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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

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Page 1: 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

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

Page 2: 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)

Page 3: 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

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.

Page 4: 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

Part I

Probability and StatisticsRelated Work

Page 5: 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

Fibonacci (1170-1250) Numbers and the Golden Ratio

Page 6: 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

Golden Ratiohttp://en.wikipedia.org/wiki/Golden_ratio

Page 7: 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

Construction of the Golden Ratiohttp://en.wikipedia.org/wiki/Golden_ratio

Page 8: 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

a b a

a b

11

Page 9: 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

Fibonacci Numbers and the Golden Ratio1, 1, 2, 3, 5, 8, 13,…………..

http://en.wikipedia.org/wiki/Golden_ratio

Page 10: 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

The Mona Lisahttp://www.geocities.com/jyce3/leo.htm

Page 11: 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

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.

Page 12: 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

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

Page 13: 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

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

Page 14: 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

Convergencehttp://www.geocities.com/jyce3/intro.htm

Page 15: 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

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

Page 16: 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

Part II

Applied StatisticsApplication of Linear Regression

Page 17: 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

Wolfgang Amadeus Mozart (1756-1791)http://w3.rz-berlin.mpg.de/cmp/mozart.html

Page 18: 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

Franz Joseph Haydn (1732-1809)http://www.classicalarchives.com/haydn.html

Page 19: 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

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.

Page 20: 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

Bar Lines

Page 21: 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

Scatterplot of the Data

Page 22: 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

Mozart’s datar= 0.969

Page 23: 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

Haydn’s Datar= 0.884

Page 24: 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

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

Page 25: 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

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

Page 26: 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

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

Page 27: 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

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

Page 28: 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

Normal Probability Plot of the Residuals of Mozart’s Data

Page 29: 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

Residuals Vs Fitted ValuesMozart’s Data

Page 30: 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

Residual Vs Predictor VariableMozart’s Data

Page 31: 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

Histogram of the ResidualsMozart’s Data

Page 32: 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

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

Page 33: 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

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

Page 34: 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

Normal Probability Plot for the Residuals of Haydn’s Data

Page 35: 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

Normal Probability Plot for the Residuals of Haydn’s Data after Removing the Two Outliers

Page 36: 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

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

Page 37: 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

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.

Page 38: 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

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.

Page 39: 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

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