Exam 2 Review Solutions

Math, Music and Memory

Solutions for Exam 2 Review Questions

The brain can memorize and store millions of sounds and recall them instantly when needed. The brain also helps us adapt to new sounds, (e.g., speech recognition in a foreign country). Finally, the brain helps us filter out unwanted sounds, (e.g., background noise) and allows us to distinguish between competing sounds, (e.g., different instruments in an orchestra).

Amplitude is 4, period is 2, frequency is 1/2, phase shift is 1/2. Graph will have roots (zeros) at 1/2, 3/2, 5/2, 7/2, etc.

sin(3A) = 3 sin(A) - 4 sin³(A). Use sin(3A) = sin(2A + A) = sin(2A) cos(A) + cos(2A) sin(A), and then apply the double-angle formulas from HW #5 along with cos²A + sin²A = 1.

110 Hz. This is the A two octaves below A440. It will be 10 white keys below middle C on the piano.

Perfect Fifth (the ratio 3/2 is a P5 in either Pythagorean Tuning or Just Intonation). Major Third (the ratio 5/4 is a M3 in Just Intonation).

The precise value is 3¹²:2¹⁹. This represents the gap between two half steps and a whole step. It is also the gap between B♯ and C on the spiral of fifths. Going up by 12 perfect fifths will result in a pitch that is a Pythagorean comma higher than the one obtained by going up by 7 octaves. Problems with Pythagorean Tuning include: can't change keys very easily, B# does not equal C, two half steps do not equal a whole step, and the Circle of Fifths does not close up.

The musical notes, in ascending order, are G, G', D', G'', B'', D'', F'', G''', A''', B''', C♯''', D''', where the ' sign signifies a higher octave. The successive intervals are Oct., P5, P4, M3, m3, m3, M2, M2, M2, M2, m2.

Pythagorean Tuning: 371.3 Hz.
Just Intonation: 366.7 Hz.
Equal Temperament: 370.0 Hz.

Just intonation was designed to exploit the resonances inherent in the overtone series. Its strength lies in the fact that many of its intervals have low-numbered ratios such as 2/1, 3/2, 4/3, and 5/4, so the higher partials for notes sounding together in these frequency ratios will match up, creating harmony that sounds more pleasing to our ears. The major chords built on the first, fourth, and fifth scale degrees are each in the nice ratio 4:5:6. Problems with just intonation include that there are two different whole steps (9:8 and 10:9; off by the syntonic comma 81:80), two half steps do not equal either whole step, the circle of fifths does not close up, B♯ ≠ C (enharmonic equivalence fails), it is hard to change keys, and melodic drift.
Equal temperament was created to solve many of the problems arising with just intonation. By making a consistent half step throughout the scale and lower/higher octaves, it became feasible to change keys and play more chromatic melodies and harmonies. Now, two half steps equals one whole step and the circle of fifths closes up properly (enharmonic equivalence is restored). Weaknesses include that the just 3:2 perfect fifth is lost and that some intervals (e.g., the major third and sixth) are far from their just values, sometimes leading to difficulties with tuning and harmonizing. Other than the octave (2:1) or unison (1:1), all frequency ratios in equal temperament use irrational numbers.

Just intonation. Since we are staying in one key and using the major scale, we don't need to worry about how to sharp or flat notes to move into a new key. Since we are playing mostly the I, IV and V chords, all of which have the 4:5:6 ratio, the harmony will re-enforce the overtone series and sound the best.

4:5:6. In a minor triad, the bottom interval is a minor third, which is 6:5 in just intonation. This can be seen from the overtone series, or by noticing that a minor third is equivalent to going up a perfect fifth and then down by a major third (up a P5 is 3/2 and then down a M3 is 4/5, and 3/2 times 4/5 = 6/5). Thus, the ratios 1/1, 6/5, 3/2 for a minor triad are equivalent to 10/10, 12/10, 15/10, which gives the desired 10:12:15. This helps explain the difference in perception between a major and minor chord.

There are 300 cents in a minor third in equal temperament (3 half steps). In just intonation, a minor third is obtained using the ratio 6/5. Thus, there are 1200*log₂(6/5) cents in a minor third, or approximately 315.6 cents. The syntonic comma, 81/80, is equivalent to 1200*log₂(81/80), which is approximately 21.5 cents.