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 ( ) 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 ( ) in his study.

Part I Probability and Statistics Related Work

Fibonacci ( ) Numbers and the Golden Ratio

Golden Ratio

Construction of the Golden Ratio

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

The Mona Lisa

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

Proof

Convergence

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.cadenceSanskrit grammarianPingala BCProsodyIndian mathematicianVirahankametresJain Hemachandra1150Gopāla

Part II Applied Statistics Application of Linear Regression

Wolfgang Amadeus Mozart ( )

Franz Joseph Haydn ( )

Units 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

Interaction Model The regression equation is y = x z xz Predictor Coef SE Coef T P Constant x z xz S = R-Sq = 89.5% R-Sq(adj) = 88.9% Analysis of Variance Source DF SS MS F P Regression Residual Error Total

Model with the Indicator Variable Z The regression equation is y = x z Predictor Coef SE Coef T P Constant x z S = R-Sq = 89.5% R-Sq(adj) = 89.1% Analysis of Variance Source DF SS MS F P Regression Residual Error Total

Model for Mozart’s Data The regression equation is y = x Predictor Coef SE Coef T P Constant x S = R-Sq = 93.8% R-Sq(adj) = 93.6% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations Obs x y Fit SE Fit Residual St Resid R R

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? Model: Hypotheses: Test Statistic: Reject if or Do not reject

Model for Haydn’s Data The regression equation is y = x Predictor Coef SE Coef T P Constant x S = R-Sq = 78.2% R-Sq(adj) = 77.6% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations Obs x y Fit SE Fit Residual St Resid

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 = x Predictor Coef SE Coef T P Constant x S = R-Sq = 87.9% R-Sq(adj) = 87.5% Analysis of Variance Source DF SS MS F P Regression Residual Error Total

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),

Homework for Students Fibonacci numbers Edouard Lucas ( ) 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