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L 10 The 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). - PowerPoint PPT Presentation
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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