Tatum definition (Fabien Gouyon )

Subject: Tatum definition
From:    Fabien Gouyon  <fgouyon(at)IUA.UPF.ES>
Date:    Thu, 11 Apr 2002 10:36:54 +0200

James, >From Bilmes writings, "the shortest note present" is not really an accurate definition for the tatum. It is not measured within the scope of successive events, but rather within a larger scope. Although it does take into account the inter-onset intervals, it does not seem restricted to the shortest interval between successive onsets. For instance, a sequence of 10 eighth-notes, 1 sixteenth-note and another 10 eighth-notes would probably have a tatum of an eighth-note, not of a sixteenth-note. Bilmes writes: "We use it to judge the placement of _all_ musical events." Also, the tatum would be "the time division that most highly coincides with _all_ note onsets." (p22) In the previous example, "judging" 20 eighth-notes with a regular grid of a sixteenth-note gap seems more "expensive" than "judging" solely 1 sixteenth- note with an eighth-note gap grid. One way to formalize the preceding could be to seek an equilibrium between (1) how well a regular grid explains the onsets, and (2) how well the onsets explain this grid. For instance, considering a grid with a very small gap, all the onsets would be close to an element of this grid (i.e. the grid "explains" well the onsets), on the other hand, many grid elements would not be attributed to any onset (i.e. the onsets "explain" badly the grid). The contrary for a grid with a large gap. So, from this point of view it seems rather close to the two-way mismatch procedure introduced by Maher and Beauchamp for measuring a sound fundamental frequency. Actually, to measure the tatum in audio signals, I personally use a two-way mismatch algorithm, not directly on the signal onsets, but rather on a histogram of the IOIs. But I'm afraid we're far from perceptual issues here... What about the perceptual emergence of a high-frequency pulse in non- isochronous sequences? ____________________________________________ Fabien Gouyon Music Technology Group IUA-UPF Barcelona http://www.iua.upf.es/~fgouyon tel : (00 34) 93 542 28 64 ____________________________________________

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