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

vice versa. In other words, dissonances >need< consonances but consonances >do not need< dissonances. In particular do the idempotent multiplication factors $0 = 0.0$ , $1 = 1.1$ , $4 = 4.4$ , $9 = 9.9$ only >need< themselves. They correspond to the perfect consonances prime and fifth as well as to the major and minor thirds. These four intervals traditionally have the capability to close a counterpoint. The consonant factors $3$ and $8$ (corresponding to the sixths) >do need< the thirds, because of $3 .3 = 9$ and $8.8 = 4$ .

Riemanns concept of relative consonance >generates< the entire consonant part $K$ of a dichotomy $K/D$ of $º A$ , which coincides with the >literal translation< of the traditional dichotomy $Kons/Diss$ of intervals.

Minor triads $M int = {t,t+ 1,t+ 9}$ are relatively dissonant with respect to major ones $M ajs = {s,s+ 1,s+ 4}$ (and vice versa), because of the fact that the (8-elemented) sets $ºA(M ajt,M ins)$ always contain a dissonant inversion.

$ºA(X)$ contains dissonant tone perspectives for almost all chords $X$ . To be more precise, besides the poor chords (c.f. definition 10) the only consonant chords are those of the class which contains the major and minor triads as well as their fifth-circle transforms.

4.2 Bigeneric Morphemes

Our second bridge to Hugo Riemann is more directly connected with the idea of harmonic morphemes. We use pairs of sixth-perspectives, $m 3$ and $n 8$ (i.e. two tone perspectives corresponding to major and minor sixths via $P$ ) and generate morphemes from them:

Definition 13 The morpheme $m n m n Mm,n = (Int(Ext( 3, 8),Ext( 3, 8))$ is called the bigeneric morpheme generated from the major sixth perspective $m 3$ and the minor sixth perspective $n 8$ . The constant tone perspective $m n n m 5m+2n 3o 8- 8o 3 = 0$ is called the Pseudo-Lie-Brackett and $5m + 2n$ the commutation characteristics of $m 3$ and $n 8$ .

Depending on the particular choices of $m$ and $n$ the chord $|Mm,n |$ may contain 1, 2 or 3 tones. The commutation characteristics classify the bigeneric morphemes $Mm,n$ with respect to the transposition classes of these chords $|Mm,n|$ :

Proposition 7 Let $Mm,n = Int(Ext(m3,n8)$ denote the intension of the bigeneric morpheme $Mm,n$ . If the commutation characteristics $5m + 2n$ is either a unit modulo 12, (i.e. $= ± 1$ or $= ±5$ ) or if it is equal to $± 2$ then the chord $|Mm,n|$ is 3-elemented. Furthermore, if $5m + 2n$ is a unit then $Mm,n$ is fully consonant and the chord $|Mm,n |$ belongs to the 48-elemented isomorphy class represented the the C-Major-triad ${0,1,4}$ . If $5m + 2n = ± 2$ then the (consonant) monoid $<m3, n8>$ has a dissonant saturation $Mm,n$ and the chord $|Mm,n|$ belongs to the 24-elemented isomorphy class of the >shortended< C-Major-seventh-chord ${0,4,10}$ . In all other cases $|Mm,n|$ has only two tones or just one tone. The table lists commutation characteristics and representatives of the corresponding transposition classes of the 3-elemented chords $|Mm,n |$