Rhythmic Research > Eigenrhythms > 2. Eigenrhythms
2. EIGENRHYTHMS
We propose to bridge this gulf using the standard dimen-sionality reduction tool of Principal Component Analysis (PCA) [3]. An ensemble of data that can be represented as points in a high-dimensional space can be approximated as the weighted sums of a few basis vectors in that space; the covariance matrix of the ensemble provides informa-tion about which dimensions are correlated (i.e. exhibit co-ordinated changes), and by finding the eigenvectors of the covariance matrix with the largest eigenvalues PCA finds the basis functions that minimize the distortion of a lower-dimensional representation. Each point in the original high dimensional space is represented by a smaller number of coefficients, which are the weights applied to each of the principal component vectors to approximate that point. Individual principal components, ordered according to their contribution to the overall distortion, can be interpreted as the main dimensions of variation among the examples in the set.
In this work, we represent drum patterns as a simple two-dimensional surface. The horizontal dimension is time, densely sampled to provide a fine resolution of drum-note events (for the results below, we used 5 ms sampling). The vertical dimension corresponds to the different instruments: we caricature popular music drum tracks as con-sisting of three instruments, bass drum, snare, and hi-hat, and have one row for each. The values in this surface are pseudo energy envelopes: each beat event is represented by a brief, decaying pulse in the surface. We use half Gaussians with a standard deviation of 20 ms, which, un-like single impulses, can gloss over small amounts of jitter in the timing of individual beats, while retaining a sharp, welldefined onset. The principal components of these surfaces, the two-dimensional surfaces that can be com-bined to approximate the entire set, constitute our "eigenrhythms".
Our goal is to produce systems that can be applied to actual recordings, but to simplify the investigation of un-derlying rhythmic information we sidestepped the stage of extracting drum events from audio by working directly from MIDI files in which note event times and drum voice identities are provided explicitly. We do not consider this a serious limitation given the success reported in auto-matic drum beat extraction from audio: Realtime extraction of bass drum and snare events was reported by Goto ten years ago [5], and more recent work has included adaptive learning of the different drum sounds [17] among many other refinements.
Our principal motivation is to investigate the viability of low-dimensional descriptions of rhythm patterns, but in order to motivate and evaluate our results we apply the representations to a genre classification task. However, we do not pay much attention to making a careful and comprehensive description of the rhythmic pattern in each individual piece: Since we are interested in the gross be-havior of a large collection of different drum patterns, the main thing we want is a large number of diverse patterns extracted from different peices. If, for a particular piece, we only extract one of several patterns used, or even if our extraction algorithm fails and returns a nonsense pattern, it is not of great concern as long as the bulk of the database - the material that the most significant principal components describe - is valid. This is another reason for starting with MIDI: there are tens of thousands of MIDI files readily available on the internet, and we would like to be able to model rhythmic information from collections of this scale without the computational load of processing an equivalent amount of audio. By limiting ourselves to pulling the pattern from a single, small excerpt taken from the middle of each piece, we also avoid the issue of tempo tracking – we assume the tempo is constant over the 10 s excerpts we use, and estimate a single value. The next section describes in more detail our method for finding the "eigenrhythm" drum pattern principal com-ponents. Section 4 presents the results of our preliminary application to 100 pieces, giving both the eigenrhythms and describing the classification experiments. We discuss some other possible applications, including resynthesiz-ing patterns from arbitrary points in rhythm space, and present our conclusions in the conclusions section.
Eigenrhythms
Eigensynth: Derivative Beat Box
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Phase Vocoder
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