Metric Analysis of Tapping Accuracy
for Triple Meter in Chopin Mazurkas

Craig Stuart Sapp <craig@ccrma.stanford.edu>
9-10 Jan 2006

This Mathematica notebook examines the accuracy at which each type of metrical beat is tapped when reverse conducting to particular performances of Chopin Mazurkas.  The tapping accuracy for four performances of two mazurkas are analyzed below:
    (1) Mazurka in A minor, Op. 7, No. 2 performed by Frederic Chiu in 1999
    (2) Mazurka in A minor, Op. 7, No. 2 performance by Ignaz Friedman in 1930
    (3) Mazurka in F minor, Op. 7, No. 3 performed by Ignaz Friedman in 1930
    (4) Mazurka in F minor, Op. 7, No. 3 performed by Charles Rosen in 1989

Necessary packages to load first:

Tapping accuracy for each beat (20-trial average)

Mazurka in A minor, Op. 7, No. 2; Chiu 1999

Calculate the differences between the average tap times and the corrected data, removing the extremities:

First display a histogram of all metrical beat displacements.

Now display separate histograms for individual metrical position in the meter:

Here is a table of the average deviation from the correct beat position, and the standard deviation for each beat:

Mazurka in A minor, Op. 7, No. 2; Friedman 1930

Calculate the differences between the average tap times and the corrected data, removing the extremities:

First display a histogram of all metrical beat displacements.

Now display separate histograms for individual metrical position in the meter:

Here is a table of the average deviation from the correct beat position, and the standard deviation for each beat:

Mazurka in F minor, Op. 7, No. 3; Friedman 1930

Calculate the differences between the average tap times and the corrected data, removing the extremities:

First display a histogram of all metrical beat displacements.

Now display separate histograms for individual metrical position in the meter:

Here is a table of the average deviation from the correct beat position, and the standard deviation for each beat:

Mazurka in F minor, Op. 7, No. 3; Rosen 1989

Calculate the differences between the average tap times and the corrected data, removing the extremities:

First display a histogram of all metrical beat displacements.

Now display separate histograms for individual metrical position in the meter:

Here is a table of the average deviation from the correct beat position, and the standard deviation for each beat:

Accuracy summary for all beats of the four performances of two mazurkas:

Here is a table of the average deviation from the correct beat position, the absolute average deviation, and the standard deviation for each beat:

1	6.655712050078253`	50.861459288664`	68.736875976145`
2	1.8082942097026669`	42.0290899791017`	56.57681414794406`
3		45.91551934085868`	64.07037524839832`

Tapping accuracy for individual trials

Examine the accuracy of beat tapping for Charles Rosen's 1989 peformance of Mazurka Op. 7, No. 3.  Raw data is from http://mazurka.org.uk/info/revcond/pid52932-05/pid52932-05-abs.ma

Mazurka in F minor, Op. 7, No. 3; Rosen 1989

Calculate the differences between the average tap times and the corrected data, removing the extremities:

First display a histogram of all metrical beat displacements.

Now display separate histograms for individual metrical position in the meter, doing one beat at a time.

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Here is a table of the average deviation from the correct beat position, and the standard deviation for each beat:

101 102 103	10.773462783171464` 3.5695792880258366`	64.58035630125354` 58.019825933955495` 60.8729066515853`	80.37779630574715` 81.5882683716494` 86.43192704414385`
201 202 203	12.980582524271847`	60.85125836553871` 57.96361579790744` 62.13809030068812`	73.38378886582395` 76.56143756280213` 84.82897206844618`
301 302 303	6.59870550161813` 1.8802588996763823`	63.82753636849212` 61.56442643038928` 56.53696546956985`	82.67809751603578` 78.52614236811823` 77.17985033377727`
401 402 403	10.440129449838183` 1.8867313915857549`	58.29057089892231` 50.5123008766142` 60.75580481980709`	74.15639520589826` 75.06964847790758` 83.07567782880957`
501 502 503	13.139158576051724` 0.41100323624591534`	60.827787727401166` 52.16957300405313` 60.65457001916592`	76.41805924762645` 68.34251817569995` 80.92406539104181`
601 602 603	8.524271844660237` 1.223300970873824`	52.30106513337745` 51.02460175322813` 58.64643227448385`	68.00097422623261` 66.35043740964146` 78.99147397164595`
701 702 703	14.300970873786364`	53.54538599302476` 49.22132151946495` 57.418229804882856`	68.6335185645479` 62.89035418515022` 78.02487192962579`
801 802 803	13.776699029126178` 1.1747572815533565`	52.23027618060102` 50.63012536525605` 55.50589122443216`	65.85841090897007` 67.07059196183393` 75.25300051950275`
901 902 903	11.485436893203934` 1.2912621359223804`	59.45546234329379` 54.89047035535859` 59.511358280704904`	71.7764670498543` 72.47409325955036` 82.74539676213234`
1001 1002 1003	14.029126213592244` 0.7281553398058359`	52.83900461871991` 52.12545951550565` 55.01140541050048`	69.53210681249436` 64.7917823637335` 75.21919109208433`
1101 1102 1103	10.236245954692514` 2.0711974110031868`	49.76180601376188` 48.22364658937384` 57.61061362993702`	65.19100054797605` 67.34618096532981` 78.39732815759832`
1201 1202 1203	10.433656957928818` 3.1423948220064837`	50.328463254469476` 53.87692839413077` 53.0138875797279`	62.531227302594814` 68.01735061563782` 72.0236704116234`
1301 1302 1303	11.689320388349525` 0.4951456310679672`	51.550193232161334` 54.2123668583278` 58.613347158073296`	66.18273337964648` 66.10304101340768` 79.97289422770397`
1401 1402 1403	7.074433656957962` 3.5695792880259387`	53.28350142960389` 46.890344675904124` 49.00930027963658`	67.13876484213097` 57.94084762686635` 69.07957046056505`
1501 1502 1503	8.190938511326845` 1.5113268608414243`	60.760517799352776` 55.01941747572819` 58.006472491909406`	77.38987680926249` 68.53663072123325` 76.95198277347924`
1601 1602 1603	7.300970873786374` 2.009708737864044`	51.65048543689309` 46.281553398058215` 54.66990291262143`	66.45163875455421` 59.85398821144943` 73.51497501526059`
1701 1702 1703	7.113268608414277` 3.190938511326898`	51.95268168536156` 43.31397869733251` 55.419737958337144`	70.48682122564024` 59.22135794311619` 73.55852018543715`
1801 1802 1803	8.72168284789648` 1.012944983818821`	51.330034247651426` 50.18522009614481` 53.28230747478543`	63.70140821015959` 66.63551908959928` 70.03310250454105`
1901 1902 1903	8.249190938511285`	55.51516008420515` 49.536871209978926` 52.95249316617981`	69.91512762906007` 65.9726347805057` 70.15251124107088`
2001 2002 2003	3.896440129449842` 1.906148867313919`	46.23860244446541` 46.57168441888966` 54.00782354604584`	62.53853181171626` 62.18437456571747` 75.38967744119117`

Checking the relation to pre-beat event durations

Macurka in A minor, Op. 7, No. 2

Mazurka in F minor, Op. 7, No 3

Beat 1:

Beat 2:

Beat 3:

Beat durations for each beat

This section measures the average beat duration for each beat in the triple meter of the Mazurkas.  Averaging for the "exact" performance by the pianists, and the average uncorrected reverse conducted taps of a listener.  Durations are averaged over one measure to compensate for long-term tempo changes, but short-term tempo changes within a measures are more difficult to remove so they are ignored.

Mazurka in A minor, Op. 7, No. 2; Chiu 1999

corrected timing data

uncorrected average tapping data

000oooo0?ooo`0]0?ooo`040000003o

ool0oooo00000080oooo00@000000?ooo`3oool000000P3oool010000000oooo0?ooo`0000090?oo

o`00C@3oool2000000@0oooo0P0000040?ooo`800000=03oool2000003P0oooo0P0000040?ooo`80

0000103oool200000340ooo

Mazurka in A minor, Op. 7, No. 2; Friedman 1930

corrected timing data

uncorrected average tapping data

Mazurka in F minor, Op. 7, No. 3; Friedman 1930

corrected timing data

uncorrected average tapping data

Mazurka in F minor, Op. 7, No. 3; Rosen 1989

corrected timing data

uncorrected average tapping data

All pieces

Manually Corrected Data

Raw Tapping Data

Looking at the relative metric fractions of each beat in sequence

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

Manually Corrected Data

Raw Tapping Data

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

Manually Corrected Data

Raw Tapping Data

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

Manually Corrected Data

Raw Tapping Data

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

Manually Corrected Data

Raw Tapping Data