All-Trichordal Saturated M-Chain Cycles

ALL-TRICHORDAL SATURATED M-CHAIN CYCLES1 KYLE QUARLES AND DANIEL SIEBURTH 1. M-CHAINS OF LENGTH FOUR Suppose we want to write a short pitch-class segment L with elements such that {l1, l2, l3} is a member of set-class X, and {l2, l3, l4} is a member of M(X).2 This is an example of an M-chain.3,4 A procedure for constructing L can be given by taking any pitch-class set {x | x ∈ X} , and then determining if pcset {y | y ∈ M(X)} has exactly two common tones with pcset x. If it does, then we can form a pcseg L which fulfills our conditions, because we can place the two common pcs in the center of L as elements , while the elements exclusive to x xor y can be placed at either end as elements {l1} and {l4 }. We can generalize: DEFINITION 1: For any pitch class sets x ∈ SC(X) and y ∈ SC(Y) iff M(X) = Y, then is an M-chain.5 Consider the pcseg , a member of pcset {026}. We want to know if there is any member of M(026) which shares two common tones with {026}. In this case, M(026) = (026), so we need another 172 Perspectives of New Music member of SC(026). As an example consider the pcset {268}. The pcsets of which pcsegs and are members, are themselves both members of SC(026), so is an M-chain. But would not also work? Indeed it would, and we can see that in fact any pcseg that fulfills the intersection condition will have another form that also meets it. Thus Definition 1 holds for any internal ordering of the set , which is necessarily a subset of L. Assured that pcsegs like this can exist, can we get a list of all of them for each trichordal SC and its M-transformation? To set about finding them, let us finish all the forms of (026) we were just considering. We know that and both satisfy our stated conditions. Let us further narrow our list by counting R/T/I equivalent pcsegs as one object. So if we succeed, for each trichordal SC we will create an object analogous to the prime-form of a SC: the set of all R/T/I transformations of a chain of any length constitutes a chain-class, represented by a prime form always starting with pc 0.6 Any 4-place M-chain will itself be a member of some tetrachordal pcset. We gain traction by listing which tetrachordal SCs have both trichord x and M(x) as subsets. So to find all the tetrachordal SCs that can form a pcseg L when x = (026), we isolate all tetrachordal SCs which have two members of SC(026) as subsets. They are (0246), (0248), and (0268). We can find all the relevant pcsegs constructible from each tetrachordal SC by testing the number of ways in which the two Mrelated trichordal subsets can form that tetrachord. So for example, with (0246) we notice that the only forms of SC(026) available to us as subsets are {026} and {046}. These two pcsets have two common tones, so we are assured of being able to construct a pair of pcsegs with our desired intersection condition, with the forms and . All other pcsegs which fulfill the intersection condition for this tetrachord will be R/Tn/In transformations of these two forms. Example 1 records all possible forms of L (in other words, all the Mchain classes), each categorized by its parent tetrachord and intersecting trichords.7 All-Trichordal Saturated M-Chain Cycles 173 M-related trichordal SCs; M(x) = y Parent tetrachord M-chain classes Outer intervals M(012) = (027) {0127} 5-1 5-1 M(013) = (025) {0235} 3-3 1-5 2-2 1-5 1-5 2-2 1-5 3-3 {0135} and {0245} 1-2 2-5 2-1 5-2 {0136} and {0356} 1-3 3-5 3-1 5-3 M(014) = (037) {0347} 4-4 5-1 3-3 5-1 3-3 1-5 4-4 1-5 {0147} and {0367} 1-3 3...