introduction to musical scales

10

1.0

1.5

2.0

consonance

0

0.8

0.2

0.6

0.4

0.4

0.6

0.2

0.8

0

0

0.2

0.4

0.6

x critical bandwidth

0.8

1.0

1.0

dissonance

1.0

800

nd Ba C

rit

ic al

600 500

e(9

ird

(6

(3/2

or

Th

le

ho

n To

W

M

in

300

/8)

/5 )

)

400 Fifth

Frequency Difference f2 - f 1

fC

B

700

200 100

1000

2000

3000

Center Frequency (Hz)

4000

5000

0

1:2

1:1 2:3 3:5

1 3:4 5:6 4:5

dissonance

2 3 4 5 6 250

300

350

400

frequency in cps

450

500

Fifths up

A D

E

G

C F#

Db

F Fifths down Ab

Bb Eb

}

Comma

Gb

Pythagorean

B

8

9 F(13

)

F(1)

1 ( 2)

2

3

)

F(1 2

27 16

4/9

128/243 64/81 16/27

8/9

(per octave equivalence)

625/256

625/512

125/64

125/64

25/16

25/16

5/4 2/3

5/4 3/2

9/4

27/8 81/16 243/32

2/3 4/5

3/2 4/5

9/8

27/16 81/64 243/128 n Intervalratio (3/2) mod1 +1

16/25

16/25

64/125

64/125

256/625

512/625

n Intervalratio (3/2)

(per octave equivalence)

Fifths

32/243 16/81 8/27

n Intervalratio (5/4) mod1 +1

n Intervalratio (5/4)

Thirds

Thirds

Fifths

Thirds

Fifths

Thirds

Fifths

Thirds

Fifths

Thirds

Fifths

7

20 0

7

6

40

15 10 5 0

5 4

7 6

3

5 4 third

6 5 4

7 6

fifth

3

5 4

2

3

2

3

third 2

2

1

1

1

1 0

0

0

7 30

6

20 5

10 0

4

7 6

3

5 4 third

2

3 2

1 1 0

0

fifth

0

fifth

7 30 20

6

3

5 4 third

5

5 0

4

7

6

10

5

10 0

7

6

4

7 6

fifth

3

5 4

2

3

2

3

third 2

2

1

1

1

1 0

0

0

7 20

6

10 0

5 4

7 6

3

5 4 third

2

3 2

1 1 0

0

fifth

0

fifth

7 10

6

6

4

5 0

7

6

5

6

3

5 4 third

5

2 0

4

7

4

7 6

fifth

3

5 4

2

3

2

3

third 2

2

1

1

1

1 0

0

0

7 8 6 4 2 0

6 5 4

7 6

3

5 4 third

2

3 2

1 1 0

0

fifth

0

fifth