J.S.Bach appreciation in Duvall, WA

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# Beats and the Critical Region - PowerPoint PPT Presentation

J.S.Bach appreciation in Duvall, WA. The University of Oregon Lectures Vladimir Chaloupka Professor of Physics Adjunct Professor, School of Music Adjunct Professor, Henry M. Jackson School of International Studies University of Washington October 29, 2009

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The University of Oregon Lectures

Professor of Physics

Adjunct Professor, Henry M. Jackson School of International Studies

University of Washington

October 29, 2009

A physicist reflects on Music: why is Bach’s last fugue unfinished?

Selected topics from Physics of Music (consonance and dissonance, tuning and temperament, studies of absolute pitch, musical acoustics) will be followed by some reflections on Physics and Music, culminating in considerations of the role of Music in the overall Human Civilization.

Physics of Music illustrated by examples:

I. Pythagoras (500 BC) on Consonance and Dissonance: first ever mathematization of human experience

II. Vladi’s Opus 0.5: A Little Harmonic Labyrinth

An illustration that the problems of temperament are NOT limited to keyboard or fretted instruments

Math: 3 3M up, the 4 3m down: 3*4=4*3

But: 1200 log2[(5/4)3/(6/5)4)] = 104 cents

III. Absolute (and relative) pitch studies. Non-intrusive studies of the hearing mechanism.

IV. The role of imperfections in creating the perception of perfection.

V. Acoustics at Saint Mark’s Cathedral

Beats and the Critical Region
• When two coherent sound waves superimpose, they will go in phase and out of phase at a rate corresponding to the difference of the two original frequencies, producing “beats” with a frequency f(beats) = |f1 – f2|
• A simple DEMO varying the beat frequency demonstrates the existence of a “critical region” where your brain is no longer able to count the beats, yet the frequency difference is not yet large enough for you to perceive two independent sounds. Two sounds with such frequency difference produce a rough, unpleasant sensation.

f1

f2

f1 and f2

Consonance and Dissonance
• Combination of the two above ingredients cannot but remind you of the Pythagoras’ recognition that tones with frequencies in ratio of small integers are consonant.
• Example: musical “fifth”: an interval with the frequency ratio of the fundamentals of the two tones 3:2 From the well known mathematical theorem:

3 times 2 = 2 times 3

• we conclude something quite non-trivial: every 3rd harmonic of the bottom tone will coincide with every second harmonic of the upper tone. Even when the “fifth” is slightly mistuned, this will results in slow beats, not the unpleasant roughness. And the other harmonics (5th, 7th etc) will be so far from each other that they will be “out of the critical region”, and therefore they will not produce any roughness either.

a) Note C

intensity

intensity

b) Note G

frequency

the smaller the integers involved, the more justified is the above reasoning.
• Therefore, the “unison” (frequency ratio 1:1) is the most consonant (and also quite boring) interval, followed by the fifth (3:2), fourth (4:3), Major 3rd (5:4) and minor 3rd (6:5).
• That just about does it for the consonant intervals (the Major and minor 6ths are just complements of the minor and Major 3rds).
• The “theoretical” frequency ratios for the dissonant intervals (such as 16:9) should be taken with a (large) grain of salt.
Need for a Well Tempered Violin
• An illustration that the problems of temperament are NOT limited to keyboard or fretted instruments
• In Equal Temperament:

3 major thirds up, the 4 minor thirds down: 3*4=4*3

• But: 1200 log2[(5/4)3/(6/5)4)] = 104 cents
• How Well Tempered was Bach’s Clavier? Experimental investigation with audience particip.
First documented evidence for medication-induced shift of absolute pitch perception

(see the Acoustical Journal publication, on web page)

• non-intrusive way of investigating human pitch perception, and of the hearing mechanism in general
• investigation of the phenomenon of categorical perception
• Story of non-musical[sic] patient from Australia[sic] : volunteers (AP/RP) wanted!
• “absolute pitch” is not “perfect pitch”
• I used a “physicist’s approach”- consider the human head (ears / cochlea / brain) as a “frequency-measuring instrument”
• Similar approach to perception of color: new !

Dependence of the difference between stimulus and response frequency on time. Days 17-31 correspond to Carbamazepine, preceded and followed by placebo.

Two examples of absolute pitch perception

Correlation of stimulus and response for two different possessors of absolute pitch.

More: measurements of relative pitch; remote measurements; measurements of color perception; …

Acoustics in Saint Mark’s Cathedral
• RT60 = 6 seconds => good for (most of) organ music; terrible for spoken word
• Solution: a line-array of speakers maximizing the direct sound and minimizing reverberant sound
• Measurements: “minimally intrusive” Maximum Length Sequence method
Exuberance and Humility in Music and Science

Left: The pipe organ at the St. Marks Cathedral in Seattle

Above: the 1743(Bach was just composing the Art of Fugue then!) instrument at the College of William and Mary in Williamsburg.

Bach as a genetic phenomenon

--------------------------

And above all: Bach as a phenomenon that we don’t understand (and probably don’t deserve).

Recall counting buttons on Bach’s portraits (and “finding” 14 = B+A+C+H)

The ultimate: a Music Doctoral Thesis explaining, with a great feeling of a DISCOVERY, why the Unfinished Fugue breaks off at bar 239

Johann Sebastian Bach (yes, he was young once, too …)

So why is the last KdF Fuge unfinished?
• 35,000 notes on a single, simple Theme in d
• KdF should be played on a pipe organ, but it should not sound “organy”
• Special importance of the Pedal Point
• KdF – Art of Fugue – as a TRIPLE art:

art of composing a fugue

art of playing a fugue

art of listening to a fugue

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To attempt to complete the life of a genius is worse than silly …

Music and Science: Goedel Escher Bach, and more
• Pythagoras’ integers
• Kepler’s Harmonia Mundi
• Superstring Theory: all elementary particles as “modes of vibration” of the same string

(ergo: “Princeton String quartet”)

• Laser Interferometer Space Antenna: “listening to the gravitational Symphony of the Universe”
• Music as an example of emergent complexity: parts of Art of Fugue “sound like parts of the Mandelbrot set”
Einstein as Scientist, Musician and Prophet
• Einstein as scientist: In 2005 we celebrated the Centenary of Einstein’s Annus Mirabilis
• Einstein as musician: from a review: “Einstein plays excellently. However, his world-wide fame is undeserved. There are many violinists who are just as good.”
• Einstein as prophet: “Nuclear weapons changed everything except our way of thinking.”

How important is Music?

Senator John Pastore: “what is the value of Elementary Particle Physics for National Defense?”

Dr. Robert R. Wilson: “none, except it makes the Nation worth defending”

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question: “what is the value of J.S.Bach’s Art of Fugue ?”

Vladi: “none, except it makes the Civilization more worthy of preserving”

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To solve the Basic Problem, we will have to find a way to combine Exuberance and Humility – and (Bach’s) Music can help!