22b Lesson - Pitch-class integer notation

Class discussion

Welcome to post-tonal analysis!

The importance of pitch notation in tonal harmony

• In this example, we talked about determining implied key and progression just from looking at the accidentals, harmonic intervals, and resolution
• Both measure contain a a tritone, and they each resolve in a different way: one outward, one inward. This results in resolutions to different keys from enharmonically equivalent chords

Happy Birthday in G Major (a la R. Strauss)

• Welcome to pitch class integer notation!!
  • Remember we only use 0-e in this form of notation. 0 is also 12 (and 1 is also 13, 2 is also 14, etc…)
• In this system, there is no difference between C# and Db. Both are written as 1. While it’s important to distinguish these notes in tonality, we’re post-tonal! The notes don’t matter
  • All enharmonic equivalents are called a pitch class. This also includes double sharps and double flats. For example, C#, Db#, and Bx are all in the same pitch, class, represented as 1
• What reasons are there to use this system?
  • Quickly identifying pitches (or pitch classes)
  • Identifying non-diatonic scales?
• Is C always 0?
  • In fixed zero notation, yes! There is also movable zero notation. Much like fixed and movable do, both of these notations have their uses.

Scales labeled using multiple zeros in pc integer notation

• Notice each scale is made up of a repeating 4-number pattern. In this example we can see a major scale is based around 0245, Dorian mode is based around 0235, and HW octatonic is based around 0134
• Movable zero is helpful for this kind of thing because we can quickly identify patterns in smaller things, like the scales here. It identifies the specific intervallic framework of a section, based around one note

A pitch-class set (pcs) is a collection of pitch classes. Cardinality is the number of pitches within a pcs. 0245 is a pcs, and since it has 4 pitch classes in it we call it a tetrachord. We run into problems with some of these cardinality terms because some of the words are the same as ones we associate with tonality

What is the application of an empty set?

• We need it for when we start inverting things later. An empty set is the inverse of an aggregate, containing all twelve pitch classes

Our “normal” scales and modes would be called septachords, because they contain seven pitch classes.

Can you apply this system of notation to tonal music?

• You can! The example Dr. Butterfield pulled up in class was Hindemith’s Trumpet Sonata, which he called “a mess to play” but went through using pc integer notation to look at the piece intervallically and not have to transpose as much

Further Reading

From Open Music Theory

Tonality is highly charged system where scale degrees are endowed with a magnetic or gravitational pull towards other tones. Within the key of C major, B-natural is attracted to the tonic C, other members of the dominant triad are attracted to the tonic triad, and other scale degrees have functions as well. Spelling is extremely important within common-practice tonality. A-flat — as lowered scale degree 6 — leads to G, but G-sharp — as raised scale degree 5 — leads to A. Underlying “scale degree” and “spelling” are two important concepts that will influence our study of post-tonal music.

Octave Equivalence

”Scale degree” implies an equivalence between pitches that are spelled the same but any number of octaves apart. C4 is the same as C3 is the same as C9, and so on. The concept of scale degree, then, has the idea of octave equivalence embedded within it.

Enharmonic Equivalence

Though octave equivalence is central to our understanding of tonal music, enharmonic equivalence often is not. In the key of C major, A-flat and G-sharp are not equivalent, though in isolation they sound the same. Spelling often indicates tendency: A-flat falls to G and G-sharp rises to A.

In post-tonal music, enharmonic equivalence is often assumed — with exceptions of course. Because many composers no longer felt constrained by a tonal center, the same gravitational relationships amongst tones that we find in tonal music aren’t important. A-flat and G-sharp, therefore, can be treated as representations of the same thing.

Pitch

Pitches are discrete tones with individual frequencies. The concept of pitch, then, does not imply octave equivalence. C4 is a pitch, and it is not the same pitch as C3.

Pitch class

Pitch classes are pitches under octave equivalence that are also spelled the same. A4, A3, A2, etc. are all members of the pitch class A.

Pitch Space

Integer notation

When analyzing post-tonal music where assuming octave equivalence and enharmonic equivalence is appropriate, we can use integers to represent pitch class. All C’s and any notes that are enharmonically-equivalent to C (B-sharp, for example) are pitch class 0. All C-sharps’s and any notes that are enharmonically-equivalent to C-sharp (D-flat, for example) are pitch class 1. And so on: C = 0, C-sharp = 1, D = 2, D-sharp = 3, E = 4, F = 5, F-sharp = 6, G = 7, G-sharp = 8, A = 9, B-flat = 10 (T), and B = 1 1 (E).

This type of pitch-class, which assumes octave and enharmonic equivalence is easily visualized on a clock-face diagram, like the one below.

Pitch-Class Space

Disclaimer!

Post-tonal music is extremely various. Composers have individual compositional styles, aesthetic goals, and unique conceptions of pitch. All this is to say that you must approach a composition with flexibility. For example: because it is quasi-tonal, Debussy’s music often benefits from a view that does not assume enharmonic equivalence. But sometimes it does. You must rely on your musical intuitions when analyzing this music, and you should also be willing to approach pitch in these compositions from multiple perspectives until you find one that seems most appropriate.

Atonal glossary

this glossary is far from complete, in the very early stages of being built

collection – The general term for treating multiple pitch classes as a single entity. Sets, set classes, scales, simultaneities, chords, and intervals are all specific kinds of collections.

interval class – The number of semitones between two pitch classes, counted as the shortest distance between them on a clock face. For instance, C and E make an interval class of 4. This is always the case, no matter which pitch is higher or lower, because interval class is concerned only with pitch classes. Interval classes are labeled ic1, ic2, . . . ic6. (There are none smaller than ic1 or larger than ic6.)

interval vector – The interval vector of a set class describes all of the interval classes present in a set class. There are six interval classes (1–6). The interval vector gives the number of each of those intervals in order from 1 to 6, within angle brackets. An interval vector of <101102> means that the set has one ic1 (semitone/major seventh), no ic2s, one ic3 (minor third/major sixth), one ic4 (major third/minor sixth), no ic5s (perfect fourths), and two ic6s (tritones).

ordered pitch interval – The number of semitones from one pitch (not pitch class) to the next. Ascending intervals are denoted by positive numbers, descending intervals by negative numbers. Examples: B4–G5 would have an ordered pitch interval of 8 (eight ascending semitones). B3–G5 would be 20. B4–G4 would be –4.

ordered pitch-class interval – The number of ascending semitones from one pitch-class to another. G–B is four semitones, for an ordered PC interval of 4. B–G is eight ascending semitones, for an ordered PC interval of 8.

The ordered pitch-class interval is also the modulo12 version of the ordered pitch interval. For example, B4–G4 is –4 semitones (4 semitones down). mod12(–4) = 8. C3–D4 is 14 semitones. mod12(14) = 2.

pitch – A pitch class in a specific register, such as C4 (middle C).

pitch class – One of the twelve steps on the chromatic scale, summarized by a note name (C, D-sharp, B-flat, etc.) or a number 0–11 (C = 0, C-sharp = 1, . . . B = 11). In atonal music, spelling rarely matters except to make performance easier, so enharmonically equivalent pitch classes are considered identical (C = B-sharp = D-double-flat = 0).

pitch-class set – An unordered collection of pitch classes, usually grouped into curly brackets: {C, E, G}, {D, E-flat, G}, or {4, 5, 9}.

pitch-class set class or simply set class – A category of pitch-class sets that are all related by transposition or inversion. For example, the 12 major triads are all related by transposition. While each major triad is a different pitch-class set, they all belong to the same set class (the same category of sets). Note that minor triads are upside-down major triads (minor third–major third, instead of major third–minor third). Thus since major and minor triads can be related by inversion, they belong to the same set class. Set classes are typically named according to their prime form (see prime form in this glossary).

prime form – Since set classes come in as many as 24 different forms (12 transpositions times 2 inversions), one of those forms is chosen as its name or referential form, for ease of categorization. That form is the prime form. The prime form is, in a nutshell, the inversion and rotation of the set class that keeps the pitch classes most tightly packed on and above C (0).

For help finding the prime form of a set, Jay Tomlin’s set theory calculator can be helpful. The following video demonstrates how to use it.

SetTheoryCalculator from Kris Shaffer on Vimeo.

simultaneity – Any collection of more than one pitch (class) that sound at the same time. This includes dyads/intervals, chords, clusters, and “salami slices” of contrapuntal textures.

unordered pitch-class interval – A regular simple chromatic interval: the number of half steps between two pitches. Compound intervals (larger than an octave) are typically reduced to their corresponding simple interval. They are labeled with a lower-case i: i4 is a major third, for example.