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Is the Ratio of Development and Recapitulation Length to Exposition
Length in Mozart’s and Haydn’s Work Equal to the Golden Ratio?
Ananda Jayawardhana
Introduction
• Author: Dr. Jesper Ryden, Malmo University, Sweden
• Title: Statistical Analysis of Golden-Ratio Forms in Piano Sonatas by Mozart and Haydn
• Journal: Math. Scientist 32, pp1-5, (2007)
Abstract
• The golden ratio is occasionally referred to when describing issues of form in various arts.
• Among musicians, Mozart (1756-1791) is often considered as a master of form.
• Introducing a regression model, the author carryout a statistical analysis of possible golden ratio forms in the musical works of Mozart.
• He also include the master composer Haydn (1732-1809) in his study.
Part I
Probability and StatisticsRelated Work
Fibonacci (1170-1250) Numbers and the Golden Ratio
Golden Ratiohttp://en.wikipedia.org/wiki/Golden_ratio
Construction of the Golden Ratiohttp://en.wikipedia.org/wiki/Golden_ratio
a b a
a b
11
Fibonacci Numbers and the Golden Ratio1, 1, 2, 3, 5, 8, 13,…………..
http://en.wikipedia.org/wiki/Golden_ratio
The Mona Lisahttp://www.geocities.com/jyce3/leo.htm
Example from Probability and Statistics
• Consider the experiment of tossing a fair coin till you get two successive Heads
• Sample Space={HH, THH, TTHH,HTHH,TTTHH, HTTHH, THTHH, TTTTHH, HTTTHH, THTTHH, TTHTHH, HTHTHH, …}
• Number of Tosses: 2, 3, 4, 5, 6, 7, …• # of Possible orderings: 1, 1, 2, 3, 5, 8, … • Number of possible orderings follows Fibonacci
numbers.
Probability density function:
where or
or
1 , x 22xx
Ff x
0 1 1 20, 1, for 2n n nF F F F F n
1
1
1 1 for 2
1 0
nn n
n n
F Fn
F F
1 1 5 1 5
2 25
n n
nF
1
nn
n
FLim
F
Proof
1 2 0 1
20 1 2
0
20 0 1 1 2
20 2 3
1 00
, 2, 0, 1
...
....
= ....
=
=
n n n
nn
n
F F F n F F
F x F x F F x F x
F x xF x F F F x F F x
F F x F x
F x F x FF
xF x x
x
2 1 11
x xF x
x xx x
2
1, 1
11
1 0
1 5 1 5 and
2 2
2 2 2 2
0
0
1 1
1 1 1
1 1
1 = 1 ... 1 ...
5
1 =
5
1 1 5 1 5 =
2 25
1 1 5 1 5
2 25
n n n
n
n n
n
n
n n
n
xF x
x x
x x
x x x x
x
x
F
Convergencehttp://www.geocities.com/jyce3/intro.htm
Origins
• The Fibonacci numbers first appeared, under the name mātrāmeru (mountain of cadence), in the work of the Sanskrit grammarian Pingala (Chandah-shāstra, the Art of Prosody, 450 or 200 BC). Prosody was important in ancient Indian ritual because of an emphasis on the purity of utterance. The Indian mathematician Virahanka (6th century AD) showed how the Fibonacci sequence arose in the analysis of metres with long and short syllables. Subsequently, the Jain philosopher Hemachandra (c.1150) composed a well-known text on these. A commentary on Virahanka's work by Gopāla in the 12th century also revisits the problem in some detail.
• http://en.wikipedia.org/wiki/Fibonacci_number
Part II
Applied StatisticsApplication of Linear Regression
Wolfgang Amadeus Mozart (1756-1791)http://w3.rz-berlin.mpg.de/cmp/mozart.html
Franz Joseph Haydn (1732-1809)http://www.classicalarchives.com/haydn.html
Units http://www.dolmetsch.com/musictheory3.htm
• Bars/Measures and Bar lines • Composers and performers find it helpful to 'parcel up' groups of notes into bars,
although this did not become prevalent until the seventeenth century. In the United States a bar is called by the old English name, measure. Each bar contains a particular number of notes of a specified denomination and, all other things being equal, successive bars each have the same temporal duration. The number of notes of a particular denomination that make up one bar is indicated by the time signature.
• The end of each bar is marked usually with a single vertical line drawn from the top line to the bottom line of the staff or stave. This line is called a bar line.
• As well as the single bar line, you may also meet two other kinds of bar line. • The thin double bar line (two thin lines) is used to mark sections within a piece of
music. Sometimes, when the double bar line is used to mark the beginning of a new section in the score, a letter or number may be placed above its.
• The double bar line (a thin line followed by a thick line), is used to mark the very end of a piece of music or of a particular movement within it.
Bar Lines
Scatterplot of the Data
Mozart’s datar= 0.969
Haydn’s Datar= 0.884
Regression Model
Length
Length
1 if the composition is by Mozart
0 if the composition is by Haydn
y a
x b
Z
0 1 2 3y x z xz
2~ 0, iid N
Interaction ModelThe regression equation is
y = 7.27 + 1.53 x - 4.04 z - 0.032 xz
Predictor Coef SE Coef T PConstant 7.271 5.194 1.40 0.167
x 1.5310 0.1285 11.91 0.000z -4.036 7.275 -0.55 0.581xz -0.0319 0.1540 -0.21 0.837
S = 10.9993 R-Sq = 89.5% R-Sq(adj) = 88.9%
Analysis of VarianceSource DF SS MS F PRegression 3 61706 20569 170.01 0.000Residual Error 60 7259 121Total 63 68965
0
Test for interaction
: There is no interaction (same slope for both)
: There is interaction
- vlaue=0.837a
H
H
p
Model with the Indicator Variable Z
The regression equation isy = 8.11 + 1.51 x - 5.41 z
Predictor Coef SE Coef T PConstant 8.109 3.230 2.51 0.015
x 1.50884 0.07024 21.48 0.000z -5.406 2.996 -1.80 0.076
S = 10.9126 R-Sq = 89.5% R-Sq(adj) = 89.1%
Analysis of Variance
Source DF SS MS F PRegression 2 61701 30851 259.06 0.000Residual Error 61 7264 119Total 63 68965
0
0
Test for the intercept
: Reg. lines for both have the same intercept
: is not true
- value=0.076a
H
H H
p
Model for Mozart’s DataThe regression equation is
y = 3.24 + 1.50 x
Predictor Coef SE Coef T PConstant 3.235 4.436 0.73 0.472
x 1.49917 0.07389 20.29 0.000S = 9.57948 R-Sq = 93.8% R-Sq(adj) = 93.6%
Analysis of VarianceSource DF SS MS F PRegression 1 37781 37781 411.70 0.000Residual Error 27 2478 92Total 28 40258Unusual ObservationsObs x y Fit SE Fit Residual St Resid 24 74 93.00 114.17 2.27 -21.17 -2.27R25 102 137.00 156.15 3.90 -19.15 -2.19R
1.49917 1.61803 1.608
0.07389t
Normal Probability Plot of the Residuals of Mozart’s Data
Residuals Vs Fitted ValuesMozart’s Data
Residual Vs Predictor VariableMozart’s Data
Histogram of the ResidualsMozart’s Data
Is the Slope equal to the Golden Ratio for Mozart’s data?
• Model:• Hypotheses:
• Test Statistic:• Reject if or
Do not reject
0 1y x
0 1
1 1
:
:
H
H
1
11
~ n kt tSE
0H 0.5 , 1n kt t value > p
.025,271.49917 1.61803
1.608 2.0520.07389
t t
value 0.119p 0H
Model for Haydn’s DataThe regression equation is
y = 7.27 + 1.53 xPredictor Coef SE Coef T PConstant 7.271 5.684 1.28 0.210x 1.5310 0.1406 10.89 0.000S = 12.0370 R-Sq = 78.2% R-Sq(adj) = 77.6%
Analysis of VarianceSource DF SS MS F PRegression 1 17175 17175 118.54 0.000Residual Error 33 4781 145Total 34 21956
Unusual ObservationsObs x y Fit SE Fit Residual St Resid 24 37.0 106.00 63.92 2.04 42.08 3.55 25 62.0 79.00 102.20 3.97 -23.20 -2.04
1.5310 1.6180 0.619
0.1406t
-value 0.54p
Normal Probability Plot for the Residuals of Haydn’s Data
Normal Probability Plot for the Residuals of Haydn’s Data after Removing the Two Outliers
New Regression Model for Haydn’s Data
y = 3.50 + 1.62 x
Predictor Coef SE Coef T PConstant 3.501 4.270 0.82 0.419x 1.6174 0.1076 15.03 0.000
S = 8.82003 R-Sq = 87.9% R-Sq(adj) = 87.5%
Analysis of VarianceSource DF SS MS F PRegression 1 17582 17582 226.01 0.000Residual Error 31 2412 78Total 32 19994
1.6174 1.6180 0.006
0.1076t
-value 0.99p
Conclusion
• The ratio of development and recapitulation length to exposition length in Mozart’s work is statistically equal to the Golden Ratio.
• The ratio of development and recapitulation length to exposition length in Haydn’s work is statistically equal to the Golden Ratio.
References
• Ryden, Jesper (2007), “Statistical Analysis of Golden-Ratio Forms in Piano Sonatas by Mozart and Haydn,” Math. Scientist 32, pp1-5.
• Askey, R. A. (2005), “Fibonacci and Lucas Numbers,” Mathematics Teacher, 98(9), 610-615.
Homework for Students
• Fibonacci numbers• Edouard Lucas (1842-1891) and his work• Original sources of Indian mathematicians and
their work
• Possible MAA Chapter Meeting talk and a project for Probability and Statistics or History of Mathematics