7.3: Musical Scales
Introduction
The first thing musicians must do before they can play together is "tune.” For musicians in the standard Western music tradition, this means agreeing on exactly what pitch (what frequency) is an "A,” what is a "B flat," and so on. Other cultures not only have different note names and different scales, they may even have different notes - different pitches - based on a different tuning system. In fact, the modern Western tuning system, which is called equal temperament, replaced (relatively recently) other tuning systems that were once popular in Europe. All tuning systems are based on the physics of sound. But they all are also affected by the history of their music traditions, as well as by the tuning peculiarities of the instruments used in those traditions.
Many musical instruments, including pianos, guitars, and violins, make sound when strings are plucked, hit, or set in vibration in some other way. This vibration causes the air to move, creating an audible wave, which eventually reaches our ears. We call this wave “sound.” The pitch—how high or low the sound is—is determined by various factors such as the length of the string, its thickness, and its tension. A guitarist may play an open string, say G, and then place a finger on a fret to create sound with a higher pitch, say G#, A, or B. What is happening is that the player is shortening the vibrating part of the string to produce a higher pitch. This is based on the fundamental fact that shorter (or skinner or tighter) strings produce higher pitches. Similar facts are true with woodwind, percussion, and other types of instruments in general: the smaller the vibrating portion gets, the higher the sound becomes. That is why, for instance, a tuba is bigger than a French horn, and a contrabass flute is longer than a soprano flute.
In this section, we focus our attention on string vibrations. In particular, we look at the relation between the pitch and the string length, assuming that the material, thickness, and tightness of the string remain constant.
Suppose we pluck a string of a certain length and create a wave, a vibration of the air. The string will move up and down in oscillation. Each wave, whether it is a simple water wave, sound, earthquake, radio wave, microwave, or light wave, has a frequency, which measures how many times the oscillation occurs in a given time period, such as one second. For sound, the frequency is measured by “cycles per second” (abbreviated “cps”), sometimes called “hertz” (denoted “Hz”). Frequency is what determines the pitch of the sound generated by the vibration.
When a string is plucked and set in motion, the entire string vibrates, as shown in the picture above. The frequency created by the whole string is referred to as the “fundamental frequency” of the string. It generates the lowest sound the string can possibly make. This is equivalent to playing an open string on the guitar. If the guitar is tuned in the standard way, the sixth string (the thickest one, located on the top) is the E string, meaning that its fundamental frequency corresponds to the pitch we call an “E.” This is the lowest sound a guitar can play.
Now, if we take that same string, press the midpoint of it and pluck it, only half of the string would vibrate, shortening the wave length to half and doubling the frequency. This, of course, results in a higher pitch. But how much higher? When the frequency is doubled, we call the interval between those two pitches an “octave” (meaning 8). For instance, if the open string has a frequency of 440 cps, then the half string would have a frequency of 880 cps, and that sound is called an octave higher than the first. One can, of course, keep making the vibrating portion shorter and shorter to create higher and higher sounds.
Pythagoras: Numbers, Music, and Harmony
Although music has existed in different cultures for millennia, it was Pythagoras, an ancient Greek mathematician and philosopher, who explicitly linked sound (pitch) with frequency (number), laying a foundation of music theory based on mathematics. It was known to Pythagoras that two notes (sounds with different frequencies) sound nice together (harmonious, pleasant) when the ratio of the two frequencies is a simple fraction. For instance, two pitches that are an octave apart have the ratio of 2 to 1 as described above. 2/1 (or its reciprocal ½) is a simple fraction, and these two notes sound nice if played simultaneously. In fact, if two people sing a song one octave apart, it could sound almost like a unison. These simple fractions give rise to well-known interval names in music theory:
2 = 2.00: octave
5/3 = 1.666…: major sixth
3/2 = 1.50: perfect fifth
4/3 = 1.333…: perfect fourth
5/4 = 1.25: major third
Pythagoras once said, "There is geometry in the humming of the strings, there is music in the spacing of the spheres."
These simple ratios became a foundation for various musical scales and musical chords. For this reason, sometimes Pythagoras is called the father of music theory. In fact, as a religious and philosophical leader, he taught that ratios create harmony which forms beauty in nature; therefore, ratios are divine. His followers, called the Pythagoreans, thus did not believe the existence of numbers that are not ratios of integers. It is then quite ironic that Pythagoras himself later proved that square root of 2 (the length of the hypotenuse of a right isosceles triangle whose two sides are 1 unit long) is irrational, contradicting his own teaching that every number is rational. (By the way, this number, 2, came about by a very theorem named after him—the Pythagorean Theorem!)
Regardless of this devastating discovery, numerical ratios played a very significant role in the composition of musical scales. A scale is an ordered list of music tones that covers an octave; for instance, today the standard C-major scale consists of 8 notes starting with a C and ending with the next high C. I am certain that readers who are piano students have practiced these scales thousands of times (and your family may be quite tired of hearing them).
Throughout history, there have been many musical scales in various cultures, traditions, genres, and generations. Many authors have documented a wide variety of scales and methods of tuning instruments. Interested readers can look up topics such as Pythagorean intonation, mean-tone system, and temperaments. For simplicity, we now focus on the current music scale structure, consisting of 12 notes and their frequencies. This scale may not have been so widespread (or so standard) until J. S. Bach around the beginning of the 18th century.
Here are some facts about today’s standard musical scale:
- A scale begins and ends with two pitches with the same name, one octave apart (meaning that the frequency of the last note is exactly twice that of the first note).
- This whole interval, an octave, is divided into 12 half-steps, in such a way that the frequency ratio of any consecutive half-steps is the same (described as “equal distance apart”).
Let’s take a look at these requirements mathematically now.
Suppose \(P_0\) represents the frequency of the first note in a scale. Every time you move half-step to the next note, there must be a fixed ratio increase, i.e., growth by a fixed percentage, say, \(r\) (percentage written as a decimal). So the second note has the frequency
\[P_{1}=P_{0}(1+r)\]
Then, to go to the next note by moving half-step up, we need to multiply this new number by the same multiplier, giving us
\[P_{2}=P_{1}(1+r)=P_{0}(1+r)(1+r)=P_{0}(1+r)^{2}\]
Repeating, we get
\[P_{3}=P_{0}(1+r)^{3}\]
\[ P_{4}=P_{0}(1+r)^{4}\]
and so on. This look quite familiar, correct? This is what we saw in exponential growth. This is what we saw in compounding interest. Yes, it turns out that the mathematics that rules musical scales is the same as the mathematics that describes compound interest in finance and any exponential (fixed-percentage) growth in natural or social science!
So what is this magic number \(r\), the common ratio that separates every half-step on the musical scale? Remember that after 12 half-steps, we have traveled an octave, doubling the original frequency. In other words, this is the number r so that the following equation is satisfied:
\[P_{12}=P_{0}(1+r)^{12}=2 P_{0} \nonumber \]
If this equation were to appear in the context of finance, then this r would be the annual interest rate such that the original principal is doubled after 12 years. After dividing both sides by P0 and taking the 12th root of both sides, we get
\[1+r=\sqrt[12]{2} \approx 1.059463 \nonumber \]
We finally conclude that the fixed ratio between any two half-steps is approximately 0.059463, which is close to 5.946%. So, if you get this interest rate in a bank account compounding annually, you will double your balance in exactly 12 years. If you start your musical scale with a note with a certain frequency, after 12 half-steps, you will double your frequency, resulting in the note exactly one octave higher. (Think of the first two notes in the song “Somewhere Over the Rainbow.”) Below is a graph showing what happens to your bank balance if you begin with the original principal of $440 with the annual rate 5.946% for 24 years; your money doubles twice, i.e., quadruples to $1,760. This is like hitting a note two octaves higher.
Why did I start this graph with $440? Well, in standard tuning, the middle A is set to 440 cps. Tuning forks are designed to vibrate at this frequency (if you know what a tuning fork is). Electronic tuners allow you to adjust this a bit, depending on the music you play (or how you want your music to sound). Increasing this frequency, 440 cps, by \(r=0.059463\) each time (for 12 times), you will get the following notes in the standard A scale. For some notes, the ratio of the frequency and 440 is also shown.
|
Note |
Frequency (Hz) |
Ratio |
|
|---|---|---|---|
|
A |
440.00 |
1 |
|
|
A# |
466.16 |
||
|
B |
493.88 |
1.122462 |
|
|
C |
523.25 |
1.189207 |
|
|
C# |
554.37 |
1.259921 |
approx. 5:4 |
|
D |
587.33 |
1.33484 |
approx. 4:3 |
|
D# |
622.25 |
||
|
E |
659.26 |
1.498307 |
approx. 3:2 |
|
F |
698.46 |
||
|
F# |
739.99 |
||
|
G |
783.99 |
||
|
G# |
830.61 |
||
|
A |
880.00 |
2 |
exactly 2:1 |
Take a look, for instance, at C#, with a frequency ratio of 1.2599, which is approximately 5:4. This note in the key of A major is called “major third”; D is called “perfect fourth,” and E is “perfect fifth.” Because of these simple (approximate) ratios, the notes A, C#, and E sound beautiful together. The set of these three notes is called the “A major” chord. All harmony is based on these simple ratios.
But each ratio is irrational and not exactly a simple ratio, right? This is the unfortunate result of equal temperament (devised in the 18th century); the common ratio r is the (not-so-secret) number that would allow a piece of music to sound good in every key, but the trade-off of temperament is that all ratios are now approximate, except for octaves, and thus, strictly speaking, “out of tune.” However, given the difficulty in tuning big instruments like the piano, this was the preferred compromise. The alternative is to have some ratios perfect while others are way off simple ratios, resulting in some music perhaps sounding great in one key but sounding terrible in another. Equal temperament is thus widely accepted and regarded as standard now. In other words, one irrational number, 122, seems to control the harmonious world of music.
Just one more thing… I mentioned that the A major chord consists of A, C#, and E. If you change that middle note and lower it by half-step, you get the notes A, C (natural), and E, which together form the A minor chord. Play these chords and hear the difference. The mood shifts dramatically, right? Most people can hear the difference between a song written in a major key and a minor key. In fact, it would make people laugh if you take a popular song written in a minor key and play it in the corresponding major key. That huge difference, we now know, is generated by one little note dropped by one half-step, i.e., a frequency difference of about 5.946%. Such a small ratio, yet such a big difference!
Contributors and Attributions
-
Saburo Matsumoto
CC-BY-4.0