Solutions to problems on Consonance and Dissonance

The first step is to write out the frequencies in the harmonic series for each complex tone:

Musical Note	1st	2nd	3rd	4th	5th
C4	262 Hz	524 Hz	786 Hz	1048 Hz	1310 Hz
F4	349 Hz	698 Hz	1047 Hz	1396 Hz	1745 Hz
A4	440 Hz	880 Hz	1320 Hz	2200 Hz	2640 Hz

To establish the relative consonance of the three chords: (1) C₄+F₄; (2) C₄+A₄; and (3) F₄+A₄; it is necessary to consider differences in frequency between all pure tones of the harmonic series for one note, with all pure tones of the second note. When the frequency differences are greater than ~10 Hz (meaning they are not fused) and less than the bandwidth of a critical band, then that pair of pure tones contributes to the dissonance of the chord. Below is a listing of the harmonics that contribute to dissonance for the different chords:

• C₄+A₄
  • 2^nd harmonic of C₄ with 1^st harmonic of A₄ gives frequency difference equal to 84 Hz
  • 3^rd harmonic of C₄ with 2^nd harmonic of A₄ gives frequency difference equal to 94 Hz
  • 5^th harmonic of C₄ with 3^rd harmonic of A₄ gives frequency difference equal to 10 Hz
• F₄+A₄
  • 1^st harmonic of F₄ with 1^st harmonic of A₄ gives frequency difference equal to 91 Hz
  • 4^th harmonic of F₄ with 3^rd harmonic of A₄ gives frequency difference equal to 76 Hz
• C₄+F₄
  • 1^st harmonic of C₄ with 1^st harmonic of F₄ gives frequency difference equal to 87 Hz
  • 3^rd harmonic of C₄ with 2^nd harmonic of F₄ gives frequency difference equal to 88 Hz
  • 5^th harmonic of C₄ with 4^th harmonic of F₄ gives frequency difference equal to 86 Hz

Overall, the differences are not as pronounced as they are for certain chords. One thing to note, is that both C₄+F₄ and F₄+A₄ have frequency differences between their fundamentals falling within the same critical band. This means that C₄+A₄ is the most consonant chord (the fundamental is the most dominant pure tone in most complex sounds).

The winner for the most dissonant chord is not so clear. The "winner" is probably C₄+F₄ because it has more pairs of higher harmonics falling within the same critical band.

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Last updated: 2 Dec 1999
Comments: bland@indiana.edu