OBJECTIVE To learn how musical scales are related to exponential functions and how exponential functions determine the pitch to which musical instruments are tuned.

Poets, writers, philosophers, and even politicians have extolled the virtues of music—its beauty, its power to communicate emotion, and even its healing power. The philosopher Nietzsche said that "without music life would be a mistake." Perhaps that’s why some musicians—from rock stars to classical pianists—can command huge fees for performing their art.

What is music, exactly? Is it just "noise" that happens to sound nice? In general, music consists of tones played by instruments or sung by voices, either sequentially (as a melody) or simultaneously (as a chord). Tones, the building blocks of music, are sounds that have one dominant frequency.

The tones that we are familiar with from our everyday listening can all be reproduced by the white and black keys of any properly tuned piano. The strings of a piano produce specific sound frequencies; for instance, middle $A$ has a fundamental frequency of $440$ cycles per second, or $440$ Hertz (Hz). The other "$A$" keys on the piano are either higher-sounding, with frequencies of $880$ Hz, $1760$ Hz, and so on, or lower-sounding, with frequencies of $220$ Hz, $110$ Hz, $55$ Hz, and so on. To get from one $A$ to the next, you just need to double or halve the frequency. Every pair of notes with the same letter name "sounds" the same to most listeners. Such notes are said to be separated by "octaves" on the musical scale. In this Exploration we learn how exponential functions allow us to create all the notes in the musical scale.

I. Frequencies of Notes

We divide the interval between two notes that are an octave apart into $12$ parts in such a way that each note’s frequency is a fixed multiple of the preceding one. This frequency interval is called a semitone in music theory. Since an octave involves multiplying the frequency by $2$, each semitone therefore involves multiplying the frequency by $2^{1/12}$. This means that the frequencies are evenly spaced on a logarithmic scale. The distance between semitones on a logarithmic scale is $$\text{log }2^{1/12} = \frac{1}{12}\text{log }2 \approx 0.025$$

So the keys on a piano are "evenly tempered" on a logarithmic scale.

1. The note immediately above middle $A$ is $A$ sharp ($A$# or $B$♭), then the note above $A$# is $B$, and so on. The frequency of each note is obtained by multiplying $440$ (the frequency of middle $A$) by an appropriate power of $2^{1/12}$. We have

   $A$	$440 \cdot \left(2^{1/12}\right)^0 = 440 \;Hz$
   $A$#	$440 \cdot \left(2^{1/12}\right)^1 \approx 466.164 \;Hz$
   $B$	$440 \cdot \left(2^{1/12}\right)^2 \approx 493.883 \;Hz$

   Complete the following table by calculating the frequencies of the notes produced by the keys in one octave on a piano, from middle $A$ (known by piano tuners as $A4$ or $A440$) to the next $A$ (called $A5$).

   Key	Frequency
   $A$	$440.000$
   $A$# $B$♭	$466.164$
   $B$
   $C$
   $C$# $D$♭
   $D$
   $D$# $E$♭
   $E$
   $F$
   $F$# $G$♭
   $G$
   $G$# $A$♭
   $A$

2. The lowest-sounding key on a piano is an $A$ (called $A0$ or Double Pedal $A$). This key produces a note four octaves below middle $A$ ($A440$).
   1. What fundamental frequency does the lowest piano key have?
   2. Many people lose the ability to hear low frequencies as they age. Elderly people often can’t hear frequencies lower than $40$ Hz. Will they be able to hear the fundamental tone produced by the $A0$ key?
3. The highest $A$ on the piano is $A7$, three octaves above middle $A$ ($A440$). The highest-sounding key of all on the piano is the $88$th, called $C8$. Its frequency is three semitones above $A7$.
   1. What is the frequency of $A7$?
   2. What is the frequency of $C8$, the highest key on the piano?
   3. All people with normal hearing can hear sounds up to at least $15,000$ Hz. Can these people hear the sound of the highest key on the piano?

II. Intervals, Frequencies, and Dissonance

The "equally tempered" tuning of modern instruments described above did not come into wide use until the late 17th century. Before that, musicians used many tunings that sounded good to their ears and to their listeners'. Johann Sebastian Bach published a set of preludes and fugues in 1722, called The Well-Tempered Clavier, which was designed to popularize the new tuning system for keyboard instruments. Each of the pieces in this work was written in one of the $24$ major and minor keys in the "well-tempered" tuning. Some listeners found the new tuning harsh and unmusical for reasons that we now explore.

1. When two or more keys on the piano are pressed simultaneously, the resulting mixed sound is called a chord. When the ratios of the frequencies of the notes in the chord involve small numbers, such as $2:1$, $3:2$, or $4:3$, the chord sounds pleasant and musical to most ears. But other ratios may sound unpleasant or dissonant.
   1. A perfect fifth is an interval in a chord between two notes whose frequencies are in the ratio $3/2$. On the piano, this is approximated by two notes that are seven semitones apart (for instance, $C$ and $G$). Use your table from Problem I.1 to determine the ratio between the frequencies of $G$ and $C$. How far does this differ from the ideal ratio of $3/2$?
   2. A perfect fourth is an interval between two notes with frequencies in the ratio $4/3$. On the piano a fourth is approximated by two notes that are five semitones apart (for instance, $C$ and $F$). What is the ratio between the frequencies of $F$ and $C$ on the piano? How far does this differ from the ideal ratio of $4/3$?
   3. Why do you think some people who are accustomed to perfect fifths and fourths might find the modern tuning of a piano to be dissonant?
2. In an octave the two notes have a frequency ratio of $2/1$. So to divide an octave into $12$ semitones, the frequency of each note is $2^{1/12}$ times the frequency of the preceding note: $$\text{frequency of note} = \left(\frac{2}{1}\right)^{1/12} × \text{ frequency of preceding note}$$
   1. In a perfect fifth the two notes have a frequency ratio of $3/2$. If we divide a perfect fifth into seven semitones, explain why the frequency of each note is $\left(3/2\right)^{1/7}$ times the frequency of the preceding note: $$\text{frequency of note} = \left(\frac{3}{2}\right)^{1/7} × \text{ frequency of preceding note}$$
   2. In a perfect fourth the two notes have a frequency ratio of $4/3$. To divide a perfect fourth into five semitones, by what factor must the frequency of each note be multiplied to get the frequency of the next note? $$\text{frequency of note} = \left(\frac{\color{red}{\fbox{   }}}{\color{red}{\fbox{   }}}\right)^\color{red}{\fbox{   }} × \text{ frequency of preceding note}$$
   3. Calculate the factors in parts (a) and (b), and note that they are not equal. Explain why this means that we can't tune a piano (with even tempering) so that it has exact perfect fifths as well as exact perfect fourths.