What’s Important in Music?

By Nicholas D. Lewis

Graphical score of Metastasis (Xenakis)

The philosopher Pythagoras (6th century B.C) and his disciples are credited with the discovery of the numerical relationships governing the basic intervals of music. Pythagoras was concerned with numbers, and it is with him that the idea of “harmony of the spheres” was born. Although many people don’t realize it, this is the very reason that astronomy and music were linked and even codified later by Plato and his followers.

Plato considered harmony to be its own branch of physics. This alone suggests the kinship of harmony and mathematics. On one micro-level, sound itself is governed by the laws of mathematics. Frequency, amplitude, and loudness – to name a few wave characteristics – all depend upon the structure and speed of a given wave in terms of oscillation and size. Rhythm is intrinsically mathematical in nature. Beats flow along time and are subdivided. Rhythms fill those pulses. Even tuning systems are based upon the divisions between certain frequencies.

Most important, perhaps, is that music has patterns. Mathematics, almost by definition (and certainly by connotation) deals with patterns. Since mathematics deals with patterns, and one of the most fundamental concepts of most world music is patterns, it stands to reason that mathematics is a good way to describe music. Nearly conversely, it seems to stand to reason that people could use mathematics to create a musical composition. And moving into the 20th century, that is exactly what composers began to think.

Set theory, a branch of mathematics that deals primary with collections of objects, is a very convenient modern way of showing the relationship between different musical sounds. Although musical set theory and mathematical set theory are different in many ways, they share a number of similarities.  While musical set theory does not deal with various sizes of infinitely large sets, it does serve to achieve the same basic goal: to organize elements. Although there are many types of musical set theories, the most well known is probably pitch-class set theory. When given pitch classes are introduced as elements into a given set, they can be related by techniques such as transposition, inversion, and complementation.  So why does this matter?

Although the music created by people using mathematics a medium for composition  is inherently structured, it rarely sounds good to most people.  This is because it does not  meet all (and sometimes none) of Dmitri Tymoczko “Five things that make music sound good”.   Those five things are the following:

“1. Conjunct melodic motion. Melodies tend to move by short distances from note to
note. Large leaps sound inherently unmelodic.
2. Harmonic consistency. The chords in a passage of music, whatever they may be,
tend to be structurally similar to one another.
3. Acoustic consonance. Some chords sound intrinsically good or pleasing. These
are said to be consonant.
4. Scales. Over small spans of musical time (say 30 seconds or so), most musical
styles tend to use just a few types of notes, between 5 to 8.
5. Centricity. Over moderate spans of musical time, one tonic note is heard as being
more prominent than the others, appearing more frequently and serving as a goal
of musical motion.”

When one of these rules is broken, people may perceive the music as “random” or “discordant” or any of the other infinite gradation of inaccuracies that I often hear about 20th century art music.

But Tymoczko’s claims in the “Geometry of Music”, while wonderful in their own right, are far too subjective to be taken as a universal truth.  In Western Art Music during the common practice period, his theory holds largely true.  Most art music does in fact make conform to the five rudimentary points of his theory.  But, a great number of people like Xenakis.  A great number of people think Stockhausen, Boulez, Carter, Ustvolskaya, Cage, Ligeti, Babbit, and other avant-garde composers sound good. 

I know I do.  Do they share similarities that make their music sound good to me?  And indeed, what are those similarities?

I will deal with a relatively famous piece by a relatively famous composer: Metastaseis, by Iannis Xenakis. This work, scored for 61 players, has nobody playing the same part. Here is a quote from the Wikipedia article about the piece:

“As Newtonian views of time show it flowing linearly, Einsteinian views show it as a function of matter and energy; change one of those quantities and time too is changed. Xenakis attempted to make this distinction in his music. While most traditional compositions depend on strictly measured time for the progress of the line, using an unvarying tempo, time signature, or phrase length, Metastasis changes intensity, register, and density of scoring, as the musical analogues of mass and energy. It is by these changes that the piece propels itself forward: the first and third movements of the work do not have even a melodic theme or motive to hold them together, but rather depend on the strength of this conceptualization of time.”

This is interesting for several reasons. One, it shows a new approach to the concept of time in music. Every time I am in a car I am reminded of Xenakis’ music by analogy. Whenever two turn signals are on, but click at different speeds, it captures my imagination. Every so often they click at the same time, but most of the time they are flashing at two different speeds… Abstractly ,they are two different universes. Another thing that is interesting about this quote is that it shows, though not outrightly, that Xenakis was relatively careful about macro-phrasing.  And that brings me to the purpose of this article.

Almost all music created by sentient beings has a phrase.  In fact, upon analysis of Xenakis’ work, one will find that it follows many of the same rules, though perhaps more abstractly, that the classical sonatas of Mozart’s time followed.  Or more accurately, that the composers followed.  Metastaseis follows all of the same overarching rules.  Although the tonality and approach are different, one aspect is the same: contours intentionally lead to phrases (or at least macro-phrases).  There are, at least abstractly, cadences, phrases, and even melodies in the music.  Does this suggest that the phrase is the most important of all elements of music?  It is essentially the only thing that almost all music, and all non-chance music, shares.  And indeed, I must ask a question that seems to have an obvious answer at first: is that all that’s important in music?  Do think about it.

3 thoughts on “What’s Important in Music?

  1. mrG

    I’ll take this on, mostly because it is a subject very close to my philosophies ;) but I will warn you, now part-way through the two excellent new biographies on Cage, I am inclined to pronounce John wrong in two basic assumptions, the first that all sound is equal, the second that so too all people. Anarchy, the idea of self-regulating networks emerging spontenously from chaos, is nonsense, a figment of computer-age propaganda, and I don’t blame him for falling for it, but the flaw in his thinking is that same flaw that lead him to support the theories of Chairman Mao.

    Anyway, back on topic ;) … the essential is not ratio and Pythagorus, although there is a certain intriguing leaning to mathematical formulations, as you point out, they all fail, they are like animated humans and androids, they get creepy the closer they get to the music we actually produce. the first clue is your observation that all sentient musical composition employs phrase. A second, Joseph Schillinger observed that there were many things predicted by the theory of music that were patently not musical, and many things patently musical not predicted by the theory of music. A third is the fortuitous discovery a few years back by Dale Purvis who observed diatonic intervals embedded in the vowel formants of every human language he tested, and further (apropos to Dmitri’s point #3) the relative statistical frequency of those tonal pairings maps absolutely onto the human notions of consonance.

    as Einstein said, it is not that Nature can be described by mathematics (the mathematics, says Thomas Kuhn, always limits what we can see until the edges get too fuzzy to fit the old models) but the Mystery is why Universe can be described even approximately. Dale Purvis does not find exact ratios as with Pythagorus and his subdivided perfect string, but Dale instead finds the blue notes, the bent notes, the notes a good play will play with to veer them from our mathematically dictated equal or just temperment and pull them Someplace Else.

    And thus my theory of music: it is intrinsic to the human creature. It is the sound of our being, not godly or angelic, but the primate that we are, the flesh and blood and bone of us forms an apparatus that informs us and we have learned to exploit that through our natural ability to notice pattern and abstraction, and having noticed that magical thing of the music within the spheres if you will, we changed ourselves, we became the accellerated plastic ape who rose vastly far beyond our next nearest planetary cousins. The music is music because our bodies tell us it is, just as the great art of painters is tied to the apparatus of our eyes and our primal concepts of space and indentity, just as architecture can take many forms but is only ‘interesting‘ when it serves the inhabitants. A proper theory of music, in my mind, must be a theory of Bio-Resonance.

    And it is because we are, genetically, exceedingly similar to each other (being descended from the perhaps 1000 survivors of Mt Toba 80,000 years ago) it then follows there should be certain resonances that are universal, the (natural) diatonic scales fitting the structure of our throats and ears for example, and the frame of this body that can only dance in so many ways, at so many speeds and through so many angles, it is no accident that music and art can bind us so strongly, and so spontaneously.

  2. mrG

    To get back to my aside on Cage: a bio-resonance theory of music does not permit all mathematical models, it does not automatically the buddhist attention-surprises and ambient sounds, it does not let each musician do their own thing unless that ‘thing’ is fitting in what most musicians call The Groove and as such, while there are without a doubt many interesting recordings that I love and enjoy and composers that I respect, I’m having, so far (this early in my formulations) a bit of trouble classifying them all as music simply because some text book says they are so ;)

  3. Steve Jones

    “Yes” to phrases and “you bet” to the forced marriage of Pythagorean acoustics with human biology. Implicit in the concept of a phrase of music is the elapse of a finite amount of time. For a sequence of notes to group together in the consciousness and be perceived as an entity they must clearly not be paced too slowly. A critical factor would therefore seem to be the participation of an automatic short-term memory store in which the temporal structure of sounds is preserved. Just such a pre-attentive store, once called the “echoic memory”, has been invoked to explain various phenomena, including the period of a few seconds over which auditory information can be retrospectively retrieved from an unattended source (“Yes, I was listening – you said…”).

    Of course there are other periods over which musical structure is perceived, ranging from seconds to hours. We still don’t know what music is “for”, but what’s important in music is likely to reflect what’s important in life. For me a 3-act opera with overture shouldn’t take longer to consume than a 3-course meal, with aperitif. The three-minute song seems to be paradigm of western music, and argument rages (in my head) over what biological process or social activity this might sublimate?


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