Category Archives: Harmony

Composition II – Pitch Sets

The study of set theory is one that is deep and often covered in music theory courses devoted to the analysis of 20th century music. A “nutshell” description of it is here. The typical approach to pitch set analysis involves considering pitches numerically (0=C, 1=C#, etc.), understanding the different pitch-class sets that are possible (there are 208!) considering three-note collections (trichords), four-note (tetrachords), five-note (pentachords), and so forth, with each set representing both the notes in “proper” order and under both ansposition and inversion, as is the common practice.

Why use set theory as a composer?

One point of set theory is to create a “language” for understanding music that is not based on triads as a fundamental feature of melody and harmony. Indeed, pitch sets are used like notes of a chord — they can be a framework for melodies, stacked as chords, embellished, and so forth. The order of pitches isn’t what is important (as it is with twelve-tone music).

For instance, music could be composed that features the pitch-class set 0,1,4 (it’s called “3-3,” since it is listed third on the standard table of pitch-class sets), transpositions of that set, and inversions of it:

3-3 is, in fact, a main feature of the second movement of Bartok’s String Quartet No. 2. We can most easily see this by looking at the intervals between notes of the piece; when motives are framed by a major third that is divided by a half-step on one side or the other, it is “3-3,” like in this music at rehearsal 1:

This passage also has a couple of extra notes – the G# in bar 19 helps expand 3-3 into more of a scale, as does the C# in bar 23. The C# in bar 23 also helps make another version of 3-3 (0,1,4 transposed up by a half step).

Another way that Bartók used this trichord is as main notes of the melody in rehearsal 4:

At rehearsal 7, this trichord is found alongside 3-4 (0,1,5 – imagine a perfect fourth with a half step-in the middle, like C-C#-F or C-E-F).

At bar 166, there is a very strong chord that combines 3-3 and 3-4 into a single unit.

As the music continues from this point, the cello performs a melody that also blends 3-3 and 3-4.

…and forth it continues.

Key Modulation: Elgar’s “Moths and Butterflies”

elgar-cover

Learning to follow key changes is not easy for many students, especially when they are asked to locate modulations themselves and distinguish between different methods of modulation. Here is a step-by-step guide that uses a charming example of Romantic music: Edward Elgar’s Moths and Butterflies from The Wand of Youth Second Suite. More practice is also available on this site in “Modulation Practice: Beethoven’s Violin Concerto.”

Step One: Get to know the music

Begin by hearing a recording [here’s one on YouTube] while reading the sheet music [it’s on imslp.org — Moths and Butterflies is the third movement]. Please note that this piano reduction has two mistakes! The right hand chord on beat 1 of bar 3 should have a D♯, not a C♯. In that same bar, the left hand should return to bass clef; its two notes are both E, and the bass clef continues until treble is restored in bar 5 as shown.

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Decoding the “Love On Top” Modulations

For the last few years, whenever I teach modulations, the song “Love On Top” (Beyoncé Knowles/Terius Nash/Shea Taylor) is brought up by students. It’s a modulating tour-de-force insomuch as it moves through five keys with surprising swiftness, and it doesn’t hurt that modulations like this are uncommon in R&B.

As impressive as it is, there’s just one trick used four times. Here’s how it goes.

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Modulation Practice: Beethoven’s Violin Concerto

I find that students often feel “thrown into the pool” when confronted with analysis of modulations in a full piece of music. It is one thing to describe modulations in the context of a short passage (especially when it is in a simple chord-by-chord format, as it was for most of my undergraduate theory education), and another thing altogether when in the context of real, textured music.

To bridge this gap, I offer this “play-by-play” analysis using an example from the first movement of Beethoven’s Violin Concerto. Here’s a PDF of the sheet music, which is in a piano reduction by the composer Paul Dukas. We’ll be looking at bars 39-76, which begin in the fourth bar of the third system of page 3. Here it is as a jpg, too (click to enlarge).

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Modulation: Don’t Believe What You’ve Been Told!

I love teaching modulation, but I am not satisfied with the common approaches to teaching it. We used to use Kostka & Payne at WCU (my experience with this book stops at the fifth edition, which is what I quote below), which is the text I learned from as a student. They seem to represent the consensus, that there are two kinds of modulation-by-pivot techniques:

Pivot modulations

Modulation by common chord. This is always taught first, which reveals a bias toward classical practice. It gets lots of ink in the textbook because it is something that can be broken down into steps and explained ad nauseum. The technique is to find the first harmony that works for the new key (usually its dominant) and back up to find the pivot harmony that works equally well in both keys. Students hate this… they never can remember that a common chord is common to both tonalities. But who can blame them? To a listener, the modulation begins with a chromatic shift. Continue reading Modulation: Don’t Believe What You’ve Been Told!

Chord Inversions: They’re Not Just For Labeling Anymore

A Little Background

Chord inversions are the second thing music students learn about chords, right after how to spell chords over a given root (what the pitches of D Major or F♯ Diminished is). For those who are new to the topic, follow this link.

Chord inversions are mostly taught by looking at chords in the easiest possible manner: as close-position triads or as block chords in simple four-voice chorales. When they are studied like this, students can quickly learn how to identify a chord’s position (either root position or inversion) and assign it a numeric label: 5/3 or 7 for root position, 6 or 6/5 for first inversion, and so forth.

It is much more difficult to apply the concept of inversions to music that doesn’t move in block chords, and in most music, the bass is elaborated in some way, complicating the matter. Sometimes they are ornamented with passing tones and such between “structural” tones, and when a bass line is genuinely florid (as in much classical or jazz music) it becomes very tricky indeed.

So, Why Do They Matter?

Why do we study chord inversions? To many students, it’s tedious busywork to parse the pitches that make up a chord, figure out which is lowest, and assign a numeric label. Some of the point may be to dwell a while on spelling chords. It is also an introduction to the idea of following one note of a chord to the next, which introduces the subject of voice-leading, which is often a primary concern of harmony courses (though that is increasing being considered “old-fashioned”).

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How to Interpret Chord Symbols

The ability to read a chord symbol and name the pitches of its chord is an essential skill for all musicians. I use it constantly in all of my music theory, analysis and orchestration courses to quickly describe musical harmony while dispensing of the need to suss out harmony from a written-out texture. It is, of course, also the foundation of jazz and pop improvisation.

Students of classical music often do not learn how to interpret these symbols beyond plain triads, so I am providing this lesson as an introduction for those students.

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