This is PHYS 1240 - Sound and Music Lecture 12€¦ · • Goal: make all the perfect fifths within...

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