Twentieth Century Pitch Theory:

Some Useful Terms and Techniques

 

Pitch -- a note with a specific octave position, e.g. "the A-flat below middle C", "D-natural4", etc. In tonal music, spelling counts; there is an enormous functional difference found in tonal music between B-flat and A-sharp. In much twentieth century music, by contrast, the spelling of a pitch is a matter of convenience; functional differences between enharmonically equivalent pitches such as those found in tonal music may be lacking completely.

 

Pitch Class (p.c.) -- The identity of a note without regard for its octave position or spelling. All B-flats in all octaves are members of the same pitch class; A-sharp and B-flat are members of the same pitch class.

 

Pitch Interval (ip) -- The distance between two pitches. In much twentieth century music, the distinction between dissonance and consonance (i.e. between intervals that suggest repose and intervals that suggest tension) is absent: in analyzing 20th c. music, therefore, intervals are often measured in semitones: a major third may not be functionally equivalent to a diminished fourth in tonal music, but they both can be expressed as ip4, i.e. as a pitch interval of four semitones. (ip +4 denotes an ascending ip4, ip &endash;4 a descending ip4.)

 

Interval Class (i.c.) -- The distance between two pitch classes. The identity, or size, of an interval expressed as a number of semitones without regard to (1) spelling, (2) octave compounding, or (3) inversion; the minor third C-natural4 -E-flat4, the major sixth E-flat4 - C-natural5, and the minor tenth C-natural4 - E-flat5 are all members of the same interval class (see chart).

Interval Class (i.c.)

 

Includes

 

1

 

m2, M7, m9 and their octave compounds

 

2

 

M2, m7, M9 and their octave compounds

 

3

 

m3, M6, m10 and their octave compounds

 

4

 

M3, m6, M10 and their octave compounds

 

5

 

P4, P5, P11, P12 and their octave compounds

 

6

 

A4, d5, A11, d12 and their octave compounds

Pitch Collection -- any group of pitches. If we remove octave duplicates we get a

 

Pitch Class Collection (or Pitch Class Set) -- a group of pitch classes, considered without regard to their order or to duplication of content. The p.c. membership of a scale or mode may be termed a collection, as can any other combination of pitch classes.

A p.c. set can have any number of members from 1 to 12. A collection containing six members is called a hexachord, one with five members a pentachord, one with four members a tetrachord, one with three members a trichord, and a collection or p.c. set with two members a dyad. (Sets larger than a hexachord are sometimes referred to as heptachords, octachords, nonachords, etc., but just as often will be termed "7-p.c. sets", "8-p.c. sets", etc.)

 

Aggregate -- a collection of all twelve pitch classes; a 12-p.c. set.

 

Integer Notation -- a "spelling-neutral" way of identifying the twelve pitch classes. By convention, C-natural is used as the point of origin for the chromatic aggregate; this in no way implies any special status for that (or any other) p.c. All spellings of C-natural (e.g. B-sharp, D-flat-flat) are assigned integer 0, all spellings of C-sharp (e.g. D-flat) integer 1, all spellings of D-natural (e.g. E-flat-flat) integer 2, etc. up to integer 11 (B-natural or C-flat).

 

 

p.c.

 

possible spellings

 

0

 

C-natural, B-sharp, D-double flat

 

1

 

C-sharp, D-flat, B-double sharp

 

2

 

D-natural, etc.

 

3

 

E-flat, D-sharp

 

4

 

E-natural, F-flat

 

5

 

F-natural, E-sharp

 

6

 

F-sharp, G-flat

 

7

 

G-natural, etc.

 

8

 

G-sharp, A-flat

 

9

 

A-natural, etc.

 

10

 

A-sharp, B-flat

 

11

 

B-natural, C-flat

Interval vector -- An inventory of the interval classes present in a p.c. set is called its interval vector. By convention, the interval vector is a six-digit number enclosed in [square brackets]. A set’s interval vector is reckoned by starting with the first p.c. of the set and finding the interval class of which the interval between it and each other p.c. is a part, moving to the next p.c. of the set and computing the number of half steps between it and each remaining p.c. in the set, and repeating the procedure until you reach the set’s last p.c.

Example: The pentachord C, D, F, G, A.

(1) The interval class between C and D is 2; between C and F is 5, between C and G is also 5; between C and A is 3. (See tally below.)

 

i.c. 1

 

i.c. 2

 

i.c. 3

 

i.c. 4

 

i.c. 5

 

i.c. 6

 

 

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(2) The i.c. between D and F is 3 (5 - 2 = 3); between D and G is 5; between D and A is 5 (9 - 2 = 7, which is a member of i.c. 5). (See tally below.)

i.c. 1

 

i.c. 2

 

i.c. 3

 

i.c. 4

 

i.c. 5

 

i.c. 6

 

 

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(3) The i.c. between F and G is 2; between F and A is 4. (See tally below.)

i.c. 1

 

i.c. 2

 

i.c. 3

 

i.c. 4

 

i.c. 5

 

i.c. 6

 

 

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||

 

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(4) The i.c. between G and A is 2. (See tally below.)

i.c. 1

 

i.c. 2

 

i.c. 3

 

i.c. 4

 

i.c. 5

 

i.c. 6

 

 

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The interval vector of the set, which may be expressed in integer notation as (0, 2, 5, 7, 9), is [034140], because the set contains

no intervals of i.c. 1,

3 intervals of i.c. 2,

2 of i.c. 3,

1 of i.c. 4,

4 of i.c. 5,

and none of i.c. 6.

What is the interval vector of: the whole tone scale? the major scale? a dominant seventh chord? a fully diminished seventh chord?