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Problem set 4 (complete list)
Mathematics and music
We discussed equivalence when we studied pitch classes, which are equivalence classes of
pitches sharing the same letter name. An equivalence relation is a concept that frequently
appears in music—and is important in math, too. Formally, a way of associating elements in a
set with each other (indicated by “”) is an equivalence relation if for all elements x, y, and z in
the set
 If x  y then y  x. (The relation is symmetric.)
 x  x (The relation is reflexive.)
 If x  y and y  z, then x  z. (The relation is transitive.)
Modular equivalence (a  b mod m) is only one example of an equivalence relation. If you start
with a set and an equivalence relation, you can sort the elements of the set into equivalence
classes that consist of elements that are equivalent to one another. For example, if
 our set is the integers {…, -2, -1, 0, 1, 2, …} and
 the equivalence relation is equivalence mod 4, then
the integers split into four equivalence classes:
 {…, -8, -4, 0, 4, 8, …} (integers that are equivalent to 0 mod 4)
 {…, -7, -3, 1, 5, 9, …} (all equivalent to 1 mod 4)
 {…, -6, -2, 2, 6, 10, …} (all equivalent to 2 mod 4)
 {…, -5, -1, 3, 7, 11, …} (all equivalent to 3 mod 4)
If you would like to see more examples, the Wikipedia page on “Equivalence relation” has tons
of them.
1. For each of the following, either show that it is an equivalence relation or explain why the
relation fails to be symmetric, reflexive, or transitive:
a. Set: all people; relation: “x  y if x and y have the same astrological sign.”
b. Set: all real numbers; relation: “x  y if x / y 1.”
c. Set: all musicians in bands; relation: “x  y if x plays in a band with y.”
d. Set: all frequencies; relation: “x  y if there exists a positive integer k such that
x = ky.” (In other words, x is a harmonic of y.)

2. A binary necklace is an equivalence class of
strings of black and white beads. Two strings
are equivalent if one can be rotated to resemble
the other. For example, the center and left
strings are equivalent to each other but not
equivalent to the one on the right.
a. Show that the relation “x  y if x can be rotated to resemble y” is an equivalence
relation on the set of strings of black and white beads.
b. Exactly two binary necklaces have one bead: the necklace with one black bead and
the necklace with one white bead. Draw representatives of the three binary necklaces
that have two beads.
c. Draw the four binary necklaces that have three beads.
d. Draw the binary necklaces that have four beads.
3. Here is a table representing the
binary necklaces of six beads.
Write the number of members of
each equivalence class in the center
of the corresponding necklace.
How are the numbers you wrote
related to 6?
4. A binary bracelet is an equivalence
class of strings of black and white
beads. Two strings are equivalent
if one can be rotated to resemble
the other or one can be reflected in an axis (“flipped”) to resemble the other. Use the table to
determine the number of binary bracelets with six beads. Draw arrows connecting the strings
that represent the same bracelet.
In music, binary necklaces and bracelets are used to represent equivalence
classes of scales, chords, and even repeated rhythms. Typically, you use a
black bead to indicate a pitch that is sounded. So, for example, you can
represent a chromatic scale by a string of twelve beads. The C major chord
(C-E-G = {0, 4, 7}) is represented like this:
Any rotation of this string corresponds to a different major chord. Musicians
call the action of shifting all the pitches by the same amount transposition. So, we see that
“major chords” correspond to an equivalence class of binary strings. Any reflection of this string
in some axis corresponds to a minor chord. Musicians call this operation inversion.
5. Which transpositions take C major to A major? There is more than one correct answer; be as
general as you can (hint: modular arithmetic).
6. Draw on the diagram the axis of reflection needed to take C major to C minor ({0, 3, 7}).
In general, a chord is a set of pitch classes, and a chord type is an equivalence class of chords,
where two chords are equivalent if one can be transposed so that it is identical to the other. A set
class is also an equivalence class of chords, where both transposition and inversion in an axis are
allowed.
7. Chord types are represented by binary _______________ and set classes by binary
______________.
8. How many different chords belong to the chord type of major triads? Of minor triads? How
many different chords belong to the set class of major and minor triads?
9. A three-note chord that contains two major thirds is called an augmented triad (C-E-G# is an
example). How many different chords belong to the chord type of augmented triads? List
them. How many different chords belong to the set class of augmented triads? List them.
A scalar interval is the interval between two pitch classes in a scale. For example, a second is
the interval between adjacent pitch classes, a third is the interval between two consecutive
seconds, etc. (There’s no such thing as a “first”; you would call the “interval” between a pitch
class and itself unison.) There are two different kinds of seconds in the diatonic scale; five of
these have two semitones and two are one semitone.
10. How many different thirds are there? How many are there of each kind? How many
semitones does each kind contain?
11. How many different fourths are there? How many are there of each kind? How many
semitones does each kind contain?
12. How many different fifths are there? How many are there of each kind? How many
semitones does each kind contain?
13. How many different sixths are there? How many are there of each kind? How many
semitones does each kind contain?
14. How many different sevenths are there? How many are there of each kind? How many
semitones does each kind contain?
15. What is the relationship between thirds and sixths? Fourths and fifths?
16.