Problem 68 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand the Geometric Sequence

A geometric sequence progresses by multiplying the previous term by a constant ratio. In this case, we have two known values: the frequency of middle C is 256 Hz, and one octave higher is 512 Hz.

Step 2: Determine the Common Ratio

The common ratio \( r \) can be found by dividing the frequency of the C one octave higher by the frequency of middle C. Thus, \( r = \frac{512}{256} = 2 \).

Step 3: Determine the Frequency Two Octaves Below

To find the frequency two octaves below, we need to multiply the middle C frequency by the inverse of the ratio squared (since each octave is a multiplication). So, the frequency two octaves below is \( 256 \times \frac{1}{r^2} = 256 \times \frac{1}{2^2} = 256 \times \frac{1}{4} = 64 \) Hz.

Key Concepts

Musical Frequencies

In music, different notes correspond to specific sound waves, which we measure in cycles per second, or Hertz (Hz). Each note has a unique frequency, which helps determine its pitch. This characteristic turns musical notes into a fascinating study involving mathematical patterns.
When we consider the frequencies of notes on a scale, we often see that they form a pattern known as a geometric sequence.
This means that, for each note, we multiply the frequency of the previous note by a constant factor, a pattern that helps build a harmonious sequence.

• Middle C, for instance, has a frequency of 256 Hz.
• Moving one octave higher, the frequency doubles to 512 Hz.
• This doubling is a consistent pattern in many musical scales.

This sequence ensures that each note relates harmonically to others, creating a structure that defines Western music's charm. Exploring musical frequencies through geometric sequences not only helps us appreciate music mathematically but also helps in tuning and crafting instruments.

Common Ratio

A fundamental part of a geometric sequence is the common ratio, which acts as the multiplier that takes us from one term to the next.
In the context of musical frequencies, the common ratio is the relationship between the frequencies of notes that are an octave apart. To find this ratio, we simply divide the frequency of a note one octave higher by the frequency of the reference note.

• Mathematically, for Middle C and the C an octave higher:
• \[ r = \frac{512}{256} = 2 \]

Here, the common ratio is 2, indicating that as you move one octave higher, the frequency doubles.
Understanding the concept of a common ratio provides a valuable tool for musicians and composers. It explains the systematic progression in pitch as you move through the octaves and helps in creating and adjusting scales to maintain harmony across musical notes.

Octave Relationship

The octave is a central concept in music theory, creating the fundamental structure on which much of Western music is built.
An octave spans from one note to the next with the same note name, with the frequency doubling every octave you move up. This doubling creates a direct and clear relationship within musical scales, known as the octave relationship.

• For example, as you go one octave up from Middle C (256 Hz), the frequency becomes 512 Hz.
• As you move two octaves up, the frequency becomes 1024 Hz.

Similarly, going down an octave halves the frequency, demonstrating the inverse relationship. Therefore, from Middle C, moving two octaves below leads to a frequency calculation where we divide by the common ratio squared:

• \[ 256 \times \frac{1}{2^2} = 64 \text{ Hz} \]

This octave relationship not only aids in the scientific understanding of sound but also enhances musical creativity, allowing compositions that resonate across different scales and keys.