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L 10 The Tempered Scale, Cents
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The Tempered Scale
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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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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.
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CT 2.4.5 is equal to:
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CT 2.4.5 is equal to:
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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.
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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 ?
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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 ?
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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 ?
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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 ?
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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 ?
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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
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Piano Tuning Beats and Stretched Octaves DEMO- Tuning Forks and Simulation
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DEMO Piano scale and concertos played with the Pythagorean, Just, and Tempered Scales
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