Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

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 \(\mathrm{C}, \mathrm{C} \\#, \mathrm{D}, \mathrm{D} \\#, \mathrm{E}, \mathrm{F}, \mathrm{F} \\#, \mathrm{G}, \mathrm{G}\|, \mathrm{A}, \mathrm{A}\|, \mathrm{B}\) C form a geometric sequence (progression), with \(\bar{C}\) having twice the frequency of \(\mathrm{C}\) (C# is read \(\mathrm{C}\) sharp and \(\overline{\mathrm{C}}\) indicates one octave above \(\mathrm{C}\) ). What is the ratio \(r\) between the frequencies of successive notes? If the frequency of \(A\) is 440 , find the frequency of \(\bar{C}\).

Step-by-Step Solution

Answer

The ratio \( r \) is approximately 1.059463, and \( \overline{C} \) has a frequency of 523.25 Hz.

Step 1: Understanding the Problem

The problem involves the equally tempered scale where the frequencies of musical notes form a geometric progression. An equally tempered octave has 12 notes, and the frequency of the first note (C) must be doubled by the 13th note (also C, but one octave higher). We need to find this ratio, denoted as \( r \).

Step 2: Identifying the Formula for Frequency

In a geometric sequence, each term is the previous term multiplied by a constant ratio \( r \). For the equally tempered scale, if the sequence starts at a note \( f_0 \), the note \( n \) steps away has a frequency of \( f_n = f_0 \times r^n \). For an octave, \( n = 12 \), so \( \overline{C} = C \times r^{12} = 2 \times C \).

Step 3: Solving for the Ratio \( r \)

From the equation \( \overline{C} = 2 \times C \), it follows that \( 2 = r^{12} \). To solve for \( r \), take the twelfth root of 2: \( r = \sqrt[12]{2} \).

Step 4: Calculating \( r \) Numerically

Calculate \( r = \sqrt[12]{2} \), which is approximately 1.059463. This is the constant ratio between the frequencies of successive notes.

Step 5: Finding the Frequency of \( \overline{C} \) Given \( A = 440 \) Hz

The note \( A \) is 3 positions above \( F \) in the scale from C to \( \overline{C} \), so \( A = F \times r^9 \), where \( F = C \times r^5 \). Therefore, \( A = C \times r^{14} = 440 \). Since an octave (\(12\) steps) from A ends at \( \overline{C} \), then \( \overline{C} = A \times r^{3} = 440 \times r^3 \). Substitute \( r^3 = \sqrt[4]{2} \) into the equation.

Step 6: Calculating Frequency of \( \overline{C} \)

Using \( \overline{C} = 440 \times \left(\sqrt[4]{2}\right) \), compute \( \overline{C} = 440 \times 1.189207 \approx 523.2511 \). Thus, the frequency of \( \overline{C} \) is approximately 523.25 Hz.

Key Concepts

Frequency Ratio

In the musical world, understanding the frequency ratio is essential, especially in the context of tuning and musical harmony. A frequency ratio is a comparison between the frequency of two pitches. In the equally tempered scale, all notes within an octave are spaced by the same frequency ratio. This means that each note is derived from the previous note by multiplying it by a constant, known as the frequency ratio.
For example, in the equally tempered scale, the frequency ratio is calculated in such a way that every twelfth note is exactly double the frequency of the first note, completing an octave. The formula to find this frequency ratio involves the twelfth root of 2: \[ r = \sqrt[12]{2} \]This ensures that the series of notes fits perfectly within the octave framework, providing the familiar and harmonious sound that musicians recognize.
The frequency ratio is critical for musicians and instrument manufacturers because it ensures that melodies and chords sound pleasing across different octaves. This standardized ratio allows musical instruments to be tuned consistently, ensuring that they can be played together in harmony.

Equally Tempered Scale

The equally tempered scale is a system of tuning that has been widely adopted in Western music. It divides an octave, which is a doubling of frequency, into 12 equal parts or semitones. Each semitone, therefore, represents a frequency increase by a constant factor, which is the twelfth root of 2, ensuring each step up is equal in the log scale of sound frequencies.
This tuning method permits keyboard instruments, like the piano, to play in any key and sound relatively harmonious. The invention of the equally tempered scale allowed composers to explore more complex musical pieces and modulation between keys, accommodating the dynamic range of instruments more naturally.

• Provides consistent intervals across all pitches
• Facilitates smooth modulation between keys in music compositions
• Created to accommodate the capabilities of modern instruments

While this approach isn't perfectly "in tune" when compared to other temperaments, it provides a practical compromise that is crucial for orchestras and ensembles.

Geometric Sequence Formula

Geometric sequences play a pivotal role not only in mathematics but also in music theory, particularly when concepts like progression are involved. A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio.
In the context of an equally tempered scale, the frequencies of the notes form a geometric progression. The general formula for finding the nth term in a geometric sequence is:\[ f_n = f_0 \times r^n \]where:

• \( f_0 \) is the starting frequency (such as for the note A at 440 Hz)
• \( r \) is the common ratio (typically the twelfth root of 2 for the equal temperament)
• \( n \) is the term number in the sequence

This formula is indispensable for calculating frequencies for each note in an octave, allowing musicians to determine precise pitches. It not only aids in theoretical calculations but also in the practical tuning of instruments, ensuring each note falls into its correct place in the musical scale.