Some feel that chords exist because they sound more pleasing than single tones.

a. Compare the waveforms for the major and minor harmonic chords with something that does not generally sound so pleasing like the dissonant chord, 1:1.12:1.25. What do you observe? (Use the same information as in problem 14).

b. We assume that the amplitude of all the individual tones are the same. Can we make a dissonant chord more pleasing by altering the amplitudes?

Problem 14

Sound waves can deeply influence our emotional state and certain combinations of waves sound particularly appealing. Chords in general are composed of three separate frequencies. A major chord is composed of a fundamental tone, a tone a major third above the fundamental and a tone a major fifth above. If the fundamental tone is of frequency 1 , the major chord consists of frequencies 1:1.25:1.5. A minor chord is likewise 1:1.2:1.5.

\[\delta=A \exp \left[i\left(2 \pi \omega_{i} t-k_{i} x\right)\right] \quad k_{i}=\frac{2 \pi}{\lambda_{i}}\]

Sum up the amplitudes for a major chord assuming \(A\) is the same for each, the fundamental frequency is \(220 \mathrm{~Hz}\), the speed of sound is \(343.2 \mathrm{~m} / \mathrm{s}\) and plot the waveforms you observe at \(x=0 ; 1\).