FS(C) – being the set of equivalence classes generated from the five graphs of the set-maps f,g,h,i,j within FS(F¹²³) ×FS(F¹²³). The reader may imagine chains of dominos that provide equivalences between their two ends. The dominos themselves are elements from the five graphs (x₁,f(x₁)), (x₁,g(x₁)), (x₂,h(x₂)), (x₃,i(x₃)), (x₂,j(x₂)) (x_i A_i) and can be turned into their “mirror images” as well, i.e., into (f(x₁),x₁), ..., (j(x₂),(x₂)). The three maps q₁, q₂ and q₃ of this solution are induced by the injections e_i : A_i FS(F¹²³).

In order to inspect a music-theoretical example, we study a much simpler diagram M₃, whose graph consists of just one node and one arrow. The node is loaded with the TwelveToneChord-Form and the arrow is loaded with the Minor-Third-Transposition for chords: t₃^{} : 2^{{₀,...₁₁}} 2^{{₀,...₁₁}}. The transposition t₃^{} for chords is defined by lifting the Minor-Third-Transposition for tones

to chords: t₃^{}(X) := {t₃(x)| x X}. For simplicity of notation, from now on, we use the same symbol t₃ instead of t₃^{}.

The reader might try to determine its Limit and Colimit before he or she continues reading.

The diagram M₃ has only one node, hence its Limit is a filter of the TwelveToneChord-Form. It passes exactly those TwelveToneChords which are invariant under the Minor-Third-Transposition t₃. Such transposition invariant chords are known as MessiaenChords.

Concrete examples of Messiaen₃Chord-Denotators are written as:

The Colimit of M₃ classifies those TwelveToneChord-Denotators as equivalent which can be transformed into one another through recursive minor-third-transposition. The resulting Form can be named “Trans₃ChordClass”.