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:
  2. Set: all people; relation: “x y if x and y have the same astrological sign.”
  3. Set: all real numbers; relation: “x y if .”
  4. Set: all musicians in bands; relation: “xy if x plays in a band with y.”
  5. 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.)
  1. 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.
  2. 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.
  3. 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.
  4. Draw the four binary necklaces that have three beads.
  5. Draw the binary necklaces that have four beads.
  6. 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?
  7. 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.

  1. Which transpositions take C major to A major? There is more than one correct answer; be as general as you can (hint: modular arithmetic).
  2. 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.

  1. Chord types are represented by binary ______and set classes by binary ______.
  2. 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?
  3. 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.

  1. How many different thirds are there? How many are there of each kind? How many semitones does each kind contain?
  2. How many different fourths are there? How many are there of each kind? How many semitones does each kind contain?
  3. How many different fifths are there? How many are there of each kind? How many semitones does each kind contain?
  4. How many different sixths are there? How many are there of each kind? How many semitones does each kind contain?
  5. How many different sevenths are there? How many are there of each kind? How many semitones does each kind contain?
  6. What is the relationship between thirds and sixths? Fourths and fifths?