Instead of the compound
TwelveTone-
Form one can build denotators on the basis of
a
Simple Form PiMod12 in Mazzola’s ontology (= “Pitch modulo 12” cf. [9], section
6.4):
4. A Music-theoretical Example
In his monograph on late romanic harmony Daniel Harrison presents the following table
of dual correspondences (cf. [4], p. 27), which he calles a “dual network” of harmonic
concepts:
|
|
| Major | Minor |
|
|
| | |
| 7-8 | 6-5 |
| Dominant | Subdominant |
| Authentic cadence | Plagal cadence |
| Ascending 5th semicadence | Descending 5th semicadence |
| Sharp | Flat |
| |
We discuss this table as example for a rich structure of inheritance in a “network” of
denotators, starting from a natural endomorphism of Simple Form PiMod12 and
demonstrate how Limit- and Colimit constructions can suitably explain the
correspondences described by Harrison. After a technical preparation we will revisit
them in detail.
We consider the affine symmetry e711 : 
12 
12, e711(z) := -z + 7. It induces a
natural transformation @e711 : @12 @12 of the corresponding functor, which is the
AmbientSpace of the Form PiMod12.
Now consider the following diagram DTone having two nodes loaded with the Form
PiMod12 and one arrow between them loaded with @e711:
In addition, we consider the following Forms:
