MatheMusic Analysis : Swan Lake

In this piece Tchaïkovsky uses a fixed tonal center, b minor, and lets the harmony move through different degrees.

We use the Spinnen Tonnetz, designed by H.Seress G.Baroin to illustrate the harmonic paths. A description of the Spinnen Tonnetz can be found in our pedagogic movies and by googling to retrieve the scientific articles.
For a comprehensive introduction and functions of our Spinnen-Tonnetz, please have a look, to our analyse of Paolo Conte:, Sotto le stelle del Jazz and Madeleine

Shadow Tonnetz

In order to reveal other essential or distinctive features related to a musical composition, we propose a new visualization technique, the Shadow-Tonnetz. Its goal is to unveil the “Harmonic Rhythm” of a musical piece, that is commonly understood as the rate at which the chords progress in a musical composition.


The Shadow-Tonnetz keeps traces of the chords employed and helps displaying the “musical weight” of a specific chord and paths within a composition. It enables the spectator to better apprehend the key points and harmonic structure of a composition.

Any questions or invitations to talk are welcome...
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