Problem 48 - Algebra for College Students

Question

The note middle C on a piano is tuned so that the string vibrates at 262 cycles per second, or 262 hertz (Hz). The C note one octave higher is tuned to \(524 \mathrm{Hz}\). The tuning for the 11 notes in between using the method called equal temperament is determined by the sequence \(a_{n}=262 \cdot 2^{n / 12} .\) Find the tuning for the 11 notes in between. Round to the nearest whole Hz.

Step-by-Step Solution

Answer

328 Hz, 350 Hz, 370 Hz, 393 Hz, 416 Hz, 440 Hz, 466 Hz, 494 Hz, 523 Hz, 554 Hz, 587 Hz

Step 1: Understand the Sequence Formula

The sequence formula given is \(a_{n}=262 \times 2^{n/12}\), where \(n\) represents the note number ranging from 1 to 11.

Step 2: Calculate the Frequency for Each Note

Use the formula \(a_{n}=262 \times 2^{n/12}\) to find the frequency for each note (n = 1 to 11).

Step 3: Compute the First Note

Substitute \(n = 1\) into the formula: \[a_1 = 262 \times 2^{1/12}\] Use a calculator to find the result and round to the nearest whole Hz.

Step 4: Compute the Second Note

Substitute \(n = 2\) into the formula: \[a_2 = 262 \times 2^{2/12}\] Use a calculator to find the result and round to the nearest whole Hz.

Step 5: Compute the Remaining Notes

Continue to substitute \(n = 3\) to \(n = 11\) into the formula \(a_{n}=262 \times 2^{n/12}\). Use a calculator for each calculation and round each result to the nearest whole Hz.

Step 6: List the Frequencies

After calculating, list all the results: \[a_1, a_2, a_3, \text{...}, a_{11}\]

Key Concepts

cycles per second

In music and physics, cycles per second is a term used to describe the frequency of a sound wave. It's commonly referred to in hertz (Hz), which represents the number of times the sound wave completes a cycle in one second. For example, the note middle C on a piano has a frequency of 262 Hz. This means that the string on the piano vibrates 262 times every second. Understanding this concept is essential because it helps in comprehending how different notes are derived and tuned based on their frequency. Middle C is often the reference point for tuning musical instruments, and the concept of

mathematical sequences

Mathematical sequences are ordered lists of numbers following a specific pattern or rule. In the case of tuning notes in equal temperament, we use a sequence to determine the frequencies of each note. The sequence formula given is \(a_{n}=262 \times 2^{n/12}\). Here, \(n\) represents the note number ranging from 1 to 11. Each term in this sequence represents the frequency of a note. As you move from one note to the next, the frequency increases by a factor of \(2^{1/12}\). This mathematical relationship is crucial in ensuring that each note sounds harmonically balanced with the others.

logarithmic scale

A logarithmic scale is a way of displaying numerical data over a very wide range of values in a compact way. In music, the logarithmic scale is used to measure frequencies because it better reflects how we perceive sound. For instance, the difference in perceived pitch between successive notes increases logarithmically. This is why the frequency formula for equally tempered tuning involves an exponent: \(2^{n/12}\). The exponent ensures that every step or interval between notes is consistent from a perceptual standpoint, even though the actual frequencies are based on a geometric progression.

octave calculation

An octave in music is the interval between one musical pitch and another with half or double its frequency. For example, middle C at 262 Hz has its octave at 524 Hz, which is double the frequency. This doubling of frequency aligns with our logarithmic perception of pitch. To divide an octave into 12 equal parts, each part's frequency can be calculated using the formula \(f = 262 \times 2^{n/12}\), where \(n\) ranges from 1 to 11. This ensures that the interval between each note is equal in terms of frequency ratios, providing balanced and harmonically pleasing sound intervals.

frequency rounding

Rounding frequencies to the nearest whole number makes the calculations and resulting frequencies more practical and easier to read. When we use the sequence formula \(a_n = 262 \times 2^{n/12}\), we get precise frequencies that usually have decimal points. These frequencies need to be rounded to the nearest whole number, especially when tuning musical instruments, to ensure consistency and ease of tuning. For instance, if the calculated frequency is 277.18 Hz, it can be rounded to 277 Hz. Such rounding does not significantly impact the sound quality but simplifies the tuning process.