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

3.3 Towards Enumeration of Harmonic Morphemes

Let $MON$ denote the set of all submonoids of $ºA$ . According to the lemma, it would be sufficient to run through all monoids $M (- MON$ in order to get all harmonic morphemes through $M ~ (Int(Ext(M ),Ext(M ))$ . Let further $MON s$ denote the subset of all saturated submonoids of the form $Int(Ext(M )) (_ ºA$ . The set of all harmonic morphemes is parametrized by $MON s$ .

As a preliminary step of our investigation we study two maps $int$ and $ext$ in opposite direction, that are related to single chords rather than to sets of chords.

PICT

Recall that for singleton chordsets $U = {X}$ one has $Int({X}) = ºA(X)$ , i.e., the intension of a single chord consists of its chord perspectives. We define:

int :| CH |--> MON with int(X) := Int({X}) = ºA(X).

A chord $X$ can be fully reconstructed from its constant selfperspectives, i.e., from the tone perspectives $ba (- ºA(X)$ with $a = 0$ . Obviously $b0$ is a chord perspective of $X$ if and only if $b (- X$ . This reconstruction can be expressed in terms of the evaluation at tone $t$ ,

evt : ºA --> T with evt(f) = f(t).

The restriction of $ev0$ to $T 0$ yields a bijection between the 12 constant tone perspectives $b 0$ and the 12 tones $b b = 0(0),b (- T$ . For any set $M$ of tone perspectives we set $T M0 := M /~\ 0$ and $[M ] := ev0(M0)$ . For monoids we introduce a separate symbol for this map:

ext : MON (ºA) --> |CH | with ext(M ) := [M ]

Remark 6 The set $|CH |$ can be viewed as a retract of $MON$ . The map $ext$ is surjective and the composition $exto int$ yields the identity map on $|CH |.$

The lemma suggests to study the fibers $ext-1(X)$ for all chords $X$ in order to get a systematic overview over all morphemes. These fibers are partially ordered according to the inclusion of morphemes and have an upper and a lower limit, which are characterized through the following proposition:

Proposition 4 Let $M (- MON$ be a monoid of tone perspectives. The saturated monoids $Int(Ext(M0)),Int(Ext(M ))$ and $Int({ext(M )})$ belong to the same fiber $ext-1(ext(M ))$ and one has the inclusions:

Int(Ext(M0)) (_ Int(Ext(M )) (_ Int({ext(M )})

$Int(Ext(M0))$ is the infimum and $Int(ext(M ))$ the supremum of the partially ordered set of saturated monoids $-1 MON s /~\ ext (ext(M ))$ .