Chapter 15, Problem 5SP

Using the procedure outlined in section 15.5 where the ideal ratios for a justly tuned scale are described, find the frequencies for all of the white keys between middle C (264 Hz) and the C above middle C (a C-major scale). If you have worked synthesis problem 4, compare the frequencies for just tuning to those for equal temperament.

a.    G (sol) is a fifth above C $\left(\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.\right)$.

b.    F (fa) is a fourth above C $\left(\raisebox{1ex}{$4$}\!\left/ \!\raisebox{-1ex}{$3$}\right.\right)$.

c.    E (mi) is a major third above C $\left(\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{$4$}\right.\right)$.

d.    B (ti) is a major third above G (sol).

e.    D (re) is a fourth below G (sol).

f.    A (la) is a major third above F (fa).

To determine

The ideal-ratio frequency of G.

Answer to Problem 5SP

The ideal-ratio frequency of the G is $396\text{Hz}$.

Explanation of Solution

Given Info: The frequency of the tune C is $264\text{Hz}$.

Write the formula to calculate the ideal-ratio frequency of G.

$f_G=\frac{3}{2}{f}_C$

Here,

$f_G$ is the ideal-ratio frequency of G

$f_C$ is the ideal ratio frequency of C

Substitute $264\text{Hz}$ for $f_C$ in the above equation to calculate $f_G$.

$\begin{array}{c}{f}_G=\frac{3}{2}\left(264\text{Hz}\right)\\ =396\text{Hz}\end{array}$

Conclusion:

Therefore, the ideal-ratio frequency of the G is $396\text{Hz}$.

To determine

The ideal-ratio frequency of F.

Answer to Problem 5SP

The ideal-ratio frequency of the F is $352\text{Hz}$.

Explanation of Solution

Given Info: The frequency of the tune C is $264\text{Hz}$.

Write the formula to calculate the ideal-ratio frequency of F.

$f_F=\frac{4}{3}{f}_C$

Here,

$f_F$ is the ideal-ratio frequency of F

$f_C$ is the ideal ratio frequency of C

Substitute $264\text{Hz}$ for $f_C$ in the above equation to calculate $f_F$.

$\begin{array}{c}{f}_F=\frac{4}{3}\left(264\text{Hz}\right)\\ =352\text{Hz}\end{array}$

Conclusion:

Therefore, the ideal-ratio frequency of the F is $352\text{Hz}$.

To determine

The ideal-ratio frequency of E.

Answer to Problem 5SP

The ideal-ratio frequency of the E is $330\text{Hz}$.

Explanation of Solution

Given Info: The frequency of the tune C is $264\text{Hz}$.

Write the formula to calculate the ideal-ratio frequency of E.

$f_E=\frac{3}{2}{f}_C$

Here,

$f_E$ is the ideal-ratio frequency of E

$f_C$ is the ideal ratio frequency of C

Substitute $264\text{Hz}$ for $f_C$ in the above equation to calculate $f_E$.

$\begin{array}{c}{f}_E=\frac{5}{4}\left(264\text{Hz}\right)\\ =330\text{Hz}\end{array}$

Conclusion:

Therefore, the ideal-ratio frequency of the E is $330\text{Hz}$.

To determine

The ideal-ratio frequency of B.

Answer to Problem 5SP

The ideal-ratio frequency of the B is $495\text{Hz}$.

Explanation of Solution

Given Info: The frequency of the tune G is $396\text{Hz}$.

Write the formula to calculate the ideal-ratio frequency of B.

$f_B=\frac{5}{4}{f}_G$

Here,

$f_B$ is the ideal ratio frequency of B

Substitute $396\text{Hz}$ for $f_G$ in the above equation to calculate $f_B$.

$\begin{array}{c}{f}_B=\frac{5}{4}\left(396\text{Hz}\right)\\ =495\text{Hz}\end{array}$

Conclusion:

Therefore, the ideal-ratio frequency of the B is $495\text{Hz}$.

To determine

The ideal-ratio frequency of D.

Answer to Problem 5SP

The ideal-ratio frequency of the D is $297\text{Hz}$.

Explanation of Solution

Given Info: The frequency of the tune G is $396\text{Hz}$.

Write the formula to calculate the ideal-ratio frequency of D.

$f_D=\frac{3}{4}{f}_G$

Here,

$f_D$ is the ideal-ratio frequency of D

Substitute $396\text{Hz}$ for $f_C$ in the above equation to calculate $f_D$.

$\begin{array}{c}{f}_D=\frac{3}{4}\left(396\text{Hz}\right)\\ =297\text{Hz}\end{array}$

Conclusion:

Therefore, the ideal-ratio frequency of the D is $297\text{Hz}$.

To determine

The ideal-ratio frequency of A.

Answer to Problem 5SP

The ideal-ratio frequency of the A is $440\text{Hz}$.

Explanation of Solution

Given Info: The frequency of the tune F is $352\text{Hz}$.

Write the formula to calculate the ideal-ratio frequency of A.

$f_A=\frac{5}{4}{f}_F$

Here,

$f_A$ is the ideal-ratio frequency of A

Substitute $352\text{Hz}$ for $f_F$ in the above equation to calculate $f_A$.

$\begin{array}{c}{f}_A=\frac{5}{4}\left(352\text{Hz}\right)\\ =440\text{Hz}\end{array}$

Conclusion:

Therefore, the ideal-ratio frequency of the A is $440\text{Hz}$.