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Transcript
Set 6
Let there be music
Wow! We covered 50 slides last
time! And you didn't shoot me!!
1
Stuff
 As of Saturday morning, the grades have yet to be
posted on myUCF site.
 Today we continue with our musical interlude.
 We still have to cover two basic Physics concepts:
 Energy
 Momentum
 We will return to these topics later.
2
Last time
 We looked at strings, how they vibrate and mentioned
the factors that determine the vibrational frequency of a
string.
 We also remembered the Helmholtz result (next slide)
that each note on the musical scale had a specific
frequency.
 But were those specific frequencies selected?
 Why not different ones??
 Why these PARTICULAR frequencies??
3
Helmholtz’s Results
Note from Middle C
C
D
E
Frequency
264
297
330
F
G
A
B
352
396
440
496
4
Our immediate plan
 Apply modern electronic methods, with the
assistance of a guest violinist, to answer these
questions.
 Apply these methods to the understanding of
1) The scale progression
2) The development of chords
5
We can study tones with electronics
Tone
Compare the results
From these two sources.
6
Oscilloscope
http://commons.wikimedia.org/wiki/Main_Page
7
One More Tool
Tone
Signal Generator
Electrical
8
In using these modern tools
We postpone understanding how some of these tools work
until later in the semester.
2. We must develop some kind of strategy to convince us that this
approach is appropriate.
1.
9
One more thing
• These days, the tone generation
and the oscilloscope can be
“created” on a computer.
• This will often be our approach.
10
The Violin
L
We will make some
measurements based
On these lengths.
11
Let’s Listen to the Violin
1) Let’s listen to the instrument, this time a real
one.







12
The parts
One tone alone .. E on A string
E on the E string
Both together (the same?)
A Fifth A+E open strings
Consecutive pairs of fifths – open strings.
A second? Third? Fourth? Seventh?
Guitar Tuning
13
Interval
from middle Frequency
C
major third
329.63 Hz
above
String
Helmholtz
Scientific pitch
pitch
first
E4
e'
second
B3
b
minor second
below
third
G3
g
perfect fourth
196.00 Hz
below
fourth
D3
d
minor seventh
146.83 Hz
below
fifth
A2
A
sixth
E2
E
minor tenth
below
minor
thirteenth
below
246.94 Hz
110 Hz
82.41 Hz
Consider Two Situations
For the same “x” the
restoring force is double
because the angle is
double.
The “mass” is about half
because we only have
half of the string
vibrating.
14
For the same “x” the
restoring force is double
because the angle is
double.
So…
F  kx
1
f 
2
k
m
The “mass” is about half
because we only have
half of the string
vibrating.
k doubles
m -> m/2
f doubles!
15
1
f 
2
1

2
2f
k
1

m
2
4k
1

m
2
2k
m/2
k
 4
m
Octave
1
0.5
f
0.001
0.002
0.003
0.004
0.005
0.001
0.002
0.003
0.004
0.005
0.001
0.002
0.003
0.004
0.005
-0.5
-1
1
0.5
2f
-0.5
-1
1.5
1
0.5
SUM
-0.5
-1
-1.5
Time 
16
The keyboard – a reference
The Octave
17
Next Octave
The Octave
 12 tones per octave. Why 12? … soon. Played sequentially,
one hears the “chromatic” scale.
 Each tone is separated by a “semitione”
 Also “half tone” or “half step”.
 Whole Tone = 2 semitones
18
Properties of the octave
 Two tones, one octave apart, sound well when played together.
 In fact, they almost sound like the same note!
 A tone one octave higher than another tone, has double its
frequency.
 Other combinations of tones that sound well have frequency ratios
that are ratios of whole numbers (integers).
 It was believed olden times, that this last property makes music
“perfect” and was therefore a gift from the gods, not to be screwed
with.
 This allowed Pythagoras to create and understand the musical
scale. This will be our next topic.
19
Pythagoras
20
The ratios of these lengths
Should be ratios of integers
If the two strings, when struck
At the same time, should sound
“good” together.
21
Pythagoras
 Born in Samos, Ionia
 Remembered as a mathematician.
 Well educated; learned to play the lyre, read poetry, and
could recite Homer.
 Believer that ALL relations could be reduced to number.
 All things are numbers; the whole cosmos is a scale and a
number.
 He developed the Pythagorean Theorm.
22
Pythagoras
 Each number had its own personality - masculine
23
or feminine, perfect or incomplete, beautiful or
ugly.
 Pythagoras noticed that vibrating strings produce
harmonious tones when the ratios of the lengths
of the strings are whole numbers, and that these
ratios could be extended to other instruments.
 He was a fine musician, and he used music as a
means to help those who were ill.
Pythagoras
 The beliefs that Pythagoras held were [2]:
(1) that at its deepest level, reality is mathematical in nature,
(2) that philosophy can be used for spiritual purification,
(3) that the soul can rise to union with the divine,
(4) that certain symbols have a mystical significance, and
(5) that all brothers of the order should observe strict loyalty and
secrecy.
 So it is no surprise that he looked at the lengths of strings
that sounded well together as a religious issue as well as a
scientific issue. Luckily, in this case, it worked .. sort of.
24
See you later ….
25