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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 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 ab 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 Fxx1 , 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 H 0 : There is no interaction (same slope for both) The regression equation is y = 7.27 + 1.53 x - 4.04 z - 0.032 xz 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 ,nk 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