Conchord: An Application for Generating Musical Harmony by Navigating in the Tonal Interval Space

  • Gilberto Bernardes
  • Diogo Cocharro
  • Carlos Guedes
  • Matthew E. P. Davies
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9617)


We present Conchord, a system for real-time automatic generation of musical harmony through navigation in a novel 12-dimensional Tonal Interval Space. In this tonal space, angular and Euclidean distances among vectors representing multi-level pitch configurations equate with music theory principles, and vector norms acts as an indicator of consonance. Building upon these attributes, users can intuitively and dynamically define a collection of chords based on their relation to a tonal center (or key) and their consonance level. Furthermore, two algorithmic strategies grounded in principles from function and root-motion harmonic theories allow the generation of chord progressions characteristic of Western tonal music.


Generative music Harmony Tonal pitch space 



This research is financed by National Funds through the FCT - Fundação para a Ciência e a Tecnologia within post-doctoral grants SFRH/BPD/109457/2015 and SFRH/BPD/88722/2012.


  1. 1.
    Rowe, R.: Machine Musicianship. MIT Press, Cambridge (2001)Google Scholar
  2. 2.
    Wiggins, G.A.: Automated generation of musical harmony: what’s missing. In: Proceedings of the International Joint Conference in Artificial Intelligence (1999)Google Scholar
  3. 3.
    Rohrmeier, M.: A generative grammar approach to diatonic harmonic structure. In: Proceedings of the 4th Sound and Music Computing Conference, pp. 97–100 (2007)Google Scholar
  4. 4.
    Pachet, F.: The MusES system: an environment for experimenting with knowledge representation techniques in tonal harmony. In: 1st Brazilian Symposium on Computer Music, pp. 195–201 (1994)Google Scholar
  5. 5.
    Phon-Amnuaisuk, S., Tuson, A., Wiggins, G.: Evolving musical harmonisation. In: Dobnikar, A., Steele, N.C., Pearson, D.W., Albrecht, R.F. (eds.) Artificial Neural Nets and Genetic Algorithms, pp. 229–234. Springer, Vienna (1999)CrossRefGoogle Scholar
  6. 6.
    Navarro, M., Caetano, M., Bernardes, G., de Castro, L.N., Corchado, J.M.: Automatic generation of chord progressions with an artificial immune system. In: Johnson, C., Carballal, A., Correia, J. (eds.) EvoMUSART 2015. LNCS, vol. 9027, pp. 175–186. Springer, Heidelberg (2015)Google Scholar
  7. 7.
    Pachet, F., Roy, P.: Formulating constraint satisfaction problems on part-whole relations: the case of automatic musical harmonization. In: ECAI 98 Workshop on Constraints for Artistic Applications, pp. 1–11 (1998)Google Scholar
  8. 8.
    Gang, D., Lehmann, D., Wagner, N.: Harmonizing melodies in real-time: the connectionist approach. In: Proceedings of the International Computer Music Association, Thessaloniki, Greece, pp. 27–31 (1997)Google Scholar
  9. 9.
    Manaris, B., Johnson, D., Vassilandonakis, Y.: Harmonic navigator: a gesture-driven, corpus-based approach to music analysis, composition, and performance. In: Proceedings of the 9th AAAI Conference on Artificial Intelligence and Interactive Digital Entertainment, pp. 67–74 (2013)Google Scholar
  10. 10.
    Eigenfeldt, A., Pasquier, P.: Realtime generation of harmonic progressions using controlled markov selection. In: Proceedings of ICCC-X-Computational Creativity Conference, pp. 16–25 (2010)Google Scholar
  11. 11.
    Cohn, R.: Neo-riemannian operations, parsimonious trichords, and their “tonnetz” representations. J. Music Theory 41(1), 1–66 (1997)CrossRefGoogle Scholar
  12. 12.
    Cohn, R.: Introduction to neo-riemannian theory: a survey and a historical perspective. J. Music Theory 42(2), 167–180 (1998)CrossRefGoogle Scholar
  13. 13.
    Chew, E.: Towards a mathematical model of tonality. Ph.D. dissertation, MIT (2000)Google Scholar
  14. 14.
    Chew, E., Chen, Y.: Determining context-defining windows: pitch spelling using the spiral array. In: Proceedings of the International Society for Music Information Retrieval Conference (2003)Google Scholar
  15. 15.
    Harte, C., Sandler, M., Gasser, M.: Detecting harmonic change in musical audio. In: Proceedings of the 1st ACM Workshop on Audio and Music Computing Multimedia, pp. 21–26. ACM, New York (2006)Google Scholar
  16. 16.
    Bernardes, G., Cocharro, D., Guedes, C., Davies, M.E.P.: Harmony generation driven by a perceptually motivated tonal interval space. In: ACM Computers in Entertainment. ACM, New York (2015, in press)Google Scholar
  17. 17.
    Behringer, R., Elliot, J.: Linking physical space with the Riemann Tonnetz for exploration of western tonality. In: Hermida, J., Ferrero, M. (eds.) Music Education. Nova Science Publishers Inc, Hauppauge (2009)Google Scholar
  18. 18.
    Bigo, L., Garcia, J., Spicher, A., Mackay, W.E.: PaperTonnetz: music composition with interactive paper. In: Proceedings of the 9th Sound and Music Computing Conference, pp. 219–225 (2012)Google Scholar
  19. 19.
    Gatzsche, G., Mehnert, M., Stöcklmeier, C.: Interaction with tonal pitch spaces. In: 8th International Conference on New Interfaces for Musical Expression, Genova, Italy, pp. 325–330 (2008)Google Scholar
  20. 20.
    Krumhansl, C.L., Kessler, E.J.: Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychol. Rev. 89, 334–368 (1982)CrossRefGoogle Scholar
  21. 21.
    Chuan, C.-H., Chew, E.: A hybrid system for automatic generation of style-specific accompaniment. In: Proceedings of the 4th International Joint Workshop on Computational Creativity, pp. 57–64. Goldsmiths, University of London (2007)Google Scholar
  22. 22.
  23. 23.
  24. 24.
    Bernardes, G., Cocharro, D., Guedes, C., Davies, M.E.P.: Conchord: an application for generating musical harmony by navigating in a perceptually motivated tonal interval space. In: Proceedings of the 11th International Symposium on Computer Music Modeling and Retrieval (CMMR), pp. 71–86 (2015)Google Scholar
  25. 25.
    Riemann, H.: Vereinfachte Harmonielehre. Augener, London (1893)Google Scholar
  26. 26.
    Schoenberg, A.: Structural Functions of Harmony, 2nd edn. W. W. Norton Inc., New York (1969). [1954], (Revised by Stein, L.)Google Scholar
  27. 27.
    Malmberg, C.F.: The Perception of consonance and dissonance. Psychol. Monogr. 25(2), 93–133 (1918)CrossRefGoogle Scholar
  28. 28.
    Kameoka, A., Kuriyagawa, M.: Consonance theory. Part I: consonance of dyads. J. Acoust. Soc. Am. 45, 1451–1459 (1969)CrossRefGoogle Scholar
  29. 29.
    Hutchinson, W., Knopoff, L.: The acoustic component of Western consonance. Interface 10(2), 129–149 (1979)Google Scholar
  30. 30.
    Roberts, L.A.: Consonance judgements of musical chords by musicians and untrained listeners. Acta Acustica United Acustica 62(2), 163–171 (1986)Google Scholar
  31. 31.
    Cook, N.D.: Harmony, Perspective, and Triadic Cognition. Cambridge University Press, New York (2012)Google Scholar
  32. 32.
    Rameau, J.-P.: Traité de l’Harmonie. Ballard, Paris (1722). Treatise on Harmony (trans: Gossett, P.) Dover, New York (1971)Google Scholar
  33. 33.
    Tymoczko, D.: Progressions Fondamentales, Fonctions, Degrés: Une Grammaire de l’Harmonie Tonale Élémentaire. Musurgia: Analyse et Pratique Musicales 10(3–4), 35–64 (2003)Google Scholar
  34. 34.
    Agmon, E.: Functional harmony revisited: a prototype-theoretic approach. Music Theory Spectr. 17(2), 196–214 (1995)CrossRefGoogle Scholar
  35. 35.
    Meeus, N.: Toward a post-Schoenbergian grammar of tonal and pre-tonal harmonic progressions. Music Theory Online 6(1) (2000)Google Scholar
  36. 36.
    Shepard, R.N.: Structural representations of musical pitch. In: Deutsch, D. (ed.) The Psychology of Music, pp. 335–353. Academic Press, New York (1982)Google Scholar
  37. 37.
    Kruskal, J.B.: Nonmetric multidimensional scaling: a numerical method. Psychometrika 29, 28–42 (1964)MathSciNetzbMATHGoogle Scholar
  38. 38.
    Krumhansl, C.L., Kessler, E.J.: Tracing the dynamic changes in perceived tonal organization in a spatial representation of musical keys. Psychol. Rev. 89, 334–368 (1982)CrossRefGoogle Scholar
  39. 39.
    Lerdahl, F.: Tonal Pitch Space. Oxford University Press, New York (2001)Google Scholar
  40. 40.
    De Leeuw, J., Mair, P.: Multidimensional scaling using majorization: SMACOF in R. J. Stat. Softw. 31(3), 1–30 (2009)CrossRefGoogle Scholar
  41. 41.
    Schonbrun, M.: The Everything Music Theory: Take Your Understanding of Music to the Next Level. Adams Media, Avon (2011)Google Scholar
  42. 42.
    Huron, D.: Tone and voice: a derivation of the rules of voice-leading from perceptual principles. Music Percept. 19(1), 1–64 (2001)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Gilberto Bernardes
    • 1
  • Diogo Cocharro
    • 1
  • Carlos Guedes
    • 1
    • 2
  • Matthew E. P. Davies
    • 1
  1. 1.Sound and Music ComputingINESC TECPortoPortugal
  2. 2.New York University Abu DhabiAbu DhabiUnited Arab Emirates

Personalised recommendations