A Coordinate System |

This need not be a two-dimensional system, it could as well be something like

A Cube; This time, in glorious 90s colours. One dimension might encode person, another tense, another aspect. |

Since we're dealing with a very finite number of combinations - say, this time, that we're dealing with number * case * gender or something like that - we can conveniently enough flatten this cube by mapping the elements of two of the dimensions onto one dimension, returning us to something like the coordinate system above; we need to arrange it so that one dimension is subordinate to the other, though (e.g. in the multi-dimensional representation, each dimension can keep its elements in the same order everywhere: 1, 2, 3, ... always come in that order; however, if you have a subordinate and a superordinate set of dimensions, A3 may come before B1, despite 1 < 3, if the value of A is lower than the value of B). We get this happening:

Some neighbours are preserved as neighbours: any two that are in the same column in this example, will retain their distance in the new presentation; any two that are in the same row will have their distance multiplied by four. If distance isn't interesting, this is no problem, and even if it is, it's not necessarily all that big a deal, so we'll ignore it for now. We should be aware of it, though.

But now we'l get to an interesting thing: in morphology, we have two 'spaces'/'planes'/whatever. One is the plane of possible combinations of morphemes, the other is the plane of possible combinations of meaning.

Canonical agglutination, if such a term can be used, would refer to the following situation, or higher-dimensional analogues of it:

There is a perfect correspondence between combinations of morphemes (the left coordinate system) and combinations of meanings (the right coordinate system). We find in some languages, though, that this is not the case! We can come up with a lot of things that could be going on, and this image with several forms mapped to meanings should illustrate some possibilities:

A system of correspondences that has been distorted in several ways. |

Some of the most common things in real-life languages probably are meaning-conflations (several meanings correspond to one combination of morphemes), morpheme-conflations (several morphemes express the same meaning). Direct twists might be somewhat unusual; however, if a twist/cross exists, I find it likely that more than one pair has a similar cross/twist going, and it's of course imaginable that the twist has more than just one pair of elements involved.

Would an example of this be the German article system? Or am I still not grasping the idea? https://en.wikipedia.org/wiki/German_articles#/media/File:German_definite_article_declension.png

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DeleteThe German article system is a small example of this, basically either a 4*4 system for the definite article: (number of cases times (number of genders + the plural)), or a 4*4 + 1 system ((number of genders + the plural) times number of cases) + mass noun for the indefinite article.

DeleteWhat I was really thinking about while typing this post, though, was huge TAM systems.