The Hitchhiker’s Guide to the All-Interval Twelve-Tone Rows

The Hitchhiker’s Guide to the All-Interval Twelve-Tone Rows Marco Buongiorno Nardelli (bio) Introduction Although all-interval twelve-tone series (ais) have been used in atonal music as early as the 1920s (Alban Berg’s Lyric Suite is an example), it took more than 40 years before their categorization and cataloguing. While Nicolas Slonimsky listed eighteen “invertible dodecaphonic progressions with all different intervals” in his Thesaurus of Scales and Melodic Patterns (1947), we owe Bauer-Mangelberg and Ferentz (1965) the first attempt to the generation of AIS using a computer program to explore all the combinatorial possibilities that give rise to such chords. Their approach led to a list of 1928 row “generators,” AIS that produce other AIS by inversion or transposition. Their list was later purged by David Cohen (1972–73) to an “irreducible list” of 266 chords. The final and definitive list of AIS was published by Robert Morris and Daniel Starr only a couple of years later (1974). Using a more sophisticated computer program (and a more powerful computer!) they were able to identify the full corpus of the AIS: 3856 all-interval series that are transpositionally and rotationally normal (normal form), and that reduce to 1928 once inversionally related rows [End Page 225] are eliminated. Once the AIS are identified computationally, the remaining discussion in their paper is based on a “by hand” analysis of the rows and their properties, and as such is limited by the power of human computation. In particular, they state that: It is possible that some . . . re-ordering scheme could, possibly in conjunction with other operations, link all AISs together by chains of relations, which would be an elegant description of the generation of AISs and would doubtless provide interesting compositional devices for pitch-ordering. (Morris and Starr 1974, 377) The above observation is the main motivation for this work: after introducing the computational tools used in our study, we indeed show that the “re-ordering” scheme envisioned by Morris and Starr exists in the form of complex networks and discuss the insight that can be gained by such representation. All-Interval Series Generators We generate the full corpus of the AIS using an algorithm that explores all the permutations of the vector of the sequence of intervals in an ordered pitch class set (pcs) of cardinality 12. This vector, hereby referred to as LISV (linear interval sequence vector), can be interpreted as the step-interval vector introduced by Cohn (1997) and originally proposed by Morris as the cyclic interval succession vector of stride 1 (1987). To produce the complete set of normal form generators of twelve-tone rows, we construct all the possible permutations of the LISV vector that contains all the intervals and then reconstruct the rows starting with pc 0. Note that since the computation of the LISV exploits the cyclic property of the row, we need to end the interval series with a tritone, and then remove from the list all the rows that have cardinality less than 12. In Example 1 we show the few lines of python code needed to generate all the AIS in normal form. Among the computational tools developed for this project (the IPython notebook with all the code to generate the results of this study and the final data can be found in the public repository https://github.com/marcobn/TheHichhikersGuideToAIS), we have also written a helper class to seamlessly operate on the rows with the four closed symmetry operations of inversion (I), retrograde (R), element-wise multiplication of the series by 5, mod-12 (M), and the cyclic permutation of stride w (Q), where w is chosen so that the result is again an AIS in normal form (the results of Q and R need to be [End Page 226] transposed to make them in normal form). This facilitates the classification of invariants and the identification of the irreducible normal forms, defined as “prime forms” from now on, as the subset of AIS that are not equivalent under I,R,M,Q (the subset of AIS that upon application of I,R,M,Q generate the full set of 3856 normal forms...