A Molecular Logic of Chords and Their Internal Harmony

Abstract

Chords are not pure sets of tones or notes. They are mainly characterized by their matrices. A chord matrix is the pattern of all the lengths of intervals given without further context. Chords are well-structured invariants. They show their inner logical form. This opens up the possibility to develop a molecular logic of chords. Chords are our primitive, but, nevertheless, already interrelated expressions. The logical space of internal harmony is our well-known chromatic scale represented by an infinite line of integers. Internal harmony is nothing more than the pure interrelatedness of two or more chords. We consider three cases: (a) chords inferentially related to subchords, (b) pairs of chords in the space of major–minor tonality and (c) arbitrary chords as arguments of unary chord operators in relation to their outputs. One interesting result is that chord negation transforms any pure major chord into its pure minor chord and vice versa. Another one is the fact that the negation of chords with symmetric matrices does not change anything. A molecular logic of chords is mainly characterized by combining general rules for chord operators with the inner logical form of their arguments.

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Correspondence to Ingolf Max.

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Jean-Yves Beziau deserves huge credit for supporting my work for years and for inviting me to give a tutorial on “Logic and Music” as part of the 5th World Congress and School on Universal Logic in Istanbul 2015.

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Max, I. A Molecular Logic of Chords and Their Internal Harmony. Log. Univers. 12, 239–269 (2018). https://doi.org/10.1007/s11787-018-0193-8

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Mathematics Subject Classification

  • Primary 03B99
  • Secondary 00A65

Keywords

  • Molecular logic
  • chord
  • chord operator
  • internal harmony
  • formal music theory