Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

tones of reference or - preferably - by names we write

0+ = C, 1+ = G,..., 11+ = F and 0- = c, 1-= g,..., 11- = f.

To introduce a metric on region loci we start by considering them as nodes of a Kinship Graph $G W eber$ representing direct regional kinship. This Graph $G = (Reg, Kin) Weber$ consists of 48 edges besides its 24 nodes:

Kin = {(C,G),(G,D),...,(Bb, F),(F,C)} |~| {c,g),(g,d),...,(bb,f),(f,c)} |~| {(C,a),(G, e),...,(Bb,g),(F,d)} |~| {(C,c),(G,g),...,(Bb,bb),(F,f)}

These 48 edges represent music-theoretically different types of direct (or first order) regional kinship, namely fifth kinship among Major regions and among minor regions as well as relative kinship and parallel kinship between Major and minor regions. The abstract graph $GW eber$ does not distinguish between these types. The concrete directions of edges in Figure 5 have no mathematical meaning. But note that only 37 edges out of the 48 are drawn. Me mention that the complete graph can be drawn without edge crossings on a torus.

PICT

Figure 5:

Regional kinship graph $GW eber$ according to Gottfried Weber

Lerdahl defines a metric $D : Reg × Reg-- > [0, oo )$ which quantitatively specifies and extends the kinship relation $Kin < Reg × Reg$ to all pairs of regions. To all edges $' (R,R ) (- Kin$ he attributes the same distance value

' D(R, R ) = 7.

Besides these two more types of regional kinship are selected to which larger direct distance values are attributed, namely:

Kinship to the Leittonwechsel-regions $D(R, R') = 9 for all (R, R') (- KinL := {(C,e),(G,b),...,(F,a)}$
Kinship to the Supertonic regions $D(R, R') = 10 for all (R,R') (- KinS := {(C, d),(G,a),...,(F,g)}$