Further Applications (Music, Art and Politics)
a. Using the exponential growth formula Q= Qo *fn where Q= Qo* 1.05946, n=7
We have Q= Qo *1.059467
Q= Qo *1.498
Hence, note seven half-step higher will be increased by the factor 1.498 which is equivalent to
b. Using the exponential growth formula Q= Qo *fn where Q= Qo* 1.05946 and n=2*7=14 half-steps;
Q= Qo *1.0594614
Q= Qo *2.2448
Hence it will be increased by the factor 2.2448
c. Consider Qo=260, and the formula Q= Qo *fn we shall have
Q= Qo* 1.05946, n=7 half-steps
Q= 260 *1.059467 =390cps
d. There are 7 half steps in each step. One octave will have 12 half steps, hence the total number of fifths required to complete the entire circle of fifths = 12 fifths*7=84 half steps
Number of octaves will be 84/12=7 octaves.
There are 4 quarter notes in “4/4 time.” One half note measures half unit time. Hence there are two half notes in one measure.
One eighth note measures one-eighth unit time. So one eighth note is in one eighth unit time. This implies that there are 2 one-eighth notes in one measure.
One sixteenth note measures one sixteenth unit time. Therefore, there are 2 sixteenth notes in one measure.
Further Applications: Pages 613-614 contain some really good questions to help you explore the different aspects of symmetry. Answer questions 22 and 24.
a. A four pointed star has three rotation symmetries. First, it remains the same when rotated through 90°. Similarly, it remains so through (2*90°), and (3*90°). It also have two reflection symmetries, one horizontal and one vertical.
b. A seven –pointed star has 6 rotation symmetries. It can be rotated through (360°/7) and remain the same. It can also be rotated through 102.86°, 154.29°, 205.71°, 257.14°, and 308.57°. However, it has no reflection symmetry due to the odd number of vertices.
This figure has four reflection symmetries since it can be flipped in four ways (based on the symmetry lines drawn above) without changing its shape. In addition, its shape remains the same if rotated a quarter of a circle (90°). It will also remain the same if rotated in 180° and 270°. Thus four rotation symmetries.
If candidate c withdraws, the remaining candidates will get the following votes as first choice candidates: A will get 18 first choice votes, B will get 22 votes, D will get 9 and E will get 6.
The total number of votes that can be casted=30
Plurality Method States that the candidate with the most first-place votes wins
A received 10 first place votes in column 1
B received 10 first place votes in column 2
C received 0 first place votes
D received 10 first place votes in column 3
There is a tie between A, B, D hence there is no plurality winner and no one receives a majority
Single Run off method
In this scenario, the two candidates with the most first place votes have a run-off, and the emerging winner from the fun off wins the election. But since A, B and D have same first place votes the method does not give any result
Sequential run off method
Here, a series of run offs are done, while eliminating the candidate having the fewest first place votes at each stage. The runoffs continue until one candidate with the majority of the first place votes is found and declared the winner
Step 1: C is eliminated as it has the lowest first place votes
Step 2: preference schedule without C
First
A
B
D
Second
B
A
B
Third
D
D
A
10
10
10
A, B and D have equal number of first place votes hence none can be eliminated
This method too, does not yield any result
Point System
Points are awarded according to the rank of each candidate, hence the one with most points wins.
A gets (10*4) + (10*3) + (10*1) = 80 points
B gets (10*3) + (10*4) + (10*2) = 90 points
C gets (10*2) + (10*1) + (10*3) = 60 points
D gets (10*1) + (10*2) + (10*4) = 70 points
B wins with 90 points
Pair wise comparison
Candidate with most pair wise wins is the winner
A versus B= B wins 20 to 10
B versus C= B wins 20 to 10
B versus D =B wins 20 to 10
Hence B is the winner
In brief, B is the winner in the methods that we reached a conclusion
A has the most of first-place votes (16), hence is considered the winner by plurality.
Criterion 1:
A is still the winner because it has the majority of the votes, 16 out of 10.25
Criterion 2:
Examining the pair wise races
A defeats B by 23 to 18
A defeats C by 26 to 15
A defeats D by 34 to 7
A is the winner under this criterion
Criterion 3:
Imagine a second election in which some voters move A higher in the rankings but do not change the order in which they rank B, C and D
A gets more first place votes and at the same time, cannot have its first place votes reduced. Hence by single runoff, A would still win
Criterion 4:
If in the second election, B drops out
The number of first place votes of A will be 26 in column1 and 2. By sequential runoff, A would still win.
Suppose C drops out, then first place votes for B will be 18 in columns 2 and 3, A will get 16 from column 1 and D will have 7 in column 4.
D is eliminated
In the face offs between A and B, A wins by 23 to 18
Suppose D drops out, B will have 10 first place votes –col 2, A will have 16 col-1 and C will have 15 first place votes col-3 and 4
B is eliminated, hence in the face off between A and C, A wins satisfying criterion 4.
Winner is decided by sequential runoff
A= 16 in column 1
B=10 in column 2
C= 8 in column 3
D= 7 in column 4
D is eliminated
Then
C has 15 first place votes A has 16 first place votes and B has 10 first place votes.
B is eliminated
In the face off between A and C, A will win by 26 to 15 votes.
Hence by sequential runoff, A is the winner
Using the fairness criteria
Criterion 1:
Because no candidate received a majority of first-place votes in the election, this criterion does not apply.
Criterion 2:
We examine the pairwise races
A defeats B by 23 to 18
A defeats C by 26 to 15
A defeats D by 34 to 7
A is the winner under this criterion
Criterion 3:
Imagine a second election in which some voters move A higher in the rankings but do not change the order in which they rank B, C and D
A gets more first place votes and at the same time, cannot have its first place votes reduced. Hence by single runoff, A would still win
Criterion 4:
If in the second election, B drops out
The number of first place votes of A will be 26 in column1 and 2. By sequential runoff, A would still win.
Suppose C drops out, then first place votes for B will be 18 in columns 2 and 3, A will get 16 from column 1 and D will have 7 in column 4.
D is eliminated
In the face offs between A and B, A wins by 23 to 18
Suppose D drops out, B will have 10 first place votes –col 2, A will have 16 col-1 and C will have 15 first place votes col-3 and 4
B is eliminated, hence in the face off between A and C, A wins satisfying criterion 4.
Senate has 49 democrats, 49 republicans and 2 independents. All the democrats and republicans will vote along opposing party lines, hence no party has the majority so far. The two independents may not care which way the votes go, but if they vote together, the two votes will decide the outcome of the election. The power of the two independents to influence the decision making is therefore much greater than their two votes out of 100 would imply.
