Chapter 27 Post-Tonal Theory

Procedure to determine the normal form of a PC Set

1. List the ascending orderings of the pc set. The number of ascending orderings is equal to the number of members in the set. 2. The ascending ordering with the smallest outside interval is the normal form. 3. If a tie results from the previous step, compare the next most outside intervals (first and second last, first and 3rd last, first and 4th last, etc.) 4. If there's still a tie after that then select the ordering that begins with the smallest first pc integer

All-interval series

11 different intervals make up a series.

Modulo 12 systems (mod 12)/Mod 12 arithmetic

12 equivalents labeled as integers 0-11. Subtraction and addition from 12 for a number between 0-11. Ex: clock (12 hours)

Total serialization (integral serialism)

Ordering of other musical parameters (duration, dynamics, mode of attack).

Series (aka row)

Ordering of the 12 pitch classes determined by the composition in a 12 tone composition.

Pitch class sets (pc sets)

Partitioning of atonal music into meaningful groups of related pitch classes. 1st step in analyzing a piece of atonal music. Term introduced by composer and theorist Milton Babbitt.

Retrograde inversion

Reversed order of inverted form

Retrograde

Reversed order of transposed form If go up an interval, retrograded will be go down that interval

Ascending orderings

Rewrite notes of a grouping into closest ascending order.

Interval vector (interval class vector)

Rough approximation of the quality of a sonority by listing the number of occurrences of each interval class (ic)

Inversionally symmetrical sets

Sets that map onto themselves under T(sub)nI (capital i) Have fewer than 24 distinct forms

Transpositionally symmetrical sets

Sets that map onto themselves under T(subscript)n at levels other than T(sub)0.

Outside interval

The interval between the first and last note of an ordering. If there's a tie for the smallest outside interval then look at next most outside interval (interval between first and next to last notes).

Segmentation

The meaningful groups of related pitches grouped based on relationships created by pitch, rhythm, phrasing, register, timbre, etc.

Interval class

The shortest distance between two pitch classes. ic1 m2 M7 ic2 M2 m7 ic3 m3 M6 ic4 M3 m6 ic5 P4 P5 ic6 A4/d5

Normal form/Normal order

To arrange members of a pc set into an arbitrary ordering that is most compact.

Transposition (Tn - subscript n)

To transpose a pc set: Add transposition number, n to each member of the set in turn.

Prime form

A convenient way to name a set class. Begins with 0 and are most packed to the left Used by Joseph Straus: Notates using parentheses without intervening commas or spaces Substitutes T for 10 and E for 11 in prime forms

Atonal Theory/Set Theory

A different way of analyzing music in a systematic way for the new pitch structure that composers are using.

Atonal (pantonal) works

A musical composition thata voids reference go a tonal center or centers.

Cardinality

A way of classifying a set by the number of member it contains. Empty set 0 members Monad 1 member Dyad 2 members Trichord 3 members Tetrachord 4 members Pentachord 5 members Hexachord 6 members Septachord 7 members Octachord 8 members Nonachord 9 members Decachord 10 members Undecachord 11 members Aggregate 12 members

Pointillism

Atomization of a melodic line.

Twelve-tone method

Compositional technique of post-tonal era 1. Avoidance of octave as either a melodic or harmonic interval 2. Avoidance of traditional pitch collections, such as diatonic scale, that might suggest major or minor triads and hence traditional harmonic functions 3. Use of wide-range and extremely disjunct melodies Designed to equalize all 12 pitch classes by: 1) A 12 tone composition is based on precompositional ordering of 12 pitch classes determined by composer 2) No pitch class can be used again until all other pitch classes have been sounded (exception: a pitch class can be repeated in context of common musical figures as trills, tremolos and pedal points) 3) The series may be used within confines of procedure in transposition, retrograde, inversion, or retrograde inversion, the 4 basic transformations at the foundation of 12 tone system

Klangfarbenmelodie (sound color melody)

Deliberate juxtaposition of minute melodic fragments of contrasting timbre and register. Melody in a sense is created by rapid shifting of tone colors

Inversion (TnI - subscript n and cap I)

Descending half steps from 0 instead of ascending. Not a reflection 1=11 2=10 3=9 Etc Write 0-11 then that backwards

Order numbers

Indicates the relative order position of each pitch class within a particular series form.

Transposition index

Indicates first pitch class of a series Also notated as subscript x

Inversion

Mirroring of each interval

Forte Name

More compact way to refer to a set class Theorist Allen Forte assigned each prime forms compact name based on its cardinality and relative position In analysis, a set class is usually referred to by its Forte name, prime form or both