Loading...

PPT – Is the Ratio of Development and Recapitulation Length to Exposition Length in Mozarts and Haydns Wor PowerPoint presentation | free to download - id: 23fc2b-MjUxN

The Adobe Flash plugin is needed to view this content

View by Category

Presentations

Products
Sold on our sister site CrystalGraphics.com

About This Presentation

Write a Comment

User Comments (0)

Transcript and Presenter's Notes

Is the Ratio of Development and Recapitulation

Length to Exposition Length in Mozarts and

Haydns 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 Ratiohttp//en.wikipedia.org/wiki/Golden_r

atio

Construction of the Golden Ratiohttp//en.wikiped

ia.org/wiki/Golden_ratio

(No Transcript)

Fibonacci Numbers and the Golden Ratio1, 1, 2,

3, 5, 8, 13,.. http//en.wikipedia.org/wiki/

Golden_ratio

The Mona Lisahttp//www.geocities.com/jyce3/leo.h

tm

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.

- Probability density function
- where or

or

Proof

Convergencehttp//www.geocities.com/jyce3/intro.h

tm

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

Part II

- Applied Statistics
- Application of Linear Regression

Wolfgang Amadeus Mozart (1756-1791)http//w3.rz-b

erlin.mpg.de/cmp/mozart.html

Franz Joseph Haydn (1732-1809)http//www.classica

larchives.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

Mozarts datar 0.969

Haydns Datar 0.884

Regression Model

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

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

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

Normal Probability Plot of the Residuals of

Mozarts Data

Residuals Vs Fitted ValuesMozarts Data

Residual Vs Predictor VariableMozarts Data

Histogram of the ResidualsMozarts Data

Is the Slope equal to the Golden Ratio for

Mozarts data?

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

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

Normal Probability Plot for the Residuals of

Haydns Data

Normal Probability Plot for the Residuals of

Haydns Data after Removing the Two Outliers

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

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.

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.

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

About PowerShow.com

PowerShow.com is a leading presentation/slideshow sharing website. Whether your application is business, how-to, education, medicine, school, church, sales, marketing, online training or just for fun, PowerShow.com is a great resource. And, best of all, most of its cool features are free and easy to use.

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

You can use PowerShow.com to find and download example online PowerPoint ppt presentations on just about any topic you can imagine so you can learn how to improve your own slides and presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

presentations for free. Or use it to find and download high-quality how-to PowerPoint ppt presentations with illustrated or animated slides that will teach you how to do something new, also for free. Or use it to upload your own PowerPoint slides so you can share them with your teachers, class, students, bosses, employees, customers, potential investors or the world. Or use it to create really cool photo slideshows - with 2D and 3D transitions, animation, and your choice of music - that you can share with your Facebook friends or Google+ circles. That's all free as well!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

For a small fee you can get the industry's best online privacy or publicly promote your presentations and slide shows with top rankings. But aside from that it's free. We'll even convert your presentations and slide shows into the universal Flash format with all their original multimedia glory, including animation, 2D and 3D transition effects, embedded music or other audio, or even video embedded in slides. All for free. Most of the presentations and slideshows on PowerShow.com are free to view, many are even free to download. (You can choose whether to allow people to download your original PowerPoint presentations and photo slideshows for a fee or free or not at all.) Check out PowerShow.com today - for FREE. There is truly something for everyone!

Recommended

«

/ »

Page of

«

/ »

Promoted Presentations

Related Presentations

Page of

Home About Us Terms and Conditions Privacy Policy Presentation Removal Request Contact Us Send Us Feedback

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

Copyright 2018 CrystalGraphics, Inc. — All rights Reserved. PowerShow.com is a trademark of CrystalGraphics, Inc.

The PowerPoint PPT presentation: "Is the Ratio of Development and Recapitulation Length to Exposition Length in Mozarts and Haydns Wor" is the property of its rightful owner.

Do you have PowerPoint slides to share? If so, share your PPT presentation slides online with PowerShow.com. It's FREE!