godfried toussaint - mcgill university
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Godfried Toussaint
Multisets: Applications to Music
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The clave son in convex polygon notation
Easytodiscoversymmetries.
3
3
4
4
2
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The clave son as a multiset: inter-onset-interval histogram notation
Inter‐onsetintervalvaluesMul3p
licity
Alltheinter‐onsetintervalsdetermineamul3set.
12345678
Shouldweusethismorecompleteinforma3on?
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X-ray crystallography in the 1920’s
fic3onal3‐atommolecule
Distancesbetweenatoms
Distancegeometryproblem:Givenamultisetofdistancesbetweenasetofobjects(withoutlabels),canonereconstructthespatialcon=igurationoftheobjects,andifso,isthesolutionunique?
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A. Lindo Patterson, “Ambiguities in the X-ray analysis of crystal structures,” Physical Review, March, 1944.
Differentcyclotomicsetscanhavethesamesetofdistances(homometricsets).
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PaulErdős,inpersonalcommunicationtoA.LindoPatterson,PhysicalReview,March,1944.
The infinite family of homometric pairs of Paul Erdős
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The Hexachordal Theorem
Twocomplementaryhexachordshavethesameintervalcontent.Firstobservedempirically:ArnoldSchoenberg.~1908
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Induction Proof of the Hexachordal Theorem
JuanE.Iglesias,“OnPatterson’scyclotomicsetsandhowtocountthem,”ZeitschriftfürKristallographie,1981.
p=nbb+(1/2)nbwq=nww+(1/2)nbw
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Induction Proof of the Hexachordal Theorem: �Base case when all the vertices are black.
Lemma: Eachdistancevaluedoccursntimes.
Ifd=1ord=n‐1,itsmultiplicityequalsthenumberofofsidesofann‐vertexregularpolygon.
Examplewithn=12
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Induction Proof of the Hexachordal Theorem: �Base case when all the vertices are black.
If1<d<n‐1,anddandnarerelativelyprime,themultiplicityofdequalsthenumberofofsidesofann‐vertexregularstarpolygon.
Example:n=12d=5
Lemma: Eachdistancevaluedoccursntimes.
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Induction Proof of the Hexachordal Theorem: �Base case when all the vertices are black.
Ifdandnarenotrelativelyprimethenthemultiplicityofdequalsthetotalnumberofsidesofagroupofregularpolygons.Thereareg.c.d.(d,n)polygonswithn/g.c.d(d,n)sideseach.
Example:n=12d=3
Lemma: Eachdistancevaluedoccursntimes.
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Induction Proof of the Hexachordal Theorem: �General step for each value of d.
p=nbb+(1/2)nbwq=nww+(1/2)nbw
Bothneighborsatdistancedarewhite.
Case1:
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p=nbb+(1/2)nbwq=nww+(1/2)nbw
Induction Proof of the Hexachordal Theorem: �General step for each value of d.
Bothneighborsatdistancedareblack.Case2:
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p=nbb+(1/2)nbwq=nww+(1/2)nbw
Induction Proof of the Hexachordal Theorem: �General step for each value of d.
Oneneighboratdistancedisblackandtheotherwhite.Case3:
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Induction Proof of the Hexachordal Theorem: �Corollary Step
p=nbb+(1/2)nbwq=nww+(1/2)nbw
ifp=qthennbb+(1/2)nbw=nww+(1/2)nbw
andnbb=nww
Corollary: Ifp=qthenthetwosetsarehomometric.
Proof:
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Points with Specified Distance Multiplicities
PaulErdős‐1986
Canonefindnpointsintheplane(no3onalineandno4onacircle)sothatforeveryi=1,2,...,n‐1thereisadistancedeterminedbythesepointsthatoccursexactlyitimes?Solutionshavebeenfoundforn=2,3,...,8.IlonaPalástiforn=7and8.
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Winograd-Deep Scales
Everyinter‐pulseintervalinthecircularlatticeisrealizedbyapairofonsets,anditoccursauniquenumberoftimes.
Deepscaleshavebeenstudiedinmusictheorysinceatleast1967.TerryWinogradandCarltonGamer,JournalofMusicTheory,1967.