Logic and Music The Logic of Chords and Harmony Section of Logic and Theory of Science




The tutorial provides a detailed introduction into a very new approach to a formal theory of music: the logic of chords and their internal harmony. The logic of chords is closely related to mathematical theories of music but by far not identical with them. The constitutive decisions creating our logic of chords are (a) fixing the logical space by specifying a scale, (b) indicating chords as our wellstructured basic elements and (c) introducing nary chord operators. The participants are invited to formulate and discuss advanced ideas to develop a calculus of chords. Session 1 We start with the introduction of our symbolism to describe tones, intervals, interval lengths and interval classes. We define several relations between intervals. The most important one is the relation of directly connected intervals. Chords are multidimensional sequences of directly connected intervals. The general form of chords will be explained with emphasis on the inner complexity of the pattern of intervals. We differentiate between basic and intermediate intervals (in chords with 4 and more tones) as well as the reference interval. With respect to characteristic matrices of interval lengths it is possible to characterize classes of chords solely with respect to their inner structure. No further context is needed. Chord classes with sufficient complex matrices of interval lengths contain submatrices which characterize other chord classes. Such submatrices can be connected or disconnected. Finally we discuss the interesting cases of interval length perfect (e.g., allinterval tetrachord) and interval length disjoint chords. Session 2



Session 3 The simplest form of internal harmony is the (contextfree) relation between two chords with respect to their inner formal structure alone. The application of a unary operator creates necessarily an internal harmony between its input and its output. Creating internal harmony depends on the logical behavior of the chord operator as well as the inner structure of the argument(s). It is an inspiring research question for the logic of music which aspects of tonality can be characterized as internal harmony. A known candidate is the asymmetric relation "is the parallel minor of" (but not tonic parallel). A novelly defined concept is the symmetric relation "is the Xdominant of" with X is empty ("dominant") or replaced by "sub" ("subdominant"). To determine a chord which is located between two other chords we need binary chord operators. But nary operators are insufficient to determine tonic chords as internally harmonious. To get this we have to extend our approach by a fixed point on the scale. The range of both kinds of theory is still an open question. But there is also a creative aspect here: composing new music. One possibility is the creation of familylike sequences of chords. Adapting Wittgenstein we can say that "the strength of the thread [harmony in a sequence of chords] does not reside in the fact that some one fibre [interval length] runs through its whole length [whole sequence], but in the overlapping of many fibres [crisscrossing of interval lengths]." (Philosophical Investigations 67) An audio example will be given an formally explained. Primary text for the tutorial Well enough in advance of our tutorial an extended handout will be hyperlinked here! Secondary sources
Further References (in German) [1] Max, I., 2003, Zur Familienähnlichkeit von Begriffen und Akkorden [On family resemblance of concepts and chords]; in: expressis verbis. Philosophische Betrachtungen, ed. by M. Kaufmann & A. Krause, Halle, 385415. [2] Max, I., 2010, Familienähnlichkeit als Analysemethode von Spätwerken Beethovens und Wittgensteins [Familiy resemblance as method of analysis of Beethoven's and Wittgenstein's late works]; in: Image and Imaging in Philosophy, Science, and the Arts, Papers of the 33rd International Wittgenstein Symposium, Kirchberg am Wechsel/Lower Austria 2010, ed. by Nemeth, E., H. Richard & W. Pichler, Kirchberg am Wechsel/Lower Austria, 196200. [3] Max, I., 2014, Ist „Familienähnlichkeit“ ein philosophischer, ein theoretischer Begriff oder beides? [Is "family resemblance" a philosophical or/and a theoretical concept?]; in: Analytical and Continental Philosophy: Methods and Perspectives, Papers of the 37th International Wittgenstein Symposium, Kirchberg am Wechsel/Lower Austria 2014, ed. by RinofnerKreidl, S. & H. A. Wiltsche, Kirchberg am Wechsel/Lower Austria, 184187. 
