1. L 10 The Tempered Scale, Cents

2. The Tempered Scale

3. 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).

4. 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= 2 1/12 = 1.05946.

5. CT 2.4.5 is equal to:

6. CT 2.4.5 is equal to:

7. C#/C = 2 1/12 or C# = 2 1/12 C D/C# = 2 1/12 or D= (2 1/12 )C# = 2 1/12 (2 1/12 C), or D= 2 2/12 C…. B=2 11/12 C, and finally 2C= 2 12/12 C, which was the requirement.

8. CT 2.4.5 In the twelve tone equal tempered system, suppose I want f 2 to be one whole step above f 1. What should I multiply f 1 by to get f 2 ?

9. CT 2.4.5 In the twelve tone equal tempered system, suppose I want f 2 to be two whole steps above f 1. What should I multiply f 1 by to get f 2 ?

10. CT 2.4.5 In the twelve tone equal tempered system, suppose I want f 2 to be a perfect fifth above f 1. What should I multiply f 1 by to get f 2 ?

11. CT 2.4.5 In a just system, suppose I want f 2 to be a perfect fourth below f 1. What should I multiply f 1 by to get f 2 ?

12. CT 2.4.5 Suppose we start at the note A = 440 Hz and go up a fifth to E. What is the ratio of the frequencies of E just /E equal tempered ?

16. Sound clips Pythagorean, Just, Equal temperament http://www.pyxidium.u- net.com/Acoustics/MusicMaths/MusicMaths.html# pythag http://www.phy.mtu.edu/~sui ts/scales.html

17. Piano Tuning Beats and Stretched Octaves DEMO- Tuning Forks and Simulation

23. DEMO Piano scale and concertos played with the Pythagorean, Just, and Tempered Scales