L 10The Tempered Scale,

Cents

The Tempered Scale

We wish to make a chromatic scale- 13 tones including the octave- and we want all the

intervals (ratios of adjacent notes to all be the same).

We start with , let’s say, C. Of course the octave above is 2C.

We want the ratio of the frequencies of all half steps e.g.

C#/C = D/C# = Eb/D, etc. to be the same.

Let’s pick that ratio to be 12√2= 21/12

= 1.05946.

CT 2.4.5

12 2 :A

is equal to:

212 )2( :B

121

2 :E

)2(1/ :C 12

212 )2(1/ :D

122

2

CT 2.4.5

122 :A

is equal to:

2 :B

121

2 :D

21/ :C

theseof none :E

1212 2

C#/C = 2 1/12 or C# = 21/12CD/C# = 21/12 or D= (21/12 )C#

= 21/12(21/12C) , orD= 22/12C….

B=211/12C, and finally2C= 212/12C,

which was the requirement.

CT 2.4.5

12 2 :A

In the twelve tone equal tempered system, suppose I want f2 to be one whole step above f1. What should I multiply f1 by to get f2?

212 )2( :B

121

2 :E

)2(1/ :C 12

212 )2(1/ :D

CT 2.4.5

122

2

1 :A

In the twelve tone equal tempered system, suppose I want f2 to be two whole steps above f1. What should I multiply f1 by to get f2?

122

2 :B

42 :E

121

124

2 :C 124 -

2 :D

CT 2.4.5

122

2

1 :A

In the twelve tone equal tempered system, suppose I want f2 to be a perfect fifth above f1. What should I multiply f1 by to get f2?

125

2 :B

712 2 :E

124

2 :C 127 -

2 :D

CT 2.4.5

125

2

1 :A

In a just system, suppose I want f2 to be a perfect fourth below f1. What should I multiply f1 by to get f2?

122

2 :B

43 :E

124

2 :C 34 :D

CT 2.4.5

5 23/2 :D

Suppose we start at the note A = 440 Hz and go up a fifth to E. What is the ratio of the frequencies of Ejust/Eequal tempered ?

125

2

2/3 :A 2

3/2 :B127

125

2

3/4 :C

127

2

2/3 :E

Sound clipsPythagorean, Just, Equal

temperament

http://www.pyxidium.u-net.com/Acoustics/MusicMaths/MusicMaths.html#pythag

http://www.phy.mtu.edu/~suits/scales.html

Piano TuningBeats and Stretched Octaves

DEMO- Tuning Forks and Simulation

Bach_duo_1.wmv

DEMOPiano scale and concertos played

with thePythagorean, Just, and Tempered

Scales