2. On the Methodology of the Tempo Measurements

Of course, not only can one address tempo questions normatively, but also descriptively. For example, we could ask who plays what in which tempo, if and how the shaping of tempi has changed over time and whether or not there are national or culturally specific tempo traditions. Thanks to the computer, it is now much easier to measure tempo and tempo variations. However, this still half automated, as the margin of error for fully automated measurement is too high.

The most common software for tempo measurement is the so-called Sonic Visualiser. With the aid of this software, one can set audible and visual markers while listening to a recording and then check these for accuracy through repeated listening. ( 13) It is enough when examining tempi—in contrast to examinations of rhythm or agogic—to mark the downbeat of each measure, through which one can establish the average tempo of each measure. There are also studies that show that tempi established by these means are very closely related to the perceived average tempo of the measured piece of music or passage. ( 14) Based on these values, we can create graphic representations of the tempi such as the following, which shows the tempo progression of Claudio Arrau’s 1986 recording of the first movement of Beethoven’s Piano Sonata op. 2/3 (see Figure 1). The measures are indicated on the x axis and the tempo (in BPM ‘beats per minute’ = M. M.) on the y axis. This is a logarhythmic representation representative of proportional tempo hearing. The connecting line between the individual measures serves to better display the progression of tempo from measure to measure and says nothing about the progression of tempo within an individual measure.

Figure 1: Tempo Curve op. 2/3 Claudio Arrau (1986)

In addition to the production of these kinds of tempo curves, one can also average the tempo over the course of several measures. This allows us to identify what average tempo, for example, a particular pianist plays a particular theme and what relationship this tempo has to that of another theme. One can calculate tempo variations from measure to measure from the relationship of successive tempi, average these and then say whether or not a pianist plays on average more strictly or more freely in tempo. We can also identify the tempo amplitude—the span or relation between the fastest and the slowest values of a particular recording. And of course one can also measure the actual duration of a recording and calculate the average tempo.

When establishing the tempo amplitude—the relation between the fastest and slowest tempo values of a recording—one cannot however consider all tempo values. Specific measures must be excluded: extreme ritardandi, extended measures, the endings of sections and the like. If these were included, isolated extreme ritardandi in interpretations that were otherwise strictly in tempo would result in tempo amplitudes that aren’t actually representative of those of the recording. In specific cases, decisions about which measures to exclude and which to include is no simple matter and already requires an interpretive act. Further, in order to avoid overestimating the importance of individual highs and lows of tempi, the top and bottom 2% of tempo values have not been taken into consideration.

As far as tempo variations are concerned, here we are talking about the tempo variations from measure to measure. When looking at the Sonata op. 2/3, we also experimented with measurements at the level of every two measures and at the level of formal divisions. Often the tempo variations from measure to measure are not real tempo ‘variations’ but rather elements of phrasing/phrase shaping—meaning, something that one could characterize as ‘large-scale rhythm.’ A good example of this would be the first measures of op. 2/3 (see Figure 1). The strongly diverging tempo values from measure to measure are neither intended as tempo variations nor are they perceivable as such. They are solely the result of a marked two-measure phrasing in which there is a very small extension of the rests in the 2nd and 4th measures.
If we were so inclined to approach the category of tempo variation in a manner other than from measure to measure, we would see that regarding all of the differences of detail, significant connections can be drawn between the different forms of variations. Those whose tempi vary more from measure to measure, also usually do so from two-measure group to two-measure group or from formal section to formal section. For this reason, the following presentation of our findings will focus on tempo variations from measure to measure.

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13. On the software Sonic Visualiser, see the Homepage of CHARM (The AHRC Research Centre for the History and Analysis of Recorded Music), the English institution, to which computer aided interpretation research owes so much:    http://www.charm.rhul.ac.uk/analysing/p9_0_1.html (last seen on January 3rd 2013). – A software, which allows, though not by measuring but rather by listening, to compare different recordings is the ‘Interpretation Switcher’. By temporally aligning the recordings it offers the chance to switch between them at any time. One such device containing a large number of recordings of Beethoven’s Appassionata can be found in the studio for digital collections in the permanent exhibition of the Beethoven-Haus in Bonn:    http://www.beethoven-haus-bonn.de/sixcms/detail.php?template=portal_en (last seen on December 7th 2013).

14. Compare Stefan Weinzierl and Hans-Joachim Maempel,    ‘Zur Erklärbarkeit der Qualitäten musikalischer Interpretationen durch akustische Signalmaße’, in:    Gemessene Interpretation, ed. von Loesch and Weinzierl (  as fn. 1), pp. 213–236.