L 10 the tempered scale cents
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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).

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L 10 the tempered scale cents
L 10The Tempered Scale, Cents



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
CT 2.4.5 is 2C.

is equal to:


Ct 2 4 51
CT 2.4.5 is 2C.

is equal to:


C#/C = 2 is 2C.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 52
CT 2.4.5 is 2C.

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?


Ct 2 4 53
CT 2.4.5 is 2C.

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?


Ct 2 4 54
CT 2.4.5 is 2C.

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?


Ct 2 4 55
CT 2.4.5 is 2C.

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


Ct 2 4 56
CT 2.4.5 is 2C.

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 ?


Sound clips pythagorean just equal temperament
Sound clips is 2C.Pythagorean, Just, Equal temperament

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

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


Piano tuning beats and stretched octaves demo tuning forks and simulation
Piano Tuning is 2C.Beats and Stretched OctavesDEMO- Tuning Forks and Simulation


Demo piano scale and concertos played with the pythagorean just and tempered scales
DEMO is 2C.Piano scale and concertos played with thePythagorean, Just, and Tempered Scales


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