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This is PHYS 1240 - Sound and Music
Lecture 12
Professor Patricia Rankin
TA: Tyler McMaken
Cell Phones silent
Clickers on
Physics 1240 Lecture 14
Today: Scales, Tutorial
Next time: Review for midterm
physicscourses.colorado.edu/phys1240
Canvas Site: assignments, administration, grades
Homework – HW7 Not due till Wed March 11th 5pm
Homelabs – Hlab4 Not due till March 16th
3. (avg score: 30%) What would
happen to the frequency of the
second mode (the next member of
the harmonic series after the
fundamental) of an open-open pipe
if a cap was placed on one end?
A) It would increase by a factor of 2
B) It would decrease by a factor of 2
C) It would decrease by a factor of 3
D) It would stay the same
E) It would change by some other
factor
𝑛 = 1:
open-open pipe closed-open pipe
𝑛 = 2:
𝑛 = 3:
𝑛 = 4:
𝑛 = 5:
𝑓𝑛 = 𝑛 ∙𝑣𝑠2𝐿
𝑓𝑛 = 𝑛 ∙𝑣𝑠4𝐿
HW 6 review
Review
• Consonance: tones have whole number frequency ratios
• Dissonance: harsh sound when 2 tones (or upper harmonics) produce
beats within the same critical band
• Harmonic series → Pythagorean intervals
octave
(2/1)
perfect
fifth
(3/2)
perfect
fourth
(4/3)
major
third
(5/4)
minor
third
(6/5)
major
second
(9/8)
minor
second
(16/15)…………
Questions:
1) Why does a piano have 12 notes in each octave?
2) How do we tune those 12 notes (how do we decide what frequencies to
assign to each note)?
Pythagoras of Samos
• 500s BCE
• Founded school of numerology
• Music of the spheres
• Pythagorean Hypothesis:
Consonant musical intervals are related to
low integer ratios of frequencies
Clicker 14.1
Two monochords are plucked to produce sound. One string is 50 cm long,
and the other is 40 cm long. What is the musical interval between these
plucked notes?
A) octave
B) tritone
C) perfect fourth
D) major third
E) minor third
Clicker 14.1 D
Two monochords are plucked to produce sound. One string is 50 cm long,
and the other is 40 cm long. What is the musical interval between these
plucked notes?
A) octave
B) tritone
C) perfect fourth
D) major third
E) minor third
Clicker 14.2
A note is played at 100 Hz. Then, the pitch moves up by a perfect fifth, then
it moves up by a perfect fourth. What is the new frequency?
A) 100 Hz
B) 133 Hz
C) 150 Hz
D) 180 Hz
E) 200 Hz
Clicker 14.2 E
A note is played at 100 Hz. Then, the pitch moves up by a perfect fifth, then
it moves up by a perfect fourth. What is the new frequency?
A) 100 Hz
B) 133 Hz
C) 150 Hz
D) 180 Hz
E) 200 Hz
(100 Hz) ×3
2×
4
3= 200 Hz
The Piano Keyboard
EC D F G A B
The Piano Keyboard
C# / D♭
D# / E♭
F# / G♭
G# / A♭
A# / B♭
Half step or semitone
Half step
Half step
Whole step or whole tone
whole step = two half steps
Whole step
octave
whole step
“whole tone”
half step
“semitone”
fifth
Intervals on the Piano Keyboard
fourth IntervalFrequency
ratio
# of half
steps
Octave 2/1 12
Perfect fifth 3/2 7
Perfect fourth 4/3 5
Major third 5/4 4
Minor third 6/5 3
Scale #1: Just Tuning
• Based on lowest integer frequency ratios
Ratio to C:1
1
9
8
5
4
4
3
3
2??
2
1
Scale #1: Just Tuning
• For A, go up a perfect fourth then up a major third:4
3×
5
4=
5
3
• For B, go up a perfect fifth then up a major third:3
2×
5
4=
15
8
or, from A, go up a major second:5
3×
9
8=
15
8
Ratio to C: 1
1
9
8
5
4
4
3
3
2? ?
2
1
5
3
15
8
Scale #1: Just Tuning
• Benefits: sounds pure
• Drawbacks: only works in one key (not all fifths are perfect 3/2 ratios)
Circle of fifths:
Ratio to C: 1
1
9
8
5
4
4
3
3
2
5
3
15
8
2
1
Scale #2: Pythagorean Tuning
• Goal: make all the perfect fifths within the scale pure (3/2)
• Problem: Pythagorean comma
⇒ Impossible to tune a piano perfectly with this system
Ratio to C: 1
1
3
2
9
8
27
16
81
64
243
128
4
3
2.03
1?
Scale #3: Equal Temperament
• Solution: temper the fifths (split the leftover frequency among other
intervals to make them each slightly out of tune)
• Equal temperament:
• All 12 half step intervals are the same frequency ratio
• Each half step is a factor of 122 = 2
1
12 ≈ 1.05945• Any song can be played in any key without going out of tune (since
everything is already “equally out of tune”)
Scale #3: Equal Temperament
• Each half step is a factor of 122 = 2
1
12 ≈ 1.05945
Clicker 14.3
In an equal-tempered 12-note scale, what is the
frequency ratio corresponding to a major third?
A) 5/4
B) 81/64
C) (21
12)3 ≈ 1.189
D) (21
12)4 ≈ 1.260E) 12/4
IntervalFrequency
ratio
# of half
steps
Octave 2/1 12
Perfect
fifth3/2 7
Perfect
fourth4/3 5
Major
third5/4 4
Minor
third6/5 3
Clicker 14.3 D
In an equal-tempered 12-note scale, what is the
frequency ratio corresponding to a major third?
A) 5/4
B) 81/64
C) (21
12)3 ≈ 1.189
D) (𝟐𝟏
𝟏𝟐)𝟒 ≈ 𝟏. 𝟐𝟔𝟎E) 12/4
IntervalFrequency
ratio
# of half
steps
Octave 2/1 12
Perfect
fifth3/2 7
Perfect
fourth4/3 5
Major
third5/4 4
Minor
third6/5 3
Scale #3: Equal Temperament
• Each half step is a factor of 122 = 2
1
12 ≈ 1.05945
• Now a tune can sound alright when played in any key
• Equal temperament didn’t take hold until around the time of Mozart – why
not sooner?
• Hard to tune this way with just a tuning fork
• None of the intervals are purely consonant; they’re just “good
enough”
• Just Tuning: uses only pure, harmonic intervals
• Pros: all pure consonances for intervals from same note
• Cons: can only play in one key
• Pythagorean Tuning: makes all fifths in any key pure (3/2)
• Pros: all pure consonances for fifths
• Cons: thirds are dissonant; Pythagorean comma
• Equal Temperament: same interval for all adjacent notes
• Pros: can play in any key
• Cons: all intervals are very slightly dissonant
Note
name:C D E F G A B C
Just
Frequency
ratio to C:
1
1
9
8
5
4
4
3
3
2
5
3
15
8
2
1
Pythagorean1
1
9
8
81
64
4
3
3
2
27
16
243
128
2.03?
1
Equal-
Tempered
1
1 2112
2
2112
4
2112
5
2112
7
2112
9
2112
11 2
1
Tutorial
What is the frequency ratio of a Pythagorean comma?
A) 1.01364
B) 1.1524
C) 1.5
D) 1.11111
E) 1
