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L 10 The Tempered Scale, Cents

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).

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L 10 The Tempered Scale, Cents

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  1. L 10The 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= 21/12 = 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# = 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.

  8. CT 2.4.5 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?

  9. CT 2.4.5 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?

  10. CT 2.4.5 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?

  11. CT 2.4.5 In a just system, suppose I want f2 to be a perfect fourth below f1. What should I multiply f1 by to get f2?

  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 Ejust/Eequal tempered ?

  13. Sound clipsPythagorean, Just, Equal temperament http://www.phy.mtu.edu/~suits/scales.html http://www.pyxidium.u-net.com/Acoustics/MusicMaths/MusicMaths.html#pythag

  14. Piano TuningBeats and Stretched OctavesDEMO- Tuning Forks and Simulation

  15. DEMOPiano scale and concertos played with thePythagorean, Just, and Tempered Scales

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