Question

Musical Scale: The frequency of the "A" note above middle C is, by international agreement, equal to \(440 \mathrm{Hz}\). A note one octave higher is at twice that frequency, or \(880 \mathrm{Hz}\). The octave is subdivided into 12 half-tone intervals, where cach half-tone is higher than the one preceding by a factor equal to the twelfth root of \(2 .\) This is called the equally tempered scale and is usually attributed to Johann Sebastian Bach \((1685-1750) .\) Write a GP showing the frequency of each half-tone, from 440 to 880 Hz. Work to two decimal places.

Short answer

The GP representing the frequency of each half-tone from 440 Hz to 880 Hz is: 440.00, 466.16, 493.88, 523.25, 554.37, 587.33, 622.25, 659.26, 698.46, 739.99, 783.99, 830.61, 880.00 Hz.

Step-by-step solution

Understanding the Concept

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. To find the frequencies of each half-tone from 440 Hz to 880 Hz, we need to create a GP with the first term as 440 Hz, the last term as 880 Hz, and the common ratio being the twelfth root of 2.

Calculating the Common Ratio

The common ratio (r) is the factor by which each half-tone increases. This is calculated by taking the twelfth root of 2. Mathematically, this is expressed as r = \(2^{\frac{1}{12}}\).

Writing the GP

To write the GP sequence, we start with the first term (440 Hz) and apply the common ratio by multiplying each subsequent term by \(2^{\frac{1}{12}}\). We continue this process until we reach the twelfth term, which should be an octave higher, or 880 Hz.

Rounding to Two Decimal Places

Each term calculated in the sequence must be rounded to two decimal places as required by the exercise.

Key concepts

Frequency of Musical Notes

When we talk about the frequency of musical notes, we're referring to the number of vibrations per second that a note produces. These vibrations, measured in Hertz (Hz), are what the human ear interprets as pitch. The machine-accepted standard for tuning sets the 'A' above middle C at exactly 440 Hz, a decision solidified in the 20th century. The pitch of a note determines its place on the musical scale, and as we move up or down the scale, the frequency either doubles to reach the next octave or halves to drop to the previous octave. This difference in pitch is what allows musicians to create varying sounds and melodies.

Consider a piano; it's easy to hear that the note to the right of another is higher in pitch and vice versa. To illustrate this, the note one octave above 'A' is 880 Hz – exactly double the frequency. In music, particularly Western music, the octave is usually divided into 12 half-tone steps. This approach allows composers and musicians to play a wide range of music using a standardized and uniform system.

Equally Tempered Scale

The equally tempered scale is the universally accepted system for tuning most Western instruments. The foundation of this scale is the division of an octave into 12 equal parts, or semitones, creating what is known as a 'chromatic' scale. Before the equally tempered scale, instruments were often tuned to just intonation or meantone temperaments, which sounded harmonious in certain keys but dissonant in others. The disadvantage of these systems was pronounced when musicians wanted to modulate to different keys within a piece.

The equal tempering of scales enabled instruments to play in any key, providing a consistent sound across all keys. This versatility is especially important in complex compositions and those that require modulation. Although the concept was not entirely new, Johann Sebastian Bach's 'The Well-Tempered Clavier' demonstrated its practical application and advantages, leading to wide acceptance and implementation in musical instrumentation.

Twelfth Root of 2

In order to divide an octave into 12 equally spaced notes, we rely on a bit of mathematics, specifically the concept of the twelfth root of 2. This value, approximately equal to 1.059463, is what we multiply a given frequency by to find the frequency of the note a semitone higher. The twelfth root of 2 is calculated using the formula \( r = 2^{\frac{1}{12}} \).

The twelfth root of 2 is an irrational number, meaning its decimal representation goes on indefinitely without repeating. When applied successively twelve times, the resulting frequency is precisely double the original, resulting in an octave. This mathematical concept is crucial for creating the equally tempered scale that we discussed previously. By using the twelfth root of 2 as our common ratio in a geometric progression, we are able to accurately define the frequencies of each note in an octave, ensuring that instruments can perform together in harmony.

Johann Sebastian Bach

Johann Sebastian Bach, a renowned composer and musician of the Baroque period, had a significant influence on music theory and practice. While Bach did not invent the concept of the equally tempered scale, his work, particularly 'The Well-Tempered Clavier', is often credited for demonstrating its usefulness. Composed around 1722 and later in 1742, this collection of preludes and fugues in all major and minor keys showcased the benefits of equally tempered tuning.

Bach's composition provided a practical example of a keyboard instrument that could play in every key without the need to retune. This work allowed for modulation between keys, flexibility in composition, and was pivotal in the transition from just intonations to equal temperament. Bach's legacy in refining and promoting this tuning system is a testament to his profound impact on the development of Western music.