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Home , Pythagorean comma, Third (chord)

This is PHYS 1240 - Sound and 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 HW 6 review

open-open pipe closed-open pipe 3. (avg score: 30%) What would happen to the frequency of the 푛 = 1: second mode (the next member of the harmonic series after the fundamental) of an open-open pipe 푛 = 2: if a cap was placed on one end? 푛 = 3: A) It would increase by a of 2 B) It would decrease by a factor of 2 푛 = 4: C) It would decrease by a factor of 3 D) It would stay the same E) It would change by some other 푛 = 5:

factor 푣푠 푣 푓 = 푛 ∙ 푓 = 푛 ∙ 푠 푛 2퐿 푛 4퐿 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

perfect perfect major minor major minor fourth third second second (2/1) (3/2) (4/3) (5/4) (6/5) (9/8) (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) C) D) E) 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 , 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 3 4 (100 Hz) × × = 200 Hz 2 3 The Piano Keyboard

C D E F G A B The Piano Keyboard C# / D♭ F# / G♭

D# / E♭ G# / A♭

A# / B♭ Half step or Half step Half step Whole step or whole tone whole step = two half steps

Whole step Intervals on the Piano Keyboard octave # of half fourth Interval Frequency fifth ratio steps

Octave 2/1 12

Perfect fifth 3/2 7

Perfect fourth 4/3 5

Major third 5/4 4

whole step half step “whole tone” “semitone” Minor third 6/5 3 Scale #1: Just Tuning

• Based on lowest integer frequency ratios

1 9 5 4 3 2 Ratio to C: ? ? 1 8 4 3 2 1 Scale #1: Just Tuning

• For A, go up a perfect fourth then up a major third: 4 5 5 × = 3 4 3

• For B, go up a perfect fifth then up a major third: 3 5 15 × = 2 4 8

or, from A, go up a : 5 9 15 × = 3 8 8

1 9 5 4 3 5 15 2 Ratio to C: ? ? 1 8 4 3 2 3 8 1 Scale #1: Just Tuning

• Benefits: sounds pure

• Drawbacks: only works in one key (not all fifths are perfect 3/2 ratios)

Circle of fifths:

1 9 5 4 3 5 15 2 Ratio to C: 1 8 4 3 2 3 8 1 Scale #2:

• Goal: make all the perfect fifths within the scale pure (3/2)

• Problem: ⇒ Impossible to tune a piano perfectly with this system

1 9 81 4 3 27 243 2.03 Ratio to C: ? 1 8 64 3 2 16 128 1 Scale #3:

• 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 1 12 • Each half step is a factor of 2 = 212 ≈ 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

1 12 • Each half step is a factor of 2 = 212 ≈ 1.05945 Clicker 14.3

# of half In an equal-tempered 12-note scale, what is the Interval Frequency frequency ratio corresponding to a major third? ratio steps

A) 5/4 Octave 2/1 12 B) 81/64 1 Perfect 3 3/2 7 C) (212) ≈ 1.189 fifth 1 4 D) (212) ≈ 1.260 Perfect 4/3 5 E) 12/4 fourth Major 5/4 4 third

Minor 6/5 3 third Clicker 14.3 D

# of half In an equal-tempered 12-note scale, what is the Interval Frequency frequency ratio corresponding to a major third? ratio steps

A) 5/4 Octave 2/1 12 B) 81/64 1 Perfect 3 3/2 7 C) (212) ≈ 1.189 fifth ퟏ ퟒ D) (ퟐퟏퟐ) ≈ ퟏ. ퟐퟔퟎ Perfect 4/3 5 E) 12/4 fourth Major 5/4 4 third

Minor 6/5 3 third Scale #3: Equal Temperament

1 12 • Each half step is a factor of 2 = 212 ≈ 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 C D E F G A B C name:

1 9 5 4 3 5 15 2 Just 1 8 4 3 2 3 8 1

1 9 81 4 3 27 243 2.03? Pythagorean Frequency 1 8 64 3 2 16 128 1 ratio to C:

Equal- 1 1 2 1 4 1 5 1 7 1 9 1 11 2 Tempered 1 212 212 212 212 212 212 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