science

Pitch class space

Pitch class space is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Pitch class space rather than just read about it. In short: In music theory, pitch-class space is the circular space representing all the notes (pitch classes) in a musical octave. In this space, there is no distinction between tones separated by an integral number of octaves.

Key takeaways

Pitch class space belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Pitch class space to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Pitch class space from memory before moving on to harder problems.

Reference excerpt

In music theory, pitch-class space is the circular space representing all the notes (pitch classes) in a musical octave. In this space, there is no distinction between tones separated by an integral number of octaves. For example, C4, C5, and C6, though different pitches, are represented by the same point in pitch class space. Since pitch-class space is a circle, we return to our starting point by taking a series of steps in the same direction: beginning with C, we can move "upward" in pitch-class space, through the pitch classes C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, and B, returning finally to C. By contrast, pitch space is a linear space: the more steps we take in a single direction, the further we get from our starting point.

Tonal pitch-class space Deutsch and Feroe (1981), and Lerdahl and Jackendoff (1983) use a "reductional format" to represent the perception of pitch-class relations in tonal contexts. These two-dimensional models resemble bar graphs, using height to represent a pitch class's degree of importance or centricity. Lerdahl's version uses five levels: the first (highest) contains only the tonic, the second contains tonic and dominant, the third contains tonic, mediant, and dominant, the fourth contains all the notes of the diatonic scale, and the fifth contains the chromatic scale. In addition to representing centricity or importance, the individual levels are also supposed to represent "alphabets" that describe the melodic possibilities in tonal music (Lerdahl 2001, 44–46). The model asserts that tonal melodies will be cognized in terms of one of the five levels a-e:

(Lerdahl 1992, 113) Note that Lerdahl's model is meant to be cyclical, with its right edge identical to its left. One could therefore display Lerdahl's graph as a series of five concentric circles representing the five melodic "alphabets." In this way one could unite the circular representation depicted at the beginning of this article with Lerdahl's flat two-dimensional representation depicted above. According to David Kopp (2002, 1), "Harmonic space, or tonal space as defined by Fred Lerdahl, is the abstract nexus of possible normative harmonic connections in a system, as opposed to the actual series of temporal connections in a realized work, linear or otherwise."

See also Pitch circularity Pitch interval Scientific pitch notation Pitch constellation

Sources Deutsch, Diana & John Feroe (1981). "The Internal Representation of Pitch Sequences in Tonal Music". Psychological Review. 88 (6): 503–22. doi:10.1037/0033-295X.88.6.503. Full Text Kopp, David (2002). Chromatic Transformations in Nineteenth-Century Music. Cambridge University Press. ISBN 978-0-521-80463-9.{{cite book}}: CS1 maint: location missing publisher (link) Lerdahl, Fred; Jackendoff, Ray (1983). A Generative Theory of Tonal Music. MIT Press. ISBN 9780262120944. Lerdahl, Fred (1992). "Cognitive Constraints on Compositional Systems". Contemporary Music Review. 6 (2): 97–121. CiteSeerX 10.1.1.168.1343. doi:10.1080/07494469200640161. {{cite journal}}: Cite uses deprecated parameter |citeseerx= (help) Lerdahl, Fred (2001). Tonal Pitch Space. Oxford and New York: Oxford University Press.

Further reading Straus, Joseph (2005). Introduction to Post Tonal Theory (3rd ed.). Up Saddle River, New Jersey: Prentice Hall. ISBN 978-0-13-189890-5.

Illustrations

Worked examples

Example 1 — a first encounter with Pitch class space

Start with the simplest possible case. Write down what Pitch class space claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Pitch class space before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Pitch class space ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Pitch class space

In research: Pitch class space appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Pitch class space in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Pitch class space is common in secondary-school and first-year university syllabi. It links to neighbouring topics Musical set theory, Pitch space, so understanding it makes those chapters shorter.
In everyday life: Look for Pitch class space outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Pitch class space in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Pitch class space means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Pitch class space out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Pitch class space in simple terms?

Why does Pitch class space matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Pitch class space?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Pitch class space.