Problem 76 - Precalculus Mathematics 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 \(C\) two octaves below middle C.

Step-by-Step Solution

Answer

Two octaves below middle C is 64 Hz.

Step 1: Identify the Sequence

The frequencies given, 256 and 512, form a geometric sequence. In a geometric sequence, each term is obtained by multiplying the previous term by a constant ratio, denoted as \( r \).

Step 2: Determine the Ratio

To find the common ratio \( r \) in the sequence, divide the frequency of the upper C (512 Hz) by the frequency of middle C (256 Hz):\[ r = \frac{512}{256} = 2 \]

Step 3: Use the Ratio to Find the Desired Frequency

Two octaves below middle C means finding the third term below middle C in the sequence. Use the common ratio \( r = 2 \) to calculate this frequency: \[ 256 \times \left(\frac{1}{r}\right)^2 = 256 \times \left(\frac{1}{2}\right)^2 = 256 \times \frac{1}{4} = 64 \]Hz.This calculation reflects moving down two octaves, effectively dividing the frequency by \( r^2 \).

Step 4: Verify the Calculation

Verify the calculation is correct by considering the logic: every octave down halves the frequency, so from 256 Hz, after two octaves down, the frequency becomes \(256 \times \frac{1}{2} \times \frac{1}{2} = 64\) Hz.

Key Concepts

Musical Notes Frequencies

Musical notes have specific frequencies measured in cycles per second, or Hertz (Hz). Each note doesn’t exist in isolation but fits into a structured pattern of sounds that we call a scale. In Western music, the notes repeat every so often at intervals called octaves.
Frequencies of musical notes are arranged in such a way that they form what is known as a geometric sequence. In this pattern, as you move from one note to another, you multiply by a specific factor, maintaining harmony and musical consistency. For example, the note Middle C has a frequency of 256 Hz, and this frequency changes systematically up or down the scale. ...

Octaves

The concept of an 'octave' is central to understanding musical notes. The musical notation system divides sounds into these repeating units, called octaves, which encompass eight notes on a standard musical scale.
When a note is moved up or down by one octave, its frequency either doubles or halves respectively. For instance, if Middle C is at 256 Hz, the C one octave higher is at 512 Hz, and the C one octave lower would be at 128 Hz. This doubling or halving of the frequency is a consistent rule applied across the ranges of musical notes, making it easier to calculate the frequencies when changing octaves.

Common Ratio

In a geometric sequence, the 'common ratio' is the factor by which you multiply to move from one term in the sequence to the next. In the context of musical notes and octaves, the common ratio helps find the frequency relationships between notes.
Taking the example of Middle C and the C in the next octave above it, the common ratio is determined by dividing the higher frequency by the lower one:

• If Middle C is 256 Hz and the next C is 512 Hz, the calculation would be: \[ r = \frac{512}{256} = 2 \] This ratio of 2 means that beyond identifying the sequence, you can calculate further frequencies using this multiplier, or its reciprocal for octaves below Middle C.
• For calculating a C two octaves below, the method involves using the reciprocal of the ratio twice:\[ 256 \times \left(\frac{1}{2}\right)^2 = 64 \,\text{Hz} \]Hence, understanding the concept of common ratio allows for pinpointing various octave relationships within a musical frequency scale.