Problem 68 - Precalculus: Mathematical for Calculus

Question

The frequencies of musical notes (measured in cycles per second) form a geometric sequence. Middle C has a frequency of \(256,\) and the C that is an octave higher has a frequency of \(512 .\) Find the frequency of \(\mathrm{C}\) two octaves below middle C.

Step-by-Step Solution

Answer

The frequency of C two octaves below middle C is 64 Hz.

Step 1: Understand the Problem

The frequencies of notes form a geometric sequence. This means each frequency is multiplied by a constant ratio to get the next frequency. We are given that middle C is 256 Hz and the C one octave higher is 512 Hz. We need to find the frequency of C two octaves below middle C.

Step 2: Define the Geometric Sequence

In a geometric sequence, each term is obtained by multiplying the previous one by a constant ratio, called the common ratio. We know the frequency of middle C is 256 Hz and an octave higher is 512 Hz. Thus, we need to find the constant ratio first.

Step 3: Calculate the Common Ratio

The common ratio \( r \) for moving from one C to the next octave can be calculated by dividing the frequency of the higher C by the lower C. Thus, \( r = \frac{512}{256} = 2 \).

Step 4: Apply the Geometric Sequence Formula

To find the frequency of C two octaves below middle C, we need to divide the frequency of middle C by the common ratio for each octave decrease. Two octaves below means dividing by the ratio squared, \( 256 / 2^2 \).

Step 5: Calculate the Lower Frequency

Calculate the frequency two octaves below by evaluating \( \frac{256}{2^2} = \frac{256}{4} = 64 \). Hence, the frequency of C two octaves below middle C is 64 Hz.

Key Concepts

Octave Frequency

In music, when you move an octave higher, the frequency of the note doubles. Conversely, when you move an octave lower, the frequency halves. This is because octaves are based on a geometric sequence. When you play the same note (like C) in different octaves, it sounds the same but just higher or lower in pitch.
If you start with a note at a certain frequency (like the 256 Hz of middle C), the next C one octave higher would be 512 Hz, as shown in the exercise, because of this doubling relationship.
Multiply the frequency by 2 for each step up the octave scale and divide by 2 for each step down. This consistent pattern of doubling and halving the frequencies across octaves describes a fundamental nature of sound and music perceived by the human ear.

Common Ratio

The 'common ratio' is a crucial part of understanding geometric sequences. In the case of musical notes, it tells us how much to multiply or divide the frequency as you move from one note to the next in an octave sequence.
In this exercise, you observed the ratio by dividing the frequency of C an octave higher by the frequency of the given C. For example, the ratio was calculated using 512 Hz (C above middle C) divided by 256 Hz (middle C), providing a common ratio of 2.

• This ratio determines the progression of the sequence.
• If you move up an octave, multiply by the ratio.
• If you move down an octave, divide by the ratio.

Understanding the common ratio helps in predicting frequencies quickly in geometric sequences.

Middle C Frequency

Middle C, often seen in music as a reference point, has a frequency of 256 Hz in this exercise. This note is central in many musical contexts and is used for tuning and matching pitches across instruments.
Since frequencies form a geometric sequence in music, knowing the frequency of middle C allows musicians and composers to calculate other frequencies in the scale.
Using the middle C frequency as a base, one can determine the pitch of any note simply by progressing through the sequence guided by our common ratio, doubling or halving as needed.

• This makes middle C a convenient starting point for audio calculations and a key to exploring musical scales.
• By understanding this, musicians can better grasp how scales and harmonics are structured.