Hierarchical Structures in Music

The hierarchical structure within music, especially within rhythmic passages and melodic contours, is a well-known phenomenon.  For example, in his entertaining and thought-provoking book (with an excellent bibliography), This Is Your Brain On Music , Daniel Levitin says (p. 154) in regards to musical production:

     Our memory for music involves hierarchical encoding - not all words are equally salient, and not all parts of a musical piece hold equal status.  We have certain entry points and exit points that correspond to specific phrases in the music ... Experiments with musicians have confirmed this notion of hierarchical encoding in other ways.  Most musicians cannot start playing a piece of music they know at any arbitrary location; musicians learn music according to a hierarchical phrase structure.  Groups of notes form units of practice, these smaller units are combined into larger units, and ultimately into phrases; phrases are combined into structures such as verses and choruses of movements, and ultimately everything is strung together as a musical piece.

In a similar vein, related to musical theory, Steven Pinker summarizes the famous hierarchical theory of Jackendoff and Lerdahl in his fascinating book, How The Mind Works (pp. 532 - 533):

Jackendoff and Lerdahl show how melodies are formed by sequences of pitches that are organized in three different ways, all at the same time...The first
representation is a grouping structure.  The listener feels that groups of notes hang together in motifs, which in turn are grouped into lines or sections,
which are grouped into stanzas, movements, and pieces.  This hierarchical tree is similar to a phrase structure of a sentence, and when the music has lyrics the two partly line up...The second representation is a metrical structure, the repeating sequence of strong and weak beats that we count off as ``ONE-two-THREE-four.''  The overall pattern is summed up in musical notation as the time signature...The third representation is a reductional structure. It dissects the melody into essential parts and ornaments.  The ornaments are stripped off and the essential parts further dissected into even more essential parts and ornaments on them...we sense it when we recognize variations of a piece in classical music or jazz. The skeleton of the melody is conserved while the ornaments differ from variation to variation.

Finally, although Robert Cogan in his landmark book, New Images of Musical Sound , does not explicitly point out an explicit hierarchical structuring, he does nevertheless repeatedly refer to it.  Here are four quotations as examples:

1. [In analyzing the electronic piece, Fall, on p. 111]:  Consequently, the two levels, infinite and finite (or grave and acute) are transformed in parallel ways, undergoing independent but parallel variation.  The constituent elements of both levels are progressively flattened out and lengthened in time...It enacts on the highest structural level (the relationship of the parts) that quality of spreading out and slowing down that we have found reenacted at every level.

2. [In analyzing Beethoven's Piano Sonata in E, Opus 109, on p. 56]:  The changing wave gestures follow one another ceaselessly, forming the larger stages of growth, accumulation, climax, and dissolution that constitute a single overriding wave - which is, finally, the structure of the entire movement.

3. [In analyzing the Tibetan Tantric Chant: Invocation of Mahakla, on p. 33]:  the cymbals reenact the opposition between consonant attacks and resonating sustained vowels of the men's voices.  The spectra of the cymbals and voices show, under close scrutiny, an uncanny point of similarity...the voice / cymbal alternations...reenact the initial grave/acute opposition of the voices alone on a vast scale - throughout the piece's entire range.

4. [In analyzing the Billie Holiday song, Strange Fruit, on p. 38]:  The coordinated structural forces that forged the climax of stanza 1 all join again here to create and maintain the climactic ending of the entire piece.  Together they complete the flowering of the oppositions that were presaged in the brief climax of line 1. 

Relation to Language

The hierarchical structure of music described above is similar (as pointed out in the Pinker quote) to the hierarchical structuring that is found in human language.  [This connection was first proposed by Leonard Bernstein in The Unanswered Question .]  For example, here is a classic example from Lerdahl's book, Tonal Pitch Space , (p. 11):

It exhibits the tree structure of the musical phrase that begins the passage from Bach's Christus, der ist mein Leben.  There are, in fact, English language statements that show this exact same hierarchical structuring.  Here is one unusual example:

                                              The governor of California from Illinois was then clearly named Reagan.

An important prediction was made by A.D. Patel in his paper, Language, music, syntax and the brain, that this analogy is not a superficial one.  That, in fact, some language areas of the brain that process syntactical aspects of language would also be activated when listening to music.  As a sterling example of science, experiments were done by D. Levitin and V. Menon that confirmed Patel's prediction.  Click here for a downloadable copy of their paper, Musical structure is processed in ``language'' areas of the brain: a possible role for Brodman Area 47 in temporal coherence.

   While syntax is important in language, Chomsky has shown that it is essentially independent of meaning.  To illustrate this independence, he coined the memorable sentence:

                                                               Colorless green ideas sleep furiously.

A syntactically correct English sentence that is utterly meaningless.  There are, of course, musical analogues to this.  One example would be Muzak, or ``elevator music'' or ``grocery store music.''  You can listen to it and the notes are all correctly chosen, but the music has no life; it can play in the background without you even noticing.

So where does meaning, either in language or music, come from?  One theory that carries a lot of theoretical and experimental weight is that meaning comes from metaphoric relations, at the most basic level, metaphors that carry emotional weight (related to survival and reproduction for example).  To see how metaphor relates to meaning, look again at the Chomsky sentence.  If we drop the last word, we obtain:

                                                                      Colorless green ideas sleep.

This sentence has some meaning, of a metaphoric kind.  Here green ideas refers to new ideas, and they are colorless if they are blandly expressed; those ideas will sleep in the sense that they will not excite anybody's interest.  Notice how adding the last word, furiously, destroys the metaphor and the meaning is gone with it.

    Of course, metaphors abound in music.  It is easy to think of examples.  A rising pitch connoting excitement and joy, a descending pitch in the lower registers connoting grave and mournful feelings.  The finale of the Firebird Suite (click here for audio and video analysis) clearly has metaphors to bird calls and, especially, birds taking off into flight (the phoenix rising from the ashes).   Or Beethoven's Pastoral symphony is full of metaphors to country scenes.  It is most significant, for metaphorical relations to emotional feelings, that the pathway within the brain of processing of musical signals first passes through the amygdala (related to emotions) and then through syntactical and pattern recognition areas (which then reciprocally interact with the amygdala).  For more on brain function, see "Your brain on music.")   A good article describing the basic ideas of metaphors in relation to fundamental musical perceptions such as pitch is Metaphor and Music Theory: Reflections from Cognitive Science by L.M. Zbikowski.  As an example of the insights in this article, Zbikowski states the following in regard to the up-down metaphor for pitches:

         Mapping the spatial orientation up-down onto pitch works because of the correspondences between the image-schematic structure of components of the spatial and acoustical domains.  Both space and the frequency spectrum are continua that can be divided into discontinuous elements.  In the spatial domain, division of the continuum results in points; in the acoustic domain, it results in pitches.  Mapping up-down onto pitch allows us to import the concrete relationships through which we understand physical space into the domain of music, and thereby provide a coherent account of the relationships between musical pitches.

This article also contains an excellent set of references for the relationship between metaphorical thought and music.

    An important generalization of the metaphorical theory of meaning is the blending theory of meaning.  The blending theory (see the book, The Way We Think, for a thorough description in relation to language) describes how meaning arises by blending aspects of different categories.  In music, there are many ways that such blending occurs.  For example, there is blending whenever there is an interaction of opposites (leading to growth in perception) from the musical opposites listed in the section on dialectics.  That includes the very important opposition between Repetition-Structure vs.Transformation-Surprise.  A good example of this is in the Stravinsky string piece, Three Pieces for String Quartet, Piece II, analyzed by Robert Cogan in New Images of Musical Sound (pp. 56 - 61).  Here there is a blending of opposites, silences vs. sonorities (an extreme case of the soft/loud opposition), and pizzicato violin notes blended in with sharp, percussive violin bowings.  Of course, a most salient example of blending is when human lyrical voicings are blended with instrumental notes.

Some philosophical ruminations on musical language and spoken language

In terms of potential size, a spoken language contains an infinite number of possible sentences.  To see this potential infinity, consider the sentence, "Yesterday, it rained."  We get a new sentence by adding "Jim said that", obtaining "Jim said that yesterday it rained."  And we get a another new sentence by adding "Jim said that" to the last sentence:  "Jim said that Jim said that yesterday it rained."  By continuing this process ad infinitum we can in principle obtain an infinite number of syntactically correct sentences.  (We have to add in principle since no living human could speak, or process, the gargantuanly long sentences that arise during this process.)  Of course, music is also an infinite system of signals, with language like aspects (as described above).  It is interesting mathematically that the two infinities are of different size (see Chapter 3 of the excellent book, The Heart of Mathematics, by Burger and Starbird, for a clear discussion of the idea of different sizes of infinities).  Since a spoken language system is a discrete combinatorial system (each sentence is organized, via a rooted tree diagram, from words chosen from a finite vocabulary) it follows that the potential infinity of spoken language sentences is a countable infinity.  In fact, the set of all grammatical sentences in a spoken language is a subset of the set of finite sequences of words chosen from the finite vocabulary; this latter set is countably infinite, hence the grammatical sentences are as well.  On the other hand, since music is generated from analog signals (continuous variations of loudness, pitch, vibrato, contour, etc.) it follows that the potential infinity of musical passages is an uncountable infinity - an infinity that is larger in size than a countable one.  Of course, the "meaning" of musical phrases is not nearly so precise as with spoken language (it should be mentioned in connection with this that there has been speculation that perhaps musical language developed as a precursor to spoken language - see The Singing Neanderthals for a thorough discussion).