```

```

# Metric Analysis of Tapping Accuracy

## Error histograms and deviations for each beat

The following array of plots display histograms of beat-tapping errors as compared to their manually corrected times. The errors are referenced to the average of 20 reverse conducting of the performance (an analysis of individual trials is given further below).

The first column of plots (in gray) display the tapping errors for all beats in a performance. The following three columns of plots separate beat errors according to metric position (red=1, green=2 and blue=3). The last row in the array of plots displays the histograms for all four performances.

Beneath each plot is two numbers:

1. The average displacement error (mean deviation) of taps from their manually corrected positions.
2. The standard deviation of the distribution of taps around their manually corrected positions.

Performer
Mazurka
 All Beats Beat 1 Beat 2 Beat 3
Chiu 1999
Op.7 No.2

 md=46.5; sd=62.3 md=56.0; sd=72.2 md=36.1; sd=49.4 md=47.3; sd=63.4
Friedman 1930
Op.7 No.2

 md=49.4; sd=69.3 md=64.6; sd=88.9 md=37.9; sd=50.2 md=45.7; sd=59.8
Friedman 1930
Op.7 No.3

 md=41.3; sd=58.3 md=34.3; sd=44.0 md=49.0; sd=66.3 md=40.5; sd=60.9
Rosen 1989
Op.7 No.3

 md=47.9; sd=64.5 md=47.8; sd=60.8 md=46.0; sd=60.3 md=50.0; sd=70.6
All
 md=46.3; sd=63.6 md=50.9; sd=68.7 md=42.0; sd=56.6 md=45.9; sd=64.1

Note that mazurka Op. 7, No. 2 in A minor is in a slower tempo than mazurka Op. 7, No. 3 in F minor. Also, the Friedman performance of Op. 7, No. 3 was the first manually corrected performance, and it is also a difficult recording to manually correct.

On the average over all beats in the four performances, tapping the first beat is the hardest, with a median deviation of 50.9 milliseconds. The easiest beat to tap was beat 2 which has a median deviation of 42.0 milliseconds. The 11.1 millisecond difference between the average accuracies of beats one and two is small, but probably significant.

In terms of individual performances, the most accurately placed metric beats were: 2, 2, 1, 2. The least accurate metric beats were: 1, 1, 2, 3.

## Error characteristics of individual tapping trials

The error displacement of taps from the true beat given in the graphs above are derived from the average of tapping twenty separate times to the same performance, in this case Charles Rosen's 1989 performance of Mazurka Op. 7, No. 3 in F minor. In this section, the individual tapping trials are compared to the manually corrected beat times. The meaning of the color of each plot is the same as in the previous section: gray=all beats, red=beat 1, green=beat 2, and blue=beat 3.

Here is a list of the "best beat" for the metric position with the lowest mean deviation in each of the 20 trials: 3, 3, 3, 2, 2, 2, 2, 1, 2, 1, 2, 3, 1, 3, 3, 2, 2, 2, 2, 2. The "worst beat" for each trial is: 1, 1, 2, 1, 1+3, 3, 1, 2, 1, 2, 3, 2, 2, 2, 2, 1, 1, 3, 1, 1. The mean deviation and standard deviation for the displacement error for each trial compared to the manually corrected timings can be found in the Mathematica notebook used to do the analysis calculations.

## Average duration of beats in metric cycle

In general, the durations of beats in the metric cycle of a mazurka are not equal. This section measures the average duration as a fraction of the metric cycle for each beat. The beat in each measure was converted to a fraction of the measure. If all three beats are equivalent in duration, then each would take 33.3% of the full duration of the measure. Note that changes in global tempo during a measure will confuse these measurements.
Performer
Mazurka
 All Beats Beat 1 Beat 2 Beat 3
Chiu 1999
Op.7 No.2

 33% ± 5.1% 35% ± 5.7% 31% ± 3.3% 34% ± 5.3%
Friedman 1930
Op.7 No.2

 33% ± 6.1% 31% ± 5.7% 33% ± 5.2% 36% ± 6.3%
Friedman 1930
Op.7 No.3

 33% ± 6.7% 32% ± 7.0% 35% ± 5.9% 33% ± 6.9%
Rosen 1989
Op.7 No.3

 33% ± 6.2% 32% ± 7.0% 34% ± 5.1% 34% ± 6.2%
All
 33.3% ± 6.0% 32.7% ± 6.5% 33.2% ± 5.1% 34.1% ± 6.3%

Compare the manually corrected data (above) with data derived from the average tapped durations of the beats (below). The main noticeable difference is that the standard deviations are smaller (the peaks are narrower), and also beats one and three do not show a side peak in the 40% range).
Performer
Mazurka
 All Beats Beat 1 Beat 2 Beat 3
Chiu 1999
Op.7 No.2

 33% ± 5.1% 35% ± 5.9% 31% ± 3.2% 34% ± 5.1%
Friedman 1930
Op.7 No.2

 33% ± 4.8% 34% ± 4.1% 32% ± 4.4% 34% ± 5.5%
Friedman 1930
Op.7 No.3

 33% ± 4.0% 32% ± 3.6% 33% ± 2.6% 35% ± 5.0%
Rosen 1989
Op.7 No.3

 33% ± 5.5% 31% ± 5.7% 33% ± 4.1% 35% ± 5.8%
All
 33.3% ± 4.9% 32.9% ± 5.1% 32.5% ± 3.7% 34.5% ± 5.3%

The following plots show the same data in sequence, so that the pattern of metric stress of the individual beats can be seen as they progress throughout the performance. The colors represent the same beats in the metric cycle as in the plots above: red=1; green=2, blue=3.

Mazurka in A minor, Op. 7, No. 2 performed by Chiu in 1999:

Mazurka in A minor, Op. 7, No. 2 performed by Chiu in 1999 (raw tapping):

Mazurka in A minor, Op. 7, No. 2 performed by Friedman in 1930:

Mazurka in A minor, Op. 7, No. 2 performed by Friedman in 1930 (raw tapping):

Mazurka in F minor, Op. 7, No. 3 performed by Friedman in 1930:

Mazurka in F minor, Op. 7, No. 3 performed by Friedman in 1930 (raw tapping):

Mazurka in F minor, Op. 7, No. 3 performed by Rosen in 1989:

Mazurka in F minor, Op. 7, No. 3 performed by Rosen in 1989 (raw tapping):

## Mathematica Notebook

Here is the Mathematica notebook used for calculations and to generate the plots:

```

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