Godfried Toussaint

Multisets: Applications to Music

The clave son in convex polygon notation

Easytodiscoversymmetries.

3

3

4

4

2

The clave son as a multiset: inter-onset-interval histogram notation

Inter‐onsetintervalvaluesMul3p

licity

Alltheinter‐onsetintervalsdetermineamul3set.

12345678

Shouldweusethismorecompleteinforma3on?

X-ray crystallography in the 1920’s

fic3onal3‐atommolecule

Distancesbetweenatoms

Distancegeometryproblem:Givenamultisetofdistancesbetweenasetofobjects(withoutlabels),canonereconstructthespatialcon=igurationoftheobjects,andifso,isthesolutionunique?

A. Lindo Patterson, “Ambiguities in the X-ray analysis of crystal structures,” Physical Review, March, 1944.

Differentcyclotomicsetscanhavethesamesetofdistances(homometricsets).

PaulErdős,inpersonalcommunicationtoA.LindoPatterson,PhysicalReview,March,1944.

The infinite family of homometric pairs of Paul Erdős

The Hexachordal Theorem

Twocomplementaryhexachordshavethesameintervalcontent.Firstobservedempirically:ArnoldSchoenberg.~1908

Induction Proof of the Hexachordal Theorem

JuanE.Iglesias,“OnPatterson’scyclotomicsetsandhowtocountthem,”ZeitschriftfürKristallographie,1981.

p=nbb+(1/2)nbwq=nww+(1/2)nbw

Induction Proof of the Hexachordal Theorem: �Base case when all the vertices are black.

Lemma: Eachdistancevaluedoccursntimes.

Ifd=1ord=n‐1,itsmultiplicityequalsthenumberofofsidesofann‐vertexregularpolygon.

Examplewithn=12

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.

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.

Induction Proof of the Hexachordal Theorem: �General step for each value of d.

p=nbb+(1/2)nbwq=nww+(1/2)nbw

Bothneighborsatdistancedarewhite.

Case1:

p=nbb+(1/2)nbwq=nww+(1/2)nbw

Induction Proof of the Hexachordal Theorem: �General step for each value of d.

Bothneighborsatdistancedareblack.Case2:

p=nbb+(1/2)nbwq=nww+(1/2)nbw

Induction Proof of the Hexachordal Theorem: �General step for each value of d.

Oneneighboratdistancedisblackandtheotherwhite.Case3:

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:

Points with Specified Distance Multiplicities

PaulErdős‐1986

Canonefindnpointsintheplane(no3onalineandno4onacircle)sothatforeveryi=1,2,...,n‐1thereisadistancedeterminedbythesepointsthatoccursexactlyitimes?Solutionshavebeenfoundforn=2,3,...,8.IlonaPalástiforn=7and8.

Winograd-Deep Scales

Everyinter‐pulseintervalinthecircularlatticeisrealizedbyapairofonsets,anditoccursauniquenumberoftimes.

Deepscaleshavebeenstudiedinmusictheorysinceatleast1967.TerryWinogradandCarltonGamer,JournalofMusicTheory,1967.