Math and MusicMath and Music

A Dual NatureA Dual NatureMichael RemchukMichael Remchuk

Math 552Math 552Spring 2008Spring 2008

The Basics: Music to MathThe Basics: Music to Math

The notes (“equal-tempered pitch classes”)The notes (“equal-tempered pitch classes”)

C, C♯, D, D♯, E, F, F♯ , G, G♯ , A, A♯ , BC, C♯, D, D♯, E, F, F♯ , G, G♯ , A, A♯ , B

C, DC, Dbb, D, E, D, Ebb,, E,E, F, GF, Gbb, G, A, G, Abb, A, B, A, Bbb, B, B

0 1 2 3 4 5 6 7 8 9 10 110 1 2 3 4 5 6 7 8 9 10 11

The Basics (Cont.)The Basics (Cont.)

TerminologyTerminology

The IntervalThe Interval

Int(s,t) = -Int(t,s)Int(s,t) = -Int(t,s)

Int(s,t) = Int(s,u) + Int(u,t)Int(s,t) = Int(s,u) + Int(u,t)

ExampleExample

4,24,2Int(4,2) = 2-4 =-2 = 10Int(4,2) = 2-4 =-2 = 10Int(2,4) = 4-2 = 2Int(2,4) = 4-2 = 2

10 notes up = 10

10 notes down = -10

UniquenessUniqueness

Int(s,t) = nInt(s,t) = n– Unique t (mod 12)Unique t (mod 12)

s = G = 7s = G = 7

n = 5n = 5

t-s=nt-s=n

-> t = 5+7 = 0 = C-> t = 5+7 = 0 = C

Algebra “pops up” alreadyAlgebra “pops up” alreadyZ/12Z/12

Generators:1, 5, 7, 11Generators:1, 5, 7, 11

Chromatic:Chromatic:– C, C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, B, CC, C♯, D, D♯, E, F, F♯, G, G♯, A, A♯, B, C

Circle of fourthsCircle of fourths– C, F, A♯, D♯, G♯, C♯, F♯, B, E, A, D, G, CC, F, A♯, D♯, G♯, C♯, F♯, B, E, A, D, G, C

Circle of fifthsCircle of fifths– C, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, F, CC, G, D, A, E, B, F♯, C♯, G♯, D♯, A♯, F, C

Descending ChromaticDescending Chromatic

ActionsActions

TranspositionsTranspositions

Tn := Int(s,Tn(s)) = nTn := Int(s,Tn(s)) = n

Tn(x) := x + nTn(x) := x + n

InversionInversion

IIuv uv := u -> v:= u -> v

IIuvuv = I = I(u+1)(v-1)(u+1)(v-1)

IInn(x) = -x + n(x) = -x + n

Interval PreservationInterval Preservation

Def: Int(a,b) = Int(Y(a),Y(b))Def: Int(a,b) = Int(Y(a),Y(b))

Interval ReversalInterval Reversal

Def: Int(a,b) = Int(Y(b),Y(a)) Def: Int(a,b) = Int(Y(b),Y(a)) = - Int(Y(a),Y(b))= - Int(Y(a),Y(b))

TranspositionTransposition

InversionInversion

PropertiesProperties

Tn ◦ Tm = Tn+mTn ◦ Tm = Tn+m

Tn ◦ Im = Im+nTn ◦ Im = Im+n

Im ◦ Tn = Im-nIm ◦ Tn = Im-n

Im ◦ In = Tn-mIm ◦ In = Tn-m

But Wait!!!But Wait!!!

(T(T11))n n = T= Tnn

TTnnIIoo = I = Inn

12 transpositions12 transpositions

12 inversions12 inversions

DD2424

The ObjectsThe Objects

Major Triad <x,y,z>Major Triad <x,y,z>Int(x,y) = 4Int(x,y) = 4Int(x,z) = 7Int(x,z) = 7Minor Triad <x,y,z>Minor Triad <x,y,z>Int(x,y) = 3Int(x,y) = 3Int(x,y) = 7Int(x,y) = 7C: <0,4,7>C: <0,4,7>c: <0,3,7> c: <0,3,7>

Some TriadsSome Triads

Operations on ObjectsOperations on Objects

TTnn(<x,y,z>) = <T(<x,y,z>) = <Tnn(x),T(x),Tnn(y),T(y),Tnn(z)>(z)>

IInn(<x,y,z>) = <I(<x,y,z>) = <Inn(x),I(x),Inn(y),I(y),Inn(z)>(z)>

TT99(<0,4,7>) = <9,13,16> = <9,1,4>(<0,4,7>) = <9,13,16> = <9,1,4>

= <1,4,9>= <1,4,9>

PLR ActionsPLR Actions

P(<x,y,z>) = IP(<x,y,z>) = Ix+zx+z<x,y,z><x,y,z>

L(<x,y,z>) = IL(<x,y,z>) = Iy+zy+z <x,y,z> <x,y,z>

R(<x,y,z>) = IR(<x,y,z>) = Ix+yx+y <x,y,z> <x,y,z>

ParallelParallel

Leading tone exchangeLeading tone exchange

RelativeRelative

PLR ActionsPLR Actions

P {1,3}P {1,3}

0 -> 70 -> 7

4 -> 34 -> 3

7 -> 07 -> 0

<0,4,7> -> <7,3,0><0,4,7> -> <7,3,0>

P: C->cP: C->c

Take C := <0,4,7>Take C := <0,4,7>

(0+7) / 2 = 3.5(0+7) / 2 = 3.5

9.59.5

L {2,3}L {2,3}

L(<0,4,7>)L(<0,4,7>)

(4+7)/2 = 5.5(4+7)/2 = 5.5

11.511.5

<0,4,7> -> <11,7,4><0,4,7> -> <11,7,4>

C -> eC -> e

R {1,2}R {1,2}

R(<0,4,7>)R(<0,4,7>)

(0+4)/2 = 2(0+4)/2 = 2

88

<0,4,7> -> <4,0,9><0,4,7> -> <4,0,9>

R: C->aR: C->a

The PLR GroupThe PLR Group

R then L then R then L . . .R then L then R then L . . .

C, R(C), LR(C), RLR(C) . . .C, R(C), LR(C), RLR(C) . . .

24 distinct operations24 distinct operations

(LR)(LR)1212 = 1 = 1

LR , L generate DLR , L generate D2424

L(LR)L = RL = (LR)L(LR)L = RL = (LR)-1-1

R(LR)R(LR)33(C) = c -> R(LR)(C) = c -> R(LR)33 = P = P

R(LR)^3 = PR(LR)^3 = P

Douthett and Steinbach’s Chicken Wire Torus

Summing it upSumming it up

T/I GroupT/I Group

PLR GroupPLR Group

What happens if we combine them?What happens if we combine them?

DualityDuality

We have the T/I Group and the PLR We have the T/I Group and the PLR GroupGroup

Note:Note:Centralizer (T/I) = PLRCentralizer (T/I) = PLRCentralizer (PLR) = T/ICentralizer (PLR) = T/I

T/I and PLR are dual!T/I and PLR are dual!

Examples in MusicExamples in Music

Canon in D (Pachelbel)Canon in D (Pachelbel)

Examples in MusicExamples in Music

““Grail Theme” Grail Theme”

ParsifalParsifal by Wagner by Wagner

ConclusionConclusion

ObjectsObjects– Notes (Z/12)Notes (Z/12)

Generators: Chromatic, CirclesGenerators: Chromatic, Circles

– Major / Minor Chords (Triads)Major / Minor Chords (Triads)

OperationsOperations– T/IT/I– PLRPLR

DualDual

BibliographyBibliography

http://arxiv.org/PS_cache/arxiv/pdf/0711/0711.1873v1.pdfhttp://www.jstor.org/stable/view/843478?seq=1David Benson: “Music: A Mathematical Offering”David Benson: “Music: A Mathematical Offering”