© Jan. 2017 by Michael Adduci, updated 11/13/2019.

Music Theory II

Handouts
  • Part-Writing handouts (review)

    • Guidelines for Bach-style part-writing.

    • My key to the abbreviations I use when correcting your part writing exercises.

  • Flashcards you can use to build your own practice questions for secondary dominants and leading tones. Print them out and separate into three piles as indicated. Draw one card from each pile to create a question (e.g., "what is V6 of ii in D Major?"), then answer it!

  • A diagram of the typical forms used in the four movements of a symphony.

  • Modulation handouts

    • Pivot Modulation examples, Set One (piano exercises)

    • Pivot Modulation examples, Set Two (chorale phrases)

    • Bach-style chorales illustrating pivot modulation. Note, in these "chorales" each phrase pivots to a new key. These modulations are way more frequent than would occur in real music - these examples are only intended for practice at analysis.

Theory II Topic List - click a link to scroll down to that section of the listening examples!

  1. Secondary dominants / leading tones

  2. Pivot chord modulation

  3. Modal mixture

  4. Altered predominants

Exercises
Concepts and Listening Examples
 

Topic 1: Secondary Dominants

  • In any key, the V chord is called the "dominant" and the I chord is the "tonic." We can say that V is the dominant of I - its entire purpose in life is to move to the tonic. When we begin to introduce chromaticism into our music, we discover that any chord can have a dominant of its own. The tonic (I) has a dominant (V); the supertonic (ii) can have its own dominant as well, analyzed as "five of two" - V / ii. We call this a secondary dominant - secondary, because it is not the primary dominant of the key we are in (it is sometime called an applied dominant instead). Now, why on Earth would anyone do this?

    • By giving a non-tonic chord its own dominant, the resolution of that dominant makes the non-tonic chord sound like I, just briefly. This is called tonicization (verb: to tonicize).

    • If you take a functional chord progression like vi - ii - V7 - I (in C Major: Am - Dm - GMm - CM), and tonicize the two chord, your progession becomes vi - V7/ii - ii - V7 - I (C: Am - AMm - Dm - GMm - CM). Play this on the piano - you will hear that the new progression is more ambiguous - which is the real tonic? It also sounds really good, because the strong root movement down by P5 sounds very satisfying to our common-practice ears.

    • This is the most elementary form of functional chromaticism (that is, non-diatonic notes that have a harmonic purpose, not just passing tones). Once we give ourselves permission to add chromatic elements to the music, we start making the tonic less stable.... and this leads us to the process of actually changing from one key to another within a piece of music (see Modulation, below).

  • A secondary dominant, therefore, is a dominant-qualtiy chord (that is, M or Mm), that has a root that is a P5 above the root of a regular diatonic chord in the key we are considering. Here are the rules for its usage:

    • The secondary dominant will require one or more accidentals - it is a chromatic chord (out of the key)

    • Just like the regular dominant lives to move to the tonic, a secondary dominant also lives to move to its tonic. The strongest presentation of a secondary dominant is for it to immediately resolve to its "implied" tonic. So, V /V must move directly to V - down by P5. Not doing this can be interesting (we will cover delayed/deceptive chord resolution in Theory 3), but keep in mind that failing to resolve correctly weakens the secondary dominant. If you weaken it too much, you have to ask yourself why you used it in the first place. For our purposes in Theory 2, we will assume that all secondary dominants must resolve immediately to their tonic.

    • The secondary dominant's quality must be M or Mm. Its implied tonic must be a diatonic chord that is M or m only (A and d chords do not have a strong enough root to support the weight of their own dominant).

    • Using our example above, V7/ii to ii in C Major: AMm is always the dominant of D, no matter what key we are in. The dominant-tonic relationship is preserved. So, we can use these two chords in any key that contains D. Here is how it works:

      • In C Major, D is ii, so AMm - Dm is V7/ii - ii.

      • In D Major, D is I, so AMm - DM is just regular old V7 - I (yawn)

      • In E minor, D is VII, so AMm - DM is V7/VII to VII.

      • In F Major, D is vi, so AMm - Dm is V7/vi to vi.

      • In Eb Major, D is viio, so AMm doesn't work as a secondary dominant in that key - viio is a diminished chord, so you can't tonicize it.

      • In E Major, there isn't a D! That means AMm doesn't work as a secondary dominant in that key either, because its implied tonic is not a diatonic chord. Remember: you can only tonicize diatonic chords that have M or m quality.

      • And so on. Notice that the quality of the D chord changes depending on what key we are in (that is fine as long as it is either Major or minor), while the secondary dominant (AMm) always stays the same.

  • Example 1: The coda (last 7 measures) of the slow movement from Marcello's Concerto for Oboe in C Minor (1717) contains one secondary dominant: V7/iv (CMm), resolving immediately to iv (Fm).

Concerto for Oboe in C Minor, II - Marcello
00:00 / 00:00

  • Example 2: This short excerpt from Beethoven's Symphony No. 1 in C Major, Op. 21 (1801) demonstrates the power of tonicization, the technique of briefly making a chord other than the tonic of the key sound like "I". In the introduction to the first movement, there are three tonicizations in the first four bars, none of which are in the home key of C Major. The next eight bars are a prolongation of G, the dominant of C - by alternating between GMm and CM 6/4 (tonic six-four) the tension of the dominant is maintained without ever being resolved. The first true "tonic" C Major chord appears in bar 1 of the Allegro, measure 13 of the piece.

    • Score (includes harmonic analysis)

Symphony No. 1, I - Beethoven
00:00 / 00:00
  • We can also extend this concept of "secondary harmony" and "tonicization" to include the leading tone, not just the dominant. A leading tone is a note that moves up by m2 to tonicize the next note: Ti brings us back to Do. We know this innately - try playing a Major scale on the piano, but stop on the leading tone - Do Re Mi Fa Sol La Ti...... how long can you stand to wait until you feel compelled to finish the scale? That is the power of the leading tone. In fact, the dominant triad also contains Ti (the V chord is Sol-Ti-Re), which is one reason why the dominant has such a strong pull towards the tonic.

    • Just like the tonic chord (I) has its leading tone chord (viio), other diatonic chords can have their own leading tone - a secondary leading tone.  It follows all of the rules that secondary dominants do (see above). The only difference is that it is a diminished chord (d, dm or dd), and so it resolves up by m2 to its implied tonic (the secondary dominant resolves down by P5).

      • In C Major, C is I, so Bd - CM is viio - I.

      • In F Major, C is V, so Bd - CM is viio/V - V.

      • In G minor, C is iv, so Bd - CM is viio/iv - iv.

      • In Db Major, C is viio, so Bd doesn't work as a secondary leading tone in that key - viio is a diminished chord, so you can't tonicize it.

      • In D Major, C doesn't exist, so Bd - C would never happen. You can only tonicize diatonic chords that are M or m. 

 

Topic 2: Pivot Chord Modulation

  • Also known as common chord modulation, 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.

Example 1: modulation from F Major to d minor.

Pivot Modulation - Example 1
00:00 / 00:00

Example 2: modulation from F Major to B-flat Major.

 

Pivot Modulation - Example 2
00:00 / 00:00
  • Changing key (modulating) comes with a couple of requirements. Music does not just move randomly from one key to another.

    • The general rule is: 1) establish the old key; 2) change the key; 3) confirm the new key.

      • A good way to establish a key is to have a V - I motion. This does not require a cadence - just a few chords that set Do in our ear.

    • A modulation will be foreshadowed by the presence of the same accidentals multiple times - usually Ti in the new key (for sharps) or Fa in the new key (for flats) for a simple modulation.

    • After you pivot to the new key, you must provide a cadence in the new key. That means you have to go all the way to the end of the phrase before changing keys again. All modulations must be confirmed by a cadence! Music does not change into one key for a bar, then into another key for a bar, and then again, without ever pausing. If you modulate, you must have a cadence to "prove" it. And, cadences only occur at the ends of phrases.

      • See the two examples above - the music starts with enough chords to establish the original key (in example two, there is an entire phrase in the original key), then a pivot chord, and then a cadence in the new key at the end of the phrase.

      • In example 3 below, the first phrase is in A Major; the second phrase pivots to F# minor, ending with a cadence in that key; the third phrase pivots back to A Major, ending with a cadence to confirm the modulation. It is very common to modulate away from the starting key, and then later modulate back (rather than moving on to another new key).

        • Example 3: J. S. Bach, Chorale "Du Friedensfürst, Herr Jesu Christ,"  BWV 116 (1724)

          • Score (includes harmonic analysis)

Du Friedensfürst, Herr Jesu Christ - Bach
00:00 / 00:00
  • There may be more than one possible pivot chord! Choose one that makes the most sense. Also, the pivot chord is typically a pre-dominant chord, but when in doubt I like to pivot as early as possible. Remember that the mechanics of labeling the modulation will occur earlier in the music than the point where your ear perceives the key change has started.

  • In many cases, because closely-related keys are being used, there are not many accidentals and it could be possible to analyze the entire selection in one key. However, if you observe carefully you will notice small (or large) errors in reasoning when doing this. Perhaps you might see a non-functional chord progression, like a viio chord that does not move to I as it should, or you may see a high frequency of chords that should be rare, such as iii. Maybe an incorrect cadence occurs. Use your instincts and if something doesn't feel right to you, it probably isn't. This does assume that you are paying attention, however! In analysis, as in all other activities in life, attention to detail is extremely important.

    • In example 1 above, it is technically possible to analyze the entire excerpt in F Major with no modulation. However, this gives a final cadence of V/vi - vi, which is not a deceptive cadence (V - vi) or any other type. Therefore, the chorale must not be in F Major at this point. Additionally in F Major beats 1-3 of measure 3 would show V/vi - IV, which is an incorrect resolution of the secondary dominant, so the chorale can't be in F Major at that point, either.

    • In example 2 above, analyzing the entire chorale in F Major creates two significant problems. First, in the third measure there is a Cmm chord. In F Major this would be minor v7, borrowed from the parallel minor. Technically possible, but in practice this would never happen on V in a Major key, and if it did happen it would occur somewhere very important, such as in a half cadence, and not in the middle of the phrase on a weak beat. Second, in F Major the final cadence would be V7/IV - IV, which is not a cadence.

      • Now, a mini rant on the importance of being observant: the music student faced with analyzing the above chorale may not even notice the trouble they are in. This is because they probably would have analyzed measure 4 beat 2 as a I7 chord since they forgot that I7 is a Major-Major 7th chord while the harmony shown is Major-minor (hence, V7/IV). They would then go on to construct their final cadence as I - IV and label it as a plagal cadence, completely unaware that not only is that not a plagal cadence, but the real issue is that the chorale isn't even in the original key anymore. They also missed the final warning sign: measure 4 taken as a whole is a standard tonic-6/4 - dominant 7 - tonic formula, a cliche in this style, and the formula only works in B-flat Major, not in F Major.

      • On a seemingly-unrelated side note, the famous Simpsons "monorail" episode ends with an "escalator to nowhere." Citizens of Springfield blithely ride the escalator several hundred feet into the air, only to be surprised when it drops them onto nothing and they fall to the ground. Why were they even riding it in the first place? It doesn't go anywhere! How could they spend several minutes riding the escalator and never look around and think, "something's not right here, maybe I should get off." How does this apply to the music student? Pay attention to detail, and do your homework when you are not being distracted by something else; don't overlook the obvious.

  • In more advanced usage, chromatic pivot chords are allowed: the pivot chord does not have to be diatonic, as long as is functional in both keys (for example, V/V could be a pivot chord). This will be addressed in MUSC 3A - for Theory 2, assume that all pivot modulations are between closely related keys only, using only pivot chords that are diatonic in both keys.

Further examples of pivot chord modulation:

  • Example 4: Bach, Minuet in G Major, BWV Anh. 114 (1725), mm. 9-26.

    • Score. The recording begins in measure 9; the modulation is in bar 19 (notice the repeated C#s in mm. 20-23), with a cadence in bar 23. In this example, the music immediately pivots back to the starting key in the next phrase; this is not required.

 

  • Example 6: Bach, Polonaise from Orchestral Suite No. 2 in B Minor, BWV 1067 (c. 1738), mm. 1-10.

 

Minuet in G Major, BWV Anh 114 - Bach
00:00 / 00:00
Symphony No. 6 in D Major, II - Haydn
00:00 / 00:00
Polonaise from BWV 1067 - Bach
00:00 / 00:00
Sonatina in G Major, Op. 36, No. 2 - Clementi
00:00 / 00:00

Topic 3: Modal Mixture (also known as borrowed chords / modal borrowing)

  • The concept of "modal mixture" means that composers are writing their music to emphasize a particular note as "tonic" rather than trying to establish a sense of Major or minor tonality.

  • Philosophically, this means that a particular composer (in particular, Dvorák, Brahms and Schubert are noted for their use of modal mixture) 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 the tonalities of both G Major and g minor. All of these chords are considered "diatonic" to the tonal center, G. In simple terms, there is now less of a distinction between Major and minor - there is just Tonic. Few composers take this technique to its philosophical extreme - most music remains clearly in either Major or minor despite use of modal mixture to blur those distinctions.

    • Please note that modal mixture only applies to a Major key and its parallel minor, or to a minor key and its parallel Major; not to any other "modes" of a particular tonic. Don't get confused by different uses of the word "modal."

  • In its simplest form modal mixture is a single borrowed chord. In most cases, the music is in a major key, and a chord is borrowed from the parallel minor to embellish the Major tonality. Remember that in a Major key there are only 7 diatonic chords, while in a minor key there are 13. Composers began "importing" (borrowing) the more-rich sonorities from minor to spice up their Major-tonality music. The minor mode is already contains most of the harmonies from the parallel Major key, so borrowing chords when in a minor key is less common.

     

    • The most common borrowed chords are diminished, such as iio and vio from the parallel minor. The minor iv and v chords are also popular borrowed chords.

    • Another commonly borrowed chord is the "Picardy third," a borrowed I chord at the very end of a piece written in minor. Major I can be borrowed at any point in the music, but it is only called a "Picardy third" when it is the final chord of the piece.

      • Example One: Marcello, Oboe Concerto in C Minor (1717), movement 3 - ends with a Picardy third

      • Example Two: J.S. Bach, Sonata in G Minor, BWV 1029, movement 1 - NO Picardy Third

 
Concerto for Oboe in C Minor, III - Marcello
00:00 / 00:00
Sonata in G Minor, I - Bach
00:00 / 00:00

  • In more advanced usage modal mixture 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.
       

  • ANALYSIS: When only a few chords are borrowed, analyze them with the correct Roman numeral for their quality and function. Then add (BC) to show that the chord has been borrowed from the parallel key. If an extended section is involved (as in the Dvořák example below), write the name of the new key and analyze in that key rather than writing (BC) on every chord.

    • REMINDER: Chords that already exist in a minor key because of the harmonic/melodic minor scale forms are not considered to be chords of modal mixture. Therefore, they do not need to be analyzed with the (BC) indication. For example, while the Major V chord in a minor key does exist in the parallel major, the chord is primarily created by raising the leading tone (harmonic minor), not through modal borrowing.

 

Further examples of modal mixture:

  • Example Three: Beethoven, Symphony No. 9 in D Minor, Op. 125, movement four (1824). At this point in the movement Beethoven has just concluded a long transitional section/fugato in the key of b minor. At the end of the transition there is a dominant pedal (F#) accompanying a B Major chord that Beethoven then restates as a b minor chord, before continuing on to D Major for the next section.

    • Analysis: all in b minor, with the B Major chord labelled I (BC).

  • Example Four: Brahms, Symphony No. 2 in D Major, Op. 73 (1887), movement three. 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).

    • Score (the recording starts right after the fermata in the middle of page 81)

  • Example Five: Schubert, "Die liebe Farbe" - No. 16 from Die schöne Müllerin, Op. 25, D. 795 (1823). This strophic song uses two modal inflections to color its minor key: the Major I (BC), and the minor v (making the piece sound Aeolian). Note, the score is in b minor but the recording has been transposed into a minor.

  • Example Six: Dvořák, Symphony No. 8 in G Major, Op. 88 (1889), movement two. This movement begins in E-flat major, but moves frequently between E-flat Major, c minor and C Major.

    • Dvořák's modal mixture takes a long time to develop - you need to hear the whole movement to truly appreciate it. Both the score and a recording are available at IMSLP. It is a beautiful movement (and a great piece) - well worth your time.

    • As you listen to the movement, 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.

Symphony No. 9 in D Minor, IV - Beethoven
00:00 / 00:00
Symphony No .2 in D Major, III - Brahms
00:00 / 00:00
Die liebe Farbe - Schubert
00:00 / 00:00

Topic 4: Altered Pre-Dominants (Neapolitan and Augmented-Sixth Chords)

  • The Neapolitan Chord

    • An altered version of the diatonic supertonic chord, the Neapolitan is a Major triad with the lowered supertonic as its root ("flat two"). For example, in the key of C Major, the Neapolitan is a D-flat Major triad. The Neapolitan usually appears in first inversion. This chord is most common in minor keys, but it works in Major as well.

    • It is analyzed as "N", not as "II" or "bII", and if it is in first inversion, the complete analysis is N6 ("Neapolitan Six").

    • As it is a pre-dominant chord, the Neapolitan usually resolves to V and then to I: the bass resolves up by step from Fa to Sol, and the lowered supertonic (Ra) moves to the leading tone (Ti) in the V chord, and then to tonic (Do) in the I chord.

      • Voice leading: the Ra - Ti - Do motion described above most often occurs within one voice - that is the strongest presentation of the Neapolitan chord. Not keeping these tendency notes together weakens the chord; some composers choose to do this deliberately.

 

Neapolitan examples:

  • Beethoven, Piano Sonata No. 14 in C-Sharp Minor, Op. 27 No. 2, "Moonlight" (1801), first movement.

    • The chord progression here is i - i4/2 - VI - N6 - [V7 - i6/4 - V7] - i (IAC)

    • Score

  • Elfman, "Batman Theme" from the film Batman (1989).

"Moonlight" Sonata, I - Beethoven
00:00 / 00:00
Batman Theme - Elfman
00:00 / 00:00
Overture to "Estrella de Soria" - Berwald
00:00 / 00:00
  • The Augmented-Sixth Chord

    • This is an altered version of the diatonic subdominant chord. The chord contains the interval of an Augmented Sixth (Aug6), where it gets its name. As a pre-dominant chord, the Aug6 chord typically resolves to V.

    • The Aug6 chord appears in three different varieties, which have been given nationalistic names: the Italian chord, the French chord, and the German chord. These names do not imply country of origin or sole ownership of these chords.

    • When analyzing an Aug6 chord, label it by type: It (Italian), Fr (French), or Ger (German). Avoid using a generic "Aug6" label.

    • All forms of the Aug6 chord contain three specific scale degrees: tonic (scale degree 1, or "Do"), lowered submediant (b6, or "Le"), and raised subdominant (#4, or "Fi"). The lowered submediant pitch is usually in the bass, giving the Aug6 interval between b6 and #4. Note that in a minor key, the submediant is already "flat" compared to the parallel major, and does not need to be altered further. In the keys of C Major and c minor, the Aug6 chord is spelled F#-Ab-C, with Ab in the bass.

    • In addition to the three standard notes, each form of the Aug6 chord also contains one additional "color" note:

      • The German chord adds the lowered mediant pitch (b3, "Me" not "Mi"). In C Major: F#, Ab, C, Eb

      • The French chord adds the supertonic pitch (2, "Re"). In C Major: F#, Ab, C, D

      • The Italian chord doubles the tonic pitch (Do). In C Major: F#, Ab, C, C

    • Voice Leading: Le and Fi both resolve outward to Sol. To avoid parallel fifths ("Mozart" fifths), the German chord typically resolves to I6/4, and then to V. The French and Italian chords both resolve directly to V.

Augmented-sixth examples:

  • Italian: Beethoven, Symphony No. 5 in C Minor, Op. 67 (1808), movement one.

    • Chord progression: i  ||  V  ||  i - V - i - V - i - V - i - It - V (HC)

    • Score

  • French: Beethoven, Piano Sonata No. 4 in E-flat Major, Op. 7 (1797), movement two.

    • This movement is in C Major. The recording is of measures 72-78; the French chord appears twice during this section.

    • Score

    • Harmonic analysis

  • German: Michael Haydn, Requiem in C Minor (1771), Kyrie.

    • This is a standard presentation of the German chord: it resolves to i6/4 and then moves to V for an authentic cadence on I (BC).

    • Score - shown below:

Symphony No. 5 in C minor, I - Beethoven
00:00 / 00:00
Piano Sonata No. 4, II - Beethoven
00:00 / 00:00
Requiem in C Minor, Kyrie - M. Haydn
00:00 / 00:00