Pitch Class Set Relations

Pitch Class Set Relations A Tutorial

Brian Hanson

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Pitch Class Set Relations: A Tutorial

1.1 INTEGER REPRESENTATION

[A# and B are represented with the letters T and E]

1.2 INTRODUCTORY DEFINITIONS

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

d{1→0(170)}=11

C C#/Db D D#/Eb E F F#/Gb G G#/Ab A A#/Bb B 0 1 2 3 4 5 6 7 8 9 T E

Pitch One of the twelve chromatic notes with a specific octave position. Pitch Class One of the twelve chromatic notes independent of octave displacement and

enharmonic spelling. Pitch Class Set A group of pitch class integers. Normal Form The most compact rotation of a pitch class set. Prime Form A pitch class set in normal form that is “packed to the left” and transposed to 0.

0 1 2 3 4 5 6 7 d = 1 2 3 4 5 6 7

1 2 3 4 5 6 7 8 9 10 11 12 0

d = 1 2 3 4 5 6 7 8 9 10 11

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d{7→1(701)}=6

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

d{0→1(701)}=6

(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

Inversion Formula: i{pcs(x)} = mod12c[r{pcs(x)}]

r = retrograde/ mod12c = mod12 complement/ i = inversion/ pcs = pitch class set(s)

7 8 9 10 11 12 0

d = 1 2 3 4 5 6

7 8 9 10 11 12 0

d = 1 2 3 4 5

0 1 d = 1

0 1 2 3 4 5 6 0 E T 9 8 7 6

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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.

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)

i1 + x = 0 E + x = 0 E + 1 = 0

x = 1

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

E 0 5 +1 +1 +1 0 1 6

pf(710)=(016)

#(0167) Uint(0167) 4 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)

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(01457)

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)

#(01457) Uint(01457) 5 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)

(0157)

0→1=1 1→5=4 0→5=5 1→7=6 0→7=7 5→7=2

ic1 1 (0→1) ic2 1 (5→7) ic3 0 ic4 1 (1→5) ic5 2 (0→5), (0→7)∨5=mod12c(7) ic6 1 (1→7)

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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)C= pitch class set complement/ ⇒ = then

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 1 4 +3 +3 +3 3 4 7

ic1 ic2 ic3 ic4 ic5 ic6 1 1 0 1 2 1

intν(0157)=[110121]

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Τ6(0247)=(68T1)

0 2 4 7 +6 +6 +6 +6 6 8 T 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)

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Find Τ7r(0157).

0 1 5 7 +7 +7 +7 +7 7 8 0 2

r(7802)= 2 0 8 7

Τ7r(0157)=(2087)

Find Τ3i(014).

0 1 4 +3 +3 +3 3 4 7

i=mod12c{r(347)} 5 8 9

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.

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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

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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)


i(01367)+Τ0-8{i(01367)}

(569E0) (67T01) (78E12) (89023) (9T134) (TE245) (E0356) (01467) (12578)

(12578)⊂(012578)

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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)


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)

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1.9d UNION ∪ = combination of pitch class sets without duplication

(015)∪(0158)=(0158) (02468)∪(01234)=(0123468)

1.9e SUBSET AGGREGATE 2(0145) = 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) 3 2 1 0 0 0 ⊗ ⊗ ⊗ ⊗ ⊗ ⊗

intν(0157) 1 1 0 1 2 1

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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) 3 2 1 0 0 0 ⊗ √ √ ⊗ √ √

intν(0124) 2 2 1 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) 2 2χic3{intν(0134)} 1χic2{intν(0124)} 1 0 0 √ ⊗ ⊗ √ √ √

intν(0134) 2 1χic3{intν(0124)} 2χic2{intν(0124)} 1 0 0

1.11 PITCH CLASS SET INVARIANCE inv = invariant(s)/ Uinv = possible number of invariants/ Γ = rotation(s)/ UΓ = all possible

rotations/ ⎡⎦ = matrix/ RC = row element/ CC = column element

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)}

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

UΓ(0167)=(1670)(6701)(7016)

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

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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.

CC1 CC2 CC3 CC4 RC1 RC1+CC1 RC1+CC2 RC1+CC3 RC1+CC4 RC2 RC2+CC1 RC2+CC2 RC2+CC3 RC2+CC4 RC3 RC3+CC1 RC3+CC2 RC3+CC3 RC3+CC4 RC4 RC4+CC1 RC4+CC2 RC4+CC3 RC4+CC4

⎡⎦{i(0126)}

6 T E 0 6 0 4 5 6 T 4 8 9 T E 5 9 T E 0 6 T E 0

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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)} Uinv inv Τ(0){i(0126)} 2 [0,6] Τ(4){i(0126)} 2 [6,T] Τ(5){i(0126)} 2 [6,E] Τ(6){i(0126)} 2 [0,6] Τ(8){i(0126)} 1 [T] Τ(9){i(0126)} 2 [T,E] Τ(T){i(0126)} 3 [0,T,E] Τ(E){i(0126)} 2 [0,E]

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

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1.12 APPENDIX 1: MATHEMATICAL SYMBOLS

ao = ascending order d = distance r = retrograde mod12c = mod12 complement i = inversion pcs = pitch class set(s)

i1 = first integer i2 = second integer i3 = third integer pf = prime form # = cardinal number int =interval(s)/

Uint = possible number of intervals

⇔ = if ic = interval class

intν = interval vector ∑int = the sum of all the intervals

∨ = because

Ζrp = Z-related pair ∴ = therefore :: = with * = and ≠ = not equal pcs(x)C= pitch class set

complement ⇒ = then Τx = transposition by adding

(x) to each integer of a pcs, using mod12 arithmetic.

U⎣Τ = possible number of transpositions before self-

replication 2pcs(x) = all possible subsets of

pcs(x)

ΙR = inclusion relation ℤ = integer/element

∈ = element of ∉ = not an element of ⊄ = not a subset ⊂ = subset ⊃ = superset κ = K complex/relation

κΗ = Kh complex/relation ∩ = intersection ∪ = union inv⊂ = invariant subset(s)

Uinv⊂= possible number of

invariant subsets 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) inv = invariant(s)

Uinv = possible number of invariants

Γ = rotation(s) UΓ = all possible rotations

⎡⎦ = matrix RC = row element CC = column element

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

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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

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Pitch Class Set Relations