Pattern Discovery in Old Hispanic Neume Notation Paul Rouse, University of Bristol 16th May 2021 1 Introduction The earliest surviving musical notation for medieval liturgical chant consists of neumes which do not provide the precise pitch and rhythmic information which forms the basis of much musicological scholarship. For traditions, such as the Old Hispanic chant, which were suppressed or fell out of use before pitched notation was developed, there is no later, pitched record of the same melodies,1 and they cannot be transcribed into modern notation. However, we do know how the neumes describe the direction of melodic movement, although not the exact intervals, and their shapes have a rich structure, which is clearly used in a consistent manner, even if we do not understand its meaning. Thus they do contain sufficient information for meaningful analysis, albeit using methodologies largely developed in the last decade.[1] A class of musicological questions can be phrased in terms of identifying frequently-occurring pat- terns of neumes, especially ones which are used in positions of particular musical significance, such as cadences or phrase beginnings, or settings of important words, or are used in a particular relationship with stressed and unstressed text syllables. In previous work, we have developed computer-aided methods for building a database of notated chants, for codifying the written shape and pitch contour represented by neumes, and for searching for given patterns of neumes.[2] Our Chant Editing and Analysis Program (CEAP) provides musicologists with a tool to test hypotheses concerning the use of specific neume patterns, but not, until now, to discover new patterns. The work reported here extends these techniques to discover recurring patterns automatically, presenting the musicologist with a display of those which occur frequently. The patterns are necessarily approximate. A musical formula may vary according to the context, for example to accommodate an extra syllable of text, or different accent pattern, and, while there is much consistency in the use of the notation, scribal variations do occur. In some cases one manuscript uses two separate neumes, while another runs them together to form a larger unit. Additionally, our tran- scriptions may differ from the scribe’s intent, either by using two distinct, but almost identical, shapes where the medieval musician would have seen no difference, or by making a different judgement as to whether a gap between strokes starts a fresh neume. As a consequence of such variations, the result of comparing two patterns must be a measure of similarity, not a simple yes or no answer. Before we can discuss the algorithm for discovering patterns, we must first explore the measurement of similarity between two patterns, which, in turn, involves an understanding of the information which can be gathered from neume notation. In the next section, we formulate a set of features 1While two dozen Old Hispanic chants do survive in heighted Aquitanian notation, the evidence they provide about cognates in unpitched notation is limited, and must be treated with care.[1] 1 which characterise each note within a neume, providing a new approach to encoding the neume interpretations established in our previous work. Then we show how this formulation is used to calculate a quantitative measure of the similarity, or otherwise, between two patterns. In section 4, we are then in a position to discuss how this measure is used in the discovery of recurring patterns. 2 Encoding Neume Interpretations For pattern-matching purposes, we focus on the sequence of individual notes represented by a neume, in terms of both melodic shape and the characteristics of the pen strokes used to notate them. This provides a structured way to encode the meaning of each neume, and to compare corresponding parts of the notated music without assuming that division into neumes is completely consistent. However, the beginning and ending notes are given specific markers in their encodings, allowing the division into neumes to be taken into account in measuring the quality of match. Comparison between patterns will rely on encoding this information in a particular way, which we present first below. This is a fine-grained encoding which should, nonetheless, be meaningful to musicologists, albeit not the most useful representation for everyday work. Section 2.2 shows how the new formulation is related to, and derived from, concepts and encodings developed previously for describing neume notation. 2.1 Note Features In the encoding used in this paper, the characteristics of each note within a neume are expressed as a set (in the mathematical sense) of binary features, which are explained in this section. The term “feature” is always used with this meaning, and we will use the term “feature vector” to refer to the set of features which apply to a given note. Representing neume characteristics as sets of binary features leads to a straightforward measure of similarity between patterns, presented in section 3. As shown in Fig. 1, a neume is divided into components representing separate notes. Each component can be described by the shape of the main part of the pen stroke, the shape of the connection with the previous stroke, and the direction of pitch movement relative to the preceding note. Unless there is a sharp change of direction, a connection covers a small region as the pen moves out of one stroke and into the next, and has its own direction of curvature. (a) (b) Fig. 1 After the first note of a neume, we usually know whether each note is higher than its predecessor, lower, or at the same pitch. Sometimes we know only that the pitch is unlikely to be lower, so the movement must be broadly upward, or conversely that it is unlikely to be higher, so the movement must be broadly downward. The appendix lists all of the symbols used in the present encoding; those beginning with the letter P are used for the features describing pitch relationships. Each of the 2 cases just described is encoded by a pair of features in the feature vector, chosen so that overlapping meanings share one common feature. For example, a note which is definitely higher than the one before is described by the pair of features fPh;Pphg, while a note which is merely unlikely to be lower is described by the pair fPph;Pnlg. These pairs share the feature Pph, so are considered to match each other partially. At the start of a new neume, we have no information relating the pitch to the previous neume, so no pitch-related features are included in the feature vector of the first note. Similar methods are used to encode the remaining characteristics. For example, again using feature names defined in the appendix, a continuous, smooth connection between strokes is represented by the features fCj;Csg in the feature vector, always accompanied by an additional feature which describes the direction of curvature and whether a loop is formed. The shared features Cj and Cs provide a partial match between any pair of smooth connections, even if they have different curvature directions, or one makes a loop while the other does not. Likewise, the features used to represent stroke shapes are designed to produce partial matching between the long strokes, whether curved or straight, and regardless of slope or curve direction; this happens via the feature called Sl in the appendix. Angled strokes with different directions also partially match each other because of the shared feature Sa. Only one feature is used for short, horizontal strokes, but for consistency with most strokes, which are encoded using two features, it is given double weight in the calculation of the metric in section 3. The same applies to the feature for a wavy stroke. In addition to the features arising from the interpretation of the note itself, some extra feature types show the context in which the note appears in the neume. One feature (Ns) is used to mark the first note of a neume, and another (Ne) is used to mark the last (both are present on the same note in the case of a single-note neume). Another feature is present in addition on the first note when the neume is the first on a syllable (Nsyl). When the neume is terminated with a hook, an additional feature (Nh) is present on its last note. In the future, further features may be added to encode characteristics of the text, such as whether the syllable is stressed. As an example, the complete feature vector for the third note of Fig. 1(a) is: fPh;Pph;Cj;Cs;Cw;Sl;Scrg The symbols used here are all described in the appendix, where there is also a table showing the encodings of all five notes of both example neumes. This is a low-level encoding, which would normally be derived automatically from a more user- friendly notation, such as the neume descriptions used by CEAP. Before moving on to discussing its use in constructing a measure of similarity between patterns, we briefly comment on the relationship with other notations. 2.2 Relationship with Existing Encodings The feature-based encoding rests on the same conceptual framework as the neume descriptions used in CEAP. The neumes module of MEI accommodates similar ideas.[3] The crucial change required for the pattern-matching methods of this paper is the use of binary features, since, as shown in the next section, this allows similarity between patterns to be measured simply by counting the number of features which are shared, and the number which are not shared.2 By contrast, CEAP and MEI notations both make use of attributes which can hold a range of values.