MUSIC 371 - Music Theory 4

Karen Sunabacka

Estimated reading time: 9 minutes

Table of contents

The Twilight of the Tonal System

The end of Tonality - da da daaaaa.


  • Horton & Ritchey: Chapter 30, Chapter 31: V9, V11, V13.
  • Kostka: pp 1-2 and Chapter 2 (pp17-33). Things to focus: enharmonics and scale formations (whole-tone, diatonic modes, and octatonic scales).

Charles Ives - Three Quarter-Tone Pieces

What is Music Theory?

Readings: Palisca, Claude V. & Ian D. Bent, “Theory, theorists” from Grove Music Online.

Schenkerian Analysis 1. Introduction

Jazz Music Theory



Major keys

Ionian I
Dorian II
Phrygian III
Lydian IV
MIxo Lydian V
Aeolian VI
Locrian VII

Relative minor keys

Aeolian I
Locrian II
Ionian III
Dorian IV
Phrygian V
Lydian VI
MIxo Lydian VII

Nonserial Atonality

This is based on the chapter 9 of Kostka.

Characteristics of atonal music

First, not pitch-centric.

The pitch aspect of atonal music required a new vocabulary: cell, basic cell, set, pitch set, pitch-class set, and referential sonority. The analysis of atonal music usually includes the process of identifying and labeling these important pitch sets, a process that involves segmentation.

Segmentation is in some ways much more difficult than the analysis of chords in traditional tonal music. Two problems

  • when beginning the analysis, one usually does not know which sets will or won’t turn out to be significant in the piece.
  • labeling the sets for ease of comparison

As we shall see, Forte’s system of pitch-class sets reduces the number of combination of pitches sets considerably.

or PCI

Number for each note name (C=0):

Note name Pitch class Note name Pitch class
C 0 (or 12) F#/Gb 6
C#/Db 1 G 7
D 2 G#/Ab 8
D#/Eb 3 A 9
E 4 A#/Bb 10
F 5 B 11
PCI Traditional Label PCI Traditional Label
0 P1 (unison) 6 A4, d5 (tritone)
1 m2 7 P5
2 M2 8 m6
3 m3 9 M6
4 M3 10 m7
5 P4 11 M7

Enharmonic equivalence always applies when dealing with pitch classes. Through octave equivalence, we identify C major triads. In addition, we consider C major triads and F major triads to be transpositionally equivalent. To analyze and compare the pitch-class sets in atonal music, we need a process that will reduce any set to some basic form: normal order. The normal order of a pitch-class set is that ordering of the PCs that spans the smallest possible interval.

  • begin the scale on any of the PCs,
  • leave out any duplicated PCs, but continue up to octave.
  • find largest interval between any two adjacent PCs
  • the top note of this largest interval is the bottom note of the normal order.

A complication that occasionally arises is seen in a set that has no single largest interval, but instead has two or more intervals that are tied for largest. The tie is broken by comparing the intervals between the first and next-to-last notes in both versions. The normal order is the version with the smaller interval. If the intervals between the first and next-to-last notes had been the same, we would have proceeded to the intervals between the first and third-to-last notes, and so on, until the tie was broken. In some sets, however, the tie cannot be broken, and in such cases we choose the ordering that begins with the smallest pitch-class integer.

The interval successions in the two versions are identical (because they are transpositionally equivalent), so it is impossible to break the tie. A set such as this is called a transpositionally symmetrical set, because it reproduces its own pitch-class content under one or more intervals of transposition.

Pitch-class sets that are related by inversion to be equivalent. This is called inversional equivalence.

Best normal order. This concept is important because the best normal order is the generic representation of all possible transpositions and inversions of a set.

  • first find its normal order and notate its inversion.
  • normal order of the inversion
  • compare the two normal orders: the “better” of the two is considered to be the best normal order.

A set, in which the normal order of the set and the normal order of its inversion are identical or transpositionally equivalent, is called an inversionally symmetrical set because it reproduces its pitchclass content at one or more levels of inversion.

After applying numbers to the best normal order, the resulting series of numbers is called the prime form, and it represents all of the pitch-class sets in that set class.

Because inversional equivalence is still in effect, we then have six interval classes (“interval class” is sometimes abbreviated as IC):

Interval Class Pitch-Class Interval Traditional Interval
1 1, 2 m2, M7
2 2, 10 M2, m7
3 3, 9 m3, M6
4 4, 8 M3, m6
5 5, 7 P4, P5
6 6 A4, d5

To analyze a set according to its interval content, tabulate all of the ICs between each note in the set and all of the notes above it. Consider a set (example 9-17 in the textbook). there should be a image...

The table below demonstrates the procedure:

From Up to IC
G Ab 1
G B 4
G C# 6
Ab B 3
Ab C# 5
B C# 2

It contains exactly one occurrence of each IC. This information is usually presented in the form of an interval-class vector (or ICV). So for this set, because it contains one of each IC, has an ICV of <111111>.

Pairs of sets that share the same vector (they come only in pairs) are known as Z-related sets.

This means that if we transpose a triad up or down by a minor 3rd, a major 3rd, or a perfect 4th, exactly one pitch class will be held invariant — that is, it will be retained.

An inversion matrix will allow us to predict how many PCs will be held invariant under inversion. For example, we convert (D, Eb, G, A) set to (2, 3, 7, 9), then

Invert the set, we get (3, 5, 9, 10). This is considered the 0th transposition. If you look in the matrix, you will find two occurrences of the number 0, which means that two PCs will be held invariant — in this case PCs 3 and 9. There is one occurrence of the number 2, so transposing by PCI 2 will keep one PC invariant: (5,7,11,0), and so on.

Forte Labels

Allen Forte’s Structure of Atonal Music.

Prime form of a set, then look up its Forte Label as well as its ICV.

Every time we see an “Z” in the label, for example, 4-Z29, it means its IVC is shared with another a pitchclass set.


Sometimes pitch-class sets that do not belong to the same set class may be related to each other by sharing common subsets, just as C7 and Cadd6 are closely related because they have the C major triad in common.

literal subset: literally the same pitch class. nonliteral subsets, related by transposition or inversion, but do not literally share the same three pitch classes.

A special kind of subset is the scalar subset, which is a subset that is derived from a particular scale type.


The term aggregate is used to refer to any such statement (all 12 pitch classes be heard within a fairly short period) of all 12 pitch classes, without regard to order or duplication.

Check here for a list of Post-Functional Theory Terminology.

Serialism (The 12-Tone Row)

Chapter 10 of Kostka

12tone videos:

Balinese Music

The notes are (roughly) C#, D, E, G#, A.

Core melody and melody

layout in the video

gong cycle        polo        core melody
beat keeper       sangsih     melody

melody: cannot have the same note repeated twice

  • First option: anticipate target note with the one lower neighbour note
  • second option: anticipate target note with the one higher neighbour note
  • also we can mix: one higher one lower


Polo interlocking anticipating the melody. Sangsih is based on polo.

Neo-Riemannian Theory

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