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Difficulty: MediumInferences from the Text

Unlike monophonic traditions that tolerate acoustic discrepancies in pitch intervals across octaves, early modern European polyphony demanded harmonic convergence among simultaneous vocal lines. Under Pythagorean tuning, constructed entirely from acoustically pure fifths (a frequency ratio of 3:2), major thirds were derived rather than tuned directly, yielding intervals noticeably sharper than pure thirds (5:4). While acceptable in medieval plainchant, this discrepancy generated dissonant acoustic beats in triadic harmony, prompting Renaissance instrument makers to adopt quarter-comma meantone temperament. By intentionally narrowing each fifth by a quarter of a syntonic comma, meantone temperament produced consonant, pure major thirds in central keys. However, because twelve pure or slightly narrowed fifths cannot mathematically equal seven perfect octaves, the accumulated acoustic deficit had to be concentrated into a single, highly dissonant interval—the so-called "wolf fifth." Consequently, keyboard compositions written for meantone instruments were strictly confined to harmonic progressions within adjacent keys, as modulating into remote tonalities exposed performers to jarring acoustic beats.

Based on the passage, which of the following statements can be logically inferred? (Select all that apply.)

  1. Pythagorean tuning generates dissonant acoustic beats during triadic harmony primarily because its major thirds deviate from the acoustically pure ratio of 5:4.Answer
  2. B
    Renaissance instrument makers resolved the acoustic deficit of meantone temperament by dispersing tuning discrepancies evenly across all twelve fifths.
  3. Performances of meantone-tuned keyboard music risked severe acoustic dissonance when navigating modulations that engaged the interval absorbing the cumulative pitch deficit.Answer
  4. D
    Medieval plainchant avoided harmonic instability by tuning vocal lines to equal temperament rather than purely derived intervals.
  5. E
    Monophonic musical traditions strictly require melodic pitch intervals to maintain a 5:4 ratio in order to preserve octave alignment.

Answer

The statements indicating that Pythagorean tuning produces dissonant beats in triads due to deviation from the 5:4 ratio, and that meantone keyboard performances risked dissonance when modulating into keys that engaged the concentrated deficit interval, are both logically entailed by the text.
The passage supports two inferences: first, Pythagorean tuning produced dissonant beats in triadic harmony because its major thirds were derived and deviated from the pure 5:4 ratio; second, meantone temperament concentrated accumulated pitch discrepancies into a single 'wolf fifth,' causing severe dissonance when keyboard music modulated into tonalities that sounded this compromised interval.

Step-by-Step Solution

1
Locate textual statements regarding Pythagorean tuning and acoustic beats in polyphony.
The text states that Pythagorean derived thirds were sharper than pure thirds (5:4), generating dissonant beats in triadic harmony.
This establishes textual justification for the inference concerning Pythagorean interval ratios.
2
Analyze the passage's explanation of meantone temperament, octave closure, and modulation.
The text explains that the acoustic deficit was concentrated into a single 'wolf fifth,' making modulations into remote keys produce jarring dissonance.
This confirms the inference regarding performance risks during remote harmonic modulations on meantone instruments.
3
Evaluate the remaining options against textual boundaries.
The passage contradicts claims of even deficit distribution, omits equal temperament in plainchant, and denies strict 5:4 constraints in monophony.
Eliminating unsupported logical leaps ensures only statements strictly entailed by the text are accepted.

Key Concept

Textual Inference and Deductive Entailment
Estimated Time:1m 30s
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Inferences from the Text

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