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


tone $t$		$x{0,1,4}(t)$		$x{0,4,10}(t)$


0: » $C$ «	$T$	$2 2 0 {a,b,c,f,f ,g,g , 1}$	$* T$	$2 2 0 {a,b,c,h,h,f,f ,g,g , 1}$

1: » $G$ «	$T$	$2 2 0 {a,b,c,f,f ,g,g , 1}$	$* R$	$2 {a,b,c,h,h,g,g }$

2: » $D$ «	$C$	${a,b,c}$	$* L$	$2 {a,b,c,h,h,f,f }$

3: » $A$ «	$R$	$2 {a,b,c,g,g}$	$* R$	$2 {a,b,c,h,h,g,g }$

4: » $E$ «	$T$	$2 2 0 {a,b,c,f,f ,g,g , 1}$	$* T$	$2 2 0 {a,b,c,h,h,f,f ,g,g , 1}$

5: » $B$ «	$L$	$2 {a,b,c,f,f }$	$* C$	${a,b,c,h,h}$

6: » $Gb/F #$ «	$R$	$2 {a,b,c,g,g}$	$* P$	$2 2 {a,b,c,h,h,f,f ,g,g }$

7: » $Db/C#$ «	$R$	$2 {a,b,c,g,g}$	$* R$	$2 {a,b,c,h,h,g,g }$

8: » $Ab/G#$ «	$L$	$2 {a,b,c,f,f }$	$* L$	$2 {a,b,c,h,h,f,f }$

9: » $Eb$ «	$P$	$2 2 {a,b,c,f,f ,g,g }$	$* R$	$2 {a,b,c,h,h,g,g }$

10: » $Bb$ «	$R$	$2 {a,b,c,g,g}$	$* T$	$2 2 0 {a,b,c,h,h,f,f ,g,g , 1}$

11: » $F$ «	$C$	${a,b,c}$	$* C$	${a,b,c,h,h}$

When Riemann distinguishes between consonant tones (those belonging to a prime chord of reference) and dissonant tones he makes a further refinement by introducing the notion of characteristic dissonances. He particularly calls the seventh of a dominant or the sixth of a subdominant characteristic for these tonal functions. If we make a formal anlogy between the logical $True/F alse$ - dichotomy and Riemann’s $Consonance/Dissonance$ -dichotomy as applied to tones (relative to a prime chord), we see that Riemann has a refined understanding of $F alse ~= Dissonant$ . Surprisingly, is the literal mathematical meaning of »characteristic function« very close to Riemann’s music-theoretical one. We mention this, because Riemann himself deliberately understood harmony as a kind of »Musical Logic«. With respect to the internal logics of the bigeneric morpheme $M 1,4$ the characteristic function does in fact provide five different answers to the 12 questions »Does tone $t$ match with the C-major-triad ?« and not just two. Each answer is the left-ideal of those tone-perspectives in $M 1,4$ mapping $t$ into $|M | = {0,1,4} 1,4$ . The answer is never $Ø$ , i.e. no tone is >absolutely false<. The worst answer is the set $C$ of the three constant tone perspectives $a = 00$ , $b = 10$ , $c = 40$ which project any tone $t$ into ${0,1,4}$ . It applies to the tones $t = 2,11$ (i.e. » $D$ « and » $F$ «). The best and fully true answer is the full monoid $T = M 1,4$ which applies only to the consonant chord tones $t = 0,1,4$ due to the fact that $M 1,4$ contains the identity $01$ . Between the worst and the best answer there are three more answers--being labeled $R$ , $L$ , $P$ . The parallel minor third $t = 9$ (i.e. » $Eb$ «) has the high truth value $P$ , where the only missing tone perspective is the identity $01$ . The leading tones » $B$ « and » $Ab$ « (to prime » $C$ « and fifth » $G$ « respectively) get the truth value $L$ , while the remaining three tones $t = {3,6,7,10}$ (i.e. » $A$ «, » $F#$ «, » $Db$ «and » $Bb$ «) get the relative truth value $R$ . The hierarchy of these truth values is ramified (see Hasse-Diagram) between $C$ and $P$ and yields the incomparable values $R$ and $L$ .

The picture of the dissonant case $M 10,4$ is very similar. Note, that the attribute dissonant is now interpreted on the meta level: All the five truth values $C*,R*, L*,P*,T*$ are dissonant, insofar all of them they contain the dissonant tone perspectives $h = 46$ and $h = 106$ . This is in accordance with the fact that $M10,4 = {0,4,10}$ contains the >tritonus< ${4,10}$ being the full image of $h$ and $h$ . The parallel truth value $* P$ is now represented by the tone $t = 6$ (i.e. $Gb/F #$ )