Compactness and convexity have been shown to represent important principles in music, reflecting a notion of consonance in scales and chords, and have been successfully applied to well-known problems from music research. In this paper, the notion of compactness is applied to the problem of pitch spelling. Pitch spelling addresses the question of how to derive traditional score notation from 12-tone pitch classes or MIDI. This paper proposes a pitch spelling algorithm that is based on only one principle: compactness in the Euler-lattice. Generally, the goodness of a pitch spelling model is measured in terms of its spelling accuracy. In this paper, we concentrate on the parsimony, cognitive plausibility and generalizability of the model as well. The spelling accuracy of the algorithm was evaluated on the first book of Bach's Well-tempered Clavier and had a success rate of 99.21%. A qualitative discussion of the model's cognitive plausibility, its parsimony and its generalizability is given.
BalzanoG. J. (1980). The group theoretical description of 12-fold and microtonal pitch systems. Computer Music Journal, 4 (4), 66–84.
2.
BodR. (2002a). Memory-based models of melodic analysis: Challenging the Gestalt principles. Journal of New Music Research, 31 (1), 27–37.
3.
BodR. (2002b). A unified model of structural organization in language and music. Journal of Artificial Intelligence Research, 17, 289–308.
4.
CambouropoulosE. (2001). Automatic pitch spelling: From numbers to sharps and flats. In Proceedings of the VIII Brazilian Symposium on Computer Music. Fortaleza, Brasil.
5.
CambouropoulosE. (2003). Pitch spelling, a computational model. Music Perception, 20 (4), 411–29.
6.
ChaterN. (1999). The search for simplicity: A fundamental cognitive principle?The Quarterly Journal of Experimental Psychology, 52A (2), 273–302.
7.
ChewE. (2000). Towards a mathematical model of tonality. Ph.D. thesis, Operations Research Center, Massachusetts Institute of Technology, Cambridge.
8.
ChewE. (2001). Modeling tonality: Applications to music cognition. In MooreJ. D.StenningK. (eds) Proceedings of the 23rd Annual Meeting of the Cognitive Science Society, CogSci2001, (pp. 206–11). Edinburgh, Scodand, UK.
9.
ChewE. (2002). The spiral array: An algorithm for determining key boundaries. In AnagnostopoulouC.FerrandM.SmaillA. (eds) Music and Artificial Intelligence—Second International Conference, ICMAI 2002, (pp. 18–31). Edinburgh, Scodand, UK.
10.
ChewE. (2006). Slicing it all ways: Mathematical models for tonal induction, approximation and segmentation using the spiral array. INFORMS Journal on Computing, 18 (3).
11.
ChewE.ChenY.-C. (2003a). Determining context-defining windows: Pitch spelling using the spiral array. In Proceedings of the 4th International Conference for Music Information Retrieval ISMIR.
12.
ChewE.ChenY.-C. (2003b). Mapping midi to the spiral array: Disambiguating pitch spellings. In BhargavaH. K.YeN. (eds) Computational Modeling and Problem Solving in the Networked World, Proceedings of the 8th INFORMS Computer Society Conference, ICS2003, vol. 21 of OR/CS Interfaces Series, (pp. 259–75). Kluwer Academic Publishers.
13.
ChewE.ChenY.-C. (2005). Pitch spelling using the spiral array. Computer Music Journal29 (2), 61–76.
14.
ChuanC.-H.ChewE. (2005). Polyphonic audio key-finding using the spiral array CEG algorithm. In International Conference on Multimedia and Expo (ICME), (pp. 21–24). Amsterdam, Netherlands.
15.
CollardR.VosP.LeeuwenbergE. (1981). What melody tells about metre in music. Zeitschrift für Psychologie, 189, 25–33.
16.
FokkerA. D. (1949). Just Intonation.The Hague: Martinus Nijhoff.
17.
HelmholtzH. (1954/1863). On the Sensations of Tone, (second English ed). Dover.
18.
HoningH. (2006). Computational modeling of music cognition: a case study on model selection. Music Perception, 23 (5), 365–76.
19.
HoninghA. K. (2003). Group theoretic description of just intonation. In BuzzancaG. (eds) Proceedings of the 3rd International Conference Understanding and Creating Music, vol. 3. Caserta, Italy.
20.
HoninghA. K. (2006a). Convexity and compactness as models for the preferred intonation of chords. In Proceedings of the Ninth International Conference on Music Perception and Cognition (ICMPC 9). Bologna, 22–26 August.
21.
HoninghA. K. (2006b). The Origin and Well-Formedness of Tonal Pitch Structures. Ph.D. thesis, University of Amsterdam, The Netherlands.
22.
HoninghA. K. (2007a). Automatic modulation finding using convex sets of notes. In Proceedings of Mathematics and Computation in Music (MCM2007). Berlin, Germany May 18–20.
23.
HoninghA. K. (2007b). Pitch spelling: Investigating reduction of the search space. In Proceedings of the 4th Sound and Music Computing Conference (SMC07). Lefkada, Greece, 11–13 July.
24.
HoninghA. K.BodR. (2005). Convexity and the well-formedness of musical objects. Journal of New Music Research, 34 (3), 293–303.
25.
JackendoffR. (1991). Musical parsing and musical affect. Music Perception, 9, 199–230.
26.
KrumhanslC. L. (1990). Cognitive Foundations of Musical Pitch. Oxford Psychology Series No. 17. Oxford University Press.
27.
LerdahlF.JackendoffR. (1983). A Generative Theory of Tonal Music.MIT Press.
28.
Longuet-HigginsH. C. (1962a). Letter to a musical friend. Music Review, 23, 244–48.
29.
Longuet-HigginsH. C. (1962b). Second letter to a musical friend. Music Review, 23, 271–80.
30.
Longuet-HigginsH. C. (1987). The perception of melodies. In Longuet-HigginsH. C. (ed) Mental Processes: Studies in Cognitive Science, (pp. 105–129). London: British Psychological Society/MIT Press. Published earlier as Longuet-Higgins 1976.
31.
Longuet-HigginsH. C.SteedmanM. (1987/1971). On interpreting Bach. In Longuet-HigginsH. C. (ed) Mental Processes: Studies in Cognitive Science, (pp. 82–104). British Psychological Society/MIT Press. Published earlier as Longuet-Higgins and Steedman (1971).
32.
MeredithD. (2003). Pitch spelling algorithms. In Proceedings of the Fifth Triennial ESCOM Conference, (pp. 204–07). Hanover University of Music and Drama, Germany.
33.
MeredithD. (2005). Comparing pitch spelling algorithms on a large corpus of tonal music. In WiilU. K. (ed) Computer Music Modeling and Retrieval: Second International Symposium, CMMR, (pp. 173–92). Springer, Berlin.
34.
MeredithD. (2006). The ps13 pitch spelling algorithm. Journal of New Music Research, 35 (2), 121–59.
35.
MeredithD. (2007a). Computing Pitch Names in Tonal Music: A Comparative Analysis of Pitch Spelling Algorithms. D. Phil, dissertation. Faculty of Music, University of Oxford.
36.
MeredithD. (2007b). Optimizing Chew and Chen's pitch spelling algorithm. Computer Music Journal, 31 (2).
37.
RiemannH. (1914). Ideen zu einer Lehre von den Tonvorstellungen. Jahrbuch der Musikbibliothek Peters, 21/22, 1–26. Leipzig.
38.
ShepardR. N. (1982). Geometrical approximations to the structure of musical pitch. Psychological Review, 89, 305–33.
39.
TemperleyD. (2001). The Cognition of Basic Musical Structures.The MIT Press.
40.
TheodoruG.RaphaelC. (2007). Pitch spelling with conditionally independent voices. In the 8th International Conference on Music Information Retrieval (ISMIR 2007), (pp. 201–06). Vienna, Austria.
41.
VidyamurthyG.ChakrapaniJ. (1992). Cognition of tonal centers: A fuzzy approach. Computer Music Journal, 16 (2), 45–50.
42.
WertheimerM. (1923). Untersuchungen zur Lehre von der Gestalt. Psychologische Forschung, 4, 301–50.
