Exploring the Syntonic Side of Major-Minor Tonality

  • Thomas NollEmail author
  • David Clampitt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11502)


The description of the Major and Minor modes as fillings of a triadic division of the octave offers the possibility to study them as Pairwise Well-Formed Modes. As a consequence one obtains two projections: the diatonic projection yields the well-known Ionian and Aeolian modes and provides a link between the triadic modes and the pseudo-classical modes. The syntonic projection looks unfamiliar at first sight, but closer inspection shows that these modes provide a common ground for the natural, harmonic, and melodic manifestations of both the Major and the Minor modes.


Triadic mode Diatonic and syntonic mode Tonal and modal step intervals Sturmian morphism Algebraic combinatorics on words Major/minor tonality 


  1. 1.
    Agmon, E.: Linear transformations between cyclically generated chords. Musikometrika 3, 15–40 (1991)Google Scholar
  2. 2.
    Agmon, E.: The bridges that never were: Schenker on the contrapuntal origin of the triad and seventh chord. Music Theory Online 3(1) (1997).
  3. 3.
    Clampitt, D.: Pairwise Well-Formed Scales: Structural and Transformational Properties. Ph.D. diss, SUNY at Buffalo (1997)Google Scholar
  4. 4.
    Clampitt, D.: Mathematical and musical properties of pairwise well-formed scales. In: Klouche, T., Noll, T. (eds.) MCM 2007. CCIS, vol. 37, pp. 464–468. Springer, Heidelberg (2009). Scholar
  5. 5.
    Noll, T., Clampitt, D.: Kaleidoscope substitutions and pairwise well-formed modes: Major-Minor duality transformationally revisited. J. Math. Music 12(3) (2018)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Cohn, R.: Maximally smooth cycles, hexatonic systems, and the analysis of late-romantic triadic progressions. Music Anal. 15, 9–40 (1996)CrossRefGoogle Scholar
  7. 7.
    Dahlhaus, C.: Untersuchungen über die Entstehung der harmonischen Tonalität. Bärenreiter, Kassel (1969)Google Scholar
  8. 8.
    Hauptmann, M. (W. E. Heathcote, trans.): The Nature of Harmony and Metre. Swan Sonnenschein & Co., London 1888 [1853]Google Scholar
  9. 9.
    Hindemith, P.: The Craft of Musical Composition. Schott, London (1940)Google Scholar
  10. 10.
    Mazzola, G.: The Topos of Music. Birkhäuser, Basel (2002)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Departament de Teoria, Composició i DireccióEscola Superior de Música de CatalunyaBarcelonaSpain
  2. 2.School of MusicOhio State UniversityColumbusUSA

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