Transcript Is the Ratio of Exposition Length to Development and

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 Statistics
Related Work
Fibonacci (1170-1250) Numbers and the
Golden Ratio
Golden Ratio
http://en.wikipedia.org/wiki/Golden_ratio
Construction of the Golden Ratio
http://en.wikipedia.org/wiki/Golden_ratio
ab a
 
a
b
1
1


Fibonacci Numbers and the Golden Ratio
1, 1, 2, 3, 5, 8, 13,…………..
http://en.wikipedia.org/wiki/Golden_ratio
The Mona Lisa
http://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: f  x   Fxx1 , x  2
2
where
 Fn 1
 F
 n
F0  0, F1  1, Fn  Fn1  Fn2 for n  2
n
Fn  1 1 

for n  2 or


Fn 1  1 0 
Limn 
Fn

Fn 1
or
n
n





1 1 5
1 5

Fn 

 
 
5  2   2  


Proof
Fn  Fn 1  Fn  2 , n  2, F0  0, F1  1

F  x    Fn x n  F0  F1 x  F2 x 2  ...
n 0
F  x   xF  x   F0   F0  F1  x   F1  F2  x 2  ....
=F0  F2 x  F3 x 2  ....
F  x   F1 x  F0
x
F  x  x
=
x
=F0 
F  x 
x
x

1  x  x 2 1   x 1   x 
  1,     1

1
1

 2   1  0

1 5
1 5
and  
2
2
F  x 
x
1   x 1   x 
1  1
1 

   1   x 1   x 
1
1   x   2 x 2  ...  1   x   2 x 2  ...
=
 

5 
 1
 n   n  x n
=


n 0 5


n
n
1  1  5   1  5   n

=
 
 x
2
2
n  0 5 
 
 


n
n
1  1  5   1  5  

Fn 
 
 
5  2   2  



Convergence
http://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 Statistics
Application 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 data
r= 0.969
Haydn’s Data
r= 0.884
Regression Model
y  Length a
x  Length b
1 if the composition is by Mozart
Z 
0 if the composition is by Haydn
y  0  1 x  2 z  3 xz  

 ~ iid N 0, 
2

Interaction Model
Test for interaction
The regression equation is
y = 7.27 + 1.53 x - 4.04 z - 0.032 xz
H 0 : There is no interaction (same slope for both)
H a : There is interaction
p - vlaue=0.837
Predictor
Constant
x
z
xz
Coef
7.271
1.5310
-4.036
-0.0319
SE Coef
5.194
0.1285
7.275
0.1540
T
1.40
11.91
-0.55
-0.21
P
0.167
0.000
0.581
0.837
S = 10.9993 R-Sq = 89.5% R-Sq(adj) = 88.9%
Analysis of Variance
Source
DF
Regression
3
Residual Error
60
Total
63
SS
61706
7259
68965
MS
20569
121
F
170.01
P
0.000
Model with the Indicator Variable Z
Test for the intercept
H 0 : Reg. lines for both have the same intercept
H a : H 0 is not true
The regression equation is
y = 8.11 + 1.51 x - 5.41 z
Predictor
Constant
x
z
Coef
8.109
1.50884
-5.406
SE Coef
3.230
0.07024
2.996
p - value=0.076
T
2.51
21.48
-1.80
P
0.015
0.000
0.076
S = 10.9126 R-Sq = 89.5% R-Sq(adj) = 89.1%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
2
61
63
SS
61701
7264
68965
MS
30851
119
F
259.06
P
0.000
Model for Mozart’s Data
The regression equation is
y = 3.24 + 1.50 x
t 
Predictor Coef
SE Coef
T
Constant 3.235
4.436
0.73
x
1.49917
0.07389
20.29
S = 9.57948 R-Sq = 93.8% R-Sq(adj) = 93.6%
Analysis of Variance
Source
Regression
Residual Error
Total
Unusual Observations
Obs
x
24
74
25
102
1.49917  1.61803
 1.608
0.07389
P
0.472
0.000
DF
1
27
28
SS
37781
2478
40258
MS
37781
92
F
411.70
P
0.000
y
93.00
137.00
Fit
114.17
156.15
SE Fit
2.27
3.90
Residual
-21.17
-19.15
St Resid
-2.27R
-2.19R
Normal Probability Plot of the
Residuals of Mozart’s Data
Residuals Vs Fitted Values
Mozart’s Data
Residual Vs Predictor Variable
Mozart’s Data
Histogram of the Residuals
Mozart’s Data
Is the Slope equal to the Golden
Ratio for Mozart’s data?
y  0  1 x  
• Model:
• Hypotheses:
H 0 : 1  
H1 : 1  
1  
t
~ tn  k 1
SE  1 
• Test Statistic:
• Reject H 0 if t  t0.5 ,nk 1 or
t 
p  value > 
1.49917  1.61803
 1.608  2.052  t.025,27
0.07389
p  value  0.119
Do not reject H 0
Model for Haydn’s Data
The regression equation is
y = 7.27 + 1.53 x
Predictor
Coef
SE Coef
T
Constant
7.271
5.684
1.28
x
1.5310
0.1406
10.89
S = 12.0370 R-Sq = 78.2% R-Sq(adj) = 77.6%
Analysis of Variance
Source
Regression
Residual Error
Total
Unusual Observations
Obs
x
24
37.0
25
62.0
y
106.00
79.00
P
0.210
0.000
t 
1.5310  1.6180
 0.619
0.1406
p-value  0.54
DF
1
33
34
SS
17175
4781
21956
MS
17175
145
F
118.54
Fit
63.92
102.20
SE Fit
2.04
3.97
Residual
42.08
-23.20
St Resid
3.55
-2.04
P
0.000
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
Constant 3.501
x
1.6174
SE Coef
4.270
0.1076
T
0.82
15.03
P
0.419
0.000
t 
1.6174  1.6180
 0.006
0.1076
p-value  0.99
S = 8.82003 R-Sq = 87.9% R-Sq(adj) = 87.5%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
1
31
32
SS
17582
2412
19994
MS
17582
78
F
226.01
P
0.000
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), 610615.
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