Logic and Music
-The Logic of Chords and Harmony

Ingolf Max

Section of Logic and Theory of Science
Department of Philosophy
University of Leipzig, Germany




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 well-structured basic elements and (c) introducing n-ary chord operators. The participants are invited to formulate and discuss advanced ideas to develop a calculus of chords.
Chords possess a complex interval-related inner structure. .Internal harmony is the totality of formal (context-free) relations between chords which is given without fixing any point on the scale. It can be shown that tonality needs such a fixed point.
(a) Our logical space is given by the scale of integers. Each integer will be interpreted as a (different) simple tone. Tone intervals are ordered pairs of tones. Each interval has a characteristic positive length l with l > 0.
(b) A key feature of the logic of chords is that this formal theory is not an atomic one. The basic elements are chords consisting of at least three tones, two basic intervals and one reference interval. A basic interval is the relation between directly adjacent  tones. The reference interval is the relation between the highest and the deepest tone of any chord. A chord is a molecular expression characterized not only by its tones but mainly by its matrix of interval lengths. Each chord can be uniquely identified solely by its inner structure. A class of (partially or totally tone-different) chords – e.g., the class of 3-tone-major-chords in root position – can be identified simply by knowing its characteristic matrix of interval lengths. Internal harmony is nothing else than the relation between two or more chords based solely on the inner structure of the chords. In this sense "chord" as well as "harmony" are formal concepts. Euphony is not necessary. E.g., we have of course chords and harmony in twelve-tone music (dodecaphony) and free jazz.
(c) An n-ary chord operator takes an n-tuple of chords as input and yields a chord as its output. Unary operators are negations (complete inversions of basic interval lengths relative to tone-related or interval-related fixed points), other interval permutation operators, barré operators (outputs with isomorphic matrices) and inversion operators. If it comes to more complex harmonic constructions like sequences consisting of tonic, subdominant and dominant we need at least binary operators to create them (cadence operators).

Session 1
The inner logical form of chords

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 multi-dimensional 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 sub-matrices can be connected or disconnected. Finally we discuss the interesting cases of interval length perfect (e.g., all-interval tetrachord) and interval length disjoint chords.

Session 2
Unary chord operators

We start with unary chord operators like  negations of chords. One type of chord negation is characterized by the complete inversion of the order of all basic intervals within a fixed reference interval. Another type is the complete inversion of this order with a fixed middle tone (if the chord contains an odd number of tones) or a fixed middle basic interval (if the chord contains an even number of tones). We show that these negations are analogies of the (partial) negation in the logic of first degree entailments. We will sketch a proof that either of these negations of an arbitrary major chord yield a corresponding minor chord and vice versa. Using both negations alternately we get the major and minor chords with fixed matrices of all basic tonalities. If time permits we discuss other unary chord operators like barré operators and inversion operators.





Session 3
Internal harmony, tonality, binary chord operators and family resemblance

The simplest form of internal harmony is the (context-free) 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 X-dominant 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 n-ary 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 family-like 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, 385-415.

[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, 196-200.

[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 Rinofner-Kreidl, S. & H. A. Wiltsche, Kirchberg am Wechsel/Lower Austria, 184-187.