More MOF and Mallalieu-related Rows

More MOF and Mallalieu related rows Caleb Morgan and Michael Davis IN POLYPHONIC MUSIC, SEPARATE MELODIC STRANDS combine to form a meaningful whole. No matter what the style, the combina tion is controlled and deliberate. In the history of 12-tone music, there have been many efforts to control the "harmonies" created by multiple rows sounding at once. How to create amusic inwhich vertical combi nations relate to the original series in some way?so that the harmonies are made from the same cloth as the lines?seems to be an enduring problem. Multiple-order-function series (MOF rows) are one solution to thisproblem. They also have a number of other uses. As we will see, themost impressiveMOF row is theMallalieu series. Because it is unique?to within multiplicative transforms?we are motivated to find other series that have a good deal of theMallalieu row's MOF properties. By approaching the generation of MOF rows from several angles, we can produce a number of different lists of series ?large and varied enough to satisfydifferent composer's sensibilities. MOF and Mallalieu-Related Rows 101 We extend previous work of music theorists such as Babbitt, Morris, and others in fiveways: 1) a handy algorithm for generating MOF rows; 2) a list of 73+ "limited-interval" series for use in creation and expansion of series; 3) software which performs brute-force generate and -test techniques to findMOF rows; 4) a "ranking" technique for generating "fuzzy" MOF rows from the discrete logarithms of power residue series; 5) another simple technique for deriving MOF rows from any power-residue series. Starting with the Mallalieu series?named after its discoverer, Pohlman Mallalieu?we will examine various MOF rows and their methods for generation. (We will also consider some self-similar pitch class sequences that are not 12-tone rows.) Definitions ofMOF row A succinct definition seems difficult.Morris does not produce a defini tion per se, but he implies that a MOF row X has a certain series of successive contiguous segments that are found in another transformation of the row, FX, such that each segment fromX occurs in either forward or retrograde order and the segments are merged contiguously or non contiguously in FX. Philip Batstone attempts no definition, but gives a technique for generating MOF rows. Andrew Mead has this to say about theMallalieu row: The row has the property that when it is divided into chains of equally spaced order numbers and the resulting partition isproperly concatenated, the ordered aggregate so produced is itself some transposition of the row.1 In an article by Brian Alegant, we find thisdefinition of the properties of theMallalieu row: ". . .the cyclic-interval patterns of this row generate its own transpositions."2 George Perle, in a mention of the series of Schoenberg's Quintet for Wind Instruments, Op. 26?a very early 12 tone composition perhaps employing the first MOF row?says "(it) is based upon a set that is cyclically permuted to produce an approximate transposition of the theme originally derived from the set."3 Our defini tion here is slightly broader and more pragmatic. For our purposes, a MOF row is any 12-tone series containing embedded sequences of pcs? especially sequences formed by skipping some pcs?which clearly resem ble another related series (usually in the same row-class). The 102 PerspectivesofNew Music resemblance can be imprecise and may involve sequences or motifs that are shorter than twelve notes. We are concerned with two kinds ofMOF rows: 1) Exact sequences of pcs that are found multiple times in the other related rows in the row-class with variable or irregular patterns of intervening pcs. 2) Inexact sequences of pcs (altered by small intervals) that are found multiple times in other related rows in the row-class with regular patterns of intervening pcs (or permutation). The self-similar patterns desired involve either imperfect resemblances or irregular patterns of permutation, but not both at once. Both kinds of pattern may be found in the same series, however, in different parts of the series. Both kinds of self-similarity at once is beyond the scope of thispaper. Illustration ofMOF concept with Mallalieu matrix The basic concept isperhaps more...