math and music a dual nature michael remchuk math 552 spring 2008
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Math and MusicMath and Music
A Dual NatureA Dual NatureMichael RemchukMichael Remchuk
Math 552Math 552Spring 2008Spring 2008
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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
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The Basics (Cont.)The Basics (Cont.)
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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)
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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
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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
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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
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ActionsActions
TranspositionsTranspositions
Tn := Int(s,Tn(s)) = nTn := Int(s,Tn(s)) = n
Tn(x) := x + nTn(x) := x + n
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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
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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
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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
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But Wait!!!But Wait!!!
(T(T11))n n = T= Tnn
TTnnIIoo = I = Inn
12 transpositions12 transpositions
12 inversions12 inversions
DD2424
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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>
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Some TriadsSome Triads
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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>
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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
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PLR ActionsPLR Actions
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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
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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
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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
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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
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R(LR)^3 = PR(LR)^3 = P
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Douthett and Steinbach’s Chicken Wire Torus
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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?
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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!
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Examples in MusicExamples in Music
Canon in D (Pachelbel)Canon in D (Pachelbel)
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Examples in MusicExamples in Music
““Grail Theme” Grail Theme”
ParsifalParsifal by Wagner by Wagner
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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
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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”