Music Theory, level 3A (Adduci)
MUSC 3A recommended reading list (Turek, Theory for Today's Musician):
- Ch. 11-14 - 1A Review: part writing / voice leading
- Ch. 15-17 - 2A Review: secondary dominant/LT, tonicizing groups, pivot chord modulation
- Ch. 21 - modal borrowing, modal mixture
- Ch. 22 - altered pre-dominants
- Ch. 23 - altered dominants
- Ch. 24 - advanced modulation
- Ch. 25 - extended chords
- optional material: Ch. 28-30 - Form
Everyone, please give a warm welcome to Franz Schubert, the official composer for Theory 3!
Schubert's music uses all of the techniques we are discussing this semester (modal mixture, various modulations, etc.), and none of the more advanced (theory 4-level) techniques we are not covering. This makes him the perfect composer to study if you are looking for extra material to practice outside of class. Schubert's lieder ("Art" songs for voice and piano) are compact and quick to analyze. He wrote three song cycles (Die schöne Müllerin, Winterreise, and Schwanengesang), and many individual songs as well. Visit Schubert's page at the Petrucci Archive for access to his public-domain scores. Happy analyzing!
Welcome to the MUSC 3A page!
Please choose one of the following sections:
- A collection of music theory-related links from around the Web.
- The greensheet and other administrative documents
- Handouts given in class.
- Homework assignments and projects.
- Listening examples.
Web Links
This section contains links to internet sites designed to help you with music theory and ear training. The Internet is a fantastic resource for self-driven learning!
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Greensheet
This section includes the greensheet and any other class policy documents.
Left-click to view, right-click to download.
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Handouts
This section contains copies of the informational handouts given in class. Handouts that went along with music played in class can be found under Listening Examples, below.
- (8/25/14) A packet of harmonic analysis practice exercises focusing on diatonic harmony, secondary harmony, and diatonic pivot modulations.
- The answers for the analysis practice packet.
- (9/10/14) Part Writing handouts
- Guidelines for Bach-style part-writing.
- My key to the abbreviations I use when correcting your part writing exercises.
- Self-Test Exercise: identify as many part writing errors as you can. The solution is also provided.
- (9/17/14) In-Class Activity: analysis of Schubert's lied, "Der Müller und der Bach" from Die schöne Müllerin, Op. 25.
- (9/22/14) A guide to transposing instruments and how to read their music.
- (9/24/14) Transposition Practice Exercise 1 (four-voice chorale)
- Exercise
- Solution
- Recording: this chorale is also an example of how to use an enharmonically-respelled German chord to pivot to a distantly-related key.
- (10/8/14) The Study Guide for the midterm exam.
- (10/8/14) Suggestions for harmonic analysis (a reality check).
- (10/20/14) Examples of modulation using chromatic pivot chords.
- (11/3/14) Examples of extended chords. See "Listening Examples," below, for recordings of these two excerpts.
- (11/12/14) In-Class Activity: analysis of Chopin's Prelude in E minor, Op. 28 No. 4. See "Listening Examples," below, for a recording of this piece. The second half of this prelude includes several advanced harmonies: a tritone substitution (actually a tritone exchange), sus chords, and an enharmonically-spelled German chord.
- Solution, option 1: Root-and-Quality analysis (the best way to analyze linear chromaticism)
- Solution, option 2: Roman Numeral analysis (very ineffective)
- (11/26/14) Analysis Day! Here are the scores we studied, plus the answer key.
- (11/26/14) The Study Guide for the final exam.
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Assignments
This section contains copies of the homework assignments and projects given in class.
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Listening Examples
This section contains links to musical examples referenced during class. The examples are provided for class purposes only.
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To play, left-click the excerpt title and then click "play" when the player opens.
- Pivot Chord Modulation (Review)
- This is the smoothest possible way to change keys. One chord that has a function in both the starting key and the new key is used as a doorway - that chord has a dual function, operating in both keys at the same time. In early Common Practice usage (Baroque/Classical periods) these modulations tended to be between closely-related keys only (keys only one accidental apart on the circle of fifths), and the pivot chord had to be diatonic in both keys. In later usage (Classical/Romantic periods), chromatic pivot chords were introduced (see Advanced Modulation, below), so that the pivot chord did not have to be diatonic, as long as it was functional (for example, V/V could be a pivot chord). Additionally, through the use of chromatic pivot chords it is possible to modulate to distantly-related keys. This will be elaborated on later in the semester.
- Example 1: basic pivot modulation from F Major to d minor.
- Example 2: basic pivot modulation from F Major to B-flat Major.
- Example 3: This Bach chorale ("Du Friedensfürst, Herr Jesu Christ") contains two modulations by pivot chord. Give a complete harmonic analysis.
- Please refer to the MUSC 2A website for additional handouts and listening examples related to simple pivot modulation (diatonic pivot to closely-related keys).
- Modal Mixture (Review)
- Please refer to the MUSC 2A website for the complete presentation of this topic.
- The concept of "modal mixture" means that composers may use all possible harmonies that relate to a certain tonic pitch. If the "key" is G, then composers are free to use any chords from either G Major or g minor tonalities. All of these chords are considered "diatonic." In simple terms, there is now less of a distinction between Major and minor - there is just Tonic. In most cases, modal mixture is used to enrich music in a Major key by bringing in sonorities from the parallel minor. Remember that in a Major key there are only 7 diatonic chords, while in a minor key there are 13. The minor mode is already more harmonically dense, so it is much less common to see chords borrowed from the parallel Major key when the music is in minor.
- In the 19th Century (and earlier), modal mixture only applies to a Major key and its parallel minor, or to a minor key and its parallel Major.
- In the 20th Century (and beyond), with the resurgence of modal scales in composition, it would be possible to borrow chords from any of the tonic pitch's modes, not just from Major / parallel minor.
- In compositions from the late Classical period and onward, modal mixture becomes more advanced and may involve either an extended section of music in the parallel key or many rapid alterations between Major and parallel minor (often changing the quality of the tonic chord many times: I - i - I - i).
- When changing from Major to parallel minor (and back), the tonal center is not changing, so it is not a modulation. When only the mode is changing, we call the process mutation. Since the music is not modulating, a cadence is not necessary to "confirm" the new mode. This means that modal mixture is a very flexible process and can happen very rapidly.
- Example 1: Brahms, Symphony No. 2 in D Major, Op. 73, movement three (1887). This movement begins in G Major but features many shifts between G Major and g minor before the end of the first large section. At the end of the first section the harmony changes from I to i several times in a row. Analysis: all in G major, labeling the g minor chords as i (BC).
- Example 2: Dvorák, Symphony No. 8 in G Major, Op. 88, movement two (1889). This movement begins in E-flat major, but moves frequently between E-flat Major, c minor and C Major. The excerpt ends in C Major (the end of the first large section of the movement). As you listen to the excerpt, try to hear the cadences that confirm modulations, and also listen for the moments when Dvorák mutates between C Major and c minor without a cadence.
- Altered Pre-Dominants (Review)
- Please refer to the MUSC 2A website for the theoretical background for this topic. More advanced listening examples pertinent to MUSC 3A will be added during this semester.
- The Neapolitan Chord
- Tchaikovsky, theme from Swan Lake, Op. 20 (1876). This statement of the theme is the opening to Act 2 of the ballet (the theme appears in each act). The piece is in the key of B minor, and features a prominent Neapolitan chord in measure 17.
- Score (This is the score for the entire movement; the recording is only of the first 20 bars)
- Recording
- Bach, Passacaglia and Fugue in C Minor, BWV 582, Fugue (coda). In this example, notice the insistence on V - i motion, setting up an expectation of V on the downbeat. Bach uses the Neapolitan instead, then follows with a long pause to further emphasize this unusual chord. After the caesura, the coda resumes with the expected V chord and then the concluding section emphasizes the modal mixture of I and iv. Also pay attention to the importance of the final tonic pedal and the imperfect authentic cadence, viio7 (fully diminished) - I (Picardy third).
- Score
- Recording (The recording starts at the pickup to the last 12 measures of the score excerpt)
- Bonus Example: Rachmaninov, Piano Concerto No. 2 in C Minor, Op. 18 (1901), second movement (excerpt). This is the beginning of the cadenza for the second movement, which is approached from I6/4 to N6 motion in the orchestra instead of the traditional pause on I6/4.
- Score
- Recording (The recording starts at the beginning of this score excerpt)
- The Augmented-Sixth Chord
- The German chord, in particular, can be useful as a tool to modulate to distantly-related keys, because it sounds like a Mm7 chord. Respelling one or more notes of the German chord enharmonically can allow us to stack up a tertian Mm7, which we can then use to pivot to a new key where either that Mm7 exists, or where it is V7 of a diatonic chord. The German chord contains the notes Le (b6), Do (1), Fi (#4), and Me (b3) - in C Major: Ab-C-F#-Eb. The notes with accidentals can be respelled enharmonically as follows:
- Non-Traditional Chord Resolutions
- As the common-practice system of functional harmony began to be stretched and made increasingly chromatic, it became more common for composers to set up a chord that has an expected function, and then in some way fail to realize that function. There are two main ways to accomplish this: either the chord can resolve properly but not immediately, by inserting additional chords between the tendency chord and its chord of resolution; or the chord can resolve deceptively, that is, to a chord that has some sort of close relationship to the expected chord of resolution. Note, neither of these techniques are the same as a non-functional ("embellishing") chord, that does not resolve at all.
- Delayed Resolutions. The best way to delay the resolution of a chord is to first resolve it to another (or more than one) chord that also has a tendency to resolve to the original destination.
- For example, the Neapolitan chord has a tendency to resolve to the dominant. viio/V and V/V also resolves to V, and I6/4 also resolves to V. So, a composer (Mozart, in particular, liked to do this) might write the following chord progression: N6 - viio7/V - V/V - I6/4 - V. In this way, the actual resolution (to V, in this example) is strengthened by the increasingly urgent need for the ear to reach the dominant.
- This makes good compositional sense - if you were to just insert random chords in between V/V and V, for example, when the V chord is finally reached the ear will have lost its sense of importance and arrival. This can be used deliberately to weaken a secondary dominant, usually by inserting the I chord: V/V - I - V (remember, motion to I always counts as progression in a functional harmonic system).
- Deceptive Resolutions. A resolution counts as deceptive if the tendency chord nearly resolves to its intended destination, but not quite. This is most often done by resolving to a different chord that shares some tones in common with the intended chord, or that is chromatically altered in some way from the intended version of hte chord.
- This is how a deceptive cadence works! The V7 chord (Sol-Ti-Re-Fa) has three tendency tones that want to resolve to I: Ti (leading tone) goes to Do, Fa (chord 7th) falls to Mi, and Sol (the dominant scale degree) falls to Do. That is a very strong set of tendencies, and this is why an authentic cadence sounds so powerful. Now, in a deceptive cadence, the harmony moves V-vi instead of V-I. But, Ti still rises to Do, and Fa still falls to Mi - the only difference is that Sol rises to La. The vi chord and the I chord share two tones in common (Do and Mi). Therefore, the resolution of V7-vi sounds shocking - your expectations are mostly met, but something important (Sol-Do) did not happen properly.
- Example: in the same way, this type of "almost, but not quite" resolution can be used anywhere, not just at a cadence. In the "Meditation" from Jules Massenet's opera, Thais (1894), a very deceptive resolution occurs in measures 57-58. There is a C#Mm chord in measure 57, which is V/iii in the key of D Major. This chord should resolve to F#m (iii), but instead it resolves to DM (I). The F#m and DM triads share two tones in common - F# and A. To make this resolution even more deceptive, the melody (solo violin) actually moves from C# to F# - a correct motion to the intended chord root.
- Example: Mozart, String Quartet No. 17 in B-flat Major, K. 458, movement 2. This excerpt, in B-flat major, arrives on an F#dd chord in measure 26. This is analyzed as viio7 / vi. As such, it should then resolve to vi (Gm). Instead of doing this, Mozart resolves to Edd, another secondary leading tone (viio7 / V) in bar 27. The Edd chord is spelled E-G-Bb-Db, which contains two of the notes of Gm (G-Bb-D). Additionally, the bass moves from F# to G, as it should if the chord were resolving correctly. The soprano also lands on Bb, which is one of the common notes between the proper resolution and the deceptive resolution. Having both of the outer voices go where they "should" makes the deception stronger.
- Example: Puccini, "Suor Angelica" (opera #2 of "Il Trittico", 1918). This excerpt, the last 30 seconds of the opera, sets up the expectation of a plagal cadence in C Major. Instead, IV (FM) resolves to BbMmM - a chord that contains only the C of the expected I (CM) chord. This C is resolved correctly in the soprano and the bass, giving a sense of correct resolution combined with harmonic ambiguity. After maintaining this mysterious sound for a few seconds, Puccini then resolves to I, completing the final cadence of the opera.
- Advanced Modulation examples
- Phrase Modulation
- A phrase modulation is a very simple way to change key: the composer ends one phrase in the original key, and starts the next phrase immediately in the new key with no pivot or preparation. This type of modulation is very common in highly sectionalized, repetitive forms such as the Minuet and Trio, where short sections of music are repeated. In a typical section of a minuet there may be a move to a new key (by pivot chord, perhaps) followed by a cadence in the new key. When the section repeats, the music is instantly back in the original key with no warning.
- Example One: Haydn, Symphony No. 6 in D Major, Hob. 1/6 "Le matin" (1761), third movement. This excerpt is the beginning of the Trio portion of this Minuet and Trio movement. Section One begins and ends in d minor, and is repeated. Section Two starts immediately in F Major, via phrase modulation. During Section Two, the music pivots back to d minor. When Section Two is repeated, another phrase modulation occurs when the section restarts in F Major.
- Common Tone Modulation
- Example One: Sir Edward Elgar, "Enigma Variations." The transition between variation eight and the famous ninth variation ("Nimrod") uses a modulation by common tone. Variation VIII ends in G Major. The tonic pitch, G, is sustained by the strings, and is then reinterpreted as the third of an E-flat Major chord at the beginning of variation IX, which is in E-flat Major.
- Score
- Recording (this is a very soft part of the piece, you may have to turn your volume up to hear the end of the excerpt)
- Example Two: Francaix, Theme and Variations for Clarinet and Piano (1974). There is a common tone modulation between the end of the Theme and the beginning of Variation One. It works exactly like the Enigma Variations example does. The Theme is in D Major; a D is sustained between movements, and then Variation One begins in B-flat major on the I chord.
- Example Three: Beethoven, Symphony No. 9 in D Minor, Op. 125, "Choral" (1824). In the fourth movement, before the Jannissary section, the music is in D Major. There is a I-IV-V-I approach to the end of the phrase cadence, then a tonicization of the dominant: V/V - V - V/v - V. Following these four chords, the entire orchestra sustains the dominant pitch "A" by itself (with no supporting harmony). This A pitch is then placed into an F Major context under the fermata. F is V in the key of B-flat Major, the key of the Jannissary march that follows. The "common tone" is the pitch A, which is common to both the key of A Major and the key of B-flat Major. Because the common tone modulation is so powerful, your ear "believes" that the music has modulated to F Major ... it takes the insistent emphasis on the pitch Bb in the next section to convince you that you are actually in B-flat Major.
- Beethoven was a tricky fellow. Remember, at a fundamental level music is a struggle between repetition and contrast. Beethoven lays the groundwork of function and expectation for the modulation described above back in the third movement of the symphony. Towards the end of the movement, Beethoven sets up a melodic pattern in the brass that involves emphasis on the dominant and the sustaining of a single tone (just like the pattern of the 4th movement modulation). The first time this melody appears, there is no tonal movement at all. About one minute later, the same melodic pattern reappears. This time, the single note in the brass becomes part of what sounds like a common tone gesture but is really just an extended tonicization of the dominant (Eb) using modal mixture (listen to the tonality shift from Eb Major to eb minor), before eventually returning to the tonic of Bb Major. By repeating the same figure, Beethoven sets up an expectation of repeated harmony that he finally shatters in the fourth movement, when the common tone gesture does move to a new key. Beethoven deliberately established your harmonic expectations, 15-20 minutes in advance, before breaking those expectations in the fourth movement. This makes the modulation even more powerful when it finally arrives.
- Example 2-B: third movement; no modulation.
- Example 2-C: third movement; common tone gesture, but no modulation.
- Example Four: Bernstein, "Tonight" from West Side Story (1957). This tune begins in A Major and moves to C Major. The chord progression (mm. 73-76) leading up to the modulation is F#m, G#m, GM, CM. The F#m chord is diatonic in A (vi) but not in C. The G#m is not functional in either key. The GM chord is diatonic in C (V) but not in A. This chord does have a chromatic function in A (bVII BC), but it can't serve as a chromatic pivot chord because the previous G#m harmony is non-functional - we have already "left" the original key. The CM chord is not diatonic in A, but is I in the key of C. Despite these non-functional harmonies, the modulation sounds smooth and uncomplicated, the ear is actually drawn to the singer's pitch (B), which is maintained throughout this section - a common tone between the keys of A and C.
- Score with correct and incorrect analysis of the modulation.
- Recording
- Sequential Modulation: the use of a sequence to migrate away from a tonal center and establish a new key. By repeating the same melodic/harmonic pattern at different pitch levels, the original sense of "tonic" is lost. The sequential pattern can then be altered or ended to allow a pivot into the destination key.
- Example One: J.S. Bach, Minuet II from the Cello Suite in G Major (BWV 1007). This minuet is in g minor. In measure 33 Bach begins a sequence that proceeds downward by whole step. During the second repetition of the sequence, he alters the final beat (measure 36) to produce an FMm harmony, allowing a smooth pivot into the key of B-flat Major.
- Example Two: Alessandro Marcello, Concerto for Oboe in C Minor (1717), third movement. This excerpt begins in E-flat major, and features three overlapping sequences - 1) up by step, 2) down by step), 3) up by step. By layering these three different sequences together and changing the harmonic rhythm, Marcello keeps the passage from bogging down - the ear quickly tires of sequential patterns if they are not varied. During the transition to sequence three Marcello pivots to g minor.
- Example Three: Dvorák, Symphony No. 8 in G Major, Op. 88, movement two (again). In addition to being an excellent example of modal mixture, this movement begins with a modulation from Eb Major to c minor, through the use and modification of a sequence and the enharmonic respelling of a German Aug6 chord. See the recorded excerpt and score in "Modal Mixture," above, and listen to the first ten measures.
- Non-Diatonic Pivot Chords: As mentioned above, more advanced pivot modulations use pivot chords that are functional, but not necessarily diatonic, in the old and new keys. A chord may have a diatonic function, a secondary function (e.g., V/V), or some other chromatic function (eg., an augmented sixth chord). The pivot chord may be diatonic in one key and chromatic in the other, or chromatic in both keys. The important point is that the pivot chord has a function in both keys.
- Enharmonic Modulation: a subset of pivot chord modulation that relies on enharmonic respelling of a chord to give it a function in a different key. For example, any Mm7 chord can be respelled enharmonically as a German Aug6 chord in a new key (and vice versa; the Mm7 could be "the" dominant, or a secondary dominant); any dd7 chord can be respelled enharmonically to give it a different root, making it a leading tone to a new key. Explore these possibilities on your own: take a Mm7 or a dd7 chord, and see what keys you can modulate to using it as an enharmonic pivot chord! Remember, with the surge in chromatic harmony in the late Common Practice period, pivot chords can be chromatic in one or both keys.
- Example One: an enharmonic respelling that allows a modulation to a distant key. This "chorale" is in F# minor. In measure 4, the E#dd chord is respelled as a Bdd chord, allowing a pivot to the key of C Major.
- Example Two: the transposition practice exercise provided under "Handouts," above, is also a good example of how the German chord can be respelled enharmonically to function as a dominant (secondary, in this case), allowing it to be used as a pivot chord. I have duplicated the links to that exercise here.
- Altered Dominants
- The pitches of the V chord may be altered, as follows:
- The dominant triad (no seventh chords) may have a raised fifth (analysis: V#5). This typically occurs only in major keys.
- In major or minor, the dominant seventh chord may have a lowered fifth (analysis: V7b5).
- In major or minor, the dominant triad or seventh chord may have both a lowered third and a lowered fifth (analysis: V7b5b3). This creates a dominant that sounds as either a diminished triad or a half-diminished seventh chord.
- Example: Dvorák, Symphony No. 8 in G Major, Op. 88, movement two (yet again). This movement has already been used as an example of long-term modal mixture AND sequential modulation (see above). The beginning of the movement also contains four altered dominant chords - can you hear them? They are the "mysterious" sounding harmonies played by the clarinets (for example, see measure 13, beat 2, and measure 16, beat 2). For the score and recording, see "Modal Mixture," above.
- Altered dominants may also be secondary dominants; however, composers tend to avoid using the diminished V in this context. But, see example two!
- Example 1: Schubert, String Quartet No. 13 in A Minor, D. 804 "Rosamunde" (1824). In this example, Schubert uses the V7b5 chord as a secondary dominant: V7b5 / V. This chord appears in measure 9, on beat 3.
- Example 2: Lombardo, Concerto for Oboe, measures 10-19. Two times during this phrase, the composer uses V7b5b3 (the half-diminished dominant) as a secondary dominant, and it sounds amazing! As you can see, rules are made to be broken.
- Extended Chords ("Tall" Chords)
- Major or minor chords may be extended by adding a 9th, 11th and/or 13th above the bass.
- The dominant is the most commonly extended chord (either primary or secondary dominant chords).
- Extended chords are typically in root position, with the tallest extension in the melody, treated as an appoggiatura or passing note. The extended chord usually collapses to a triad or seventh chord before resolving.
- Example One: Scott Joplin, "The Augustine Club Waltz" - examine how extensions are used to embellish a normal circle progression.
- Example Two: Giacomo Puccini, Aria: "Che gelida mannina" from La bohème (1896). The extended chord (V9 of V, in this case), occurs at the very end of the recording (the score shows the final two bars of the recording).
- Tritone Substitution
- Any chord may be replaced by a new chord whose root is a tritone away. For example, a CM chord can be replaced by an F#M or GbM chord. This new, substituted chord will function like the orignial chord.
- The purpose of a tritone substitution is to create chords whose roots move downward by half step. This is a more interesting type of chord movement than a functional bassline that leaps by P4 an P5. When done properly, that is, in a circle progression (harmonic motion by P5) such as ii - V - I, you will get this type of root movement.
- The original chord and its substitute generally have the same quality (this sounds best, but is not required).
- Mm chords are the most-often substituted chords (other chords can be replaced, but this does not sound as good). The reason that dominant-7th chords work well in tritone substitutions is that any two Mm chords that are at tritone apart actually contain the same d5 interval, enharmonically spelled. For example, AMm (A-C#-E-G) and EbMm (Eb-G-Bb-Db) contain the same tritone: C#-G (or G-Db).
- The standard form that the tritone substitute usually takes is to replace the V7 chord in a standard ii7 - V7 - I progression. In the key of C Major, the original progression is Dmm, GMm, CM. The GMm chord is replaced with a new chord a tritone away - DbMm. This gives a new, stepwise chord progression: Dmm - DbMm - Cm. The new roman numerals are ii7 - bII7 (tts) - I. "TTS" stands for "tritone substitution". The new Roman numeral for the tritone substitute should have the correct quality, and if the root of the chord is not in the key, an accidental should be applied to the Roman numeral as shown in this example.
- Please Note: bII7 (tts) should not be confused with the Neapolitan chord - the Neapolitan chord never (or rarely) has a 7th, and it resolves to V, not to I. Your analysis must be based on the chord's actual function. In other words, its identity is determined by its destination.
- For this technique to work, the chromatic, stepwise chord root movement must be present, and the original chord progression must be obvious and functional.
- Example: Jules Massenet, "Meditation" from Thais (1894), measures 58-60. In this slightly unusual example, the harmony is DM - EbMm - DM. The analysis is I - bII7 (tts) - I. The example is unusual because the first chord (m. 58) would typically be ii7, not I.
- Score
- Recording (This recording covers measures 53-60)
- There is a bonus tritone substitution here, depending on how you hear/analyze this excerpt. We have already discussed measures 57-58 as an example of a deceptive resolution (see above). However, this mysterious C#Mm chord COULD be a tritone substition for the regular IV chord (GM). If so, then Massenet's original, functional chord progression would be I - IV - I - V7 - I, or DM-GM-DM-AMm-DM (piano student everywhere recognize this). After performing two tritone substitutions, he ends up with I - VII7 (tts) - I - bII7 (tts) - I, or DM-C#Mm-DM-EbMm-DM: a much smoother, more chromatic chord progression, and a very compelling sound that does not bother our modern ears at all.
- Linear Chromaticism
- This technique, most common to the late Romantic / early 20th Century repertoire, involves the gradual "mutation" of harmonies, rather than the use of functional root movements. The goal is to create ascending or descending chromatic lines in one or more voices. To accomplish this, the composer takes a chord and alters it, one note at a time, to create new harmonies. The process is slow and usually involves chromatic changes. For example, an EbMm (Eb-G-Bb-Db) chord could be mutated to Edm (E-G-Bb-Db), and then to CMm (E-G-Bb-C) by changing one note at a time.
- Example: Chopin was a pioneer of this technique, and it appears frequently in his music (and in the music of Debussy). For example, listen to an excerpt from Chopin's Prelude in E Minor, Op. 28 No. 4. See if you can hear the individual changes being made to the chords in each bar.
- Score
- Recording (This recording is of only the first 12 measures)
- Harmonic Analysis (root-and-quality analysis, only using Roman numerals at phrase endings where clear structure is evident)
- Roman Numeral Analysis - only to show how little help this kind of analysis gives us when linear chromaticism is being used.
- Chromatic Mediants
- Take any chord, for example CEG, and add a second chord whose root is a third away (for example, EGB). This chord has two tones in common with the original chord. Now, chromatically alter one of those common tones and re-write the second chord so that it has the same quality as the first chord (now, EG#B). This new chord is a chromatic mediant.
- A chromatic mediant is usually related to the tonic (I) chord, and is a three-chord event: I, then the CM, then I again. Using our example, the chords would be CEG - EG#B - CEG, and the analysis would be C: I - III (CM) - I. The purpose of this sonority is to embellish / prolong the tonic.
- Example 1: Tchaikovsky, Chant sans paroles from Souvenir de Haspal, Op. 2, No. 3 (1867), measures 62-64. There are two very clear chromatic mediants here: I - bVI (CM) - I - bVI (CM) - I. The bVI chords could also be analyzed as modal borrowing from F minor, but in this case because they are surrounded by the tonic chord, it is more appropriate to analyze them as chromatic mediants.
- Example 2: Debussy, Sarabande from Pour le piano (1901), measures 9-10.
- A doubly-chromatic mediant has both common tones altered, so that the two chords are no longer the same quality (for example, CEG and EbGbBb). A CM - Ebm - CM chord progression in C would be analyzed as C: I - biii (DCM) - I.
- Chromatic mediant relationships can be related to single chords, but they also are used in the key relationships within a movement and/or the key relationships between movements of a multi-movement work.
- For example, compare the use of sonata form in the first movement of Beethoven's 5th Symphony, and Brahms's 3rd Symphony.
- Beethoven 5: first theme in c minor, second theme in Eb Major. Eb is the diatonic mediant to c - that is, it is the mediant that is in the original key. The recapitulation is back in c minor, giving us the mediant motion of c - Eb - c (cmin: i - III - i).
- Brahms 3: first theme in F Major, second theme in A Major. A Major is a chromatic mediant to F Major - FAC and AC#E contain two common tones, but the C has been raised to C# so that both chords have the same quality. The recapitulation is back in F Major, giving us the chromatic mediant motion of F - A - F (FMaj: I - III [CM] - I).
- Also consider the key relationships between movements in Beethoven's 5th Symphony, Beethoven's 9th Symphony, and Dvorak's 8th Symphony:
- Beethoven 5: First movement in C minor, second movement in Eb Major (the relative major - a diatonic mediant).
- Beethoven 9: First two movements in D minor, third movement in Bb Major (a diatonic mediant)
- Dvorak 8: First movement in G Major, second movement in Eb Major (a chromatic mediant), which then modulates to C Major (another further chromatic mediant away from Eb).
- Sometimes in order to correctly diagnose a chromatic mediant (or any other harmony), you have to consider the different layers of music that are occurring simultaneously. For example, consider the C-Major chorale that begins in measure 17 of the finale to Saint-Saëns's "Organ" Symphony (Symphony No. 3, written in 1886).
- The score for this excerpt.
- A recording of this excerpt.
- Discussion:
- The chorale is a set of four pairs of measures. The first measure of each pair is in 9/4 time, and the second measure of each pair is in 6/4 time. Each of the 9/4 measures features a diatonic chord progression in chorale texture, with a cadence that ends in the following 6/4 measure where the last chord of the progression is sustained for the entire bar. The chords in the 6/4 measures are I (CMaj), V (GMaj), III (EMaj), I (CMaj).
- These four chords, despite being separated from each other by an entire chord progression in each 9/4 bar, are related to each other as chromatic mediants. The ear picks up on these particular chords because they are sustained for so long, and because they are the last chord of each two-bar phrase. The E Major chord in measure 22 is a chromatic mediant to the two tonic chords in measures 18 and 24.
- This chromatic mediant relationship (CMaj: I - III [CM] - I) is interrupted by the V chord in measure 20 - this is not unusual for a chromatic mediant. However, it is interesting to note that the E Major chord is ALSO a chromatic mediant to the V chord (EG#B and GBD)!
Review of Theory Fundamentals (Cadences, Non-Harmonic Tones)
- Diatonic Triads and Seventh Chords handout.
- Harmonic Function handout ("the solar system of harmony!").
- Cadences - A cadence is a musical punctuation mark, used to give a sense of flow and pacing to music. The strength of a punctuation mark (like a comma versus an exclamation point) communicates to the reader whether the idea is complete or if the writer is just pausing for emphasis. Just like a run-on sentence, music written without cadences can sound breathless, hurried and panicky. Consider these points about how cadences are used:
- A cadences only occurs at the end of a musical phrase.
- A cadence is an event involving the final two chords or harmonies of a phrase. Chord quality does not matter - only function. In the examples below, you will see Arabic numbers (1,2,3) instead of Roman numerals (I, ii, iii). This is to show that, for example, if a cadence uses the "Five" chord (dominant), it doesn't matter if it is Major V or minor v - only that it is "5."
- When phrases are paired together (into a musical "period"), the first (antecedent) phrase usually ends with a weak cadence, and the second (consequent) phrase usually ends with a strong cadence. When thinking about periodic structure in music, think of the antecedent as a question, and the consequent as the answer. The question does not make sense unless it is immediately followed by an answer. Using a weak cadence tells the listener that the answer is coming. Using written English as an example, which of these two statements makes the most sense?
- I woke up! And went to school,
- I woke up, and went to school.
In the first example, the presence of a strong punctuation mark in the middle of the statement disrupts the flow of the argument. The lack of a strong ending to the statement leaves the listener wondering what has been left out. The second example is correctly punctuated - the first clause is a complete statement in itself, but it is not the entire argument so it ends with a comma to tell the reader to expect more afterward. Like written language, music needs to have a meaningful balance of motion and rest for it to be understandable to the listener. Cadences are one important way for the composer to communicate clearly with the audience.
- In music of the Common Practice era (approximately 1600-1900), there are four main types of cadences. Music written before and after this period has additional types of cadences. Theory 1A concentrates on Common Practice music; the four primary cadence types are listed below with examples.
- Authentic Cadence. The most common and most obvious type of cadence; in an authentic cadence, the harmony moves from dominant-function (5 or 7) to tonic (1). There are two varieties:
- Perfect Authentic Cadence (PAC) - for an authentic cadence to be perfect, three things must be true. 1) the harmony moves from 5 to 1. 2) the bass voice leaps from Sol to Do (up or down). 3) The soprano voice ends on Do. If all three of these requirements are not met, the cadence is not a PAC.
- Imperfect Authentic Cadence (IAC) - the harmony moves either 5 to 1 or 7 to 1. No other requirements are needed.
- Example 1: the final phrase of the fourth movement of Felix Mendelssohn's Symphony No. 5, "Reformation," Op. 107 (1832) ends with a perfect authentic cadence.
- Example 2: at the beginning of the same movement, the third and fourth phrases are a period. The antecedent phrase ends with an imperfect authentic cadence, and the consequent phrase ends with a perfect authentic cadence.
- Plagal Cadence (PC). The plagal cadence moves from 4 (subdominant) to 1 (tonic). This cadence is commonly called the "Amen cadence" because it is often found at the end of Protestant hymns in Major keys, accompanying the word Amen: Ah (IV) --- men (I). In classical music, the plagal cadence is often in a minor key (iv-i).
- Example 1: in Mendelssohn's Reformation Symphony, the first movement ends with a plagal cadence in the key of d minor.
- Example 2: in Johannes Brahms's Symphony No. 4 in E Minor, Op. 98 (1885), the first movement also ends with a plagal cadence, in e minor.
- Half Cadence (HC). The half cadence ends on the dominant chord, 5. Any other chord may be used in the first half of the cadence. The half cadence is a weak gesture (a musical "comma"), and so it appears at the end of an antecedent phrase of a period. A lot of Common Practice music is built using periods that arch upward to the dominant (antecedent, ending with HC) and then fall back to the tonic (consequent, ending with PAC or IAC).
- Example 1: The very first phrase of Beethoven's Symphony No. 5 in C Minor, Op. 67 (1808) ends on a half cadence.
- Example 2: The first two phrases of the Minuet (third movement) from Haydn's Symphony No. 6 in D Major, "Le matin," Hob. 1/6 (1761) are a contrasting period. The antecedent phrase ends with a half cadence, and the consequent phrase ends with a perfect authentic cadence (V-I harmony, the flute ends on Do, and the bass leaps Sol-Do).
- Deceptive Cadence (DC). In general terms, a deceptive cadence is one that moves from 5 to any chord other than 1 - our ears are programmed to expect to hear the dominant chord move to the tonic chord, and any other destination sounds misleading. However, in common usage the deceptive cadence is specifically when the harmony moves from 5 to 6 (usually V-vi in a Major key). This works best by having both chords in root position, and by resolving the leading tone up to tonic as it should, but then moving the bass up by step from Sol to La instead of leaping from Sol to Do.
- Example 1: The minuet from Haydn's Symphony No. 6 also has many deceptive cadences. In this particular example, after the deceptive cadence the phrase is then extended with an additional authentic cadence, but this is not a requirement.
- More on periods:
- A Parallel Period is one where the antecedent and consequent phrases start the same way. The consequent phrase can have a different ending - only the beginning of the phrase has to be the same as the antecedent. For example, the childrens' song "Mary Had a Little Lamb" has two phrases that both start with the same melody, forming a parallel period.
- Phrase One: "Mary had a little lamb, little lamb little lamb" (Mi Re Do Re Mi Mi Mi, Re Re Re, Mi Sol Sol)
- Phrase Two: "Mary had a little lamb whose fleece was white as snow" (Mi Re Do Re Mi Mi Mi Mi Re Re Mi Re Do).
- A Contrasting Period is one where the consequent phrase starts differently than the antecedent phrase.
- The antecedent phrase should end with a weak cadence: IAC, HC, or DC.
- The consequent phrase should end with a strong cadence: PAC, PC (or IAC - less common).
- Non-Harmonic Tones (NHT) (also called non-chord tones) - these notes are not part of the harmony. They are added for melodic embellishment, to make the music more intricate and interesting, or to smooth out music with a lot of leaps. When analyzing music, first figure out the chord being played, and then circle any notes that do not fit into the chord. These are the NHT. In classical music there are nine types of NHT; many of them are still used in today's compositions and in popular music as well. NHT may be accented, occuring on the beat, or unaccented, occuring off the beat.
- Passing Tone (PT): approached by step, resolved by step in the same direction.
- Neighboring Tone (NT): approached by step, resolved by step in the opposite direction (returns to starting note).
- A neighboring tone may be above the chord note (upper neighbor) or below it (lower neighbor).
- Changing Tones (CT): Also called a Cambiata, this is a pair of notes: one upper neighbor tone and one lower neighbor tone.
- For example, think about the beginning of the song "Some Enchanted Evening" from the musical South Pacific. In this example the melody is Do-Ti-Re-Do Do Sol. The harmony during this tune is I (Do-Mi-Sol). The Ti and Re in the melody would be analyzed together as CT.
- Appoggiatura (APP): approached by leap, resolved by step (usually in the opposite direction). The appogiatura is usually accented (occurs on the beat).
- The "formula" for an appogiatura is to leap up then step down. This does not always have to be the case, however. In the song "Maria" from Leonard Bernstein's West Side Story, at the start of the chorus the "Ri" in the word "Maria" is an appogiatura on the downbeat. It leaps up by an A4, then resolves up by a m2 (Ma-ri-a = Do-Fi-Sol)
- Escape Tone (ET): the opposite of an appogiatura: the escape tone is approached by step and resolved by leap. Remember - to "escape" a burning building, "step" to the window, and "leap" out.
- Suspension (SUS): approached by sustaining a note from the previous chord, then resolved downward by step.
- The suspension is a three-part event:
- In the first chord, the note fits into the chord.
- In the next chord, the harmony changes, but one voice retains a note from the previous chord. This is the "suspended" note.
- In the final chord, the suspended note resolves down by step, once again fitting into the harmony.
- There are four types of suspension, measured by the interval distance of the suspended note above the lowest chord note. The four types are 9-8, 7-6, 4-3, and 2-3 (bass suspension). For example, in a 4-3 suspension, the suspended note is a fourth above the bass, and it resolves to a third above the bass.
- Retardation (RET): the physical opposite of a suspension. It works in exactly the same way, except that the resolution is up by step instead of down.
- Anticipation (ANT): the temporal opposite of a suspension. In an ANT, one voice resolves early to the next chord. This often happens at the final cadence of a chorale.
- Phase one: all voices sing the first chord.
- Phase two: one voice moves early (often on an eighth note) to a note from the next chord.
- Phase three: all remaining voices move to the next chord.
- Example: The first phrase of the last movement chorale from J.S. Bach's Cantata No. 140 "Wachet Auf, Ruft uns die Stimme" ends with an anticipation in the Alto voice.
- Pedal Tone (PED): Also called a pedal point, this is a long-term event, not a single note. Pedal points often occur in the bass, where they used to be played on the pedals of the pipe organ (hence the name "pedal tone"). One voice, usually the bass, sustains a note for a very long time. This note is sometimes part of the harmony, and sometimes not. Pedal points are often on the dominant or tonic pitches of the key. The purpose of a pedal point is to draw the listener's attention to the dominant (after a period of harmonic uncertainty), to prepare them for the final resolution back to tonic. Examples of pedal points:
- Beethoven, Symphony No. 5 in C Minor, third movement: there is a tonic pedal in the timpani for the last minute of the movement, which helps link the third movement directly into the fourth movement.
- Bach, Prelude in C Major from The Well-Tempered Clavier - in this excerpt from the end of the piece, the bassline starts out with "Do - Fa - Fi - Le" before stabilizing on Sol. The use of Fi and Le emphasizes the eventual resolution to the dominant, by approaching Sol by half step from both above and below. Once the bassline arrives on Sol (the pitch G, in this case), it becomes a dominant pedal for eight measures. The constant repetition of the dominant pitch (sometimes part of the harmony, sometimes not) builds tension and intensity into the piece. This tension is finally released in the ninth bar when the bass falls from Sol to Do at the beginning of the coda. The piece then ends on a C Major chord.
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