Chapter 14 Exercises: Harmony & Counterpoint — When Physics Meets Composition

Part A: Consonance, Dissonance, and Frequency Ratios

1. The interval of a perfect fifth has a frequency ratio of approximately 3:2. The interval of a major third has a ratio of approximately 5:4. Without using formulas, explain in your own words why the perfect fifth tends to sound more "stable" or "consonant" than the major third. What does the ratio tell you about how the overtone series of the two notes will interact?

2. Helmholtz proposed that dissonance arises from beating between overtones. Consider two notes a semitone apart (e.g., C and C#, with approximate frequencies 261 Hz and 277 Hz). Describe in plain language what "beating" means physically, estimate the beat rate between these fundamentals, and explain why this produces the characteristic roughness of a minor second.

3. The history of "dissonance" is a history of changing definitions. In medieval polyphony, the third (C-E) was classified as a dissonance requiring resolution. By the Renaissance it had become a standard consonance. Today, jazz musicians treat the minor seventh as stable home-base harmony. Construct an argument that explains why the definition of "dissonance" changes over time without abandoning the claim that consonance/dissonance has a physical basis. Where does physics end and culture begin?

4. Compare the diminished triad and the augmented triad. Both are built by stacking intervals of equal size. The diminished triad stacks two minor thirds; the augmented stacks two major thirds. Using the concept of frequency ratios and harmonic series alignment, explain why both sound tense and unstable, but in different ways. (Hint: think about symmetry and directional tendency.)

5. The critical bandwidth model of consonance predicts that two tones within the same critical band will sound rough. This band is wider (in musical interval terms) at low frequencies than at high frequencies. Given this, would you expect a major chord voiced in a low register (e.g., C2-E2-G2) to sound more or less "muddy" than the same chord voiced in a high register (C5-E5-G5)? Explain your reasoning, and describe a practical musical situation in which this physical fact constrains compositional or orchestrational choices.

Part B: Functional Harmony and Chord Progressions

6. The dominant seventh chord (V7) in the key of C major is G-B-D-F. Identify the following within this chord: (a) the leading tone and its expected resolution, (b) the chordal seventh and its expected resolution, (c) the tritone formed within the chord, and (d) how the tritone resolves to the tonic chord. Explain why this chord is so effective at creating tension that demands resolution.

7. Consider the progression I - IV - I in C major (C major - F major - C major). This is called a "plagal cadence" or "Amen cadence." Compare it to the "authentic cadence" V - I. Using the concepts of harmonic function and acoustic physics, explain why the authentic cadence feels more decisive and conclusive, while the plagal cadence feels softer and more "devotional." What physical differences between the IV and V chords account for this?

8. In jazz, the II-V-I progression (e.g., Dm7 - G7 - Cmaj7 in C major) is the fundamental cadential formula. Compare this to the classical IV-V-I. What is similar in physical/harmonic terms? What additional acoustic richness does the II-V-I provide compared to IV-V-I? Why might jazz musicians prefer the II chord to the IV chord for creating pre-dominant tension?

9. Modal mixture involves borrowing chords from the parallel minor key into a major key progression. For example, in C major, the chord built on the sixth scale degree borrowed from C minor would be Ab major (the "flat-VI" chord). This chord has been used extensively in rock music (e.g., the "I - bVII - bVI - V" progression in many classic rock songs). Describe the acoustic effect of this borrowed chord. How does it temporarily shift the tonal center of gravity, and why does returning to V after it feel particularly dramatic?

10. The so-called "deceptive cadence" occurs when the expected resolution from V to I is redirected to VI instead (in C major: G7 → Am). Analyze this in terms of voice leading: which voices resolve as expected, and which voice "deceives" the listener by moving to an unexpected destination? Why does the VI chord sound like a temporary resting place rather than a complete arrival?

Part C: Voice Leading Principles and Analysis

11. Here is a chord progression described in terms of voices (soprano, alto, tenor, bass). Identify any voice-leading problems and explain, in physical and musical terms, why each is problematic: - Chord 1 (C major): Soprano G, Alto E, Tenor C, Bass C - Chord 2 (G major): Soprano D, Alto D, Tenor G, Bass G - (Hint: look carefully at what the soprano and tenor are doing simultaneously.)

12. The "rule of the octave" in 18th-century thorough-bass practice specified specific chord positions and voicings for each scale degree when that scale degree appeared in the bass. The rule was designed to produce smooth voice leading above any bass line. Using your understanding of the principle of harmonic efficiency (minimize melodic motion), explain why having a fixed set of default harmonizations for each bass note would be useful, and identify one potential weakness of the rule (situations where it might produce suboptimal voice leading).

13. Contrary motion — where the soprano and bass move in opposite directions — is generally preferred in four-voice writing. Explain this preference from both a voice-independence perspective (each voice as a separate acoustic entity) and from the perspective of balanced frequency distribution (the acoustic spectrum of the chord). Are there musical situations where parallel or similar motion would be preferable, and why?

14. The technique of the pedal point involves sustaining a single pitch in the bass (usually the tonic or dominant) while harmonies that do not necessarily include that pitch move above it. The bass note creates dissonance against some of the chords above it, then resolves when the harmony returns to a chord containing that pitch. Describe the acoustic effect of a dominant pedal point (sustaining G in the bass while harmonies move above it) in terms of the physics of sustained dissonance and anticipation. Why is the pedal point so effective at building large-scale harmonic tension?

15. Voice-leading efficiency can be measured by counting the total number of semitones moved by all voices in a chord change. Consider two different voicings of the progression C major → F major, and calculate which requires less total motion: - Option 1: Soprano E→F (1 semitone), Alto C→C (0), Tenor G→A (2), Bass C→F (5). Total: 8 semitones. - Option 2: Soprano C→C (0), Alto E→F (1), Tenor G→A (2), Bass C→F (5). Total: 8 semitones. - Option 3: Soprano G→F (2), Alto E→F (1), Tenor C→A (3), Bass C→F (5). Total: 11 semitones.

Based solely on voice-leading efficiency, which option(s) are most preferred? Are there other musical factors (range, voice crossing, doubling) that might affect your answer?

Part D: Counterpoint, Fugue, and Wave Transformations

16. The fugue subject from Bach's Fugue in C Minor (BWV 847, from the Well-Tempered Clavier, Book I) begins with a descending stepwise line. When this subject is inverted (turned upside down), all descending intervals become ascending by the same amount. Describe in plain language what the inverted version of a descending stepwise line would sound like, and explain what "inversion" means both musically and physically (as a transformation of the melodic "waveform").

17. Augmentation in fugue writing means presenting the subject with all note values doubled (quarter notes become half notes, etc.). Diminution means halving all note values. Explain these operations in terms of the physics of time-scaling. If you think of the fugue subject as a pattern in time, what does augmentation do to the "temporal frequency" of the pattern? In what sense is a fugue that uses both the original and augmented forms simultaneously a study in musical "heterodyning"?

18. The stretto technique involves overlapping entries of the fugue subject — a new voice begins the subject before the previous voice has finished. Describe the acoustic experience of a stretto passage using the concept of wave superposition. What makes stretto particularly challenging to compose well? (Consider: the overlapping versions of the subject must form acceptable harmonies and counterpoint with each other at every moment, not just at their beginnings and endings.)

19. Bach's The Art of Fugue includes "mirror fugues" in which the entire score (all voices simultaneously) can be read both right-side-up and upside-down (with all intervals inverted and the bass and soprano voices swapped) to produce a different but equally valid fugue. This property is called invertible counterpoint at the octave. Explain what constraints must be satisfied for counterpoint to be invertible at the octave. Which intervals are "safe" for invertible counterpoint (they remain consonant when inverted) and which are problematic?

20. Describe the structure of a simple three-voice fugue exposition, from the first entry of the subject through the completion of all three entries. Identify: (a) where the answer typically appears (in pitch), (b) what the first voice does while the second voice states the answer, (c) where the exposition ends. Then explain why the convention of answering the subject a fifth above (or fourth below) is physically motivated, not arbitrary.

Part E: Non-Western Harmony, Jazz, and Atonality

21. Indian classical music uses a continuous drone (tonic + fifth, or sometimes tonic + fourth) throughout a performance, against which a melody unfolds through the raga structure. Compare this to the Western concept of a "tonic chord" as a gravitational center. In what sense does the drone create "harmonic" structure even without chord changes? What does the raga system provide that the drone cannot, and what does the Western progression system provide that the drone-raga system cannot?

22. The Indonesian gamelan uses two tuning systems: pélog (a seven-note scale with approximately equal spacing between some intervals but not others) and sléndro (a roughly five-note scale with approximately equal spacing). Neither system uses the simple-integer frequency ratios that Western music theory identifies as the basis of consonance. Research question: If Helmholtz's beating theory of consonance is correct, would you expect the harmonic coincidences in gamelan music to produce more or less "roughness" than in Western equal temperament? What might this tell us about whether the gamelan aesthetic seeks to minimize or exploit roughness?

23. Jazz musicians use a technique called reharmonization — replacing the original chords of a standard with new, harmonically richer substitutions while keeping the melody intact. The most common substitution is the tritone substitution: replacing a dominant seventh chord with the dominant seventh chord a tritone away (e.g., replacing G7 with Db7 in a G7-Cmaj7 progression). Explain why this substitution works in terms of voice leading and common tones. (Hint: G7 contains B and F; Db7 contains F and Cb=B. What do these two chords share?)

24. Arnold Schoenberg's twelve-tone method requires that all twelve chromatic pitches appear before any is repeated (avoiding establishment of a tonal center). Imagine you are a composer working in this system. Design a twelve-tone row that has interesting internal properties — perhaps it contains an embedded perfect fifth, or its retrograde is the same as its inversion. Describe the properties of your row in words, and explain how different transformations of the row (original, inversion, retrograde, retrograde inversion) would create variety while maintaining the "equality" of all twelve pitch classes.

25. We have argued that the rules of tonal harmony emerge partly from acoustic physics and partly from cultural convention. Construct a thought experiment: imagine that the speed of sound in air were twice as fast as it actually is (keeping other physical properties constant). Would this change the physics of consonance and dissonance? (Hint: consider whether the harmonic series — which depends on the physics of string vibration and air column resonance, not the speed of sound — would be affected.) What does your analysis suggest about which aspects of musical physics are "universal" and which might vary with fundamental physical constants?