IX. Post-tonal music
- A pitch class set (pc set) is a group of pitch classes.
- Normal order is a way of naming a pitch class set.
- Normal order is the smallest possible arrangement of pitch classes, in ascending order.
- To transpose a set by Tn, add n to each integer of the set.
- To invert a set by In, first, invert the set (take each integer’s complement mod 12), then transpose by n.
- The clock face may help you perform any of these tasks.
Pitch class sets
When we talk about a group of pitch classes as a unit, we call that group a , often abbreviated pc set. Any group of pitch classes can be a pitch class set.
is the most compressed way to write a given collection of pitch classes, in ascending order. Normal order has a lot in common with the concept of . Root position is a standard way to order the pitch-classes of triads and seventh chords so that we can classify and compare them easily. Normal order does the same, but in a more generalized way so as to apply to chords containing a variety of notes and intervals.
Following are a mathematical and a visual method for determining normal order.
G♯4, A2, D♯3, A4
|1. Write as a collection of pitch classes (eliminating duplicates) in ascending order and within a single octave. There are many possible answers.||8,9,3|
|2.Duplicate the first pitch class at the end.||8,9,3,8|
|3. Find the largest ordered pitch-class interval between adjacent pitch classes.||8 to 9: 1
9 to 3: 6
3 to 8: 5
9 to 3 is the biggest interval.
|4. Rewrite the collection beginning with the pitch class to the right of the largest interval and write your answer in square brackets.||[3,8,9]|
Occasionally you’ll have a tie in step 3. In these cases, the ordering that is most closely packed to one side or the other is the normal form. If there is still a tie, choose the set most closely-packed to the bottom.
Visual method (clock face method)
If you don’t like the processes described above, this video clearly explains how to use the clock face to quickly find normal order.
In post-tonal music, transposition is often associated with motion: take a chord, motive, melody, and when it is transposed, the aural effect is of moving that chord, motive, or melody in some direction. That’s the effect in, in two disconnected passages from Claude Debussy’s La cathédrale engloutie. The opening motive [B, D, E] or [11, 2, 4] is transposed four semitones higher in m. 18, representing the cathedral’s slow ascent above the water.
Transposing something preserves its intervallic content, and not only that, it preserves the specific arrangement of that thing’s intervals. When we hear the passage at m. 18 above, we recognize its relationship to the passage in m. 1 because the same intervals return, but starting on a different pitch.
Transposition is an operation—something that is done to a pitch, pitch class, or collection of these things. Alternatively, transposition can also be a measurement—representing the distance between things.
Transposition is often abbreviated Tn, where n represents the ordered pitch-class interval between the two sets. n is called the of this transformation.
Transposing a set
To transpose a set by Tn, add n to every integer in that set ().
Given the collection of pitch classes in m. 1 above and transposition by T4:
The result is the pitch classes in m. 18.
T4 [11, 2, 4] = [3, 6, 8]
Identifying transpositions and calculating the index number
To determine the transpositional relationship between two sets, subtract the first set from the second. If the numbers that result are all the same, the two things are related by that Tn. For example, to label the arrow in, an analyst would “subtract” the pitch class integers of m. 1 from the pitch-class integers in m. 18. Note that both sets should be in .
[3, 6, 8] and [11, 2, 4] are related by T4.
If you are asked to invert a set and are not given an index number, assume you are inverting the set mod 12. This means taking the of each number . The complement of each integer x is the number y that is the difference between x and 12. For example, the complement of 4 is 8: 4+8=12. The complement of 6 is 6: 6+6=12. The complement of 0 is 0: 0+0=0, which is 12 mod 12.
Inverting [2, 4, 7] in this way would yield [5, 8, 10].
In: Invert-then-transpose method
But sometimes, sets are both inverted and then transposed, as in. The abbreviation for this is In.
In, the first set [2, 4, 7] is inverted by I8. To invert a set by I8 follow this process, in this order:
- Invert: [2, 4, 7] becomes [5, 8, 10].
- Transpose: adding 8 to every number in [5, 8, 10] yields [1, 4, 6].
In: Subtraction method
You can calculate the new set created by In by subtracting all the pitch classes of your first set from n.
What is I8 of [2, 4, 7]?
I4 [0, 2, 7] = [1, 4, 6].
Identifying inversions and their index numbers
Any two pitches related by inversion can be added together to form the . This makes sense as a logical extension of the subtraction method above: if the inverted pitch y is the result of n–x, then it is also true that n = x + y.
Another way to visualize this is on the clock face.If you have two sets that are 1) both in normal order and 2) related by inversion, the notes within each set will map onto one another in reverse order, as shown inbelow. Write the two sets in normal form on top of one another, then add the opposing integers of each set together as illustrated in to yield the index number of the I relation. If the sum of each number pair is 12 or more, subtract 12 so that your n is in .
If you prefer a more visual method for transposing and inverting, watch the video below.
- Straus, Joseph Nathan. 2016. Introduction to Post-Tonal Theory. 4th ed. Upper Saddle River, NJ: Prentice Hall.
- Worksheet on normal form and transformations (.pdf, .docx). Asks students to calculate normal form of various sets, and to calculate Tn/In relationships in “Nacht” by Arnold Schoenberg.
- Composition prep worksheet (.pdf, .docx). Prepares students for the set class composition by asking them to find sets and transformations.
- Inversion 1 © Megan Lavengood is licensed under a CC BY-SA (Attribution ShareAlike) license
- inversion-2 © Megan Lavengood is licensed under a CC BY-SA (Attribution ShareAlike) license
- Inversional pairs © Megan Lavengood
- Cross-addition for inversion © Megan Lavengood is licensed under a CC BY-SA (Attribution ShareAlike) license
A group of pitch classes.
The most compressed way to write a given collection of pitch classes.
Ordering the notes of a chord so that it is entirely stacked in thirds. The root of the chord is on the bottom.
In a transformation (Tn or In), n is the index number. n represents the interval of transposition in semitones.
Mod-12 is short for modulo 12, where numbers wrap around upon reaching 12. Arithmetic in mod-12 is most familiar through clock time: after 12-o-clock, the time becomes 1-o-clock again.
An integer x's complement mod 12 is the number y that would sum to 12. For example, 11's complement mod 12 is 1.