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

The following table contains a few particular cases of homology groups and their ranks.

Corpus ListNo. Paradigm.Group k rk(SHG ) -Schaf-frath---------39------------T---------6-----6-k-- Schaf frath 13 CP 5 11 Schaf frath 48 CP 5 4 Schaf frath 35 CP 5 3 -Bach----------------2-----------CP---------3-----3---- Bach 3 CP 3 3 Bach 15 CP 3 3 Bach 59 CP 3 2 Bach 19 CP 3 1 Bach 25 CP 3 1 Bach 44 CP 3 1 -Baroni/Jacoboni----12-----------CP---------3-----5---- Baroni/Jacoboni 18 CP 3 3 Baroni/Jacoboni 2 CP 3 2 Baroni/Jacoboni 11 CP 3 2 Baroni/Jacoboni 36 CP 3 1

As an information concerning the total amount of homology, the sum of the ranks of all symmetric homology groups was computed for each complex. The following table displays the corresponding mean values for the three samples with respect to both paradigmatic groups.

-Corpus-----------Paradigm.Group--Sk->=0rk(SHGk)- Schaffrath T 22.1 Bach T 22.3 -Baroni/Jacoboni--------T--------------25.1------ Schaffrath CP 54.6 Bach CP 19.0 Baroni/Jacoboni CP 21.8

Apart from the sum of the Betti numbers $Rank(SHG ) k$ , their alternating sum $xG = S (-1)kRank(SHG ) k>=0 k$ , the Euler characteristic, is considered an overall measure of combinatorial complexity. The corresponding mean values are listed below.

G -Corpus----------P-aradigm.Group------x--- Schaffrath T -7.2 Bach T -19.7 -Baroni/Jacoboni---------T----------22.5-- Schaffrath CP 31.6 Bach CP 14.5 Baroni/Jacoboni CP 15.1

Though there is some homology in degrees larger than 2, the main effect here comes from the dominance of first homology if the paradigmatic group consists