Set Theory

Question 1

Set Theory

Answer 1

A new set of tools that lets us talk about pitch collections (“sets”) in a new way

Can be used on any group of notes, whether simultaneous (chords) or not (melodies)

Puts intense focus on the intervals between pitches

Has no limits on which intervallic collections it can describe

Question 2

Pitch Class Notation

Answer 2

Set Theory uses numbers to represent all versions of a pitch (in every octave, in every voice)

C is 0, C sharp is 1, etc.
10 and 11 are often represented with “t” and “e”

Question 3

Enharmonic Equivalence

Answer 3

In set theory, there is no distinction between enharmonic spellings of a pitch.

This is different from tonal theory! (Consider Ger+ 6 vs. V7)

Question 4

Pitch Class Intervals

Answer 4

Intervals still describe the space between two pitches 
In set theory, we use numbers to describe intervallic space
A pitch-class interval counts UPWARD from one pitch class to another pitch class
To get the interval, we subtract the number of the lower pitch class from the number of the higher pitch class

Example

D note to F# upward
So it is D or 2 and F# or 6
Therefore higher note minus lower note 6-2=4

Question 5

Intervals with Mod 12

Answer 5

Sometimes interval subtraction will yield a negative number
We can use “modular arithmetic” to help us here
Set theory uses “mod 12” because it only has 12 values (0 through 11)
We can add or subtract 12 to out pitch class numbers to help work out some of our set theory math

Question 6

Mod 12 clock

Answer 6

Use the clock face with 12 numbers but remember 10 and 11 is represented by t and e and 0 is basically 12.

Question 7

Normal order

Answer 7

To compare different pitch class sets we want to simplify them
The first step to doing this is to find the normal order
Normal order is the ordering of pitch classes that results in smallest interval span from first to last note

Question 8

Finding normal order

Answer 8

Fit the all the chord or phrase into single octave than compare the rotations until it create the lowest outer octave look at every grouping until it is smallest outer interval
Than subtract the higher minus lower interval for smallest number

Question 9

Normal order is notated in brackets

Answer 9

Example 
0 2 9 e = 11-0 =11
2 9 e 0 = 12-2= 10
9 e 0 2 = 14-9 =5
e 0 2 9 = 21-11= 10 
Smallest outer interval is the 3rd example with 5 and it is written as [9,e,0,2] -- this is normal order

Question 10

Normal order breaking ties

Answer 10

Sometimes more than one order will have the smallest outer interval so we go to the next interval in looking for the smallest interval
This set will be our normal order it will have pitched grouped more closely at the beginning of the set

Question 11

Transpositions symmetry

Answer 11

Like breaking ties however the entire set have the same interval in every rotation
These sets display transposition all symmetry they are transpositionally symmetrical pitch classes so we can use any from of these sets for our normal order
Example [0,4,8] [4,8,0] [8,0,4]

Question 12

Equivalences

Answer 12

In tonal theory we are used to treating some different things as the same or equivalent for analytical purposes
We observe octave equivalences - we treat notes in different octaves as the same for analytical purposes
We observe transpositional equivalences - we treat transpositions of a chord as being the same type for analytical purposes.

Question 13

Set theory inversional equivalences

Answer 13

• set theory observes those equivalences but also observes inversional equivalence
• inversions of pitch class set are considered to be the same for analytical purposes
• here inversion refers to switching the direction of the intervals

Question 14

In set theory

Answer 14

The major triad and he minor triad are considered to be equivalent because we care most about the intervals that are there. 
Example 
C E G upward 0 4 7 interval +4 + 3
C AbF downward 0 8 5 interval -4 -3
C EbG upward 0 3 7 interval +3 +4 
All the same

Question 15

Best Normal Order

Answer 15

After we find Normal Order, We must compare that set to its inversion (also in normal order)
Then compare these sets in the same way we compared to find normal order- look for the smallest intervals from the outside in.
Example
5 7 e 0 F G B C upward interval +2 +4 +1
5 6 t 0 F bG bB C upward interval +1+4+2
Both intervals are same as they both contain 4 1 2 in differing order so sets are equivalent

Outer interval are unchanged F To C or 5-0 is 7 and F to C is or 5 -0 is 7 remember 7 is P5
Next inner interval F to B or 5 to e is 6 and F to bB or 5 to t is 5 remember pitch class interval inversion of 5 and 6 is P4 and TT

Question 16

Check your inversions

Answer 16

Make sure that the inversion of your normal order is in normal order itself it usually will be but not always

Question 17

Prime Form

Answer 17

The final step in classifying a set of pitch classes is to find its prime form
To find prime form we transpose the best normal order so that it begins on 0 (C)
To transpose we simply add or subtract to our pitches
Prime form is the equivalent of the tonal harmon’s major and minor triads

Question 18

Prime form order

Answer 18

Take best normal order and transpose the first pitch class to 0

Example
7 8 9 1
GG#AC#

Minus 7

0. 1. 2. 6. C C#D F#

Result minus 7 / down 5

We notate prime form using a bracket and numbers with no spaces or commas between them [0126] - we are not looking for a specific grouping of notes that have a 0126 but instead we are referring to all 4 note collections that have this particular collection of intervals in them meaning 1 pitch than 1 pitch above than another pitch above than 4 pitches above that
Prime form represents all the possible ways of these collection of intervals that repeat these intervals

Question 19

Evaluating a set of pitches start to finish

Answer 19

Fun the normal order group the pitches so they fit in the smallest space with small intervals grouped forward the left
Find the best normal order - compare your normal order with its inversions normal order for grouping small intervals to the left
- then find the prime form by transposing your set so that it begins on C/0

Question 20

Pitch Class Intervals and Interval Class

Answer 20

Remember they use numbers to indicate the distance between different pitch classes, and we always subtract the first pitch class from the second one or the higher - lower pitch

Question 21

Pitch class intervals and interval class

Answer 21

Since we treat inversions as equivalent in set theory we can reduce the number of intervals in music by simplifying pitch class intervals to their interval class
Both of these intervals belong to the same interval class because they are inversions of one another

Question 22

Interval classes

Answer 22

There are six interval classes 1,2,3,4,5 and 6

Question 23

IC 1

Interval Class 1

Answer 23

Pitch Class 1, 11 = m2, M7

Question 24

IC 2

Interval Class 2

Answer 24

PCI
pitch class interval 2,10 = M2, m7

Question 25

IC3 | Interval Class

Answer 25

Pitch Class Interval PCI 3,9 = m3,M6

Question 26

IC 4 | Interval Class 4

Answer 26

PCI 4,8 = M3, m6

Question 27

IC 5 | Interval Class 5

Answer 27

Pitch Class Interval PCI 5,7= P4,P5

Question 28

IC 6 | Interval Class 6

Answer 28

PCI Pitch Class Interval 6 = TT

Question 29

Interval class vector

Answer 29

``` The interval class vector helps us understand the intervallic content in a pitch class set and to calculate the interval class vector we count the number of times each interval class occurs inside the set Example C. D. Eb. G 0. 2. 3. 7 Between c and d is 2 interval class Between c and eb is 3 interval class Between c and g is 7 interval class ``` Step 1 from C/0:IC1=0,IC2=1,IC3=1,IC4=0,IC5=1,IC6=0 ``` Next Between d and eb is 2 interval class Between d and g is 5 interval class ``` Step 2 from D/2 IC1=1,IC2=0,IC3=0,IC4=0,IC5=1,IC6=0 Next Between eb and g is 4 interval class Step 3 Eb/3 IC1=0,IC2=0,IC3=0,IC4=1,IC5=0,IC6=0 ``` Final we tabulate the info in this format where each number represents the amount of times an interval class appears. The numbers go in order - the first shows IC1, then IC2.... The interval-class vector for our set [0237] would be Triangle bracket 111120 triangle bracket which means for for 1st number 1 interval class appears 2nd number 1 interval class appears so on and so forth ```

Question 30

12 possible PC inversions

Answer 30

PC 0. 1 2. 3. 4. 5. 6. 7. 8. 9. 10. 11 12pc0. 11 109 8. 7 6. 5. 4 3. 2. 1 Tn. TnI Tn add variable onto set TnI perform pc inversion then add number

Question 31

Pc inversion

Answer 31

Pc inversion knocks the original set out of normal form so instead of square brackets we use round parenthesis

Question 32

Cardinality

Answer 32

``` One of the important ways we classify a set is by its cardinality Or the number of members it contains C 0 empty set 1 monad 2 dyad 3 trichord 4 tetrachord 5 pentachord 6 hexachord 7 septachord 8 octachord 9 nonachord 10decachord 11 undecachord 12 aggregate ```

Question 33

Procedure to find prime form

Answer 33

Determine the normal form of a set example (B,G#,G) Noemal form is [7,8,11] Determine which Tn operation will transpose the set so that its first member is 0 and apply it to its set Example T5 [7,8,11] = [0,1,4] Invert the set and repeat the previous two steps for the inverted set Example [7,8,11] inverted is (5,4,1) normal form is [1,4,5]. T11[1,4,5]=[0,3,4] Compare the forms instep 2 and 3. Select the form that is most compact to the left as prime form example compare [0,1,4] and [0,3,4]. Therefore prime form is (014)