Pitch Class Set Relations by Brian Hanson

Pitch Class Set Relations A Tutorial Brian Hanson

Pitch Class Set Relations: A Tutorial 1.1 INTEGER REPRESENTATION C 0

C#/Db 1

D 2

D#/Eb 3

E 4

F 5

F#/Gb 6

G 7

G#/Ab 8

A 9

A#/Bb T

B E

[A# and B are represented with the letters T and E] 1.2 INTRODUCTORY DEFINITIONS Pitch Pitch Class

One of the twelve chromatic notes with a specific octave position. One of the twelve chromatic notes independent of octave displacement and enharmonic spelling. A group of pitch class integers. The most compact rotation of a pitch class set. A pitch class set in normal form that is “packed to the left” and transposed to 0.

Pitch Class Set Normal Form Prime Form

1.3 PRIME FORM CONVERSION PROCESS ao = ascending order/ d = distance Step One: Convert pitch class set into ascending order. ao(710)=(017) Step Two: List all of the possible rotations of the pitch class set. (017) (170) (701) Step Three: Find the rotation with the smallest distance from the first integer to the last integer using mod12 arithmetic. [This step calculates normal form] d{0→7(017)}=7 0 d=

1 1

2 2

3 3

4 4

5 5

6 6

7 7

d{1→0(170)}=11 1

12 0 11

d{7→1(701)}=6 7

12 0 5

If there is a tie between two rotations, use the distance from the first integer and the penultimate integer. Continue using the penultimate integer, until there is no tie. Step Four: Determine if the chosen rotation is “packed to the left.” [A set is “packed to the left”, if the smallest interval is the distance between the first integer and the second integer, rather than the penultimate integer and the final integer] d{7→0(701)}=5 7

12 0 5

d{0→1(701)}=6 0 d=

1 1

(701) is “packed to the right”, because the smallest interval is between the penultimate integer and the final integer. If the chosen rotation is “packed to the right,” it must be inverted to make it “packed to the left.” If the chosen rotation is “packed to the left,” no inversion is needed. 1.3a PITCH CLASS SET INVERSION To invert a pitch class set, express the set in retrograde using mod12 complements. Mod12 Complements 0 0

1 E

2 T

3 9

4 8

5 7

Inversion Formula: i{pcs(x)} = mod12c[r{pcs(x)}] r = retrograde/ mod12c = mod12 complement/ i = inversion/ pcs = pitch class set(s)

6 6

r(701)=(107) mod12c{r(701)}=(E05) i(701)=(E05) (E05){i(701)} is now “packed to the left”. Step Five: After the set is “packed to the left”, it must be transposed to 0. Transposition to 0 Formula: [i1 + x = 0], [i1 + x = pfi1], [i2 + x = pfi2], [i3 + x = pfi3] i1 = first integer/ i2 = second integer/ i3 = third integer/ pf = prime form First, find x, which creates a solution of 0 when added to the first integer. Second, add x to each of the integers of the pitch class set (using mod12 arithmetic) to the find the prime form integers. i1 + x = 0 E+x=0 E+1=0 x=1 E +1 0

i1 + x = 0 i2 + x = 1 i3 + x = 6

0 +1 1

5 +1 6

pf(710)=(016) 1.4 CARDINAL NUMBERS # = cardinal number/ int =interval(s)/ Uint = possible number of intervals/ ⇔ = if The cardinal number reveals the number of elements a set contains. (0167) has a cardinality of 4, #(0167)=4. The cardinality of a set also reveals the possible number of intervals formed, (Uint), by a pitch class set. (0167) #(0167) 4

Uint(0167)

0+1+2+3=6 Uint⇔(#=4)=6/Uint(0167)=6

1(0→1)/2(0→6)/3(0→7)/4(1→6)/5(1→7)/6(6→7)

(01457) #(01457) 5

Uint(01457)

0+1+2+3+4=10 Uint⇔(#=5)=10/ Uint(01457)=10

1(0→1)/2(0→4)/3(0→5)/4(0→7)/5(1→4)/6(1→5)/7(1→7)/8(4→5)/9(4→7)/10(5→7) 1.5 INTERVAL VECTORS ic = interval class/ intν = interval vector/ ∑int = the sum of all the intervals/ ∨ = because While the cardinality of a set reveals the number of intervals formed, the interval vector reveals which intervals are formed. The interval vector is a bracketed 6-integer figure. The position of each integer represents that particular interval class, while the numbers represent the quantities of each interval formed by the pitch class set. [ic1, ic2, ic3, ic4, ic5, ic6] If the interval is larger than 6, it is represented by its mod12c. 8=4ic ∨ 4=mod12c(8) Find intν(0157). #(0157)=4/ Uint(0157)=6/ ∑int(0157)=25(1+5+7+4+6+2) (0157) 0→1=1 0→5=5 0→7=7

1→5=4 1→7=6 5→7=2

ic1 ic2 ic3 ic4 ic5 ic6

1 (0→1) 1 (5→7) 0 1 (1→5) 2 (0→5), (0→7)∨5=mod12c(7) 1 (1→7)

ic1 1

ic2 1

ic3 0

ic4 1

ic5 2

ic6 1

intν(0157)=[110121] 1.5a THE Z-RELATED PAIR Ζrp = Z-related pair/ pcs = pitch class set(s)/ ∴ = therefore/ :: = with/ * = and/ ≠ = not equal Z-Related Pair Formula: Ζrp = 2(pcs) :: =(intν)*≠(pf) A pair of pitch class sets is Z-related, when both pitch class sets (irreducible to the same prime form) have the same interval vector. intν{(013467)*(012369)}=[324222]∴{(013467)*(012369)}=Ζrp intν{(0137)*(0146)}=[111111]∴{(0137)*(0146)}=Ζrp 1.6 PITCH CLASS SET COMPLEMENTS pcs(x) = pitch class set complement/ ⇒ = then C

Pitch Class Set Complement Formula: ⇔pcs(x)=(024579E) ⇒(024579E)C = (1368T) If pcs(x) has a cardinality of 5 {#pcs(x)=5}, then the pitch class set complement, pcs(x)C, is a set with the cardinality equal to the mod12c of the cardinality of the original set, which would be 7 ∨{#pcs(x)=5}. The pcs(x)C is composed of every chromatic integer not used in the original set. (014)C=(256789TE)*(256789TE)C=(014) (0123456)C=(789TE)*(789TE)C=(0123456) 1.7 TRANSPOSITION Τx = transposition by adding (x) to each integer of a pcs, using mod12 arithmetic. Transposition is the process of a adding a fixed integer to each integer of a pitch class set. Τ3(014)=(347) 0 +3 3

1 +3 4

4 +3 7

Τ6(0247)=(68T1) 0 +6 6

2 +6 8

4 +6 T

7 +6 1

1.7a LIMITED TRANSPOSITION U⎣Τ = possible number of transpositions before self-replication Certain pitch class sets have the quality of limited transposition, meaning there are a limited number of transpositions possible before the original pitch class set is replicated. To find U⎣Τ, continue Τ1(x), until the original set is replicated. Τ1(02468T)=(13579E) Τ1(13579E)=(2468T0)=original pcs∴U⎣Τ=1 (02468T)=whole tone scale Τ1(024579E)=(13568T0) Τ1(13568T0)=(24679E1) Τ1(24679E1)=(3578T02) Τ1(3578T02)=(4689E13) Τ1(4689E13)=(579T024) Τ1(579T024)=(68TE135) Τ1(68TE135)=(79E0246) Τ1(79E0246)=(8T01357) Τ1(8T01357)=(9E12468) Τ1(9E12468)=(T023579) Τ1(T023579)=(E13468T) Τ1(E13468T)=(024579E)=original pcs∴U⎣Τ=12 (024579E)=diatonic major scale [Notice there are only 12 unique transpositions, 1 for each chromatic pitch] 1.8 RETROGRADE r = retrograde Retrograde is the reversal of order of the integers in a pitch class set. r(0167245ET398)=(893TE5427610) r(014)=(410)

Find Τ7r(0157). 0 +7 7 r(7802)= 2

1 +7 8

5 +7 0

7 +7 2

Τ7r(0157)=(2087) Find Τ3i(014). 0 +3 3 i=mod12c{r(347)} 5

1 +3 4

4 +3 7

Τ3i(014)=(589) 1.9 THE INCLUSION RELATION ΙR = inclusion relation/ ℤ = integer/element/ ∈ = element of/ ∉ = not an element of/ ⊂ = subset/ ⊃ = superset/⊄ = not a subset/ κ = K complex/relation/ κΗ = Kh complex/relation/∩ = intersection/ ∪ = union/2pcs(x) = all possible subsets of pcs(x) The inclusion relation states that pcs(x)⊂pcs(y)⇔every ℤ{pcs(x)}∈ pcs(y) Every ℤ (015)∈(0158)∴(015)⊂(0158)*(0158)⊃ (015)∴(015)*(0158)=ΙR (014)⊄(0158)∨ every ℤ (014)∉ (0158) 1.9a THE SET COMPLEX K The Set Complex K Formula: κ = pcs(x) or pcs(x)C[⊂ or ⊃]pcs(y) A K relationship occurs between a pair of pitch class sets, in which pcs(x) OR pcs(x)C is interconnected to pcs(y), by virtue of the inclusion relation. The inclusion relation can be standard, as in (024)⊂(0246)*(0246)⊃(024), or the relation can be based on the process of transposition or inversion. The following problem will display the process of using transposition and inversion to see if two pitch class sets are interconnected by the inclusion relation.

Find ⇔ κ = (0347)*(01368) It is apparent that (0347) is not a standard subset of (01368); therefore, each transposition and inversion of (0347) must be evaluated. Τ0-11(0347) (0347) (1458) (2569) (367T) (478E) (5890) (69T1) (7TE2) (8E03) (9014) (T125) (E236) Τ0-11(0347) does not provide a valid subset of (01368). Each inversion must be evaluated. i(0347)+Τ0-11{i(0347)} (5890) (69T1) (7TE2) (8E03) (9014) (T125) (E236) (0347) (1458) (2569) (367T) (478E) i(0347)+Τ0-11{i(0347)} does not provide a valid subset of (01368); therefore, the set complement must be evaluated. (0347)C=(12589TE) pf(12589TE)=(01345689) (01345689)⊃(01368)∴(0347)*(01368)=κ

1.9b THE SET COMPLEX KH The Set Complex Kh Formula: κΗ = pcs(x) and pcs(x)C[⊂ or ⊃]pcs(y) Notice the formulas for the set complexes K and Kh are nearly identical. The set complex Kh is more significant and rare, because it requires pcs(x) AND pcs(x)C to be interconnected to pcs(y), by virtue of the inclusion relation. Find ⇔ κΗ = (01367)*(012578) It is apparent that (01367) is not a standard subset of (012578); therefore, each transposition and inversion of (01367) must be evaluated. Τ0-11(01367) (01367) (12478) (23589) (3469T) (457TE) (568E0) (67901) (78T12) (89E23) (9T034) (TE145) (E0256) Τ0-11(01367) does not provide a valid subset of (012578). Each inversion must be evaluated. i(01367)+Τ0-8{i(01367)} (569E0) (67T01) (78E12) (89023) (9T134) (TE245) (E0356) (01467) (12578) (12578)⊂(012578)

In order for the relationship between (01367) and (012578) to be characterized as the Kh variety, the (01367)C must also be interconnected to (012578), by virtue of the inclusion relation. (01367)C =(24589TE) pf(24589TE)=(0123679) (012578)C=(3469TE) pf(3469TE)=(012578) Τ0-11(012578) (012578) (123689) (23479T) (3458TE) (4569E0) (567T01) (678E12) (789023) (89T134) (9TE245) (TE0356) (E01467) Τ0-11(012578) does not provide a valid subset of (0123679). Each inversion must be evaluated. i(012578)+Τ0-2{i(012578)} (457TE0) (568E01) (679123) (679123)⊂(0123679)*(12578)⊂(012578)∴(01367)*(012578)=κΗ 1.9c INTERSECTION ∩ = common elements of pitch class sets (015)∩(0158)=(015) (02468)∩(01234)=(024)

1.9d UNION ∪ = combination of pitch class sets without duplication (015)∪(0158)=(0158) (02468)∪(01234)=(0123468) 2

(0145)

1.9e SUBSET AGGREGATE = All possible subsets of (0145)

2(0145)=(01)(04)(05)(14){(15)=(04)}{(45)=(01)}(014)(045){(145)=(014)} 2(0145)=(01)(04)(05)(14)(014)(045) 1.10 SIMILARITY RELATIONS Smax= maximum similarity/ Smin = minimum similarity/ Rp = Rp relation/ R0 = R0 relation/R1 = R1 relation/R2 = R2 relation/ χ = interchangeable/ √ = ic match(es)/ ⊗ = ic mismatch(es) Maximum Similarity Formula: Smax = 4/6ic{pcs(x)}= 4/6ic{pcs(y)} Minimum Similarity Formula: Smin = 0/6ic{pcs(x)}= 0/6ic{pcs(y)} 1.10a THE RP RELATION The Rp Relation Formula: Rp⇔⊂pcs(x)=⊂pcs(y)*#(⊂)=#{pcs(x or y)-1} The Rp Relation occurs between 2 pitch class sets that have a common subset with a cardinality that is equal to the cardinality of the original sets minus 1. Rp=(0156)*(0157)∨(015)⊂(0156)*(015)⊂(0157)*#(015)=#{(0156) or (0157)-1} 1.10b THE R0 RELATION The R0 Relation Formula: R0 = 0/6ic{pcs(x)}= 0/6ic{pcs(y)} Notice how the R0 Relation formula is identical to the Minimum Similarity formula. When the interval vectors of two pitch class sets have 0 matches in each interval class, there is a R0 Relation. R0=(0123)*(0157)∨0/6ic{pcs(0123)}= 0/6ic{pcs(0157)} intν(0123)=[321000] intν(0157)=[110121] intν(0123) intν(0157)

3 ⊗ 1

2 ⊗ 1

1 ⊗ 0

0 ⊗ 1

0 ⊗ 2

0 ⊗ 1

1.10c THE R2 RELATION The R2 Relation Formula: R2 = 4/6ic{pcs(x)}= 4/6ic{pcs(y)} Notice how the R2 Relation formula is identical to the Maximum Similarity formula. When the interval vectors of two pitch class sets have 4 matching interval classes, there is a R2 Relation. R2=(0123)*(0124)∨4/6ic{pcs(0123)}= 4/6ic{pcs(0124)} intν(0123)=[321000] intν(0124)=[221100] intν(0123) intν(0124)

3 ⊗ 2

2 √ 2

1 √ 1

0 ⊗ 1

0 √ 0

1.10d THE R1 RELATION The R1 Relation Formula: R1 = 4/6ic{pcs(x)}= 4/6ic{pcs(y)}*⊗ic{pcs(x)}χ⊗ic{pcs(y)} Notice how the R1 Relation formula is nearly identical to the R2 Relation formula. Differing from the R2 Relation, in the R1 Relation, the 2 interval class mismatches are interchangeable. R1=(0124)*(0134)∨4/6ic{pcs(0124)}= 4/6ic{pcs(0134)}*⊗ic{pcs(0124)}χ⊗ic{pcs(0134)} intν(0124)=[221100] intν(0134)=[212100] intν(0124) intν(0134)

2 √ 2

2χic {intν(0134)} 1χic {intν(0124)} ⊗ ⊗ 1χic {intν(0124)} 2χic {intν(0124)} 3

1 √ 1

0 √ 0

1.11 PITCH CLASS SET INVARIANCE inv = invariant(s)/ Uinv = possible number of invariants/ Γ = rotation(s)/ U = all possible rotations/ ⎡⎦ = matrix/ RC = row element/ CC = column element Γ

Pitch class set invariance examines the unchanged elements of a pitch class set after the process of permutation (rotation, transposition, inversion). 1.11a INVARIANCE UNDER ROTATION Possible Number of Invariants under Rotation Formula: Uinv under Γ =#{pcs(x)}

Under the process of rotation, every element remains invariant. Uinv=4∨#{pcs(0167)}=4 U (0167)=(1670)(6701)(7016) Γ

1.11b INVARIANCE UNDER TRANSPOSITION Possible Number of Invariants under Transposition Formula: ⇔{ic(x)}=A⇒Uinv under Τ(x)=A Under the process of transposition, every element will not remain invariant. Determining the possible number of invariants under the process of transposition will require an examination of the pitch class set’s interval vector. intν(015)=[100110] The 1 in ic1 reveals that when (015) is transposed by 1, 1 element will remain invariant. [⇔ ic1=1⇒Uinv under Τ1=1] Τ1(015)=(126) inv=[1] Uinv=1 The 1 in ic4 reveals that when (015) is transposed by 4, 1 element will remain invariant. [⇔ ic4=1⇒Uinv under Τ4=1] Τ4(015)=(459) inv=[5] Uinv=1 1.11c INVARIANCE UNDER INVERSION Under the process of inversion, every 0 and 6 will remain invariant, because the mod12c(0)=0 and the mod12c(6)=6. Remember the inversion formula: i{pcs(x)}= mod12c[r{pcs(x)}]. Also, if ℤ (x) and mod12c(x) are present in any given pitch class set, then both ℤ (x) and mod12c(x) will remain invariant. [⇔ℤ (x)*mod12c(x)∈pcs(x)⇒ℤ (x)*mod12c(x)=inv(under inversion)] i(01678E)=(1456E0) inv=[0,1,6,E] Uinv=4

i(01457)=(578E0) inv=[0,5,7] Uinv=3 The process of inversion can be combined with the process of transposition. i(0126)=(6TE0) Τ0-11{i(0126)} [with invariants bracketed in bold] ([6]TE[0]) (7E[0][1]) (8[0][1][2]) (9[1][2]3) (T[2]34) (E345) ([0]45[6]) ([1]5[6]7) ([2][6]78) (3789) (489T) (59TE) Uinv under Τ0-11{i(0126)}=16

The inversion of pcs(x) can be used to create a matrix, which will reveal what elements will remain invariant and the total number of invariants. RC1 RC2 RC3 RC4

CC1 RC1+CC1 RC2+CC1 RC3+CC1 RC4+CC1

CC2 RC1+CC2 RC2+CC2 RC3+CC2 RC4+CC2

CC3 RC1+CC3 RC2+CC3 RC3+CC3 RC4+CC3

CC4 RC1+CC4 RC2+CC4 RC3+CC4 RC4+CC4

E 5 9 T E

0 6 T E 0

⎡⎦{i(0126)} 6 T E 0

6 0 4 5 6

T 4 8 9 T

The 3 T’s reveal that 3 elements [0,T,E] remain invariant when ΤT{i(0126)}. The 2 4’s reveal that 2 elements [6,T] remain invariant when Τ4{i(0126)}. The following graph will detail the complete revelation of the matrix. Τ (x){i(0126)} Τ(0){i(0126)} Τ(4){i(0126)} Τ(5){i(0126)} Τ(6){i(0126)} Τ(8){i(0126)} Τ(9){i(0126)} Τ(T){i(0126)} Τ(E){i(0126)}

inv [0,6] [6,T] [6,E] [0,6] [T] [T,E] [0,T,E] [0,E]

Uinv

2 2 2 2 1 2 3 2

1.11d INVARIANT SUBSET inv⊂ = invariant subset(s)/ Uinv = possible number of invariant subsets ⊂

An invariant subset is a subset that remains unchanged after the process of permutation. Find inv⊂*Uinv (01257)⇔T5(01257) T5(01257)=(567T0) inv⊂=[(05),(07),(57),(057)] Uinv = 4 ⊂

⊂

Find inv⊂*Uinv (01257)⇔i(01257) i(01257)=(57TE0) inv⊂=[(05),(07),(57),(057)] Uinv = 4

Find inv⊂*Uinv (01257)⇔Τ6{i(0126)} i(01257)=(57TE0) Τ6(57TE0)=(E1456) inv⊂=[(15)] Uinv = 1

⊂

Find inv⊂*Uinv (01257)⇔Τ2{i(0126)} i(01257)=(57TE0) Τ2(57TE0)=(79012) inv⊂=[(01),(02),(07),(12),(17),(27),(012),(027),(127),(0127)] Uinv = 10 ⊂

⊂

1.12 APPENDIX 1: MATHEMATICAL SYMBOLS ao = ascending order mod12c = mod12 complement i1 = first integer pf = prime form Uint = possible number of intervals intν = interval vector Ζrp = Z-related pair * = and ⇒ = then

2pcs(x) = all possible subsets of pcs(x) ∈ = element of

⊂ = subset κΗ = Kh complex/relation inv⊂ = invariant subset(s) Smin = minimum similarity R1 = R1 relation √ = ic match(es)/ Uinv = possible number of invariants ⎡⎦ = matrix

d = distance i = inversion i2 = second integer # = cardinal number ⇔ = if

r = retrograde pcs = pitch class set(s) i3 = third integer int =interval(s)/ ic = interval class

∑int = the sum of all the intervals ∴ = therefore ≠ = not equal

∨ = because

Τx = transposition by adding (x) to each integer of a pcs, using mod12 arithmetic. ΙR = inclusion relation

:: = with pcs(x) = pitch class set complement U⎣Τ = possible number of transpositions before selfreplication C

ℤ = integer/element

∉ = not an element of ⊃ = superset ∩ = intersection Uinv = possible number of invariant subsets Rp = Rp relation R2 = R2 relation ⊗ = ic mismatch(es) Γ = rotation(s)

⊄ = not a subset κ = K complex/relation ∪ = union Smax= maximum similarity

RC = row element

CC = column element

⊂

R0 = R0 relation χ = interchangeable inv = invariant(s) U = all possible rotations Γ

Several of the mathematical symbols were created for the use of this paper and do not reflect traditional usage.

1.13 APPENDIX 2: FORMULAS Inversion Formula: i{pcs(x)}= mod12c[r{pcs(x)}] Transposition to 0 Formula: [i1 + x = 0], [i1 + x = pfi1], [i2 + x = pfi2], [i3 + x = pfi3] Z-Related Pair Formula: Ζrp = 2(pcs) :: =(intν)*≠(pf) Pitch Class Set Complement Formula: ⇔pcs(x)=(024579E) ⇒(024579E)C = (1368T) The Inclusion Relation Formula: ΙR = pcs(x)⊂pcs(y)⇔every ℤ{pcs(x)}∈pcs(y) The Set Complex K Formula: κ = pcs(x) or pcs(x)C[⊂ or ⊃]pcs(y) The Set Complex Kh Formula: κΗ = pcs(x) and pcs(x)C[⊂ or ⊃]pcs(y) Maximum Similarity Formula: Smax = 4/6ic{pcs(x)}= 4/6ic{pcs(y)} Minimum Similarity Formula: Smin = 0/6ic{pcs(x)}= 0/6ic{pcs(y)} The Rp Relation Formula: Rp⇔⊂pcs(x)=⊂pcs(y)*#(⊂)=#{pcs(x or y)-1} The R0 Relation Formula: R0 = 0/6ic{pcs(x)}= 0/6ic{pcs(y)} The R2 Relation Formula: R2 = 4/6ic{pcs(x)}= 4/6ic{pcs(y)} The R1 Relation Formula: R1 = 4/6ic{pcs(x)}= 4/6ic{pcs(y)}*⊗ic{pcs(x)}χ⊗ic{pcs(y)} Possible Number of Invariants under Rotation Formula: Uinv under Γ =#{pcs(x)} Possible Number of Invariants under Transposition Formula: ⇔{ic(x)}=A⇒Uinv under Τ(x)=A