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Music Theory IV


  • Wagner handouts:

  • Debussy's ideas about harmonic progression. In his view all of these chord series represent progression (forward harmonic motion):

  • Examples of polychords. See the listening section, below, for recordings and discussion. The first page of the handout is not covered below - play these examples on the piano and analyze.

Progressions - Debussy
functional chord progressions written by Claude Debussy

Concepts and Listening Examples


Topic 1: Impressionism

  • Like Impressionist painting, Impressionist music focused on texture, light and color. Pieces of music had descriptive, but vague titles; composers tried to show their feelings about the subject, rather than trying to describe or portray the subject in music.

  • We begin with a few samples of Impressionist music, and then turn our attention to examples of some of the technical concepts these composers used: expanding the role of Augmented triads in music; moving from common practice cadences to modal and linear cadences; and the concept of planing.

General listening examples:

  • Nuages gris (1881), by Franz Liszt. Like Beethoven did in his day, late in Liszt's life he began trying to push the boundaries of musical structure. His music had a profound influence on Wagner, and also on Debussy. Nuages gris ("Grey Clouds") is written in the key of g minor, but it uses Impressionist-type devices to obscure the sense of G as tonic. It sets a mood of an ambiguous, cloudy day, without telling a particular story.

    • Score (first three systems only)

  • Claude Debussy's Les Préludes (1909-1913):

    • Les tierces alternées (Alternating Thirds)

    • Feux d'artifice (Fireworks)

    • Le vent dans la plaine (Wind on the Plain)

    • Des pas sur la neige (Footsteps in the Snow)

  • Debussy's Sarabande from Pour le piano (1901) - mm. 3-12

Examples of Augmented triads:

  • Hugo Wolf, Das verlassene Mägdlein (The Forsaken Maiden, 1888) is an example of augmented triads used to blur / suspend the sense of tonality.

  • Debussy, Prelude from Pour le piano (1901). In this case, augmented chords are used to make a fairly standard functional chord progression sound exotic and uncertain, and to heighten the emotional reward of the two "V-I" tonicizations of C Major (cadences?).

Examples of modal / linear cadences:

  • Modal cadence: the Gloria from Johannes Ockeghem's (c. 1410-1497) Missa Cuiuvis Toni.

    • Modal cadences are "modal" because they contain melodic/harmonic gestures that don't fit into Major or minor tonalities. We name them based on the mode that contains those notes. Examples:

      • Phrygian cadence: Ra - Do at the cadence (b2 - 1)

      • Mixolydian cadence: an authentic cadence using the minor dominant and Major tonic (v - I); alternatively, a melodic gesture using Te - Do at the cadence (b7 - 1).

    • This movement of Ockeghem's Mass is in the Phrygian mode. The excerpt is the final 25 seconds of the movement - listen for the Phrygian cadence (Ra-Do) at the end.

  • Linear cadence: Debussy, Le vent dans la plaine (again).

    • A linear cadence approaches the end of the phrase with melodic motion by step in one or more voices (often in contrary motion), regardless of the implied harmonies (if any) - it is strictly a melodic device. This example comes from the very end of the piece.

    • This same excerpt could be analyzed as a third-relationship cadence, meaning that the chord roots of the cadence move by third. In this case, the final two chords of the piece are DM (over a B-flat pedal) and Bb (unknown quality). Because of the pedal tone, the DM sonority is not as obvious, so in this particular case I think a linear cadence is the better explanation.

    • Score and analysis

Examples of planing (study the score for Debussy's Sarabande, both of these examples):

  • Diatonic planing: the melodic equivalent of a tonal sequence; the interval classes are the same, but not necessarily the quality. The planing stays within the "key" ("pitch set" would be a better term) of the music.

    • Example: Sarabande, mm. 1-2:  the “melody” actually interferes with deciphering the harmony – it is a series of parallel 4ths/5ths/octaves – this is diatonic planing. Or, look at it another way - regardless of the initial voicing, when one chord moves to another, each voice moves equally in terms of class - all down by step, or by 3rd, for instance.

  • Chromatic planing: the equivalent of a real sequence; interval classes and qualities are exactly in parallel. This may require notes outside of the key/mode/scale/pitch set.

    • Example: Sarabande, m. 11: a series of parallel Mm7 chords.

  • Summary of planing: how do you tell if planing is diatonic or chromatic? It has to do with the idea of a “diatonic” vs. “chromatic” half step – diatonic refers to only the letter names of notes. So, diatonic planing means the letter names change by the same amount. In chromatic planing, the interval qualities are also preserved. In this way, you can consider it strictly in terms of melodic motion; or, you can think of it in terms of being in or out of the "key."

Les tierces alternées - Debussy
Le vent dans la plaine - Debussy
Nuages gris - Liszt
Feux d'artifice - Debussy
Des pas sur la niege - Debussy
Sarabande - Debussy
Das verlassene Mägdlein - Wolf
Prelude - Debussy
Gloria in Mi, from Missa Cuiuvis Toni - Ockeghem
Le vent dans la plaine - Debussy
Sarabande, Ex. 1 - Debussy
Sarabande, Ex. 2 - Debussy
tertian embellishments

Topic 2: Embellishments and Alternatives to Tertian Harmony

  • Embellishments to traditional triads (these techniques also appear in Impressionism, but they are not specific to that style). These embellishments need to be labeled, as shown below; the label can be added to a chord root or to a Roman numeral.

    • Extended ("Tall") chords - see the MUSC 3A page for details.

    • Add-note chords ("soft" dissonances) - a triad with an additional note, either a second, a fourth or a sixth above the root (add2, add4, add6). We tend to add notes to Major and minor triads only. The add4 generally only occurs in minor triads - it is too dissonant against a Major third.

      • Analysis: put the addition in parenthesis. Examples: BbM(add6), IV(add2)

    • Suspended chords - a triad with a second or a fourth, but no third. Examples: F(sus2), D(sus4)

    • Indeterminate chords - a triad with no third, sometimes called a "power chord" in popular music. It has no quality; instead it assumes the quality of the surrounding music.

      • Analysis: root + (ind), for example C(ind) for a C indeterminate chord.

    • Split-note chords - a chord with two chromatic versions of one of the pitches.

      • As an example, let's consider a C chord. Here are the modifications you can make:

        • Split the root up (so the chord contains both C and C#)

        • Split the fifth down (so the chord contains both G and Gb)

        • Split the third in either direction, so that the chord contains both a Major 3rd and a minor 3rd (E and Eb in our example).

        • Split the seventh in either direction, so that the chord contains both a Major 7th and a minor 7th (B and Bb in our example).

      • The two notes of the split need to be at least an octave apart, to avoid harsh clashing between them. For a C chord with a split third: you could voice it C-E-G-Eb, or C-Eb-G-E (spelling up from the bass).

      • Analysis: put the split note in parenthesis with an exclamation point. Examples: C(3!), V7(3!)

      • This example from Gustav Holst's The Planets, Op. 32 (1916), movement seven ("Neptune, the Mystic") has two interesting features - a minor-major seventh chord in the harps (mM7, the "Hitchcock" chord), and also a Cmaj7 chord with a split third in the rest of the orchestra. In this case, the E is closest to the root, so the overall analysis would be CMM(3!)

The Planets - Neptune, the Mystic - Holst
  • Alternatives to tertian (triadic) chord structures:

    • Quartal/Quintal Harmony

      • Chords based on stacks of 4ths or 5ths, instead of 3rds. To be perceived as "quartal" or "quintal" the chord must contain at least three notes.

      • Analysis (see examples at right): find the "root" (the lowest note when the stack is spelled in 4ths or 5ths), and label the root and either Q4 for quartal or Q5 for quintal.

        • Quartal/Quintal chords can be pure (all 4ths/5ths) or impure (containing other intervals). Pure is assumed, and not notated. If the chord is impure, analyze the interval content in parentheses, moving up from the root.

        • Pure Q4/Q5 chords can additionally be consonant (all perfect intervals) or dissonant (containing non-perfect 4ths or 5ths). Also, a pure, consonant Q4/Q5 chord that contains more than 5 notes becomes dissonant automatically. Consonant is assumed; notate dissonant status by abbreviating diss. after the chord root.

      • Aaron Copland's music frequently uses these kinds of harmonies.

    • Secundal Harmony (Tone Clusters / Cluster Chords)

      • Chords based on stacks of seconds.

        • Determinate: the exact pitches are notated.

        • Indeterminate: a general area of the staff is indicated - play chromatically within that area. These are also called cluster chords.

      • Listening example: Arvo Pärt, collage über b-a-c-h (1964), movement 2. This Sarabande movement juxtaposes a common-practice, Bach-style obbligato aria with a set of secundal harmonies (tone clusters). The contrast is shocking, to say the least. Keep in mind that these particular clusters are chromatic (all m2), and also large (spanning an octave), so they are more dissonant than they would be if only few notes were involved.

        • Score. The recording starts in measure 5 (the beginning of system 2); you can see the clusters notated in the piano part starting at measure 10. Warning - the clusters are loud!

        • Pärt uses the "B-A-C-H" motive in this piece: using the German note spellings, Bach's name becomes Bb-A-C-B (down m2, up m3, down m2). In this case, Pärt transposes it to start on F instead of on Bb (see measure 1)

          • Many composers including Schumann and Shostakovich use this kind of solfege "cryptography" in their music. The B-A-C-H motive, in particular, is a way for modern composers to anchor their work to the masterworks of the past.

Examples of quartal and quintal harmonies
examples of secundal harmonies
Sarabande - Pärt
  • 20th Century Scale Resources

    • Pentatonic (five-note) scales. Examples: Major pentatonic (Do-Re-Mi-Sol-La), minor pentatonic (Do-Me-Fa-Sol-Te), and the other three modes of the pentatonic scale.

    • Hexatonic (six-note) scales. Example: the Whole-Tone scale.

    • Heptatonic (seven-note) scales. Examples: Major, minor, and the diatonic "church" modes.

    • Octatonic (eight-note) scales. Scales with more than seven notes involve one or more chromatically "split" notes - two notes with the same letter name in one scale. For example, the scale C-D-E-F-F#-G-A-B is an octatonic scale, containing two notes that use the letter F (but one is chromatically altered).

      • Diminished Scales - a sub-category of octatonic scales, built of a repeated pattern either whole step - half step or vice-versa (WHWHWHWH or HWHWHWHWH). These scales are built by combining two fully-diminished seventh chords whose roots are a step apart. They are frequently used by Hungarian composer Béla Bartók. Below are two examples from his pedagogical work, Mikrokosmos (1926-1939).

        • Bartók, "From the Isle of Bali" (No. 109)

        • Bartók, "Diminished Fifth" (No. 101) - this piece uses two different diminished scales.

    • 9-note, 10-note, and 11-note scales - adding additional split notes.

    • The chromatic scale (12-notes) - this scale is fully split - all possible half steps are included.

    • Synthetic scales - any scale new to 20th century music - new tetrachord combinations, split note scales, etc. Basically, any scale not from the common-practice period.

Diminished Fifth - Bartók
From the Isle of Bali - Bartók

Topic 3: Pandiatonicism

  • Pandiatonicism is a sub-category of the harmonic language we call "free tonality" (see below). Pandiatonic music limits itself to a defined set of pitches, which could be a Major or minor key, a modal key, or any pitch set the composer chooses. The composer then uses all of these "diatonic" pitches freely to create whatever sounds are desired. Very few, if any, pitches from outside the chosen diatonic set are tolerated.

  • Pandiatonic music is firmly anchored around a particular tonic pitch, which can be established using traditional functional methods (not preferred), or through linear or repetitive tonicization, or however the composer chooses.

  • Pandiatonicism (along with quartal/quintal harmony) is a primary part of Aaron Copland's harmonic language. Here are several examples from the original chamber version (13 instruments) of Copland's Appalachian Spring (1944), that illustrate his usage of these techniques.

  • Example 1 - This excerpt (reduced to four-part texture) from the transition to the Shaker Tune is pandiatonic in the key of A-flat Major.
    • I have included both chord root/quality analysis and traditional harmonic analysis (Roman numerals), primarily to show that use of pandiatonicism creates a lot of chords that are non-functional. This limits the utility of traditional analytical language.
    • It is interesting to note that while the 2nd, 3rd, and 4th chords from the end are all D-flat add2 chords, they each sound very different simply by clever re-voicing each time. In fact, they do not sound at all like they are using the same harmony.
    • Score

  • Example 2 - This example is significant because the pitch collection used includes both C Major and c minor - blending the concepts of pandiatonicism and modal mixture. Copland's method of establishing the tonic is fascinating - he uses four different harmonizations of the V chord to create his "Sol-Do" references. Can you find and analyze all four?

  • Example 3 - This excerpt is from the beginning of the piece. The music here is pandiatonic in A Major.
    • At rehearsal #2, however, something even more impressive is also going on. Copland is building a stack of triads, each repeated only in their own voice: D Major in the cello, A Major in the viola, and E Major in the 2nd violin.
    • Together, these three triads contain all 7 notes of the A Major scale. However, when considering the chord roots expressed by each triad, we see that the prevailing harmony in this passage is actually a D quintal chord (DQ5: D-A-E).
    • This harmony is confirmed by the melodic line (flute and 1st violin), which ends with the arpeggio D-A-E just before rehearsal #3.
    • Copland is working seamlessly at three different textural levels at the same time: pandiatonic texture, tertian/triadic melodic gestures, and quintal harmony.
      • Recording: Excerpt 3A covers measures 1-12 - the generally pandiatonic section. Excerpt 3B covers mm. 13-22 - starting at rehearsal 2.
      • Score
Ex. 1 - Appalachian Spring - Copland
Ex. 2 - Appalachian Spring - Copland
Ex. 3A - Appalachian Spring - Copland
Ex. 3B - Appalachian Spring - Copland
  • Igor Stravinsky also made use of pandiatonicism, particularly in his earlier style periods. This example from the 1911 version of his ballet Petrushka, Scene 1 (The Shrovetide Fair) is pandiatonic in the key of G Minor, and also uses polychords (more polychord examples below).

Petrushka, Scene 1 - Stravinsky

Topic 4: Polychords, Polytonality and Bimodality

  • Polychords
    • A concise definition of a polychord is more than one harmonic structure sounding at the same time. These chords (simultaneities, verticalities) may be of any type (secundal, tertian, quartal, etc) and quality. The individual chords tend to be in close position (whether inverted or not), and spaced widely apart from each other. They also tend to be different enough to keep the ear from "adding" them together: for example, a polychord of gm / CM would probably sound to the ear like a CMmM harmony rather than two distinct chords.

Examples of polychords:

  • Example 1: Igor Stravinsky, Petrushka (1911), Scene 2 (Petrushka's Room). This is the famous "Petrushka chord," which demonstrates one of the most important concepts in twentieth century music - anything that can be done vertically (as a chord or simultaneity) can also be done horizontally as a melody or linear structure. This polychord is C Major / F# Major, but is expressed as two simultaneous chord arpeggiations.

  • Example 2: Igor Stravinsky, The Rite of Spring, (1913) "Danses des adolescentes." The violins and violas play an E-flat Mm7 chord, while the celli and basses play an F-flat Major triad. There is one octave between the two chords, but it is really the physical separation between the violins and celli on stage that creates the sense of simultaneous harmonies (polychord) rather than a single dense chord. The eight French horn parts provide accents for both chords.

  • Example 3: William Schuman, Three Score Set (1943), movement two. This is the clearest example of polychords in all of Western music.

Petrushka, Scene 2 - Stravinsky
Danses des adolescentes - Stravinsky
Three Score Set, II - Schuman
  • Polytonality

    • Polytonality is an extension of the concept of polychords. Instead of having a momentary clash between two different harmonies, polytonal music has more than one tonal center active at the same time. In practice this means two tonal centers (that is, bitonality), because it is extremely difficult to establish more keys than that at the same time. Even two tonics are hard to maintain, so polytonal music tends to be more polyphonic than usual, and also tends to be very simple both harmonically and melodically.

Examples of polytonality:

  • Example 1: Maurice Ravel, Violin Sonata (1927). The second movement, "Blues: Moderato" is polytonal: it is in G Major and A-flat Major at the same time. The violin begins in G, and continues in G after the piano joins in A-flat. Later, they trade keys.


  • Example 2: Béla Bartók, "Song of the Harvest," No. 33 of 44 Duos for 2 Violins (1931). This piece is polytonal, with each violin part playing within a different octatonic scale. The actual tonal centers and form of this piece are fairly complicated - see this article by composer Richard Trythall for a very detailed analysis.

Violin Sonata, II - Ravel
Song of the Harvest - Bartók
  • Bimodality

    • Bimodality is a related concept. In this approach, two different modes of the same tonic are used; so the harmonic content of each is different, but the actual tonal center is the same. Bimodality is a much easier technique to use than polytonality, because only one tonal center is being emphasized.

    • Don't confuse bimodality and modal mixture. The use of bimodality (or polytonality) implies a large degree of polyphonic texture, where two discrete, independent and equally important melodic lines are present at the same time, each one using a different mode (or tonal center, in the case of polytonality). Modal mixture simply means that the composer is borrowing freely between Major and parallel minor to allow more harmonic options.

  • Example: Béla Bartók, Major and Minor, from Mikrokosmos (1926-1939), Book 2 No. 59. In this case, the right hand is in a minor mode (either F Dorian or F Aeolian), and the left hand is in a major mode (F Lydian).

Major and Minor - Bartók
free tonality

Topic 5: Free Tonality

  • Free tonality is a post-Impressionist musical style where the music has a clearly identifiable tonal center, but the composer may achieve that tonality through any methods desired. All pitches of the chromatic scale can be used in whatever combinations are required. Also, functional relationships are allowed but not necessarily emphasized. Pandiatonicism is a type of free tonality that is constrained to a smaller, defined pitch set; free tonality as a broad genre can use any pitches, whether in or out of a key or whether closely related to the tonic or not. Two composers representative of the free tonal genre are Aaron Copland and Paul Hindemith.

  • Works of Aaron Copland: Copland's harmonic language is primarily tertian, with quartal/quintal elements. Through his use of linear melodic lines and chord voicings featuring open fourths and fifths, his harmonies end up sounding much more "quartal" than they truly are. Pandiatonicism is also a major component of Copland's use of free tonality.

Examples of free tonality in Copland's music:

  • Example 1A: Fanfare for the Common Man (1942), the clearest and most obvious use of quartal/quintal harmonies in Copland's body of work. As in Appalachian Spring (see example 3 under "Pandiatonicism," above), Copland's quartal "chords" are often horizontal, not vertical - they play out in a single melodic line.

  • Example 1B: Now compare that phrase from the original Fanfare to the same phrase in this version, which Copland wrote into his Third Symphony (1946). The fourth movement begins with a woodwind chorale version of the Fanfare, followed by a re-orchestrated version of the original brass/percussion Fanfare. The symphony continues with a beautiful chamber music interlude in the woodwinds, still based on the Fanfare melody for a time before the actual theme of the fourth movement emerges.

  • Example 2: Rodeo - Four Dance Episodes (1942). Copland orchestrated this suite using music from his ballet score by the same name.

    • Ex. 2A: Movement Three: Saturday Night Waltz (opening)

    • Ex. 2B: Movement Four: Hoe-Down. "Beef - it's what's for dinner! (bum bum bum)" Madison Avenue leveraged the Cowboy/Wild West connotations of this movement to sell steak. They knew we didn't need to have this music explained to us - this is what the American West sounds like. While you are listening, pay attention to how important the percussion section (including piano) is to Copland's sound landscape.

  • Example 3: The finale from another ballet score, this time the concert version of Billy the Kid (1938). This excerpt is the final movement, "The Open Prairie Again." To demonstrate the open spaces of the American prairie, Copland uses a lot of open space in his harmonic texture by emphasizing fourths and fifths. Note: as a general rule for Copland's music, despite the openness and emphasis on fourths or fifths, in many cases the chords he uses are tertian triads - they just sound quartal/quintal because of their voicing.

  • Example 4: These two excerpts from the score to the movie Hidalgo (2004), composed by James Newton Howard, demonstrate the enormous impact Copland's music has had on American popular culture.

    • Ex. 4A: "Main Title"

    • Ex. 4B: "Don't Waste our Money"

Fanfare for the Common Man - Copland
Symphony No. 3, IV - Copland
Saturday Night Waltz - Copland
Hoe-Down - Copland
Billy the Kid - Copland
Hidalgo: Main Title - Howard
Hidalgo: Don't Waste our Money - Howard
  • Works of Paul Hindemith: Hindemith's harmonic language features gradually increasing dynamics and dissonance, suddenly resolved by a recognizable tonal harmony (such as a major triad). He uses a tonal center for his compositions, but does not use a "key." For him, only the tonic note was important for establishing tonality, not whether the music was in Major or minor or any other type of modality.

Examples of free tonality in Hindemith's music:

  • Example 1: Fuga Prima in C, the second movement of Ludus Tonalis (1942). You will notice right away that Hindemith's music can sound much more angular and dissonant than Copland's but it is still tonal. Listen for the references to the pitch C.

  • Example 2: the Sonata for Oboe and Piano (1938) is typical of Hindemith's chamber music writing.

    • Ex. 2A (movement one): this excerpt is at the very end of the exposition of the movement's sonata form. The tonal center is E; listen for the E Major triad that ends the exposition.

    • Ex. 2B (movement two): this excerpt comes from the end of the A section, and shows Hindemith's technique of starting with a short melodic fragment, then making it longer and more dissonant until reaching a final tertian resolution.

  • Example 3: Symphony: Mathis der Maler (1934). This piece was created from music for Hindemith's opera by the same name. Each movement of the symphony describes a scene painted by Matthias Grünewald on the Isenheim Altarpiece at the Monastery of St. Anthony at Isenheim (near modern Alsace, France).

    • Ex. 3A - Movement One: "Concert of Angels" - this excerpt is from the very beginning of the movement. Although the harmony is different, the melody here is exactly the same as Danny Elfman's Batman Theme (1989)! Coincidence?

    • Ex. 3B - Movement Two: "The Entombment" - this excerpt is from the middle section of the movement. It starts quietly and builds to massive proportions, with a major triad at the resolution.

    • Ex. 3C - Movement Three: "The Temptation of St. Anthony" - this excerpt from the exposition. The melodies and harmonies are much more angular in this movement than in the other two, and the famous tertian phrase resolutions are absent.

Fuga Prima in C - Hindemith
Oboe Sonata, I - Hindemith
Oboe Sonata, II - Hindemith
Mathis der Maler, I - Hindemith
Mathis der Maler, II - Hindemith
Mathis der Maler, III - Hindemith

Topic 6: Serialism or "Twelve-Tonality" (also "Dodecaphony" or "Serial Atonality")


Listening examples for serialism:

  • Luigi Dallapiccola, "Simbolo" from Quaderno musicale di Annalibera (1953). This excerpt (mm. 1-8) shows how important repetition became for serialists as a way to slow down the progress of the tone row. Also, the melody at the beginning shows the B-A-C-H motive very clearly.

  • Arnold Schoenberg, Piano Piece, Op. 33a (1928).

    • Score

    • A handout showing how Schoenberg breaks his tone rows down into smaller "cells" made up of 3-4 notes.

    • A handout showing the process of finding the P-0 row for Schoenberg's Op. 33a.

Simbolo - Dallapiccola
Op. 33a - Schoenberg
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