Tatum definition

[Fabien Gouyon <fgouyon@xxxxxxxxxx>]

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?

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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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