Chapter 11 Exercises: Pitch, Frequency & Musical Scales

Part A: Frequency and Pitch Mathematics

A1. Octave Calculation Concert A is defined as 440 Hz. Calculate the frequencies of: a) A3 (one octave below A4) b) A5 (one octave above A4) c) A2 (two octaves below A4) d) A7 (three octaves above A4) Show your work and identify the pattern. What single mathematical operation gives you the frequency of any octave?

A2. The Logarithmic Ladder A musician hears two pairs of notes: (200 Hz, 400 Hz) and (1000 Hz, 2000 Hz). Both pairs are an octave apart. a) What is the frequency difference in each pair? b) What is the frequency ratio in each pair? c) Which measure (difference or ratio) correctly captures that both pairs are the same musical interval? Why? d) Explain why pitch perception is "logarithmic." What does this mean practically for how we experience music?

A3. Harmonic Series A violin string vibrates at a fundamental frequency of 196 Hz (the note G3). a) List the frequencies of the first eight harmonics. b) Which of these harmonics correspond to the note G (in any octave)? c) Which harmonic is closest to a D (approximately 293 Hz, the fifth above G)? d) What is the ratio between this D-frequency harmonic and the fundamental?

A4. Interval Ratios Using the frequency ratios given in Box 11.2 (the pentatonic scale), calculate the actual frequencies of the major pentatonic scale starting from A4 = 440 Hz. - C (using ratio 1:1 from C; hint: A4 is the sixth scale degree of C major) - Alternatively, simply apply the ratios 1:1, 9:8, 81:64, 3:2, 27:16 to a root of your choice and compute the resulting frequencies. Then compare your results to the equal-tempered frequencies from Box 11.3. Where is the difference greatest?

A5. Equal Temperament Formula In 12-TET, the frequency of any note n semitones above a reference pitch f₀ is: f = f₀ × 2^(n/12). a) Calculate the frequency of every note in one chromatic octave from C4 (261.63 Hz) to B4. b) For each note, calculate the ratio to C4. c) Which note has a ratio closest to 3:2 (the just fifth)? What is its actual ratio? d) By how many Hz does the equal-tempered fifth differ from the just fifth at this octave?

Part B: Scale Construction and Analysis

B1. Build the Pentatonic Scale Using the method described in Section 11.4 (stacking perfect fifths with 3:2 ratios), construct a major pentatonic scale starting from D (146.83 Hz in equal temperament). a) Calculate each note by multiplying the previous by 3/2, then fold into the octave by dividing by 2 whenever the frequency exceeds 2× the starting pitch. b) Order the resulting five frequencies from lowest to highest. c) Calculate the interval (ratio) between each adjacent pair of notes. d) Compare your results to the D major pentatonic scale in 12-TET (D, E, F#, A, B). How close are the just-intonation frequencies to the equal-tempered ones?

B2. The Diatonic Scale as Chord Network The major scale can be constructed from three overlapping major triads (ratios 4:5:6). a) Starting from C (ratio 1:1), construct the tonic triad (C, E, G) using the 4:5:6 ratio. b) Construct the dominant triad starting from G (ratio 3:2): what are the three note ratios? c) Construct the subdominant triad starting from F (ratio 4:3): what are the three note ratios? d) List all unique ratios from these three triads, order them, and verify that they produce the C major scale. e) Which note appears in two of the three triads? Which appears in only one?

B3. Counting Scale Possibilities Suppose you want to construct a scale using exactly 5 notes per octave, drawn from the 12 notes of the chromatic scale. a) How many mathematically possible pentatonic scales are there? (This is a combination problem: C(12,5) — look this up if needed and calculate.) b) Of these, how many will have the specific pattern of the major pentatonic (whole step, whole step, minor third, whole step, minor third)? c) The major pentatonic pattern can start on any of 12 notes. Why then is only one of those considered "C major pentatonic"? d) How does this exercise illustrate the relationship between physical constraints (consonance favors certain intervals) and the enormous remaining space of cultural choice?

B4. Analyzing a Non-Western Scale The Arab maqam Rast (a common maqam from which many others derive) uses the following pitches above its tonic: Tonic, whole step, neutral second (¾ step), whole step, whole step, whole step, neutral second (¾ step), octave. a) How many pitches does this maqam use per octave? b) What is a "neutral second" in terms of equal-tempered semitones? c) What is the total sum of intervals? (It should equal 12 semitones, or 1 octave.) d) Maqam Rast is often compared to the Western major scale. What are the similarities and differences? e) Why can't the neutral second be played on a standard Western piano?

B5. Scale Symmetry The tritone divides the octave into two equal halves (each is √2:1). a) What makes this the "most symmetrical" division of the octave? b) A scale called the "whole-tone scale" consists entirely of whole-tone (2-semitone) steps. How many notes does it have? c) A scale called the "diminished scale" alternates half-step and whole-step intervals. How many notes does it have? d) Both the whole-tone and diminished scales have perfect symmetry — they look the same starting on multiple notes. How many distinct starting points give the same whole-tone scale? The same diminished scale? e) What does this symmetry mean practically for a composer or improviser?

Part C: Cross-Cultural Comparison

C1. Comparative Scale Analysis Research (using library resources or reliable web sources) the following scale systems and fill in a comparison chart:

For each system you research, note: (a) the smallest interval size, (b) whether intervals are equal or unequal, (c) whether the system is based on just ratios or equal division.

C2. The Universals Test Section 11.8 lists aspects of scales that appear to be universal versus culturally variable. a) For each item below, categorize it as "universal," "near-universal," or "culturally specific" and provide evidence: - Using an octave as the repeating unit - Using exactly 12 notes per octave - Emphasizing the perfect fifth (3:2 ratio) - Using unequal step sizes - Having a "home" pitch (tonic) that feels stable - Using the major/minor distinction b) What would it mean for a musical system to truly have NO universals? Do you think such a system is possible?

C3. The Blues Note Investigation The "blue note" is a pitch between the equal-tempered major third and minor third (or between the minor seventh and octave). Research the blues tradition and answer: a) What are the two most common blue note positions (in terms of scale degrees)? b) Why doesn't the blue note exist on a standard piano? c) How do guitarists, violinists, and vocalists produce blue notes? d) The blue note corresponds approximately to the 7th harmonic of the harmonic series (ratio 7:4 above the root for the flatted seventh). Calculate this frequency if the root is A (110 Hz). How does it compare to the equal-tempered minor seventh (A to G, ratio 16:9)? e) Some theorists argue that the blues scale represents a retention of West African tuning traditions that don't fit 12-TET. Research this argument and evaluate its plausibility.

C4. Raga Analysis Raga Yaman (Hindustani) uses the following pitches: Sa, Re, Ga, Ma# (raised fourth), Pa, Dha, Ni, Sa (octave). In Western terms, this is approximately: C, D, E, F#, G, A, B, C. a) How is Raga Yaman similar to the Western Lydian mode? b) What additional elements define a raga beyond the pitch set? Name at least four. c) Raga Yaman is traditionally performed in the evening. What does this suggest about the relationship between musical systems and social/temporal context? d) In Carnatic music, the melakarta system organizes 72 parent scales. Compare this to the Western system of 7 modes derived from the major scale. What does the difference in number suggest about the two systems' approaches to scale diversity?

C5. Gamelan Tuning Philosophy Indonesian gamelan orchestras are deliberately tuned to be unique — no two gamelan ensembles use the same pitch set, and a gamelan's unique tuning is considered part of its identity and spiritual character. a) How does this philosophy contrast with the Western ideal of standardized pitch (A440)? b) What are the practical advantages of standardized tuning? What aesthetic or philosophical values does gamelan tuning embody instead? c) Gamelan instruments are tuned in pairs, with each pair slightly detuned from the other to produce deliberate beating (called ombak — "waves"). This beating is intentional and aesthetically valued. How does this contrast with Western classical music's treatment of beating (as something to eliminate)? d) What does the gamelan philosophy suggest about the relationship between "in tune" and cultural aesthetics?

Part D: Listening and Ear Training

D1. Interval Recognition Listen to the following intervals (use a piano, piano app, or online interval trainer) and describe your perceptual response to each: a) Perfect octave (C4 to C5) — Does it sound like "the same note" or "different notes"? b) Perfect fifth (C4 to G4) — Describe the quality: open, hollow, stable? c) Major third (C4 to E4) — Does it sound stable or unstable? Bright or dark? d) Tritone (C4 to F#4) — Describe your physical and emotional response. e) Minor second (C4 to C#4) — How does this compare to the tritone in terms of dissonance? After listening, rank these intervals from most consonant to most dissonant and compare your ranking to the predictions of physics (simpler ratios = more consonant).

D2. Scale Listening Challenge Find recordings of the following scales (online search: "[scale name] ascending piano" or "[scale name] example"): a) C major b) C minor (natural) c) C blues scale (C, E♭, F, F#, G, B♭) d) Japanese in scale (C, D♭, F, G, A♭) e) Whole-tone scale from C For each scale, describe: (1) your emotional response, (2) whether it sounds "stable" or "in motion," (3) whether it sounds "Western" or "foreign" to your ear (and what that distinction even means).

D3. The Octave Experiment Working with a partner (or using a recording): a) Play a note, then play a note 7 semitones above it (a perfect fifth). Write down your perceptual impression. b) Play a note, then play a note 6 semitones above it (a tritone). Write down your perceptual impression. c) Now play the same note with the note an octave above it simultaneously. Does it feel like one note or two? d) Have your partner play random notes on a keyboard while you hum along. Do you naturally gravitate to the unison, octave, or fifth above their note? What does this tell you about the hierarchy of intervals?

D4. Cultural Listening Immersion Choose ONE non-Western musical tradition and listen to at least 30 minutes of representative music. Suggestions: - Hindustani classical (sitár or bansuri) - Carnatic classical (veena or violin) - Arab classical maqam (oud) - Turkish makam (ney) - Indonesian gamelan

After listening, write a 300-word response addressing: a) What aspects of the pitch system sound "strange" or unusual to your Western-trained ears? b) After 30 minutes, do some pitches begin to feel "normal"? What does this suggest about categorical perception and cultural training? c) Can you identify any moments of consonance or dissonance as defined by Western music theory? Do they occur in the same musical contexts?

D5. Tritone Hearing The tritone's role in Western music changed dramatically over time — from strictly forbidden to essential harmonic resource. a) Find a recording of a Gregorian chant and identify any moments where the tritone is carefully avoided. (Hint: the famous "mi contra fa" rule prohibited B and F from appearing consecutively.) b) Find a jazz recording featuring a dominant seventh chord. Identify the tritone within the chord (between the third and seventh). c) Find a heavy metal recording featuring a tritone riff (classic examples: "Black Sabbath" by Black Sabbath, the opening of "Purple Haze" by Jimi Hendrix). d) Describe how the same interval functions differently in each context. What does this suggest about the relationship between acoustic properties and musical meaning?

Part E: Creative and Synthesis Exercises

E1. Design Your Own Scale Design an original 6-note scale (hexatonic) with the following requirements: a) It must fit within one octave (12 semitones total) b) It must not be identical to any existing named scale c) It must contain at least one interval of 3 semitones or more (to avoid a chromatic character) d) It must contain at least one semitone (to allow for tension) Describe: - The intervals in semitones between adjacent scale degrees - Three adjectives describing the emotional quality of your scale - What musical genre or context it would suit - Whether it can be easily transposed (played in different starting notes)

E2. The Compromise Matrix Create a "compromise matrix" for scale construction. List the following properties across the top: - Maximum consonance within key - Easy transposability (play in any key) - Simple to learn/remember - Microtonal detail available - Compatibility with other instruments

Down the side, list these scale systems: - 12-TET - Just intonation (fixed key) - Arab maqam - Indian raga - Indonesian gamelan

For each cell, rate the system from 1-5 on that property and briefly justify your rating. Discuss: is there any system that scores high on all properties? What does this suggest about the nature of musical compromise?

E3. The Alien Scale Thought Experiment Extended Continuing the thought experiment from Section 11.11, design a complete musical system for a species with a hearing range of 10 Hz to 10,000 Hz (shifted lower than humans) and a temporal resolution three times faster than humans. a) What would the "octave" feel like to this species? (The 2:1 ratio would still apply physically.) b) How many octaves are available? (Calculate: log₂(10000/10) = ?) c) Would the concept of "melody" (sequential pitches) still apply? Would "harmony" (simultaneous pitches)? d) If this species uses a 12-note scale, what would the lowest and highest notes be in Hz? e) Design a 5-note pentatonic scale for this species, using the stacking-fifths method, starting from 20 Hz.

E4. Equal Temperament Debate Write a 400-word position paper arguing either FOR or AGAINST the following proposition: "Equal temperament should be replaced as the standard tuning system for Western music education, and students should first learn to sing and hear just intonation before encountering equal temperament." Your argument should address: - The physical properties of just intonation vs. equal temperament - The practical challenges of each system - Historical precedent (when did equal temperament become dominant?) - The goals of music education (ear training vs. instrumental flexibility) - Personal values about acoustic purity vs. practical compatibility

E5. Scale Construction from Non-Western Physics The Indian shruti system recognizes 22 microtonal positions within the octave, each defined by a specific just intonation ratio. Research the 22 shrutis and: a) List any 10 of the 22 shruti ratios. b) Calculate the frequency of each of your 10 shrutis if the tonic is A at 110 Hz. c) Identify which of your 10 shrutis are closest to the 12 equal-tempered pitches. For each, calculate the discrepancy in cents. d) Are there any shrutis that don't correspond closely to any equal-tempered note? What musical function might these serve? e) Given that the Indian system has 22 available positions but any raga uses only 5-7, how does the shruti system combine the benefits of a large pitch vocabulary with the practicality of small scales?