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

As we can appreciate from Figure 1, we have the form Pianoscore whose type $(TF )$ is Limit ( $prod$ ), although in this case the limit is a product. This way, the coordinator of the form $(CF )$ is a list of forms (Bibinf, Signatures, Tempi, Lines, General notes, Articulations, Dynamicres, Absloudsigns, Arpeggi) that also have their own types and coordinators. We should understand at first the example of the sub-form »Signatures« and the corresponding part in the denotator Träumerei, as it is illustrative (and not very long, as is the case of »General notes« where the 354 notes needed to be represented!). To study the complete form and denotator consult Montiel (1999).

As we mentioned, »Signatures« is a component of the list of 9 forms corresponding to the coordinator of the form $(CF )$ Pianoscore. As »Signatures« is also a form, it has its type, which also is Limit. This way, the coordinator of the form $(CF )$ is also a list of forms (Keysigs, Timesigs), both of which have the Power type. This is explained by the fact that we can have a set of key signatures and of time signatures; the pianistic literature is full of pieces, normally with a certain length of duration, which change key signature and/or time signature one or several times. As the coordinator of the Power type form is also a form, corresponding to »Keysigs« we have the coordinator (form) »Keysigevent«, which has the Limit type. Its coordinator is a list of two forms (Onset, Keysigss) with types Simple and Power respectively. »Onset«, which is only a way of dividing the measure and counting where we are in the piece, finishes in the module of reals $R$ . Moreover, »Keysigss«, which is of Power type because we will have a set of sharps or flats (which could be the empty set, as in the case of C major or a minor), with coordinator »Keysig«, ends in the module $Z$ . It ends with the integers because a number is assigned to each note, with its respective change represented by the signs »+« o »-« (for example, -7 is b-flat). The construction of the other coordinator of the product of Signatures, »Timesig«, is analogous. How do we see the recursivity in the construction Signatures and, in particular, »Keysigs« in the actual Denotador Träumerei? It »boils down« to myKeysigs:@Keysigs({(0, {-7})}), where the parenthesis indicate the Limit type, and the brackets denote the Power type.

9 Examples for Morphisms of Local Compositions

We will finish with some examples of symmetries in music, because the theory of morphisms between local compositions is based on them. With this accomplished, we will have given examples of all the general aspects covered in the theoretical development of our exposition.

9.1 Example: Transposition (Key Change)

Consider a translation vector $t (- Q3$ , an affine module mapping, $et : Q3 --> Q3$ and a natural transformation of functors $@et : @Q3 --> @Q3$ that takes any denotator $x : A ~> EulerM odule(x)$ to its $t$ -translate

t t e (x) : A ~> EulerM odule(e .x).

As an example, let $modo(x) : 0 ~> o$ - $Zo EulerClass(e .x)$ , where the translates are $no e ,n (- Z$ be the “octave class”, in which we add a member of the octave (an integer from 0 to 11) to x. This is represented by:

$o- Cr : 0 ~> o$ - $ClassChord(c1,c2,...,ck)$ , whose translate would be:

modo modo(x) modo(x) e .o- Cr = {e .c1,...,e .ck}.

If we designate the diatonic scale as a seven note»chord«, then to our example of $Cmaj = {0,2,4,5,7,9,11}$ we apply $emodo(7)$ to get

Gmaj = {7,9,11,0,2,4,6}.