IV. Diatonic Harmony, Tonicization, and Modulation

John Peterson

Key Takeaways

  • This chapter provides an introduction to the harmony section of Open Music Theory, then begins a discussion of how to create a sense of ending in a . We focus exclusively on using I and V for now.
  • Two larger concepts inform the way we present harmony here: and the .
  • Endings are often marked by , of which there are two primary types: and .
    • Authentic cadences involve the progression V–I. They are perfect when both harmonies are in root position and do [latex](\hat{1})[/latex] is in the soprano over tonic. If either of these conditions is not met, the authentic cadence is imperfect.
    • Half cadences involve the progression x–V, where “x” is any of a variety of harmonies.

Chapter Playlist

Introduction to Harmony

In this section, we’ll focus on how composers of common-practice Western classical music use harmony in a . Harmony is one important component (among others) of creating a phrase’s sense of forward motion toward a goal. As we study harmony, it’s important to keep two larger related concepts in mind: harmonic function and the phrase model.

 refers to three categories of chords:

  1. : Chords that sound stable, providing a sense of home or center. In Western classical music, the only chord that belongs to this category is I (in minor: i).
  2. : Chords that transition away from tonic function toward dominant function. This category can be split into two groups:
    1. Strong predominants signal that a dominant function chord is imminent: IV and ii (in minor: iv and iio).
    2. Weak predominants transition away from tonic, typically moving to a stronger predominant: iii and vi (in minor: VII, III, and VI).
  3. : Chords that provide a sense of urgency to resolve toward the tonic chord: V and viio (the same in minor).

The  refers to the typical order and flow of harmonic functions in a phrase. The principle of the phrase model is that a phrase needs at least tonic function and dominant function harmonies in order to exhibit a sense of trajectory. More often, phrases also include a predominant harmony to heighten the sense of trajectory toward closure. Phrases almost always progress from left to right in the phrase model, not from right to left, although we’ll discuss some exceptions later. This is summarized in Example 1.

diagram
Example 1. The phrase model: (a) the minimum functions needed to create forward motion, (b) more common functional layout for a phrase (the final tonic is optional), (c) harmonic functions do not progress right to left.

Introduction to Cadences

One way to create a sense of closure at the end of a is through —melodic and harmonic patterns that create goals, kind of like punctuation marks in literary sentences. They’re important not only because they can help you determine phrase endings, but also because they help establish a key. As a result, listening for potential cadence points is typically a good first step in analysis.

Example 2 shows two phrases and the two main categories of cadence: (1) inconclusive and (2) conclusive. The first phrase ends in m. 4 with a (HC) that sounds inconclusive. The second phrase begins in m. 5 and ends in m. 8 with an that sounds conclusive. The label marks this as a particular kind of authentic cadence called a perfect authentic cadence, discussed below.

Example 2. Two cadences in Joseph Boulogne, Chevalier de Saint-George’s “Ballet No. 6” from L’amant anonyme, Act II (0:00–0:07).

Authentic Cadences (they sound conclusive!)

An occurs when the harmonic progression V–I (or V–i in minor) marks the end of a phrase. There are two kinds of authentic cadence:

  • A occurs when both of the following conditions are met (Example 2, m. 8):
    • Do [latex](\hat{1})[/latex] is in the soprano over the tonic chord
    • Both V and I are in root position
  • An occurs if either of the above two conditions isn’t met, but V–I is still involved (Example 3). For instance, an IAC can occur when:
    • Mi (me in minor) [latex](\hat{3})[/latex] is in the soprano over the tonic chord (as in Example 3), or
    • V, I, or both harmonies are inverted

Example 3. An imperfect authentic cadence (IAC) in Fanny Hensel, “Ferne” Op. 9, No. 2 (0:00–0:14).

Half Cadences (they sound inconclusive!)

A (HC) occurs when a phrase ends on V (Example 2, m. 4). A variety of chords can precede V, so we often refer to the harmonic progression that marks HCs as “x–V.”

For now, we’ll restrict our vocabulary to only I and V chords (or i and V in minor) so we can learn some basic techniques of voice-leading.

Cadential Strength and the IAC

PACs are the strongest kind of cadence available to a composer because of the sense of finality they can create, and HCs are the weakest kind of cadence because of their unfinished sound. IACs are special because they occupy the space between HCs and PACs in terms of cadential strength (Example 4).
Diagram

Example 4. Spectrum of cadential strength.

Since there are many ways to compose an IAC (a composer can theoretically use various combinations of inverted chords and scale degrees in the melody), a composer can choose to make an IAC more or less strong.

The cadence in Example 3 above is relatively strong: it uses root position V–i with me [latex](\hat{3})[/latex] in the soprano, it comes at the end of a sentence in the lyrics, it uses a half note, it’s followed by a rest, and the music that follows sounds like a new phrase.

The potential IAC in Example 5 is comparably weaker: it uses V6–I, the moment goes by relatively quickly, and it’s followed by material that could easily be understood as continuing a phrase that’s under way or as referencing the beginning of a phrase that has just concluded, something that would strengthen reading m. 5 as an IAC. Performers may disagree on whether Example 5 contains an IAC or not, and they may adjust their playing accordingly.

Example 5. A potential IAC in Schubert’s Piano Sonata D. 845, III.

An additional complication is that composers often avoid creating a PAC using progressions that fit the description of an IAC, but that don’t mark the end of a phrase. In Example 6, m. 8 contains a potential IAC, but when m. 9 begins to repeat the material from m. 5 (compare the bass voice in m. 9 with the soprano in m. 5) it sounds like the passage is trying the run-up to the cadence again to achieve something stronger than the potential IAC at m. 8. We indicate that the cadence in m. 8 is by crossing the label out on the score.

Example 6. Avoiding a cadence in Beethoven’s Piano Sonata Op. 2, no. 3, I, mm. 1–13

With so many possibilities, how can we determine what counts as a true IAC? The next section (“Hearing Cadences”) offers some help.

Hearing Cadences

While there is certainly a degree of intuition involved with hearing cadences, it’s also a skill that can be honed over time. We recommend listening for cadences following this process:

Summary: steps for hearing cadences

  • Mark potential cadence points
  • Analyze the harmonies involved
  • Listen for a sense of beginning after each potential cadence point
  • At first, listen and mark potential points of rest, goal, or closure.
    • It’s common for students to initially mark too many cadences. One rule of thumb is that unless an excerpt’s tempo is quite slow, it’s not common for phrases to be two measures long.
  • Next, check the harmonies involved in these potential cadence points:
    • x–V = potential HC
    • V–I = potential AC
      • If do [latex](\hat{1})[/latex] is in the soprano over I and both harmonies are in root position, it’s a potential PAC.
      • If something else like mi [latex](\hat{3})[/latex] or sol [latex](\hat{5})[/latex] is in the soprano over I, it’s a potential IAC.
      • If V, I, or both harmonies are inverted, it’s a potential IAC.
    • If it doesn’t involve one of the above two progressions, then it’s not a potential cadence. (Note that this doesn’t mean it doesn’t represent some kind of ending. It just means it’s not a cadence.[1])
  • Finally, listen for what happens after each potential cadence point.
    • Since true cadences mark the end of a phrase, it’s very common for cadences to be followed by a sense of . This “beginning” may be a repetition of the beginning of the phrase that just ended, or it may be new material that starts a new phrase. Hearing a sense of beginning following a cadence point is a great way to help verify that what you’ve marked as a potential cadence is indeed a true cadence point.
    • If, instead, you hear repetition of material from the middle of the previous phrase, your potential cadence may not be a true cadence. Instead, it likely represents a potential cadence point that has been .

Writing Authentic Cadences (with triads only)

The steps for writing a PAC or IAC are summarized in the box below and illustrated in Examples 7–10.

Writing a PAC or IAC

  1. Determine the key
  2. Write the entire bass: soldo [latex](\hat5-\hat1)[/latex]
  3. Write the entire soprano:
    1. PAC: redo or tido [latex](\hat2-\hat1[/latex] or [latex]\hat7-\hat1)[/latex]
    2. IAC: remi or solsol [latex](\hat2-\hat3[/latex] or [latex]\hat5-\hat5)[/latex]
  4. Fill in the inner voices by asking:
    1. What notes do I already have in the bass and soprano?
    2. What notes do I need to complete the chord?
    3. What note will I double? (Remember, in root position chords, it’s common to double the bass.)
  5. If you’re writing in minor, remember that you need to use the leading-tone, ti [latex](\uparrow\hat7)[/latex], in your V chord (making it a major triad, not minor) to give it momentum toward the tonic.


Example 7. Writing a PAC in a major key.

Example 8. Writing a PAC in a minor key.

Example 9. Writing an IAC in a major key.

Example 10. Writing an IAC in a minor key.

Writing Half Cadences (using I and V only)

The steps for writing a HC are summarized in the box below and illustrated in Examples 11 and 12.

Writing a HC:

  1. Determine the key
  2. Write the entire bass: dosol [latex](\hat1-\hat5)[/latex] (note: for now, we’ll use only I and V chords, although we’ll see later that other chords more commonly precede V)
  3. Write the entire soprano: doti or mi–re [latex](\hat1-\hat7[/latex] or [latex]\hat3-\hat2)[/latex] (note: sol–sol [ [latex]\hat5-\hat5[/latex] ] is possible, but not common)
  4. Fill in the inner voices by asking:
    1. What notes do I already have?
    2. What notes do I need to complete the chord?
    3. What note will I double? (Remember, in root position chords, it’s common to double the bass.)
  5. If you’re writing in minor, remember to use the leading-tone, ti [latex](\uparrow\hat{7})[/latex], in your V chord, making it a major triad, not minor.

Example 11. Writing a HC in a major key.

Example 12. Writing a HC in a minor key.

Assignments
  1. Introduction to harmony, cadences, and phrase endings (.pdf, .docx). Asks students to write and identify cadences using only I (or i) and V chords in major and minor.

Media Attributions

  • Phrase_Model
  • Example_3_Cadence_Strength_Spectrum

  1. Sometimes, for example, the melody suggests an ending, but the harmony doesn't participate. Or, sometimes the harmony suggests an end, but the melody refuses to close. Cadences are special goal points because they represent places where both the melody and harmony agree on a phrase ending.

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