Performance correlation scapes of beat tempo

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How to interpret the graphs

The triangular pictures shown on the webpages listed above are plots displaying similarity amongst individual performances. Each performer is assigned a particular color. Black has the special meaning of a synthetic average performance. Sorry that I have not come up with a set of colors which are easy to distingusish when the number of individual performances goes above 20.

The base of the triangular plots represents time in a performance. The bottom left represents the start of the performance, and the bottom right represents the end of the performance. Each pixel at the bottom of the picture represents one beat in the performance. If the score has 100 measures and the time signature is 3/4, then there would be 300 pixels at the bottom of the triangle (unless the pictured was scaled after being produced).

The height of the triangle represents the amount of beats which are being examined at each point in the triangle. At the bottom of the picture, only the local beat is examined. In the second row, every two adjacent beats are examined, and so on until you reach the top of the triangle. At the top of the triangle, there is only one point. This point represents an analysis which includes all beats in the performance from start (left) to finish (right).

What is the analysis being done for each pixel in the triangular plots? The answer is Pearson correlation. In the case of the performance correlation timescapes, correlation is used to measure how similar the shape of one perforance tempograph relates to another (by a different performer, or the same performer recorded at a different time).

At the bottom of the triangular plots, small snippets of the performance are compared to other performances. As you get higher in the triangle, the snippets of tempi get larger and larger, until at the apex of the triangle, the entire beat-tempo sequence of a performance is compared to another performance.

In each plot, what do the colors represent? To generate a plot for a particular performance, each analysis pixel compares the same location in the score amongst all of the other performances on the page. As a result, there is a list of correlation values underlying each pixel in the plot. If there are 40 performances on the page, then there would 39 correlation values for each point in the plot: a correlation measurement between the current performance being analyzed, and the other 39 performances on the page.

It is hopeless to list all of the 39 correlations for every point in the plot. That would potentially be millions of numbers and would fill many pages of paper. So the numbers are simplified and made more manageable. First, all but the most important correlation value is thrown out, so that each pixel represents only one correlation value. Next, instead of displaying the actual numerical value of the correlation measurement, a color is used instead. In this case a color which represents the performance with the highest correlation to primary performance being analyzed.

Now you should know basic navigation of the triangular plots. How to interpret them? Look at a particular plot. It will have a name and date benath it which signifies the performer and performance date. Along with this information is a rectangle of color. This is the color assigned to this particular performance.

Note that you cannot find the color of the performance in the actual plot for that performance. Instead, colors representing the strongest correlation to other performances are displayed in its plot.

If there are two performanes which are identical, such as the same performance released on different labels, that pair of plots will contains a solid color matched to its twin performance. For example in Mazurka in A minor, Op. 17, No. 4, the 1977 Magaloff performance was originally released on a vinyl record. However, we have two releases of the same performance, both re-released by Philips on different CD series.

After looking at the pair of Magaloff performances for Mazurka 17/4, now look at the interesting pair for Hatto and Indjic [Note that the Indjic performance date is 1988, but is incorrectly dated in the Calliope 3321 re-release from which his performance tempos were extracted]. The Hatto/Indjic pairing show the same large field of cross-correlation. It is quite impossible for two performers to match their tempos beat-by-beat throughout the entire performance. This is especially true of mazurkas, where the tempo between successive beats can vary widely.

The purpose of these plots was not intended to find exact matches between performances. Instead they were in part designed to identify the same performer playing throughout their career. For example, Examine the Rubinstein performance on each of the pages listed above. There are three performances of each mazurka played by Rubinstein throughout his career. One from approximately 1939, 1952, and 1966 (he did not record them all in one or two sessions like Hatto, but instead sometime spread them out over a year or two).

Notice that the blotches of colour in the Rubinstein for all example mazurkas above mostly reference the other two Rubinstein performances, or the "average" performance. The average performance is litereally calculated by averaging the beat-by-beat tempo of all performances on the page. It is a useful analytical tool, because it often hides less important correlations, so the true influences between performances are easier to extract.

Now you are ready to explore other relationships between the performances by looking at the color blotches in the pictures. In general, the larger the blotch, the more likely that it represents a significant relationship between the two performances. Mutual color references where one plot shows the color of another plot which in turn has the color of the first plots in the analygous locations are also usually sigificant, event if they are smaller sized blotches. I will work on making that last sentence more intelligible...