not, they are said to be z-related. Example 3a shows (1124⤸↔(4211⤸ by reflection; Example 2 shows (2123⤸↔(2123⤸ by rotation (as well as reflection). (1133⤸ and (1214⤸ in Example 3c are not related by any (known) transformation, but because they are complements, by the theorem their ic-vectors are nevertheless identical – z-related.
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| Example 3a Example 3b Example 3c |
So far I have presented nothing new. It's been no more than a review, a generalization – to point to basic concepts beneath applications. Now, with one very slight adjustment in nomenclature, I can present the puzzle. The generalized hexachord theorem above is based on the idea of complementation which assumes a bifurcation – placing all the elements of a set into one or the other of two baskets. The Babbitt Theorem for pairs of tetrads only applies in K8. But could it be that the Babbitt Complement Theorem is a special case of a more general partition theorem such that z-related tetrads appearing in K-spaces other than K8 can be arranged as a partition of the space in which they are found?
First, the simplest case. What happens if we double the size of the space in the previous example from K8 to K16? Obviously, any figure in K8 will appear as a congruence (simply multiplied by 2) in K16, and any z-relation in K8 will be carried over to K16. So: (1214⤸ → (2428⤸; (1331⤸ → (2662⤸; and (2428⤸ & (2662⤸ will be z-related (but only indirectly because of the Babbitt Theorem), sharing the same ic vector [02010201]. But looking back at Table 1, we see that there is also a new pair of z-related figures, (1357⤸ and (3418⤸, not congruent to any figure in K8. Inverting (3418⤸ to (1438⤸ and experimenting with the placement of both z-pairs in K16, we discover (empirically) that we can partition K16 with its z-related tetrads (Example 4).[6]
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| Example 4 |
Now this is getting interesting. Tripling K8 to K24 and looking back at Table 1 again, we find K24 has exactly three pairs of z-related figures, and all of these taken together can partition K24 (Example 5).
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| Example 5 |
(363C⤸ & (3993⤸ are the result of tripling from K8, and (157B⤸ & (165C⤸ are new, but this partition brings in another player, the familiar pair of all-interval tetrads found in K12 appear doubled in K24: (4215⤸ → (842A⤸ & (2316⤸ → (462C⤸.[7]
But this is as far as you can go using the tetrad relationships in Table 1. For one thing, the only z-related tetrads in K12, the all-interval tetrad pair, can't partition K12. But this glitch actually opens up to an even more interesting possibility. As it happens, the well-known octatonic string in K12, (12121212⤸, acts just like the entire space in which it is embedded with respect to the Babbitt Complement Theorem (see "Z-Related Sets as Dual Inversions," PNM, 39.1 (1995), §3.18). Choose any 4-element substring in the octatonic, and its complement with respect to the octatonic will have the same ic vector. This includes the all-interval pair:
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| Example 6 |
The complement of the octatonic string is (3333⤸ – in 12tET, the diminished 7th chord – indicated in Example 6 as a gray square. So K12 can be partitioned by two z-related all-interval tetrads and the maximally even tetrad (3333⤸. This suggests a remote possibility that any K-space might be partitioned by combinations of z-related and maximally even sets. It becomes surprisingly less bizarre when we note how all the tetrads possible in K20 can be situated around a dilation of that same maximally even square:
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| Example 7 |
Actually, this might work if you consider the previous Examples 1, 4 and 5 to be z-sets situated around a maximally even set of order zero. Ah, but what about that one z-related structure in Table 1 that we've ignored – the pair in K13. Well, it works with a different max even set (32323⤸. Here it is:
[Correction & remarks coming soon]
To be continued.........
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[1] While it is tempting here to point to the recent important (and surprising) music–crystallography connection noted by others in the music theory literature by replacing "z-related" with the much more descriptive "homometric" used in crystallography, graph theory & elsewhere, unfortunately "homometric" comes with its own set of baggage, not all of which do I understand adequately enough to reference with confidence. But the real problem that Forte, evidently unwittingly, set up by calling this phenomenon the "Z-relation" is that it immediately becomes confused with Zn, the set of all congruence classes of the integers Z for the relation congrunce mod n. This forces tortured locutions such as "the set of all Z-related sets in the set Zn." My solution here (which I remain unhappy with) is to keep the "z-relation" designation in its current meaning in music theory (but in an expanded sense and use) and refer to pitch–/rhythm– (event–) space as K–space or more specifically as Kn–space. Kn–space can be represented visually as the 1-dimensional space defined by n nodes equally spaced around the circumference of a circle. The nodes in Kn–space may be labelled, as in the venerable circle of fifths (C, G, D, ..., Bb, F, (C)), or unlabelled, as it is throughout this blog entry.
[1] While it is tempting here to point to the recent important (and surprising) music–crystallography connection noted by others in the music theory literature by replacing "z-related" with the much more descriptive "homometric" used in crystallography, graph theory & elsewhere, unfortunately "homometric" comes with its own set of baggage, not all of which do I understand adequately enough to reference with confidence. But the real problem that Forte, evidently unwittingly, set up by calling this phenomenon the "Z-relation" is that it immediately becomes confused with Zn, the set of all congruence classes of the integers Z for the relation congrunce mod n. This forces tortured locutions such as "the set of all Z-related sets in the set Zn." My solution here (which I remain unhappy with) is to keep the "z-relation" designation in its current meaning in music theory (but in an expanded sense and use) and refer to pitch–/rhythm– (event–) space as K–space or more specifically as Kn–space. Kn–space can be represented visually as the 1-dimensional space defined by n nodes equally spaced around the circumference of a circle. The nodes in Kn–space may be labelled, as in the venerable circle of fifths (C, G, D, ..., Bb, F, (C)), or unlabelled, as it is throughout this blog entry.
[2] vid. note [1]. For the curious: I'm using "K" for two personal reasons. First, a reference I once read (I believe it was by Nick Collins) referred to these as "Krenek circles" but I rather doubt Krenek was the first to use them in a music theory context. The other, more whimsical, reference is to the Greek κύκλος (kyklos) from Archimedes' last words to the Roman soldier who was about to kill him: "μή μου τοὺς κύκλους τάραττε" – Do not disturb my circles! (Probably apocryphal, but I like to believe it.)
[3] First used by Arthur Lindo Patterson to illustrate homometric structures (our z-related sets) in the phase problem in X-ray crystallography. [Cite: ______]
[4] For those who need an immediate "real" application in music, the diagram can be interpreted rhythmically this way:
[3] First used by Arthur Lindo Patterson to illustrate homometric structures (our z-related sets) in the phase problem in X-ray crystallography. [Cite: ______]
[4] For those who need an immediate "real" application in music, the diagram can be interpreted rhythmically this way:
Where the symbols "(" and "⤸" indicate that the measure can be "recycled" to begin on any of the eight notes, just as the same symbols were previously used to indicate the equivalence class of cyclically related permutations.
[5] – or dilation mod 2n (along with rotation to achieve a complement relationship), a kind of quasi-homothetic transformation that warps the K–space into itself. (Music's special case for dilation in 12tET is multiplication by 5 or 7 mod 12). But there is no need to introduce this complication here.
[6] NB: For all of these cases there is likely more than one way to partition any space with z-sets. In this first case, for example, the entire partition can be mirrored.
[7] A=10, B=11, C=12, etc.
