Topology for artists - workshop sketch 3
One of my little dreams is to make and give a workshop about topology for artists. It might be called:
What you didn't know about spaces. Topology for artists. These are preliminary notes:
YouTube topology,
Styles of topology and Spaces and maps.
I started reading popular and introductory texts about topology two years ago. Since then I've become more sensitive to spatial metaphors. That's why I think a topology workshop might be interesting for artists. See for example the concept of a mapping between two complicated spaces.
Let's use music and language as examples of large, complex spaces. In his translator's note to Dante's Inferno John Ciardi writes:
When the violin repeats what the piano has just played, it cannot make the same sounds and it can only approximate the same chords. It can, however, make recognizably the same "music," the same air. But it can do so only when it is as faithful to the self-logic of the violin as it is to the self-logic of the piano.
Language too is an instrument, and each language has its own logic. I believe that the process of rendering from language to language is better conceived as a "transposition" than as a "translation," for "translation" implies a series of word-for-word equivalents that do not exist across language boundaries any more than piano sounds exist in the violin.
What must be saved, even at the expense of making four strings do for eighty-eight keys, is the total feeling of the complex, its gestalt.
This is an interesting metaphor and it can be translated into a spatial (and set-theoretic) model, that does not feel threateningly mathematical.
Some observations:
The spaces of piano and violin possibilities are both huge. But they are smaller than the space of all musical possibilities. But somehow they must fit within this space, they are subsets embedded in the larger space.
The mainstream parts of these spaces have been exhaustively explored by composers like Mozart and Beethoven.
The virtuoso parts of the spaces have been explored by composers like Liszt and Paganini.
The experimental parts of the spaces have been explored by composers like Cage and Ferneyhough.
Large parts of the piano and violin spaces are as yet unexplored.
Both spaces must have some shape and must have boundaries, because there are constraints on both the piano and the violin (neither can play infinitely loud of infinitely fast, neither can play an infinite number of notes at the same time etc.).
Most likely there exists a distance measure in this space, because the distance between Mozart and Beethoven should be smaller than the distance between Liszt and Cage.
As far as I know there are few transcriptions (mappings) from piano to violin.
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The mapping idea becomes easier in these musical spaces:
Again these spaces must be huge and must have some shape. And there exist many mappings between those spaces, in both directions. There exist transcriptions of orchestral pieces for piano but also arrangements of piano pieces for orchestra.
John Ciardi continues:
The notion of word-for-word equivalents also strikes me as false to the nature of poetry. Poetry is not made of words but of word-complexes, elaborate structures involving, among other things, denotations, connotations, rhythms, puns, juxtapositions, and echoes of the tradition in which the poet is writing. It is difficult in prose and impossible in poetry to juggle such a complex intact across the barrier of language.
These are pictures made by the
Arsuaga Vazquez Lab (Topological Molecular Biology lab at UC Davis) and for me they illustrate how the complicated structure of such poetic spaces might look. The first one uses
knots and the second one uses
simplical complexes. Both are topological devices.
And finally John Ciardi writes:
I have foregone the use of Dante's triple rhyme because it seemed clear that a rendering into English might save the rhyme or save the tone of the language, but not both.
It requires approximately 1,500 triple rhymes to render the Inferno and even granted that many of these combinations can be used and re-used, English has no such resources of rhyme. Inevitably the language must be inverted, distorted, padded, and made unspeakable in order to force the line to come out on that third all-consuming rhyme.
In Italian, where it is only a slight exaggeration to say that everything rhymes with everything else or a variant form of it, the rhyme is no problem: in English it is disaster.
This shows that some mappings (functions) between certain spaces are impossible. There are constraints on spaces and constraints on the mappings between them.
Sources:
