In the equally tempered scale to which keyed instruments...

In the equally tempered scale to which keyed instruments have been tuned since the days of J.S. Bach (1685-1750), the frequencies of successive notes $\mathrm{C}, \mathrm{C} \#, \mathrm{D}, \mathrm{D} \#, \mathrm{E}, \mathrm{F}, \mathrm{F} \#, \mathrm{G}, \mathrm{G} \#, \mathrm{A}, \mathrm{A} \#$ B, $\bar{C}$ form a geometric sequence (progression), with $\bar{C}$ having twice the frequency of $\mathrm{C}(\mathrm{C} \# \text { is read } \mathrm{C} \text { sharp). What is the ratio } r$ between the frequencies of successive notes? If the frequency of $A$ is $440,$ find the frequency of $\bar{C}$

In the equally tempered scale to which keyed instruments have been tuned since the days of J.S. Bach (1685-1750), the frequencies of successive notes $\mathrm{C}, \mathrm{C} \#, \mathrm{D}, \mathrm{D} \#, \mathrm{E}, \mathrm{F}, \mathrm{F} \#, \mathrm{G}, \mathrm{G} \#, \mathrm{A}, \mathrm{A} \#$
B, $\bar{C}$ form a geometric sequence (progression), with $\bar{C}$ having twice the frequency of $\mathrm{C}(\mathrm{C} \# \text { is read } \mathrm{C} \text { sharp). What is the ratio } r$ between the frequencies of successive notes? If the frequency of
$A$ is $440,$ find the frequency of $\bar{C}$

Key Concepts

12-Tone Equal Temperament

This is a tuning system in which the octave is divided into 12 equal parts. Each semitone step in the scale is spaced equally in terms of the logarithm of frequency, which means that the ratio between successive notes is constant. This system allows instruments to play in any key with relatively consistent harmonic relationships.

Geometric Sequence

In the context of musical frequencies, a geometric sequence means that each note’s frequency is obtained by multiplying the previous note’s frequency by a constant ratio. This property ensures that the relationship between any two notes is determined solely by the number of steps separating them, not by their absolute frequencies.

Frequency Ratio

The frequency ratio is the constant factor by which the frequency increases from one note to the next in a geometric progression. In the 12-tone equal temperament system, this ratio is the twelfth root of two, which guarantees that 12 successive multiplications of this ratio result in doubling the frequency, thereby reaching the octave.

Octave Frequency Doubling

An octave in music is defined as the interval where the higher note has twice the frequency of the lower note. This concept is fundamental in tuning systems and acoustics, and in equal temperament, the division of the octave into 12 equal parts ensures that the high note of the octave is exactly twice the frequency of the low note.

Standard Pitch

Standard pitch refers to the conventional reference frequency for tuning instruments. Typically, the note A above middle C is set at 440 Hz. This standard provides a basis for calibrating instruments and ensuring harmony across different performances and recordings.

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