Varieties of Music Theory

Choosing that title, I'm reminded of a wonderful book by William Empson, published in 1930 called Seven Types of Ambiguity. Or maybe Varieties of Religious Experience by William James. But never mind! The thing is that we normally think of music theory as being a single thing. At the very rudimentary level that seems reasonable, but as soon as you get into the details you should start noticing that "music theory" as we find it in most books only applies to a part of music. Theorists get around this embarrassing fact by saying that music theory applies to the "common practice" period in music, that stretch of time from about 1600 to around 1900. The implication is that the rest of music history is uncommon or something.

The truth is that each epoch in music history, which at its maximum extent begins with the earliest written music and discussion thereof by the Greeks in the 4th century BC, has its own characteristic theory of music. With the Greeks about all we have left is the theory as the music itself has either disappeared or is only sketchily represented in notation. But beginning in the so-called Dark Ages, around 600 AD, we find the earliest chants of the Catholic Church, later called "Gregorian Chant" after Pope Gregory who was traditionally credited with many of the chants (through divine inspiration, of course). Here is an early offertory, Iubilate deo universa terra:

The kind of music theory used to talk about this kind of music is quite different from other kinds. You run across terms like authentic and plagal modes, reciting tones, ambitus and the mysterious process known as "centonization". The reason you need a different kind of music theory for chant is that the musical structures are completely different from what we find in music from other times and places.

By the high Middle Ages, music was composed in entirely different ways using an entirely different kind of notation and we need a different kind of music theory to talk about it. Characteristic terms for this period are isorhythm, Franconian notation, rhythmic modes and the formes fixes. Here is some of the notation:

And here is a little of the music:

As each variety of music theory has its characteristic terms, just hearing them tells us what time and place we are talking about. The terms "RO", "interval vector" and "combinatoriality" are associated with serial music of the Second Viennese School. The "RO" means that the tone row is being heard in retrograde at the original transposition. Interval vector is a concept borrowed from mathematical set theory and combinatoriality refers to the way part of the row can mirror another part of the row. I think! Here is an example of serial music; a piano piece by Arnold Schoenberg:

Sometimes a single word or term can bring to mind a certain kind of music. For example, if you use the word "phasing" in talking about music, you can really only be talking about the music of Steve Reich. Phasing is a technique he used in some earlier works where two instruments playing the same rhythmic pattern slowly diverge with one slightly speeding up. Here is what that sounds like:

You see why you need completely different kinds of theory? Then there are musics that probably have no theoretical discussion, perhaps because no-one has really figured out what is going on:

So we might need a lot more theories of music. Or maybe we give up on theory entirely and just let the music wash over us as we drift into a sonic stupor.

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"Radical New Musical Languages"

Reading Alex Ross' blog this morning I notice that he is giving several talks inspired by his book The Rest is Noise.

On January 19th, The Rest Is Noise, a year-long festival inspired by the book of the same title, opens at Southbank Centre in London. I'll give four allied lectures over the course of the year; the first of these, "The Big Bang," is at noon on the 19th, in Queen Elizabeth Hall. It will survey the emergence of radical new musical languages in the late nineteenth and early twentieth centuries. Expect audio snippets of Debussy, Schoenberg, Stravinsky, Bartók, and many others, though Salome, featured that night at the London Philharmonic, will be the center of attention.

I think it may be time to have a closer look at the book, which seems to be having quite an influence on the way people look at music. But can you pick out the words in the above quote that caused my hackles to rise? Sure you can; I picked them for my title: "radical new musical languages." What's wrong with that?

Underneath the cool exterior and mellifluous writing style, Alex Ross is, I suspect, deeply conventional. He is not really a musicologist, but rather a journalist. But since musicologists these days are chasing one another down various rabbit holes, Alex Ross tends to fulfill the function of a public intellectual in music. Richard Taruskin is another, but as he sometimes expresses unusual, contrary opinions, he is less popular with the mass media. Alex Ross, however, seems more and more popular, so I think I will take a run through The Rest is Noise to see what assumptions he is operating under. In the meantime, though, let's just deconstruct that phrase "radical new musical languages."

That was most certainly not what was happening in the late nineteenth and early twentieth centuries. You may recall that on several occasions I have made the seemingly trivial point that music is NOT a language? Have a look here and here and here for some examples. Most of the time when people refer to music as a language they are simply making a metaphor and there is no real problem with it. But if someone is not aware that it is nothing but a metaphor, then some strange conclusions appear. You might start thinking that a composer can simply invent a new musical language. If music were a language, then that would simply not be possible for the very same reason that I cannot invent a new language. (Let me qualify that: of course, I can invent a new language. People have done so and Esperanto is an example. But these invented 'languages' are not languages in the sense that they are used by any community of humans. The most you could say is that someone invented the framework for a language--which was then ignored. Languages evolve naturally, they are not 'invented'.)

The philosopher Ludwig Wittgenstein came up with a very famous argument about why you can't invent your own private language, known as the private language argument. The concept of a language understood by only one person is incoherent: it violates the very notion of a language, a central concept of which is shared meaning. A language is a language if it has words and a dictionary you can look them up in. Now of course there are languages that are not even written, but they have the equivalent of dictionaries in tribal elders: "Grandma, what does "spadoinkle" mean?" If the idea of a private language is incoherent, then it would follow that all language is essentially public: that language is at its core a social phenomenon.

Music is also, at its core, a social phenomenon. Think about what that means for a moment. One consequence is that you cannot simply invent a new musical 'language'. While music is not a 'language', it has some language-like qualities. One of these is that it is a social phenomenon, perhaps a bit more so than the other arts. Take for example two fundamental kinds of music-making: singing and dancing. I talked about this here. Both song and dance are social. We sing words to other people to communicate with them. We prefer to dance with other people rather than alone for the same reason. Just what you communicate when you sing or dance with another might be hard to put into words, but the communication itself is undeniable.

So I think the consequence of all this is that a composer cannot simply invent a radical new musical language, though he can do something perhaps a bit less radical in re-shaping the way music is structured in some fundamental way. This is what those composers Alex Ross mentioned were doing. If you take a close look at Debussy you will see the music absolutely permeated on every level with the fundamental elements of tonal music. But you will also see how he has re-thought and re-shaped these elements.

Well, enough for today on this. Let's end with some Debussy so you can see what I mean.

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Phrase, Motif and Theme

I've taken a lot of music courses and quite a few of them were theory courses, but I can't recall any time when anyone ever tried to define these terms. We use them all the time, but the simple fact is that most talk about music, even among musicians, is very loose, almost like gossip. When you are playing or composing music, you do things without thinking how they would be labelled if you were talking about them. Things like playing or composing phrases, motifs and themes. I'm going to look them up now, but before I do, let me just see how I would define them off the top of my head:

• Phrase is the big, generic term that at its most general simply means the sections that music tends to fall into that are very roughly analogous to sentences or phrases in language--which is where the term came from. Phrases are typically four, eight or sixteen measures long. What distinguishes a phrase are things like the shape of the melody (rising then falling, for example), harmonic closure at the end, and rhythmic structure (typically long notes at the beginning and shorter notes at the end, or the other way around).
• Motif is a short melodic and/or rhythmic cell that is typically used as a structural device. The opening four notes of Beethoven's 5th Symphony are an iconic example. Those anapests and dactyls I keep finding in the Shostakovich quartets are another.
• Theme refers to a melodic unit that is used structurally. A good example would be the subject to a Bach fugue. In some theories of classical style, a number of different types of themes are distinguished by the phrase and harmonic structure (see William Caplan's books)

Now let's see what Wikipedia says.

http://en.wikipedia.org/wiki/Phrase_(music)

http://en.wikipedia.org/wiki/Motif_(music)

http://en.wikipedia.org/wiki/Theme_(music)

That was pretty much what I said, even with some of the same examples, only with a lot more words. The problem with all these definitions, and why you don't need to spend a lot of time on them, is that they are very general and very abstract. Music, on the other hand, is very specific and concrete! So you can define 'phrase' however you like, but you either have to avoid any specifics or find that the very first phrase you look at won't fit the definition. Bummer!

Here is the important paragraph from the Wikipedia article on "phrase" that quotes a number of different definitions:

John D. White defines a phrase as, "the smallest musical unit which conveys a more or less complete musical thought. Phrases vary in length and are terminated at a point of full or partial repose, which is called a cadence."^[7] Edward Cone analyses the "typical musical phrase" as consisting of an "initial downbeat, a period of motion, and a point of arrival marked by a cadential downbeat".^[8] Charles Burkhart defines a phrase as "Any group of measures (including a group of one, or possibly even a fraction of one) that has some degree of structural completeness. What counts is the sense of completeness we hear in the pitches not the notation on the page. To be complete such a group must have an ending of some kind … . Phrases are delineated by the tonal functions of pitch. They are not created by slur or bylegato performance … . A phrase is not pitches only but also has a rhythmic dimension, and further, each phrase in a work contributes to that work's large rhythmic organization."

OK. So, what's a complete musical thought? What's a point of repose? What if you don't have a cadence? What if the phrase starts on an upbeat, not a downbeat? What does "structural completeness" mean? The thing is that all these definitions sort of work if you are talking about classical period music, but the further away from that period, the less they fit. In Shostakovich, for example, even though his music is certainly tonal, it doesn't have traditional cadences so any attempt to understand phrasing in his music has to rely on some other principle.

And don't even start with Gregorian chant...

Now let's have some music. How about some non-Gregorian chant? This is a Mozarabic chant:

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How to Analyze Music

The first question is why would anyone ever want to? Sounds dreary, sounds tiresome and maybe even difficult. And of course, there is no need for any ordinary listener to ever analyze a piece of music. Somewhat oddly, there is not much need for most performers to analyze music either. Let me tell a little story. I once knew a very fine French horn player who was gifted enough to win an audition for first horn in an orchestra at age eighteen. He continued on, playing principal horn in various orchestras and in a famous brass quintet. When I had decided to go back to graduate school in musicology and was looking over the entrance requirements I happened to be staying with him. I mentioned one day about having to review my augmented sixth chords as that was going to be on an entrance exam. He looked at me with a puzzled look and said, "I guess they would have an augmented sixth, right?" You see he had never taken a theory course in his life! All he knew about theory was just picked up by exposure and intuition. In someone who is gifted in music, this is really all you need. A lot of performers think that "analysis is paralysis" and it is certainly true that too much thinking can hamper performance.

So it sounds as if I have proved that no-one needs analysis--or theory either! Not quite true. Ordinary listeners can enjoy music quite well without knowing the theory behind it and most performers can play music very well without knowing much about theory. There are also some interesting philosophical problems with the relationship between theory and composition that I talked about in this post.

But theory and analysis are still very useful and valuable activities. You can skate around on the surface of music forever without delving into it, but that does not mean that a deeper understanding is not desirable. The French horn player I was talking about before had to play individual lines in an orchestral texture and also oversee the French horn section, but the deeper understanding of the whole musical structure was more the responsibility of the conductor. If you are a solo pianist or guitarist, with responsibility for the whole musical texture, you really need to know more about it than just where to put your fingers and where to play loud and soft. So pianists (and even the occasional guitarist) often study the history and theory of music in much greater depth.

The music school I graduated from had courses in harmony and counterpoint in the first two years and in the last most students had a choice of 19th Century Theory and Analysis or 20th Century Theory and Analysis. Honors students had to take both. In counterpoint you study first modal counterpoint which covers music up to around 1600, then tonal counterpoint from then until 1750 (the death of Bach). Harmony, which means tonal harmony, is basically about music from 1700 to 1900, the so-called "common practice" period when nearly every composer and musician worked within the same understanding of the tonal structure of music. Late in the 19th century this structure got so burdened with deviations from the basic tonal structure that some composers decided to invent a new way of structuring music. The most important of these was Arnold Schoenberg who, in the 1920s, developed a way of composing, called 'serialism' that avoided tonal structure entirely. Since then music composition has diverged into a thousand different camps. Every composer is more or less expected to invent their own musical 'language'. There is no longer such a thing as a "common practice" which makes things very dicey indeed for audiences who usually have no idea what they are hearing or what to listen for.

Partly as a result of this, some composers have resorted to all sorts of non-musical means to reach out to the audience. These include elements of theater (even in chamber music), references to mysticism, inclusion of elements from folklore, world music, or popular music, writing music with text such as opera and other means.

That little history of music theory starts to answer the question of my title: how to analyze music. You should start by taking into account when the music was written. For example, here is the opening of a simple two-voice bicinium by Orlande de Lassus. There is no text and no instruments are specified so you could play it on any instrument or combination that can play the notes.

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You do need to know the basics of notation before you can begin such as what the clefs indicate. A set of five lines is called a staff and the top staff, the cantus, has a treble clef. This is a very stylized letter G that curls around the second line from the bottom telling you that this note is 'G'. The lower staff has what is called a 'C' clef. It is not really a stylized 'C', but the center of the bracket, pointing to the middle line, tells you that this note is 'C'. The other thing at the beginning is the flat sign which, on both staves, indicates that every time there is a 'B' note you play B flat instead. Now, since Lassus' dates are 1532 - 1594 we know that this falls in the counterpoint stage of music history. What does this mean? We should look at this music melodically and see if we can see relationships between the voices. Bearing in mind that the clefs are different, what we see here is that the Cantus has exactly the same notes as the first two measures of the Altus. The Altus starts with D, Bb, A then a scale from G to F and the Cantus has exactly the same notes (an octave higher). This is what is known as imitative counterpoint. The music is written so that the second voice comes in after the first voice, but with the same notes and the melody is written so that it will go together with itself harmoniously. There are lots of refinements and complications but if you understand this, then you understand the fundamentals of how music was structured in the 16th century.

Now, that wasn't so hard, was it? Here is a bicinium by Lassus played on two viols. It is not the same one as the example above, but it is structured the same way: the second voice imitates the first one measure later:

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Theory of Pop

I don't know who is behind Hooktheory, but a post is up doing what seems to be a rather broad analysis of harmony in popular songs. That seems interesting, so I had a look. Here is the post. This will make more sense if you go have a read. Don't bother with the comments, though.

Two things strike me about this. Firstly, it is a statistical analysis of how often chords appear, though they don't seem to measure how long chords are held for, which seems pretty important. Secondly, it is a probability analysis of the likelihood of one chord following another. If you scroll down you will see a chart titled "Chords following em [E minor]". That looks pretty wacky too. 59% of the time E minor is followed by F major in the key of C major? That's a pretty weak progression. But maybe that is, statistically, true. The problem is that all this seems quite oblivious to the function of harmony.

I've been raging about pseudo-science lately and how researchers doing research into music who know absolutely nothing about music are just embarrassing. This site is a little different. For one thing, it is digging into the music itself instead of coming at if from left field somewhere. Remember the yellow-bellied marmot in this post? Hooktheory has a page where they put all their analyses. Here that is. Here is the analysis of "California Gurls" by Katy Perry.

The first thing I did was look for a Beatles song to see how that came out. Take the analysis of "Hey Jude" for example. The authors know something about music theory, which is refreshing, but not quite enough. Look at the "outro" (I think "coda" might be the better word) for "Hey Jude". They give the progression as I, IV of IV, IV, I. The idea of a secondary dominant is a long-standing one in music theory, but it really can't be extended to the idea of a secondary subdominant which is what is being claimed so blithely here. IV of IV? S'existe pas!

The idea of a secondary dominant is a powerful one because the relationship of V or better, V7, to I is so strong that it can be transferred to any chord. There is no strong relationship of IV to I. IV, in traditional theory, prepares V, that is its role. So the progression I, IV of IV, IV, I just isn't plausible because the relationships claimed are not plausible. The progression is actually I, bVII, IV, I. This is interesting because it is so very coda-like. In common-practice harmony, the coda has the role of lowering the tension and it very commonly uses subdominant harmony to do so. Interesting that this song, with its huge coda, does the same. But what is also happening here is the creation of an ambiguity. Any time you have an ambiguous progression that fades out, it can be an undecided question just what the final chord is. For example, suppose that the key is not F but Bb? We then have the progression V, IV, I, V. That's a lot more probable than the one they suggest and just as probable as the one I first suggested. What is actually happening here is something that always happens when you have IV I progressions. It either sounds like a plagal cadence, which is inherently weak, or it sounds like the IV is actually the I. The coda to "Hey Jude" is interesting, I suggest, because it floats between both possibilities.


Well, that's my stab at doing some popular music theory. I would suggest to the authors of Hooktheory that instead of doing a statistical analysis of 1300 songs (and not coming up with much), that they pick a few good songs and try and figure out how the harmony functions.

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The Tritone

The tritone is one of those fascinating problems that music theory has dealt with from the very beginning. It is an interval, the distance between two notes, and it has been a problem because it pops up in any scale or mode. The problem is that it has a very harsh sound--is dissonant. The tritone is just three whole tones, hence the name. Like this:

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The problem comes up when you try to do a primitive kind of polyphony with two voices in parallel motion.

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Several solutions emerged. One is to lower the B natural to a B flat, which is why in the earliest written music the only accidental is a flat on B. This was also the earliest key signature. Another is to avoid parallels and from that came the rule about parallel fifths. Instead, have the voices move in oblique or contrary motion. Another solution which came later was to only move in parallel thirds or sixths, but that had to wait until these intervals, dubbed 'imperfect', became acceptable as consonant. At first the only intervals called consonant were the 'perfect' ones: the unison, fourth, fifth and octave.

Much, much later the tritone became prized as a special kind of intense dissonance and is an essential part of the most important progression that defines tonality: the V7 - I cadence.

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It is the very intensity of the dissonance built into the V7 chord with the tritone from F to B, that makes the cadence powerful because this tension resolves into the consonant tonic harmony.

The intensity of the tritone interval can be used in a melody. Leonard Bernstein used it in the song "Maria" from West Side Story. You can hear it in this excerpt right at the 35 - 36 second mark. The tritone is between "Ma" and "ri".

Sometimes the tritone is used just for shock value as in this song by George Harrison, "I Want to Tell You". Listen for the piano part between the 26 and 32 second mark and similar places. Yes, it's that same pesky F to B natural.

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Guido of Arezzo

The Italian monk Guido of Arezzo (~990 - ~1033) is probably the most important figure in Western music history that most people have never heard of. He is credited with two innovations without which the whole history of music would have been quite different. The first of these is the staff, the set of five lines that tell us exactly what note to sing or play. Before then the predecessors to the round note-heads were squiggly neumes and they gave only a very sketchy idea of what the notes were. The idea of drawing lines to create a spacial metaphor for pitch was so astonishingly effective and simple that we have used it ever since. To my knowledge, this breakthrough was made by no-one else, no-where else, at any time.

Guido's second contribution I will introduce with a little musical number:

Now where did those stupid syllables come from? Not from the pen of Oscar Hammerstein II, the lyricist for the musical. No, they come right from Guido of Arezzo, a thousand years earlier. He spent much of his time training choir singers and developed a method for teaching them to sing--from his newly-invented staff notation--melodies they had never heard before. Sight-singing it is called and every music student still learns how to do it. In order to teach the techniques, he used this hymn with words by Paul the Deacon and the music probably by Guido himself:

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If you look at the first word of each phrase you see the following syllables: ut, re, mi, fa, sol, la. This doesn't include the last phrase, Sancte Johannes. What is interesting about this tune is that each phrase starts one note higher: C, D, E, F, G and A. Very useful for teaching about the notes of the scale (or mode, as it was at this point). Later on, so as to have an open, not closed syllable, the first syllable, ut, was changed to doh. Also later on, when the idea of the scale was developed, they added one more syllable for B and they made use of that Sancte Johannes phrase so the syllable was si. This is how I learned solfege in Quebec: doh, re, mi, fa, sol, la, si, doh. In other places, like the set of The Sound of Music, they use the syllable ti instead of si.

So now we know the horrible truth: Rogers and Hammerstein stole the basic idea for the hit song "Do-Re-Mi" from Guido of Arezzo's hymn "Ut queant laxis." Here is the original:

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The Mesopotamian Manner

I was just over at Greg Sandow's blog reading about his music criticism course and leaving a comment. One of his reading assignments was this essay by George Bernard Shaw in his role as one of the great music critics. He makes fun of a new (at the time) book on music theory by H. H. Stratham called Form and Design in Music. Shaw doesn't like it for a couple of reasons. One is that Stratham chooses a melody by Wagner as an example of bad melodic writing and Shaw, of course, is a famous Wagnerian. The other is that Stratham engages in what Shaw calls the "Mesopotamian manner", that is, he talks about music using technical vocabulary such as "the dominant of D minor". Shaw delivers a brilliant satire of this by subjecting Hamlet's soliloquy to the same technique. A sample: "Shakespeare, dispensing with the customary exordium, announces his subject at once in the infinitive..." Brilliant, funny and almost telling. But it is really a straw man, isn't it? It is usually quite unnecessary to discuss Shakespeare in this manner because he is writing in what we almost recognize as our own tongue (though more and more remote from us with every decade) and there is hardly a need to point out that "To be or not to be" is in the infinitive, because we know it by virtue of speaking English. But the dominant of D minor is not necessarily obvious until it is pointed out, and it has a very definite function and role that may be worth mentioning.

Unlike Shaw, we are living in a time when fewer and fewer books on music contain what I would recognize as talk about music. If I buy a book on Shostakovich symphonies and concertos, I hope very much that it will inform me about them. I hope to find the important themes in musical notation, discussion of the harmonic structure and so on. But that is no longer the case. A book on Shostakovich symphonies and concertos contains no musical examples whatsoever, but merely the author's attempt to communicate, in metaphor, his impressions of those themes. You can talk all you like about passionate reveries and juggernaut marches, but at the end of the day it is just vague metaphor and I know very little more than I did before. Instead of ten pages of rambling metaphor, I would much prefer a line or two of musical examples.

Shaw delivers some scathing criticism of Brahms in this review and while I think he is actually too kind, I think the argument would have benefited from a couple of specific examples. Having strong opinions is all very well,  but it is even nicer to mention why. The internet provides us with some amazing resources. Not only can we put up thoughts in writing, but we can also put up musical notation and even performances to demonstrate our points:

Sometimes, we can do both simultaneously! I leave you to find on your own the other seven (7!!!) clips that make up the rest of the piece.

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Music Theory

Now those are two words to send half my readers into a deep sleep and the other half into an anxiety attack! But music theory, looked at in the right way, is absolutely fascinating. Metaphorically speaking it is grammar, plot and pacing. It helps us figure out why something is working. As a composer, when I listen to a piece of music it usually either grabs me or doesn't. Sometimes it takes a while to grab me. Or after a while, I realize it no longer grabs me. But the reasons why are often mysterious. I don't say that theory can always figure it out; but sometimes, often, it can.

The thing is that we need a lot of different theories, but what we have, mostly, is a pretty well developed theory that deals with music from about 1600 to 1900. This we call the 'common practice' period and our theory is largely based on the music of Bach, Haydn, Mozart, Beethoven and extends to roughly fit the music of Schubert, Chopin, Schumann, Brahms, Liszt, Wagner and Mahler. Before Bach, we try to look at things a bit historically, taking our cue from theoretical treatises of the time. But the workings of modal counterpoint and harmony are not as clear to us as we would like. Since 1900 what we have are a lot of manifestos and propaganda about various progressive or avant-garde composers' methods. They are biased by the needs of promotion so not terribly useful. In the case of, for example, serialism, the theory is really a workshop in how to construct a piece, but is of little use in trying to decide whether the piece is successful or not. I think that valuations are a significant goal of theory. We can certainly use theory to investigate why, say, a particular piece by Beethoven has the kind of effect it does. We should be able to do so with a piece by Webern, if we had a workable theory. But I don't think we do. Vector analysis is not really that helpful as it seems it is neutral between a very good piece and a very bad piece. Yes, I believe that modernism produced quite a number of very bad pieces and it would be interesting to talk about how and why.

I don't think that any music is beyond being examined critically and theoretically, though not all music is worth the effort. But, given a significant repertoire, whether it be of Indian ragas, Javanese gamelan, mbira from Zimbabwe, motown from Detroit or pop from Liverpool, we ought to be able to look at it and see how it works. Same should be true of Ives, Partch or Cage. Though I can see a lot of headaches in the attempt! Heck, I would be happy if someone would figure out how some of Shostakovich's or Stravinsky's music works. Yes, I know that there are various attempts, but a lot of them seem rather unconvincing. It took a long time to really come to understand the Beethoven late quartets, I'm sure it will take just as long to figure out what is happening in the Shostakovich symphonies.

For example, I don't think you can just explain away this piece, the first movement of Shostakovich's 9th Symphony, by calling it "extended tonality". I think you really need to figure out the structure and talk about things like how and why the little duet between piccolo and trombone is so funny. Shostakovich was a funny guy, not the least because he decided to make his 9th Symphony so comic, going against the long tradition, started by Beethoven, of making one's 9th Symphony particularly portentous.

I guess what I'm trying to say is that good music theory includes aesthetic judgment.

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