Math/Music: Structure and Form
Partial Solutions for Exam 2 Review Questions
3 + -3 = 0.
b = 10^4 = 10,000.
10 log_10 (50) which is approximately 17 dB.
200 Hz and 203 Hz. The "note" we hear has the average frequency of
201.5 Hz and the number of beats is 3 per second.
See HW #3, problem #2.
Amplitude is 4, period is 2, phase shift is 1/2. Graph will have roots at 1/2, 3/2, 5/2, 7/2, etc.
22 Pi. cos(22 Pi) = 1, sin(22 Pi) = 0.
sin(3A) = 3 sin(A) - 4 sin^3(A).
Perfect 5th (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 3^(12)/2^(19). 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. 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.
150, 300, 450, 600, 750, 900, 1050, 1200 Hz
See HW #4, problem #5 or your class notes. The musical notes, in ascending order, are G, G', D', G'', B'', D'', F'', G''', A''', B'''
where the ' sign signifies different octaves.
Pythagorean Tuning: 391.1 Hz.
Just Intonation: 391.1 Hz.
Equal Temperament: 392.0 Hz.
See class notes and Ch. 1 of the course text. Also, see HW #4, problem #6.
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.
Pythagorean Tuning: 27/16
Just Intonation: 5/3
Equal Temperament: 2^(3/4)
2^(5/12) (P4 is five half-steps.) To show this number is irrational, set 2^(5/12) = p/q, with
p and q integers. Raise both sides to the 12th power to obtain 2^5 = p^12/q^12. Cross multiplying
gives 2^5 q^12 = p^12. Raising an integer to the 12th power will multiply the exponents in its prime
factorization by 12, making those exponents become even. Thus,
the left-hand side of this equation (2^5 q^12) is an integer which when factored into
its prime factorization will have an odd number of 2's (5 + even or 0 = odd). But the right-hand side (p^12) has a prime
factorization with an even or zero number of 2's. Hence the number of 2's on each side cannot be equal.
This violates the Fundamental Theorem of Arithmetic
which states that a positive integer has a unique factorization into its prime factors.
Thus, 2^(5/12) is irrational.
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. 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, there are
1200*log_2 (6/5) cents in a minor third, or approximately 315.6 cents. The syntonic comma, 81/80, is equivalent to
about 21.5 cents.