Proceedings ICMC|SMC|2014 14-20 September 2014, Athens, Greece An Idiom-independent Representation of Chords for Computational Music Analysis and Generation First Author Second Author Third Author Affiliation1 Affiliation2 Affiliation3 [email protected] [email protected] [email protected] ABSTRACT given some fundamental idiom features, such as pitch hierarchy and consonance/dissonance classification, can In this paper we focus on issues of harmonic representa- automatically encode pitch simultaneities in a pertinent tion and computational analysis. A new idiom- manner for the idiom at hand? independent representation is proposed of chord types Before attempting to answer the above question one that is appropriate for encoding tone simultaneities in any could ask: What might such a ‘universal’ encoding sys- harmonic context (such as tonal, modal, jazz, octatonic, tem be useful for? Apart from music-theoretic interest atonal). The General Chord Type (GCT) representation, and cognitive considerations/implications, a general allows the re-arrangement of the notes of a harmonic chord encoding representation may allow developing simultaneity such that abstract idiom-specific types of generic harmonic systems that may be adaptable to di- chords may be derived; this encoding is inspired by the verse harmonic idioms, rather than designing ad hoc sys- standard roman numeral chord type labeling, but is more tems for individual harmonic spaces. This was the prima- general and flexible. Given a consonance-dissonance ry aim for devising the General Chord Type (GCT) repre- classification of intervals (that reflects culturally- sentation. In the case of the project C-------T (name con- dependent notions of consonance/dissonance), and a cealed for peer reviewing) [3], a creative melodic harmo- scale, the GCT algorithm finds the maximal subset of nisation system is required that relies on conceptual blending between diverse harmonic spaces in order to notes of a given note simultaneity that contains only con- generate novel harmonic constructions; mapping be- sonant intervals; this maximal subset forms the base upon tween such different spaces is facilitated when the shared which the chord type is built. The proposed representa- generic space is defined with clarity, its generic concepts tion is ideal for hierarchic harmonic systems such as the are expressed in a general and idiom-independent man- tonal system and its many variations, but adjusts to any ner, and a common general representation is available. other harmonic system such as post-tonal, atonal music, In recent years, many melodic harmonisation systems or traditional polyphonic systems. The GCT representa- have been developed, some rule-based [4,5] or evolution- tion is applied to a small set of examples from diverse ary approaches that utilize rule based fitness evaluation musical idioms, and its output is illustrated and analysed [6, 7] others relying on machine learning techniques like showing its potential, especially, for computational music probabilistic approaches [8,9] and neural networks [10], analysis & music information retrieval. grammars [11] or hybrid systems (e.g. [12]). Almost all of these systems model aspects of tonal harmony: from 1. INTRODUCTION “standard” Bach–like chorale harmonisation [4,10] among many others) to tonal systems such as “classic” There exist different typologies for encoding note simul- jazz or pop ([9,11] among others). These systems aim to taneities that embody different levels of harmonic infor- produce harmonizations of melodies that reflect the style mation/abstraction and cover different harmonic idioms. of the discussed idiom, which is pursued by utilising For instance, for tonal musics, chord notations such as the chords and chord annotations that are characteristic of the following are commonly used: figured bass (pitch classes idiom. For instance, the chord representation for studies denoted above a bass note – no concept of ‘chord’), in the Bach chorales include usually standard Roman popular music guitar style notation or jazz notation (abso- numeral symbols, while jazz approaches encompass addi- lute chord), roman numeral encoding (relative to a key) tional information about extensions in the guitar style [1]. For atonal and other non-tonal systems, pc-set theo- encoding. retic encodings [2] may be employed. For tonal computational models, Harte’s representa- A question arises: is it possible to devise a ‘universal’ tion [13] provides a systematic, context-independent syn- chord representation that adapts to different harmonic tax for representing chord symbols which can easily be idioms? Is it possible to determine a mechanism that, written and understood by musicians , and, at the same time, is simple and unambiguous to parse with computer Copyright: © 2014 XXXXXXXXXXXXXXXXXXXXX et al. This is an programs. This chord representation is very useful for open-access article distributed under the terms of the Creative Com- annotating manually tonal music - mostly genres such as mons Attribution License 3.0 Unported, which permits unrestricted use, pop, rock, jazz that use guitar-style notation. However, it distribution, and reproduction in any medium, provided the original author and source are credited. - 1002 - Proceedings ICMC|SMC|2014 14-20 September 2014, Athens, Greece cannot be automatically extracted from chord reductions monies). Neither of them can be readily extended to other and is not designed to be used in non-tonal musics. idiosyncratic harmonic systems. In this paper, firstly, we present the main concepts Harmonic consonance/dissonance has two major behind the General Chord Type representation and give components: Sensory-based dissonance (psychoacoustic an overall description, then, we describe the GCT algo- component) and music-idiom-based dissonance (cultural rithm that automatically computes chord types for each component)[18]. Due to the music-idiom dependency chord, then, we present examples form diverse music component, it is not possible to have a fixed universal idioms that show the potential of the representation and model of harmonic consonance/dissonance. A classifica- give some examples of applying statistical learning on tion of intervals into categories across the dissonance- such a representation, and, finally, we will discuss prob- consonance continuum can be made only for a specific lems and future improvements. idiom. The most elementary classification is into two basic categories: consonant and dissonant. For instance, 2. REPRESENTING CHORDS in the common-practice tonal system, unisons, octaves, perfect fifths/fourths (perfect consonances) and thirds and Harmonic analysis focuses on describing the harmonic sixths (imperfect consonances) are considered to be con- content of pitch collections/patterns within a given music sonances, whereas the rest of the intervals (seconds, sev- context in terms of harmonic labels, classes, functions enths, tritone) are considered to be dissonances; in poly- and so on. Harmonic analysis is a rather complex musi- phonic singing from Epirus, major seconds and minor cal task that involves not only finding roots and labelling sevenths may additionally be considered ‘consonant’ as chords within a key, but also segmentation (points of they appear in metrically strong positions and require no harmonic change), identification of non-chord notes, met- resolution; in atonal music, all intervals may be consid- ric information and more generally musical context [14]. ered equally ‘consonant’. In this paper, we focus on the core problem of labelling Let’s examine the case of tonal and atonal harmony; chords within a given pitch hierarchy (e.g. key); thus we these are probably as different as two harmonic spaces assume that a full harmonic reduction is available as in- may be. In the case of tonal and atonal harmony, some put to the model (manually constructed harmonic reduc- basic concepts are shared; however, actual systematic tions). descriptions of chord-types and categories are drastically Our intention is to create an analytic system that may different (if not incompatible), rendering any attempt to label any pitch collection, based on a set of user-defined ‘align’ two input spaces challenging and possibly mis- criteria rather than on standard tonal music theoretic leading (Figure 1). On one hand, tonal harmony uses a models or fixed psychoacoustic properties of harmonic limited set of basic chord types (major, minor, dimin- tones. We intend our representation to be able to cope ished, augmented) with extensions (7ths, 9ths etc.) that with chords not only in the tonal system, but any harmon- have roots positioned in relation to scale degrees and the ic system (e.g. octatonic, whole-tone, atonal, traditional tonic, reflecting the hierarchic nature of tonal harmony; harmonic systems, etc.). on the other hand, atonal harmony employs a flat mathe- Root-finding is a core harmonic problem addressed matical formalism that encodes pitches as pitch-class sets primarily following two approaches: the standard stack- leaving aside any notion of pitch hierarchy, tone centres of-thirds approach and the virtual pitch approach. The or more abstract chord categories and functions. It seems first attempts to re-order chord notes such that they are as if it is two worlds apart having as the only meeting separated by (major or minor) third intervals preserving point the fact that tones sound together (physically sound- the most compact ordering of the chord; these stacks