Theory V

MUSC 3560Unit #3

Review Slides

The “Mod-12 Squad”✦ Modulo (mod) 12 arithmetic

✦ Integers 0-11 inclusive

✦ Add or subtract 12 repeatedly to #s outside of range(Think military / international time: 23:00 - 12 = 11:00)

C

Ds

CsD

EFFsG

GsA

As B

Atonal analysis✦ Pitch class (pc)

✦ Any note in any 8ve (a C is a C is a C is a C...)

✦ Fixed-zero integer notation vs. moveable integer notation(C=0, C#/Db=1, etc.) vs. (F=0, F#=1, etc.)

✦ Enharmonic equivalence applies (C# = Db)

Atonal analysis✦ Pitch class set (pc set)

✦ Collection of pitches/integers

✦ Segmentation = intentional grouping of PCs into sets

✦ Grouped together via motive, melody, harmony, rhythmEx: (B, G#, G) = (11, 8, 7)

✦ Important: note the use of parentheses & commas between PCs

✦ Inherently unordered (PC sets may be ascending, descending, or a combo)

Atonal (a/k/a Set) Theory✦ Transposition (Tn)

✦ Add the transposition # to each member of set

✦ Inversion (Tn I)✦ Step 1: SUBTRACT the inversion # from mod 12

✦ Step 2: ADD the transposition # to each member & reorder

WARP ZONE SHORTCUT:

✦ Skip steps 1 & 2 and subtract the transposition # from each pc

I see...interval classes!✦ Interval class (ic):

✦ Shortest distance between 2 pitch classes

✦ Seven interval classes within octave*:m2/M7=ic1M2/m7=ic2 m3/M6=ic3 M3/m6=ic4 P4/P5=ic5A4/d5=ic6*Note: unison/P8 not normally included; it does exist, however as ic0

Atonal (a/k/a Set) Theory✦ Normal Order (Just say N.O.!) – a/k/a Normal Form

✦ Most compact ordering of notes from left to right

✦ 2 requirements:1) Pitches must be notated in ascending order2) Distance from 1st to last PCs = smallest possible distance

✦ Outside Interval (OI!)✦ Interval framing the set (spanning from 1st to last PC)

✦ SOI (Smallest Outside Interval)– smallest possible interval between 1st and last PC

Atonal (a/k/a Set) Theory✦ Next Most Outside Interval (NMOI)

✦ Tie-breaker used if/when needed(multiple SOIs vying for top spot)

✦ Interval between 1st and next-to-last notes

Secrets of the Normal Order✦ WARP ZONE SHORTCUT – How to quickly find the N.O.:

✦ Notate the set as a scale (i.e., double the 1st note at the 8ve)

✦ Find the largest interval (LI) between 2 adjacent PCs

✦ The upper note of the LI pair = the first note of the N.O.

N.O. shortcut example

N.O. begins here (upper PC of LI)

m2 m2 P4 m3 M2

Pitch classes

for analysis:

NO:

[9,0,2,3,4]

Notate PCs in any

ascending pattern,

starting & ending

with 8ve doubling

of the 1st PC

Atonal Analysis: Set Theory✦ Cardinality

✦ The number of pitch classes within a given PC set

✦ Cardinality type = important for classifying/organizing PC sets

0 Empty set1 Monad2 Dyad3 Trichord4 Tetrachord5 Pentachord6 Hexachord7 Septachord8 Octachord9 Nonachord10 Decachord11 Undecachord12 Aggregate

What’s your interval vector, Victor?

✦ Interval vector (I.V.), a/k/a interval-class vector (I.C.V.)

✦ Number of occurrences of each interval class in PC set

✦ Listed as string of 6 numbers without commas or parentheses ex: 321000

✦ Since there are 6 commonly used interval classes (ICs), there are 6 blanks to fill in when determining the I.V.:

✦ ____ ____ ____ ____ ____ ____ ic1 ic2 ic3 ic4 ic5 ic6 m2/M7 M2/m7 m3/M6 M3/m6 P4/P5 +4/º5

✦ Therefore, the I.V. for the set (B, G#, G) would be...

✦ ...101100

Prime Form✦ Prime Form (P.F.)

✦ Single category for ALL equivalent PC sets

✦ PC sets may have equivalence relations via

✦ Transposition (Tn)

✦ Inversion (Tn I)✦ 2 requirements for Prime Form:

✦ Set begins with integer 0 (like moveable-integer notation)

✦ Set must be the most packed (compact) to the left (see next slide)

✦ Notation of Prime Form: no commas, brackets, or spaces; only numbers & parentheses – Ex: (014)

Reducing PC sets to Prime Form✦ Step 1: Determine the N.O. of the PC set

✦ N.O. of PC set (B,G#,G) or (e, 8, 7) = [G,G#,B] = [7,8,e]

✦ Step 2: Find the transposition that will make the 1st PC = 0

✦ 7+5 = 0 – add this Tn to each PC: 8+5 = 1 11+5 = 4

✦ Now set aside this group of transposed PCs [0,1,4] for a moment...

✦ Step 3: Invert the original set (subtract from 12), then rinse, lather, and repeat steps 1&2 (determine N.O., transpose to 0)

✦ Inversion of [7, 8, e] = (5,4,1); N.O. = [1,4,5]; P.F. = [0,3,4]

✦ Step 4: Compare the two N.O.s; the most compact set = Prime Form

✦ (014) = Prime form (more compact to the left than [0,3,4])

“YTMND!”Finding Forte

Set Class / Forte Name✦ Set Class (a/k/a Forte name) (see Appendix C, pp. 548-551)

✦ “Family” of PC sets related by transposition or inversion

✦ Usually 24 Distinct Forms (DF) of any given PC set

✦ 12 possible transpositions + 12 possible inversions = 24 DFs(no retrogrades or retrograde inversions included!)

✦ Some symmetrical sets will have less than 24 DFs...

✦ Whole Tone scale set class = only 2 DFs!

✦ Octatonic scale class = only 3 DFs