Chapter 12 Exercises: Tuning Systems — The Mathematics of Consonance and Compromise

Part A: The Pythagorean Comma and Ratio Mathematics

A1. Stacking Fifths Starting from C = 1.000, calculate the frequencies of twelve successive perfect fifths (multiply by 3/2 each time). After each multiplication, if the result exceeds 2.000, divide by 2 to fold it back into the octave. a) List all twelve resulting frequency ratios. b) What frequency do you arrive at after the twelfth fifth? c) The target is 2.000 (the octave). How far short or over do you arrive? d) Express this discrepancy as a ratio (arriving value / 2.000). This is the Pythagorean comma. What is it? e) Convert the Pythagorean comma to cents. (Hint: cents = 1200 × log₂(ratio).)

A2. The Syntonic Comma Just intonation recognizes a second important comma: the syntonic comma (ratio 81:80). a) Calculate 81/80 as a decimal. b) Convert 81:80 to cents using the formula: cents = 1200 × log₂(81/80). (Use the approximation log₂(1.0125) ≈ 0.01795.) c) The syntonic comma arises from the difference between the Pythagorean major second (9:8) and the "small" just major second (10:9). Verify this by calculating (9/8) / (10/9). d) How many syntonic commas approximately equal one Pythagorean comma? (Use your cent values to calculate this.) e) Why does the syntonic comma create practical problems for just intonation? (Hint: think about what happens when you play the same note as both a "large" second and a "small" second in the same piece.)

A3. Meantone Fifth Calculation Meantone temperament adjusts the fifth so that four fifths produce a pure major third (5:4). a) If four fifths equal one pure major third (plus two octaves), write the equation: (meantone fifth)⁴ = 5/4 × 4. Solve for the meantone fifth. b) What is this value numerically? (It's the fourth root of 5.) c) How does this compare to the pure fifth (3/2 = 1.5000)? Express the difference as a percentage. d) Convert this difference to cents. Is it larger or smaller than the 2-cent difference in equal temperament? e) If the meantone fifth is slightly flat, what accumulates after many meantone fifths? Where does the Pythagorean comma "go" in meantone temperament?

A4. Equal Temperament Calculations In 12-TET, every semitone = 2^(1/12). a) Calculate 2^(1/12) to four decimal places. b) Calculate the equal-tempered fifth: 2^(7/12). Express as a decimal and in cents. c) How many cents flat is the equal-tempered fifth compared to the just fifth (702 cents)? d) Calculate the equal-tempered major third: 2^(4/12). Express as a decimal and in cents. e) How many cents sharp is the equal-tempered major third compared to the just major third (386 cents)? f) Verify the Pythagorean comma: calculate how many cents 12 equal-tempered fifths exceed 7 octaves. (12 × 700 cents = ? cents; 7 × 1200 cents = ? cents; difference = ?)

A5. Comparing Systems Using Box 12.3 (the frequency comparison table), answer the following: a) For the note E (the major third), which system is closest to just intonation? Which is furthest? b) For the note G (the fifth), which system is closest to just intonation? Which is furthest? c) If you were a composer writing for a pipe organ that was fixed in one key (say C major), which tuning system would you prefer, and why? d) If you were a composer writing a piece that modulated through all 24 major and minor keys, which system would you need, and why? e) Calculate the frequency of E4 in each of the four systems, starting from C4 = 261.63 Hz. By how many Hz do the four values differ?

Part B: Tuning History and Philosophy

B1. The Wolf Fifth The wolf fifth is the "bad" fifth that absorbs the Pythagorean comma in Pythagorean tuning (and other systems). a) In standard Pythagorean tuning arranged on the "circle of fifths," the wolf fifth typically occurs between G# and D# (enharmonically Ab to Eb). Why does it occur there rather than, say, between C and G? b) The wolf fifth in Pythagorean tuning has a frequency ratio approximately equal to the pure fifth (3:2) minus the Pythagorean comma. Calculate this ratio: (3/2) / (Pythagorean comma ratio). Express in cents. c) How many cents flat is the wolf fifth compared to the pure fifth? Would you describe this as "slightly" or "grossly" out of tune? d) A Renaissance keyboard composer wants to use Pythagorean tuning. Which five chords (specify the root notes) would sound best, and which chord would be avoided at all costs? Explain why. e) Design a simple melody in C major (just the scale degrees C, D, E, F, G) that would sound good in Pythagorean tuning. Then write the same melody transposed to F# major. Which version sounds better in Pythagorean tuning, and why?

B2. Historical Systems Timeline Create a timeline of tuning systems from 500 BCE to the present. a) Research and place the following on your timeline: Pythagorean tuning (Greek), Chinese equal temperament calculation (Zhu Zaiyu), meantone temperament (major development period), well temperament (Bach era), equal temperament (widespread adoption). b) For each transition, identify: (1) the musical context that made the previous system insufficient, and (2) the musical context that the new system was designed to serve. c) Was the transition from well temperament to equal temperament driven primarily by acoustic considerations, practical/economic considerations, or aesthetic philosophy? Argue a position. d) What tuning systems do you predict will be in widespread use in 100 years, given the current trajectory of electronic music? Justify your answer.

B3. The Bach Debate Musicologists disagree about what tuning system Bach used for the Well-Tempered Clavier (see Case Study 12.1). a) Why does the question of Bach's tuning matter? If all twenty-four keys are "usable" in any well temperament, why does the specific well temperament matter? b) What evidence would definitively resolve the debate? What kinds of evidence are available (historical documents, the music itself, instrument building records, tuning descriptions)? c) Some scholars argue Bach's own spiral notation on the title page of the WTC encodes the tuning. Research this claim briefly (it is associated with scholar Bradley Lehman, who published in Early Music in 2005). What is Lehman's argument? d) If Bach had used equal temperament, how would you expect the character of the twenty-four preludes to differ from if he used well temperament? Would some keys sound identical or different?

B4. Just Intonation in Practice Just intonation is the ideal acoustic system but is "impractical for modulation." Explore the limits of this claim. a) A string quartet (violin 1, violin 2, viola, cello) performs entirely from memory and can adjust intonation continuously. Could they perform in just intonation? What would they need to do differently from an equal-tempered keyboard player? b) The barbershop singing tradition requires singers to tune their chords to just intonation by ear ("locking the chord"). How do barbershop singers learn to do this without consciously calculating frequency ratios? c) Research the "Schisma" — the difference between a just major third reached via five pure fifths versus via the pure harmonic series fifth. Is this difference audible? What does it suggest about the limits of "pure" just intonation? d) In just intonation, there are two sizes of whole tone (9:8 and 10:9). Write out the C major scale in just intonation and indicate whether each whole step is a "large" (9:8) or "small" (10:9) whole tone.

B5. Electronic Tuning Exploration Research or experiment with electronic tuning systems. a) The Scala archive contains thousands of scale definitions. Research three historical tuning systems available in the Scala archive that have not been discussed in this chapter. For each, describe: (1) its historical origin, (2) its interval structure, (3) what musical context it was designed for. b) Several commercial synthesizers (including some Korg models) allow users to specify custom tuning via MTS (MIDI Tuning Standard). What would be the practical steps for setting up a synthesizer to play in 31-TET? c) When a composer uses electronic tuning to implement just intonation for an electronic piece, they avoid the transposability problem — because they can specify different just intonation tunings for each key change. What are the remaining aesthetic challenges of this approach? d) The record producer and musician Wendy Carlos created a documentary album called Beauty in the Beast (1986) using alternative tuning systems. Research what systems she used and what prompted her to explore them.

Part C: Beating and the Physics of Consonance

C1. Calculating Beating Two strings are sounded simultaneously. String A vibrates at 440 Hz. String B vibrates at 443 Hz. a) What is the beating frequency? b) How would you describe the perceptual effect (use Box 12.4 as reference)? c) A piano tuner uses this beating to tune an octave: they want A4 (440 Hz) and A3 (220 Hz) to produce zero beats. What would a non-zero beat rate between these two indicate about the tuning? d) In equal temperament, a fifth is 2 cents flat. Calculate the beating frequency between an equal-tempered G5 (784.0 Hz in 12-TET) and a pure fifth above C4 (261.63 × 3/2 = 392.44 Hz, in the next octave 784.88 Hz). How many beats per second? e) An organ builder deliberately tunes certain pipes to produce 1 beat per second when played with their "principal" pipe. What is this technique called, and what aesthetic effect does it produce?

C2. Consonance and Dissonance Using the physical explanation of consonance (simple ratios produce less beating through overlapping overtones), explain the following: a) Why does a pure perfect fifth (3:2) sound more consonant than a pure major third (5:4)? (Hint: which ratio involves smaller integers, and what does that mean for how high up the harmonic series you need to go before the overtones start clashing?) b) Why does the tritone (45:32 in just intonation) sound more dissonant than the minor third (6:5)? Use the concept of overlapping overtones in your explanation. c) Two clarinets play simultaneously. Clarinet overtones are predominantly odd harmonics (1st, 3rd, 5th, 7th...). If one plays A4 (440 Hz) and the other plays E5 (a fifth, about 660 Hz in 12-TET), how do their respective odd-harmonic overtones compare? Do many clash, or do few clash? d) A synthesizer uses a sawtooth wave (which contains all harmonics). A second synthesizer also uses a sawtooth wave. If they play the same note (440 Hz and 440 Hz), the combined sound should be perfectly consonant — zero beats. But if one plays 440 Hz and the other plays 440.5 Hz (half a beat per second), describe what you would hear and at what rate.

C3. The Relationship Between Beating and Musical Style Different musical styles have different tolerances for beating: a) An equal-tempered major third produces about 14 cents of error (sharp). At A4 = 440 Hz, the equal-tempered major third E4 is at 554.37 Hz, while the just major third would be at 550.00 Hz. Calculate the beating rate between these two major thirds (considering the fundamental and the first relevant overtone that would clash). b) Renaissance polyphony used meantone temperament specifically because the beating of Pythagorean thirds was considered unacceptable. But Baroque organ music often uses equal temperament with its beating thirds without complaint. What does this tell us about the relationship between musical style and tolerance for beating? c) Barbershop singing is prized for "locking the chord" — achieving zero beating through just intonation. What physical phenomenon changes in the room when a barbershop quartet locks a chord? (Hint: think about standing waves and acoustic resonance.) d) The Indonesian gamelan deliberately creates beating through paired instruments. Research the gamelan aesthetic of "ombak" (waves) and describe how deliberately-produced beating differs from accidentally out-of-tune beating in terms of: (1) regularity, (2) rate, and (3) aesthetic intention.

C4. Building an Equal Temperament Ear This exercise requires access to a keyboard instrument or tuning software. a) Play a pure major chord (C-E-G) on a piano. Now, if possible, listen to a just-intoned major chord (recorded or produced by a tuning app). Describe the perceptual difference. b) Play an equal-tempered fifth (C-G). Count the beats per second (they will be slow — about 1 per second in the mid-range). Can you actually hear them? c) Play a tritone (C-F#). Count the beating rate. Compare to the fifth — which has faster beating? d) Play a major third (C-E). Count the beating rate. Is it faster or slower than the fifth? Faster or slower than the tritone? e) Based on your experience, rank the four intervals (octave, fifth, major third, tritone) from least to most beating. Does this ranking correspond to the traditional ranking of consonance to dissonance?

C5. The Quantum Beating Analogy Section 12.7 draws a parallel between acoustic beating and quantum beating. a) In acoustic beating, the beat frequency = |f₁ − f₂|. In quantum mechanics, the beat frequency = ΔE/h, where ΔE is the energy difference and h is Planck's constant. What does this formal similarity tell us about the relationship between acoustics and quantum mechanics? b) In the choir-and-accelerator analogy, just intonation corresponds to quantized energy levels. What does it mean physically for an energy level to be "quantized"? How does this parallel the acoustic "quantization" of just intonation ratios? c) Equal temperament is described as "slightly off-resonance." In physics, what happens to a quantum system that is driven slightly off-resonance? (Research: look up "power broadening" in quantum optics or "off-resonance driving.") How does this parallel the effect of equal temperament on harmonic resonance? d) The "wolf fifth" in Pythagorean tuning is described as analogous to an energy level that "doesn't fit" the system's potential well. In quantum mechanics, what happens to energy that can't fit into an allowed level? (Hint: it must go somewhere — think about scattering or emission.) Does this have a musical parallel?

Part D: Non-Western Tuning Systems

D1. The Indian Shruti System Research the 22 shrutis of Indian classical music. a) The 22 shrutis can be approximately mapped to the following cent values (from the tonic): 0, 70, 90, 112, 182, 204, 231, 274, 294, 316, 386, 408, 435, 498, 551, 590, 612, 680, 702, 772, 814, 884, 906, 996, 1018, 1088, 1108, 1200. (Note: different sources give slightly different values.) Which of these values correspond most closely to equal-tempered notes? Which do not correspond to any 12-TET note? b) The shruti system is based on "5-limit just intonation" — ratios involving only the prime numbers 2, 3, and 5. How does this differ from the 7-limit (adding the prime 7) and 11-limit (adding 11) just intonation? c) Research the historical relationship between the 22 shrutis and the concept of sruti in Indian music theory. The term means "that which is heard" — what does this suggest about the basis of the system? d) In Indian classical music, the tonic (sa) and the fifth (pa) are often considered "fixed" and immovable, while other scale degrees may be raised or lowered for different ragas. How does this compare to the Western treatment of the tonic and fifth in major/minor harmony?

D2. Arabic Maqam Tuning a) The Arab quarter-tone system approximates 24-TET. Calculate the size of one quarter-tone in cents. b) Many Arab musicians argue that their system actually uses neutral intervals that don't correspond exactly to quarter-tones — rather, the quarter-tone system is an approximation of a more nuanced just intonation system. Research the Arab neutral third (approximately 11:9 ratio). Calculate this ratio in cents. c) Compare the Arab neutral third (11:9 in cents) to the quarter-tone approximation (3 quarter-tones = 150 cents). How close is the approximation? d) Why might Arab musicians prefer to conceptualize their system as just intonation (with neutral ratios) rather than as 24-TET, even if the two are close? What does this reveal about the relationship between notation/theory and musical practice?

D3. Gamelan Comparison Two gamelan ensembles, one from Central Java and one from Bali, play the same piece of music. a) Research: what are the approximate interval sizes in Javanese slendro vs. Balinese slendro? Are they identical? b) If each gamelan has its own unique tuning, how do musicians from different gamelan ensembles collaborate? Research the practical logistics of Indonesian musical traditions that use multiple gamelans. c) The gamelan aesthetic values "ombak" (beating waves between paired instruments). How would a Javanese musician respond to a Western demand that instruments be tuned to eliminate all beating? d) Research the "sléndro-pélog" system more broadly. What does the existence of two contrasting scale systems within a single musical tradition (both used by the same ensemble) tell us about Indonesian musical aesthetics?

D4. Thai Classical Music and 7-TET Thai classical music approximately uses seven equal divisions of the octave (7-TET). a) Calculate the interval size (in cents) of one step in 7-TET. b) How does this compare to the Western major second (200 cents) and minor third (300 cents)? c) 7-TET does not approximate the fifth (3:2 = 702 cents) well. The nearest 7-TET interval is 4 steps = 686 cents. How many cents flat is this from the just fifth? d) Given that the fifth is so poorly represented in 7-TET, why might Thai musicians not care about this discrepancy? (Hint: consider what harmonic structures Thai classical music emphasizes.) e) Research one specific Thai musical form (such as piphat ensemble music) and describe how its modal structure relates to the 7-TET tuning.

Part E: Creative and Synthesis Exercises

E1. Design a Tuning System Design an original equal temperament system with a number of steps other than 12. Choose any number between 5 and 36 that has not been discussed in this chapter (avoid 7, 12, 19, 24, 31, 53). a) Choose your number and justify it with a brief mathematical argument: how well does your system approximate the just fifth (3:2), just major third (5:4), and just minor third (6:5)? (Calculate the error in cents for each.) b) What are the unique properties of your system? Does it have any intervals that are significantly better than 12-TET? c) What intervals are worse in your system than in 12-TET? d) Design a 5-note pentatonic scale within your system by selecting 5 steps that have reasonable intervals between them. e) Name your system and write a brief "manifesto" (100 words) explaining its aesthetic philosophy.

E2. The Tuning System Debate Write a 400-word dialogue between two fictional musicians: a baroque harpsichordist who performs exclusively in meantone temperament, and a contemporary jazz pianist who performs in equal temperament. The dialogue should cover: - The acoustic advantages and disadvantages of each system - The practical constraints (repertoire, ensemble, instrument) that led each musician to their system - Whether either musician has tried the other's system, and what happened - Their respective views on what "in tune" means

E3. Beating as Composition Beating — the acoustic phenomenon of amplitude modulation from near-unison pitches — can be a compositional tool rather than a tuning problem. a) Design a piece for two sine-wave oscillators (or two voices) in which the beating rate changes over time to create a rhythmic effect. Describe the frequencies you would use at three different points in the piece, the resulting beat rates, and the musical effect. b) How do composers like Alvin Lucier (research his piece "I Am Sitting in a Room") use acoustic phenomena like beating and resonance as compositional material? c) Is there a moral distinction between "using beating for compositional effect" and "accidentally playing out of tune"? What makes the difference?

E4. Historical Reconstruction Research and reconstruct the tuning dispute that occurred between 18th-century music theorists Jean-Philippe Rameau (who favored meantone) and Jean le Rond d'Alembert (who favored equal temperament). a) What were Rameau's acoustic arguments against equal temperament? b) What were d'Alembert's practical arguments for equal temperament? c) Who do you think won the argument, based on what actually happened historically? d) Who do you think made the better argument, based on the acoustic physics we've discussed?

E5. The Comma in Other Domains The Pythagorean comma problem — where repeatedly applying a simple operation fails to return you to your starting point — has analogues in other domains. a) In timekeeping: why doesn't a year of 365 days exactly divide into whole-number months or whole-number weeks? What "comma" exists in the calendar system? b) In color: why can't you mix three primary colors to create every possible hue without compromise? c) In geometry: why can't you tile a flat surface with regular pentagons? What is the "comma" in this case? d) For each of your three examples, describe the "compromise solution" that practical applications use (analogous to equal temperament in music). e) What do these examples suggest about the general relationship between mathematical ideals and practical approximations?