strausch1
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BAIN MUSC 525
Post-Tonal Theory
Straus Ch. 1 Overview, Page 1 of 2
Straus Chapter 1 Basic Concepts and Definitions
Joseph N. Straus, Introduction to Post-Tonal Theory , 3rd ed. (Upper Saddle River, NJ: Prentice Hall, 2005).
“To appreciate [a] painting fully, you have to be willing to move from place to place.
One of the specially nice things about musicis that you can hear a single object like an interval in many ways at once.”
– Joseph N. Straus, Introduction to Post-Tonal Theory
TERMS & CONCEPTS
Twelve-tone equal temperament,
abbr. 12tet
Semitone (1, the unit interval)
Equivalence
Equivalence relation1
• Octave equivalence
• Enharmonic equivalenceEquivalence class
Pitch & Pitch-Class Notation
Pitch
• Frequency
• U.S. Standard pitch notation
(C4 = middle C)
• Staff notation
Pitch class, abbr. pc
Letter notation
Integer notation
• Fixed-zero [C=0]
• Movable-zero: e.g. [A=0]
PC clockface diagram
Mod 12
• Modular arithmetic
• Modulus
• Congruence1 ! 11 + 2 (mod 12)
Pitch Spaces
Linear pitch space, abbr. p-space
Modular pitch-class space,
abbr. pc-space
Intervals
Traditional tonal interval names
Ordered (melodic)
• Direction (signed, +/-)
• Magnitude (absolute value)
Unordered (harmonic)
Interval Types
1. Ordered pitch interval (opi)
2. Unordered pitch interval (upi)3. Ordered pc interval (opci)
4. Interval class (ic)
Interval Class
Interval class, abbr. ic
Straus ic scoreboard
Interval content
Interval vector
• Unique multiplicity of ic
Mod 12 complement
INTERVAL TYPES
Space Interval Type Abbr.2 Values3 Definition
4 Description
p- Ordered pitch interval (opi) ip -87 to 87 ip< x, y> = y- x Melodic distance between two pitches
Direction (+/-) and magnitude (in semitones)
Unordered pitch interval (upi) 0 to 87 ip( x, y) = | y- x| Harmonic distance between two pitches
Magnitude only
pc- Ordered pitch-class interval (opci) i 0 to 11 y- x (mod 12) Melodic distance between two pitch classes
Clockwise distance (in semitones) on the
pc clockface diagram
INTERVAL CLASS (ic)
Unordered pitch-class interval (upci)
ic 0 to 6 The lesser of
y- x (mod 12) and
x-y (mod 12)
Harmonic distance between two pitch classes
Shortest distance (in semitones) on the
pc clockface diagram
Straus says: “Which one we use will depend on what musical relationship we are trying to describe.”
EXAMPLE
p-space pc-space
Pitch interval opi upi opci ic
A4 – Gƒ3 -13 13 11 1
1 For precise definitions of mathematical terms and concepts see MathWorld--A Wolfram Web Resource. Available online at:
<http://mathworld.wolfram.com>.2 Abbreviation used in Straus 2005 and throughout much of the atonal theory literature.3 For a linear p-space modeled by a 12tet piano keyboard (C4=0, with boundary conditions A0 = -39 & C8 = 48) and modular pc-space
modeled by a pc clockface diagram.4 |x| is the absolute value of x.
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BAIN MUSC 525
Post-Tonal Theory
Straus Ch. 1 Overview, Page 2 of 2
PITCH-SPACE VS. PITCH-CLASS SPACE
In post-tonal theory, a 12tet pitch space ( p-space) is usually assumed. The 88 keys of the piano provide
a convenient reference for envisioning this linear space. The semitone has a magnitude of 1. It’s size
may alternatively be described as:
212
=21/12
!
1.059
P-space may be collapsed into a pitch-class space ( pc-space) under octave equivalence (modeled by
mod 12). The pc clockface diagram provides a convenient reference for envisioning this modular space.
See Morris 1987, pp. 23-26 for more information.
QUOTABLE
“[Pitch] is the intersubjective correlate of frequency.”
“Most musical systems quantize the pitch-continuum into a
finite number of pitches ordered from low to high.”
“[Relative pitch]...allows equal frequency ratios to be heard as equal pitch differences,
different intervals can be distinguished and grouped into classes.”
– Robert Morris, Composition with Pitch Classes5
“We locate pitches in an extended pitch space,
ranging in equal-tempered semitones from the lowest to the highest audible tone.”
“A broad consensus has emerged among music theorists regarding the
basic musical elements of post-tonal music–pitch, interval, motive, harmony, collection....”
“The quality of a sonority can be roughly summarized by listing all the intervals it contains.”
“Even though the interval-class vector is not as necessary a tool for tonal music as for post-tonal music,
it can offer an interesting perspective on traditional formations.”
– Joseph N. Straus, Introduction to Post-Tonal Theory
* * *
“Varèse’s octave is symmetrically partitioned by the tritone…
instead of asymmetrically by the fifth degree of the diatonic scale
into a perfect fifth and perfect fourth.”
– George Perle, The Listening Composer6
* * *
“I have chosen to define a good ear as one that perceives and retains musical structures
and understands their role in a musical transformation or other compositional process….
A good ear is an ideal toward which all musicians work in some fashion.”
– Michael L. Friedmann, Ear Training for Twentieth-Century Music7
5 Robert Morris, Composition with Pitch Class (New Haven: Yale University Press, 1987).6 George Perle, The Listening Composer (Berkeley: University of California Press, 1990).7 Michael L. Friedmann, Ear Training for Twentieth-Century Music (New Haven: Yale University Press, 1990).