L 8-9Musical Scales, Chords , and

Intervals,The Pythagorean and Just Scales

History of Western ScalesA Physics 1240 Project

by Lee Christy 2010

References to the History

Musical Intervals (roughly in order of decreasing consonance)

Name of Interval

Notes (in key of C major)

Pythagorean Frequency Ratios

Just Frequency Ratios

# Semitones (on equal-tempered scale)

Octave C ↔ C 2 2 12

Fifth C ↔ G 3/2 6/4 = 3/2 7

Fourth C ↔ F 4/3 5

Major Third C ↔ E 81/64 5/4 4

Minor Third E ↔ G 3

Major Sixth C ↔ A 27/16 9

Minor Sixth E ↔ C 8 Tonic C 1 4/4 = 1 none * a semitone interval corresponds to a frequency difference of about 6%

* The white notes of the piano give the seven notes of the C-major diatonic scale.

C D E F G A B C

The ratio of the frequency of C4 to that of C2 is:

a) 2b) 3c) 4d) 8

One octave of the diatonic scale including the tonic and the octave

note contains:a) 5 notesb) 6 notesc) 7 notesd) 8 notes

One octave of the chromatic scale (including the octave note)

contains:a) 8 notesb) 10 notesc) 11 notesd) 12 notese) 13notes

A musical scale is a systematic arrangement of pitches

Each musical note has a perceived pitch with a particular frequency

(the frequency of the fundamental)

Going up or down in frequency, the perceived pitch follows a pattern

One cycle of pitch repetition is called an octave.

The interval between successive pitches determines the type of scale.

Note span Interval Frequency ratioC - C unison 1/1C - C# semitone 16/15C - D whole tone (major second) 9/8C - D# minor third 6/5C - E major third 5/4C - F perfect fourth 4/3C - F# augmented fourth 45/32C - G perfect fifth 3/2C - G# minor sixth 8/5C - A major sixth 5/3C - A# minor seventh 16/9 (or 7/4)C - B major seventh 15/8C3 - C4 octave 2/1C3 - E4 octave+major third 5/2

Intervals12-tone scale (chromatic) 8-tone scale (diatonic)

Consonant intervalsOverlapping harmonics

tonic 120 240 360 480 600 720 840 960 1080

fifth 180 360 540 720 900 1080

fourth 160 320 480 640 800 960

M third 150 300 450600 750 900 1050

m third 144 288 432 576 720 864 1008

octave 240 480 720 960

Dissonant intervals

Perceived when harmonics are close enough for beating

harmonic series

Fundamental f1

2nd harmonic f2 = 2f1 octave

3rd harmonic f3 = 3f1 perfect fifth

4th harmonic f4 = 4f1 perfect fourth

5th harmonic f5 = 5f1 major third

6th harmonic f6 = 6f1 minor third

f2f1

21

f3f2

32

f4f3

43

f5f4

54

f6f5

65

Intervals between consecutive harmonics

CT 2.4.5

What is the name of the note that is a major 3rd above E4=330 Hz?

A: GB: G#C: AD: A#E: B

IntervalsC- D, a second

C-E, a thirdC-F, a 4thC-G, a 5th,C-A, a 6th

C-B, a (major) 7th,C-2C, an octave

C-2D, a 9th

C-2E, a 10th,C-2F, an 11th, C-2G, a 12th,

C-2A, a 13th, etc.

C-Eb, a minor 3rd

C-Bb, a dominant 7th,

C-2Db, a flatted 9th, etc.

Pythagorean ScaleBuilt on 5ths

A pleasant consonance was observed playing strings whose lengths were

related by the ratio of 3/2 to 1 (demo).Let’s call the longer string C, and the

shorter G, and the interval between G and C a 5th

Denote the frequency of C simply by the name C, etc.

Since f1= V/2L, and LC= 3/2 LG, G =3/2C.

Similarly a 5th above G is 2D, and D= 1/2 (3/2G)= 9/8 C.

Then A is 3/2 D= 27/16 C.Then 2E= 3/2 A or E= 81/64 C, and

B=3/2 E = 243/128 C.

We now have the frequencies for CDE… GAB(2C)

To fill out the Pythagorean scale, we need F.

If we take 2C to be the 5th above F, then 2C= 3/2F, or

F = 4/3 C

Just Scale, Built on Major Triads

We take 3 sonometers to play 3 notes to make a major triad, e.g. CEG. This sounds consonant (and

has been the foundation of western music for several hundred years),

and we measure the string lengths required for this triad.

We find (demo) that the string lengths have ratios 6:5:4 for the

sequence CEG.

The major triad is the basis for the just scale, which we now develop

in a way similar to that of the Pythagorean scale.

F A C C E G G B D 4 5 6 4 5 6 4 5 6

Now take C to be 1

CT 2.4.5

Suppose we start a scale at E4=330 Hz. What frequency is a (just) perfect 5th above this?

A 1650 HzB: 220 HzC: 495 HzD: 660 HzE: None of these

CT 2.4.5

What is the frequency of the note that is a (just) major 3rd above E4=330 Hz?

A: 660 HzB: 633 HzC: 512 HzD: 440 HzE: 412 Hz

CT 2.4.5

Suppose we start a scale at E4=330 Hz. What frequency is a (just) perfect 5th below this?

A 165 HzB: 220 HzC: 110 HzD: 66 HzE: None of these

compound intervals

major third + minor third

54

65

3020

32 perfect fifth

perfect fourth + perfect fifth

43

32

126

21 octave

perfect fourth + major third

43

54

2012

53 major sixth

54

1615

8060

43 perfect fourthperfect fourth + whole tone

Adding intervals means multiplying frequency ratios

more compound intervals

perfect fifth + perfect fifth

32

32

94

21

98 Octave + whole tone

major seventh + minor sixth

158

85

155

31

21

32 Octave + perfect fifth

ratios larger than 2 can be split up into an octave + something