A COMPUTER-BASED IMPLEMENTATION OF BASSO CONTINUO RULES FOR FIGURED BASS REALIZATIONS Adam Wead Ian Knopke Jacobs School of Music Music Informatics Indiana University Indiana University [email protected] [email protected] ABSTRACT would be extremely desirable (from the point of view of the performer) to explore other ways to compare the origi- This paper documents an expert system for visually dis- nal source in an automated or partially-automated fashion. playing differing rule sets for producing figured bass parts Of special interest is the places where the various treatises, from a given, unfigured continuo part. The system is based often from entirely different geographical regions, differ on a decision tree procedure that is built from rules that as to performance instructions. To date, this has not been have been interpreted and broken down into a set of at- attempted in any technologically-assisted manner. tributes, with output classifications resulting in harmoniza- tions. Besides the visual and explanatory appeal, it can The project documented here explores an alternate, computer- also be used to generate new harmonizations and has the based method for answering these questions. We envi- potential to serve as a basis for a historically-accurate ac- sions several applications for it. First, it provides a way companiment system. to visually compare multiple sets of historical rules for harmonizing bass lines, a task that is difficult to do from 1. INTRODUCTION the original texts alone. Also, the rule sets and derived decision tree can be used to automatically generate new, Creating harmonizations from partially-figured or unfig- historically-accurate figured bass parts for given bass pas- ured continuo parts is an art form in itself, and is a nec- sages, and has the potential to be used in a live situation essary part of early music performance, as well as a stan- as the basis for an accompaniment system. dard topic in musicological research. One way perform- ers learn these skills is through practical experience or di- rectly from a teacher. However, those concerned with true historical accuracy in performance often turn to the orig- inal treatises that were published during the seventeenth 2. RELATED WORK and eighteenth centuries. Most of these are available in facsimile and have been translated, allowing the interested Basso continuo and its related performance aspects have performer or scholar to directly study the historical exam- been studied for decades by musicologists and music his- ples. This is not, however, a task to be taken lightly. Each torians. Some studies focus on specific performance as- of these treatises must be studied individually, and it is up pects [5], while others look at a large cross-section of mu- to the performer to distill the different rules and concepts sic from the period [18]. Perhaps the most comprehensive from each treatise, to create a comprehensive understand- body of work is that by F. T. Arnold [1]. Of all these ing that he or she would employ in the realization of a studies, this is the only one that attempts to provide an particular continuo bass. over-arching view of basso continuo practice as a whole. Some scholars have attempted to simplify this process However, it should be kept in mind that Arnold presents a through their own interpretations. Current studies of basso composite picture that he has devised and does not always continuo practice either adopt a theoretical view, or a per- draw a clear distinction as to how individual treatises dif- formance practice view. Although the two are not mutu- fer on certain practices. ally exclusive, the intended audience for such studies is either scholars in the fields of music theory or musicol- Methods for automatically generating harmonizations ogy, or a group of performers and teachers. The scholarly have also been actively studied. Some are more general view tends to focus more on the role continuo played in studies [13, 17], but most are concerned with specific styles the historic development of western music and its related such as the music of J. S. Bach [9, 10, 6, 8] or jazz har- theories, while the practical view examines how one may monizations [15]. Some excellent work has been done on learn the methods of continuo realization and use them real-time accompaniment systems [3, 4, 2]. A good intro- in the performance of works which require an extempo- duction to the basic concepts is that provided by Robert rized harmonic realization of a basso continuo part. It Rowe [14]. 188 3. METHODOLOGY Perhaps most importantly, some treatises do not contain complete sets of rules, but were instead intended to pro- One of the goals of this project is to create a coherent, vide details for improving the playing of performers who, historically-accurate rule set for determining the proper at the time, would have already had a basic understanding figured bass for a given bass line. The rule set should of accompaniment rules. The missing details need to be adhere to the constraints of various treatises, and should filled in. be able to accommodate contradictions and differences in Two treatises were chosen to form the source for our the recommended harmonizations that are recommended. initial data: a French treatise by Denis Delair published The classic machine-learning approach to solving a prob- in 1724 [11], and an Italian treatise by Francesco Gas- lem of this sort is to train a classifier on a data set and parini published in 1708 [7]. In addition to the close prox- “learn” the rules directly from the data [12]. This would imity of their publication dates, there were other reasons require a large collection of bass lines with complete sets for choosing these two treatises. First, both contained of figured bass already written in. We have not been able fairly logical, straightforward rules expressed in a similar to locate such a data set. Unfortunately, most music col- way. Secondly, both treatises contained simple instruc- lections from this time period are only partially figured, tions intended for beginners that could be used to con- which would leave us with the difficult problem of fill- struct a foundation of principles on which more compli- ing in or “guessing” at how to complete all of the missing cated rules could be constructed later. Also, both treatises parts. Some collections are more complete, but do not ex- use the motion and direction of the bass line as one of the ist in computer form. While these could be entered in a main factors for applying figures to bass notes that have format such as Humdrum, creating a large enough data set no existing figures in them. is not trivial. We began with a single chapter from a treatise on ac- Another problem is one of authority. In this case, the companiment by Denis Delair that contained an introduc- treatises are the “experts” that beginning lute players learn tory chapter on the basic application of figures to a bass at least part of their craft from. Most machine learning line. After some experimentation, it was determined that classifiers are concerned with producing the correct clas- each rule could be adequately explained using a set of sification based on the learned data set. Some pattern clas- approximately sixteen “questions” that could be asked of sifiers, such as neural nets, present major difficulties in un- each bass note to determine the correct figure that should derstanding how a particular classification, or in this case be applied. Each question can be represented by a single harmonization was arrived at. In fact, we see this as a ma- attribute in the form of a table. Some of the attributes in jor difference between the fields of Music Information Re- our table are: trieval and Music Theory. With MIR systems, the empha- sis is primarily on getting the correct answers, while the- Any figure already present with the bass note • orists and performers are often more concerned with the The scale degree of the bass note methodology that is used to produce the result. Addition- • ally, we would like to be able to provide a compact rep- Any accidentals present resentation of the rules, including comparisons between • The function of the note, ie. part of a cadence, in a rules derived from different treatises, so a performer or • theorist can more easily understand them and their place suspension within the entire practice, and perhaps serve as a guide to The intervallic direction of the bass line before and understanding the entire set of rules. • after the note in question With these considerations, it seemed natural to instead consider an expert system derived primarily from the rules The direction and scale degrees of the notes before in the treatises themselves. However, this approach has its • and after the note in question own complications. While the treatises certainly contain rules, these are not quite in a form that is ready to be used Each treatise rule was then carefully interpreted and ex- directly by a computer. pressed in terms of the above attributes, along with the For instance, consider the following quotation: proper harmonic outcome. Most rules produced a single outcome, but in some cases a single treatise example re- And when the composition calls for a sixth, sulted in several different rules. either directly or through resolution, together We then used these rules as input to build a decision with the said major third, then the following tree. The tree algorithm used here is a fairly standard ID3 note may have a fifth, and sometimes the fifth model, with basic pruning of spurious results, similar to followed by a sixth.