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

${0,4,10}$

These two interesting cases with $5m + 2n$ being a unit or $5m + 2n = ± 2$ shall now be studied with respect to their internal logics. We simply speak of the consonant and the dissonant case respectively. We calculate the left ideals (cosieves) of $M m,n$ in both cases:

Proposition 8 Let $_O_m,n$ denote the set of left ideals of the monoid $Mm,n$ with $m f = 3$ and $n g = 8$ . Further set $a = go f$ , $b = f o g$ , $c = go a$ , $m+3n h = 6$ and $m+3n+6 h = 6$ . For $5m +2n (- {1,5,7,11}$ we have $2 2 0 Mm,n = {a,b,c,f,f ,g,g , 1}$ and

2 2 0 _O_m,n = {Ø ,{a,b,c},{a,b,c,f,f },{a,b,c,g,g },Mm,n\{ 1},Mm,n}.

For $5m +2n (- {2,-2}$ we have $M = {a,b,c,h,h,f,f 2,g,g2,01} m,n$ and

_O_m,n = {Ø,{a,b,c},{a,b,c,h,h},{a,b,c,f,f2},{a,b,c,h,h,f,f2}, {a,b,c,g,g2},Mm,n\{01},Mm,n}

The following Hasse-Diagrams display the elements of $_O_m,n$ with respect to the relation of inclusion (consonant case: left, dissonant case: right).

PICT

The symbols $C, R,L,P,T$ and $* * * * * C ,R ,L ,P ,T$ are introduced to suggest a music-theoretical interpretation of these truth values interms of »Riemann Functions«. $R$ , $L$ and $P$ stand for »Relative«, »Leittonwechsel« and »Parallel«, while $C$ stands for »Constant« and $T$ for »True« (as well as for »Tonic« in our conrete example). We inspect the characteristic functions of the C-Major-triad $|M1,4|= {0,1,4}$ and of the >shortened< C-major-seventh-chord (i.e. with omitted fifth) $|M10,4|= {0,4,10}$ as subchords of the full 12-tone chord $T$ and with respect to the internal logics of $M1,4$ and $M10,4$ , respectively, i.e. we calculate the characteristic functions

x{0,1,4} : T --> _O_1,4 and x{0,4,10} : T --> _O_10,4.