Music of the Spheres - University of Miami Department of Mathematics
Transcript of Music of the Spheres - University of Miami Department of Mathematics
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Music of the SpheresDrew Armstrong
University of Miami
February 2, 2013
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The Plucked String
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The Plucked StringA plucked string with fixed ends can only vibrate at
certain resonant frequencies
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The Plucked StringA plucked string with fixed ends can only vibrate at
certain resonant frequencies
f “fundamental” frequency
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The Plucked StringA plucked string with fixed ends can only vibrate at
certain resonant frequencies
2f
f “fundamental” frequency
1st harmonic
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The Plucked StringA plucked string with fixed ends can only vibrate at
certain resonant frequencies
2f
f
3f
“fundamental” frequency
1st harmonic
2nd harmonic
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The Plucked StringA plucked string with fixed ends can only vibrate at
certain resonant frequencies
2f
f
3f
4f
“fundamental” frequency
1st harmonic
2nd harmonic
3rd harmonic
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The Plucked StringEvery possible vibration is a “superposition” of these.
2f
f
3f
4f
“fundamental” frequency
1st harmonic
2nd harmonic
3rd harmonic
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The Plucked StringEvery possible vibration is a “superposition” of these.
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The Plucked String
Plot the energy at each frequency:
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The Plucked String
Plot the energy at each frequency:Get a “fingerprint” for the sound.
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The Plucked String
Maybe this sounds like a VIOLIN
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The Plucked String
Maybe this sounds like a CLARINET
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Pythagoras (c. 570 -- 495 BC)
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Pythagoras (c. 570 -- 495 BC)Big Discovery:
•When two strings of equal tension are plucked,
•They sound good if their lengths have the ratio of small whole numbers.
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Pythagoras (c. 570 -- 495 BC)Big Discovery:
•When two strings of equal tension are plucked,
•They sound good if their lengths have the ratio of small whole numbers.
Eureka!
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Pythagoras (c. 570 -- 495 BC)
this sounds very good(“octave”)
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Pythagoras (c. 570 -- 495 BC)
this sounds very good(“octave”)
this sounds quite good(“perfect fifth”)
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Pythagoras (c. 570 -- 495 BC)
this sounds very good(“octave”)
this sounds quite good(“perfect fifth”)
this sounds rather good(“perfect fourth”)
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Pythagoras (c. 570 -- 495 BC)
this sounds TERRIBLE!
√2
1
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Pythagoras (c. 570 -- 495 BC)
this sounds TERRIBLE!
√2
√2
1
is not a fraction of whole numbers.
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Modern Explanation
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Modern Explanation
Two nearby frequencies sound ROUGH together. (“beats”)
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Modern Explanation
Plomp and Levelt (1965) measured the psychological effect.
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Modern Explanation
220 440 660 880 1320 1760Frequency (Hz)
Play the notes 220 Hz and 445 Hz together: ROUGH
220 440 660 880 1100 1320 1540 1760
445 890 1335 1780
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Modern Explanation
220 440 660 880 1320 1760Frequency (Hz)
Play the notes 220 Hz and 440 Hz together: SMOOTH
220 440 660 880 1100 1320 1540 1760
440 880 1320 1760
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Modern Explanation
220 440 660 880 1320 1760Frequency (Hz)
Play the notes 220 Hz and 330 Hz together: STILL NICE
220 440 660 880 1100 1320 1540 17601320660330 990 1650
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Modern Explanation
The Plomp-Levelt curve based on six harmonics.
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Why do we have 12 notes?
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Why do we have 12 notes?
Pythagorean tuning is based on the most consonant interval:
2:3(a “perfect fifth”)
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Why do we have 12 notes?
Two successive perfect fifths equals
23
· 23
=49
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Why do we have 12 notes?
Three successive perfect fifths equals
23
· 23
· 23
=827
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Why do we have 12 notes?
Many successive perfect fifths equals
23
· · · · · 23
· 23
· 23
=2n
3n
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Why do we have 12 notes?
Do we ever return to the original note ?
23
· · · · · 23
· 23
· 23
=2n
3n
(i.e. some multiple of an octave)
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Why do we have 12 notes?
That is, can we find and such that m n
n perfect fifths =2n
3n=
12m
= m octaves ?
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Why do we have 12 notes?
That is, can we find and such that m n
?2n+m = 3n
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Why do we have 12 notes?
That is, can we find and such that m n
?2n+m = 3n
EVEN
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Why do we have 12 notes?
That is, can we find and such that m n
?2n+m = 3n
EVEN ODD
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Why do we have 12 notes?
That is, can we find and such that m n
?2n+m = 3n
EVEN ODD
NO!
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Why do we have 12 notes?
However: We do have 219 = 212+7 ≈ 312
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Why do we have 12 notes?
However: We do have 219 = 212+7 ≈ 312
12 7perfect fifths octaves≈
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Why do we have 12 notes?
However: We do have 219 = 212+7 ≈ 312
12 7perfect fifths octaves≈
That’s why we have 12 notes!
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Why do we have 12 notes?
But PYTHAGOREAN TUNING has its issues...
the “spiral of fifths”
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Why do we have 12 notes?
These days we use EQUAL TEMPERAMENT TUNING
• divide the octave into 12 EQUAL RATIOS
• It’s a trade-off.
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Why do we have 12 notes?
These days we use EQUAL TEMPERAMENT TUNING
• divide the octave into 12 EQUAL RATIOS
• It’s a trade-off.
So instead of 2:32:2.9966...we use
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Why do we have 12 notes?
These days we use EQUAL TEMPERAMENT TUNING
• divide the octave into 12 EQUAL RATIOS
• It’s a trade-off.
So instead of 2:32:2.9966...we use
(not exactly “perfect”)
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Music of the Spheres
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Music of the Spheres
“By the assumption of what uniform and orderly motions can the apparent motions of the planets be accounted for?”
--- Plato
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Music of the Spheres
The moon, planets and stars are fastened to revolving perfect crystalline spheres.
Theory:
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Music of the Spheres
The moon, planets and stars are fastened to revolving perfect crystalline spheres.
Theory:
The radii of the spheres have small whole number ratios.
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Music of the Spheres
A Grand Synthesis:
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Music of the Spheres
Sure, it was “wrong”, but...
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Music of the Spheres
was it really so strange?
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Music of the Spheres
Thank You!