Is the Ratio of Development and Recapitulation Length to Exposition Length in Mozart

1
Is the Ratio of Development and Recapitulation
Length to Exposition Length in Mozarts and
Haydns Work  Equal to the Golden Ratio? 

• Ananda Jayawardhana

2
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)

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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.

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Part I

• Probability and Statistics
• Related Work

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Fibonacci (1170-1250) Numbers and the Golden
Ratio
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Golden Ratiohttp//en.wikipedia.org/wiki/Golden_r
atio
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Construction of the Golden Ratiohttp//en.wikiped
ia.org/wiki/Golden_ratio
8
(No Transcript)
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Fibonacci Numbers and the Golden Ratio1, 1, 2,
3, 5, 8, 13,.. http//en.wikipedia.org/wiki/
Golden_ratio
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The Mona Lisahttp//www.geocities.com/jyce3/leo.h
tm
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Example from Probability and Statistics

• Consider the experiment of tossing a fair coin
  till you get two successive Heads
• Sample SpaceHH, 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.

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• Probability density function
• where or
  or

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Proof
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Convergencehttp//www.geocities.com/jyce3/intro.h
tm
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Origins

• The Fibonacci numbers first appeared, under the
  name matrameru (mountain of cadence), in the work
  of the Sanskrit grammarian Pingala
  (Chandah-shastra, 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 Gopala in the 12th century
  also revisits the problem in some detail.
• http//en.wikipedia.org/wiki/Fibonacci_number

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Part II

• Applied Statistics
• Application of Linear Regression

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Wolfgang Amadeus Mozart (1756-1791)http//w3.rz-b
erlin.mpg.de/cmp/mozart.html
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Franz Joseph Haydn (1732-1809)http//www.classica
larchives.com/haydn.html
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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.

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Bar Lines
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Scatterplot of the Data
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Mozarts datar 0.969
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Haydns Datar 0.884
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Regression Model
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Interaction Model

• The regression equation is
• y 7.27 1.53 x - 4.04 z - 0.032 xz
• Predictor Coef SE Coef T P
• Constant 7.271 5.194 1.40 0.167
• x 1.5310 0.1285 11.91 0.000
• z -4.036 7.275 -0.55 0.581
• xz -0.0319 0.1540 -0.21 0.837
• S 10.9993 R-Sq 89.5 R-Sq(adj) 88.9
• Analysis of Variance
• Source DF SS MS F
  P
• Regression 3 61706 20569 170.01
  0.000
• Residual Error 60 7259 121
• Total 63 68965

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Model with the Indicator Variable Z

• The regression equation is
• y 8.11 1.51 x - 5.41 z
• Predictor Coef SE Coef T P
• Constant 8.109 3.230 2.51 0.015
• x 1.50884 0.07024 21.48 0.000
• z -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
  P
• Regression 2 61701 30851 259.06
  0.000
• Residual Error 61 7264 119
• Total 63 68965

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Model for Mozarts Data

• The regression equation is
• y 3.24 1.50 x
• Predictor Coef SE Coef T P
• Constant 3.235 4.436 0.73 0.472
• x 1.49917 0.07389 20.29 0.000
• S 9.57948 R-Sq 93.8 R-Sq(adj) 93.6
• Analysis of Variance
• Source DF SS MS F
  P
• Regression 1 37781 37781 411.70
  0.000
• Residual Error 27 2478 92
• Total 28 40258
• Unusual Observations
• Obs x y Fit SE Fit Residual
  St Resid
• 24 74 93.00 114.17 2.27 -21.17
  -2.27R
• 25 102 137.00 156.15 3.90 -19.15
  -2.19R

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Normal Probability Plot of the Residuals of
Mozarts Data
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Residuals Vs Fitted ValuesMozarts Data
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Residual Vs Predictor VariableMozarts Data
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Histogram of the ResidualsMozarts Data
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Is the Slope equal to the Golden Ratio for
Mozarts data?

• Model
• Hypotheses
• Test Statistic
• Reject if or
• Do not reject

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Model for Haydns Data

• The regression equation is
• y 7.27 1.53 x
• Predictor Coef SE Coef T P
• Constant 7.271 5.684 1.28 0.210
• x 1.5310 0.1406 10.89 0.000
• S 12.0370 R-Sq 78.2 R-Sq(adj) 77.6
• Analysis of Variance
• Source DF SS MS F
  P
• Regression 1 17175 17175 118.54
  0.000
• Residual Error 33 4781 145
• Total 34 21956
• Unusual Observations
• Obs 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

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Normal Probability Plot for the Residuals of
Haydns Data
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Normal Probability Plot for the Residuals of
Haydns Data after Removing the Two Outliers
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New Regression Model for Haydns Data

• y 3.50 1.62 x
• Predictor Coef SE Coef T P
• Constant 3.501 4.270 0.82 0.419
• x 1.6174 0.1076 15.03 0.000
• S 8.82003 R-Sq 87.9 R-Sq(adj) 87.5
• Analysis of Variance
• Source DF SS MS F
  P
• Regression 1 17582 17582 226.01
  0.000
• Residual Error 31 2412 78
• Total 32 19994

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Conclusion

• The ratio of development and recapitulation
  length to exposition length in Mozarts work  is
  statistically equal to the Golden Ratio.
• The ratio of development and recapitulation
  length to exposition length in Haydns work is
  statistically equal to the Golden Ratio.

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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.

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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