Fig 1.
a, Metronome No. 7 from Tony Bingham’s collection (TB 07) [5], made in Paris c.1816. b, Depiction from the 1815 English patent [6]. The metronome consists of two masses attached to a rod: the heaviest mass remains fixed at the lower end (hidden from view), while the upper mass (lighter, visible) can be moved along the rod to change the frequency of the oscillation. This way, the user can set up the desired tempo and determine its value by reading the scale behind the rod. The rod is fixed to the metronome’s shaft and can oscillate around it. To compensate for friction, an impulse force is added to the system with the aid of a spring-driven escapement wheel, which also produces the characteristic audible ticks of the metronome. All this mechanism is held in a pyramid-shaped box that amplifies the metronome’s sound and supports its scale. This is also the basic functioning of contemporary mechanical metronomes.
Fig 2.
Tempo data from symphonic recordings.
a, Representative example of raw data from the tempo extraction algorithm for 3 different conductors performing the 1st movement of the 3rd Symphony. Although the time series seem noisy on first sight, the histogram in the right panel shows a clear pattern: the algorithm not only detects the true tempo (components right below Beethoven’s mark), but also multiples (or harmonics) of this frequency (in this example, x3/2 and x3). b, Using Beethoven’s mark as a reference, harmonics in the raw data are found and rectified. c, A final smoothing ensures consistency in terms of continuity throughout contiguous samples. d, Distribution of tempo difference between conductors’ tempo choices and Beethoven’s marks. K. Böhm, at the bottom of the list, is well known among critics as one of the slowest performers of Beethoven [25]. On the other end, R. Chailly is the conductor who comes closer to the composer’s indications as he reportedly intended. But even he falls slightly behind Beethoven’s marks on average, a circumstance that has been even praised by some critics [26]. Remarkably, M. Pletnev has the most extreme and sparse distribution, reaching tempi far below and above other conductors. In fact, critics consider him an artist of contrasts, unorthodox and unpredictable [27].
Fig 3.
Performed tempo by stylistic criterion vs. Beethoven’s marks.
Each panel shows the distribution of tempo choices for each mark. The median for each distribution is shown as a dot, and the grayed line represents the 1:1 relation. On top of that, a mixed-effects regression line (in blue) for the medians, with a 95% Confidence Interval (CI), quantifies the effect of each group of conductors: all the marks are reduced on average by a fixed amount along the whole metronome range, preserving the relative discrepancy between groups. Interestingly, 72 bpm (7th Symphony, 4th movement; represented by an empty dot) seems to be the only mark that all groups accept as accurate, and therefore it was excluded from the regression model.
Fig 4.
Metronome’s ambiguous reading point.
a, Diagram of the metronome and detail of the moving weight. This weight was 15 mm high, a distance equal to 12 bpm on the tempo scale throughout all its range. 44 out of 63 marks used by Beethoven could have been mistaken by another Maelzel mark exactly 12 bpm quicker. b, Enhanced image of Beethoven’s inscription on the first page of the 9th Symphony autograph [28]: “108 oder 120 Mälzel”, where “oder” means “or” in German, and “Mälzel” refers to Maelzel’s metronome.
Table 1.
List of recordings studied in this work.
Fig 5.
Most common tempo harmonics for each kind of meter.
The tempo extraction algorithm relies on periodic patterns and rhythmic self-similarities. This explains why many of its estimated tempi are actually multiples or submultiples of the real tempo of the sample. In this work, we have called these kinds of mistaken tempi “harmonics” due to the similarity with the homonym physical phenomenon. Their most common values depend on the metric structure of the music and are displayed here. More rarely, we also detected: (i) harmonics 2 y 3/4 in compound meters; (ii) harmonics 2 y 3/4 in simple meters due to the occasional use of triplets; (iii) harmonic 2/3, in simple triple meters.
Fig 6.
a, Each dot represents a conductor, and compares the median tempo difference (tempo choice minus Beethoven’s mark) for the main and validation data sets. b, Each dot represents a metronome mark, and compares the median tempo for the main and validation data sets. Both figures show a 1:1 relation, which ensures the consistency of the main data set.
Fig 7.
a, Contemporary metronome used as a control, model Neewer© NW-707. The maximum angle of oscillation was measured by recording the metronome’s motion and creating this composite of two video frames. b, Metronome No. 6 from Tony Bingham’s collection (TB 06) [5], sold in London, but almost certainly made in Paris c.1816. Auxiliary lines were added to the photographs to locate the shaft and the maximum oscillation angle. The lower mass is estimated to hang 2 cm above the bottom of the box, according to the patent scheme. c, Metronome diagram. The metronome is based on a double pendulum, where the heaviest mass, M, remains fixed at the lower end of a rod, and the lighter mass, m, can be moved upwards and downwards to change the oscillation frequency. The distances from the shaft to each center of mass are designated by R and r. θ is the pendulum’s angle of oscillation.
Fig 8.
a, Model validation. The parametrization of a contemporary metronome is compared to its experimental oscillation frequency. It should be noted that the experimental results do not exactly follow the 1:1 relation (gray line), which means that the calibration of the scale has a small error, and our model accurately predicts it. The model by Forsén et al. (2013) [11], which uses a double pendulum without corrections, is included for completeness. b, Effect of corrections throughout the whole range for the same metronome, expressed as a percentage over the null model (frictionless, small-angle approximation for a massless rod) for each metronome mark.
Table 2.
Measurements for all the metronomes considered.
Fig 9.
Parameter estimation for all the metronomes considered.
a, Model fit for the oscillation frequency squared as a function of the position of the moving weight. b, Estimation of nondimensionalized masses μ′ (rod) and M′ (lower mass). Both controls (measuring a dismantled metronome with precision as well as measuring all the distances from a photograph) accurately estimate the true masses for the contemporary metronome, thus validating the estimation for the rest of the metronomes.
Fig 10.
Effect of different metronome distortions on its frequency compared to the average slow-down of Romantic conductors.
a, Reduction of the distance of the lower mass to the shaft, R. b, Reduction of the lower mass, M. c, Inclination of the metronome. d, Displacement of the scale relative to the moving weight.