The Harmonic Series
Cover Art: Alex Chen
This is a Harmony World article. But what is harmony? And what is this article about? Also, why is major “happy” and minor “sad”? And how come music uses the specific notes that it does? All will be revealed in due time, dear reader, if you continue reading.
The harmonic series describes the motion of a vibrating object of fixed length, such as a string in a piano, guitar, cello, etc. The string in one of these instruments vibrates in a very complex way. As you might expect, the whole string moves back and forth in one big wave. But at the same time, the string oscillates in halves, thirds, fourths, fifths, etc. The result is that when you pluck a string on a guitar or hit a key on a piano, a host of many different notes is created. The desired note, called the fundamental pitch or fundamental frequency, results from that initial oscillation of the whole string back and forth, which produces by far the loudest sound. All of the other pitches are called overtones. The first overtone, the one produced by half of the string oscillating, has a frequency double that of the fundamental. The next overtone, produced by the string vibrating in thirds, has a frequency triple that of the fundamental. The following overtone has quadruple the frequency of the fundamental, and so on to infinity, each overtone getting higher in pitch and lower in volume as we travel up the harmonic series.
To understand how the harmonic series relates to which notes are harmonious, let’s take a closer look at notes. A note is a consistent beat, and the greater the frequency of this beat, the higher the pitch of the note. For instance, ‘middle C’ has a frequency of 261.63 Hz, while the next note above it, Db, has a frequency of 277.18 Hz. When two of these beats form a recognizable pattern with one another, they are perceived as consonant or harmonious. For instance, the next highest C note has a frequency 523.25 Hz, exactly twice that of middle C, meaning for every beat emitted by a middle C, two beats are emitted by the higher C. This pattern is so recognizable to our ears that we consider the two pitches to be the same note, but in different registers. You may notice that in the harmonic series, this 2:1 relationship is the relationship between the fundamental and the first overtone. These strong, consonant relationships based on the harmonic series produce harmony, the namesake of our glorious magazine. The relationship between two frequencies is called an interval, and this specific 2:1 interval is known as an octave. So in sum, the first overtone on the harmonic series is exactly one octave higher than the fundamental.
Let’s run quickly through the important intervals on the harmonic series. Keep in mind that we are talking about intervals now, which describe the distance between two notes in terms of the ratio of their frequencies. Each step on the harmonic series brings us a unique interval, but not necessarily a new note; for example, every overtone whose frequency is 2, 4, 8, 16, etc. times that of the fundamental will be the same note as the fundamental, but in higher and higher octaves. Similarly, every overtone whose frequency is 3, 6, 12, 24, etc. times that of the fundamental will be the same note as the dominant in higher and higher octaves.
The next two tones on the harmonic series are separated by an interval of 3:2, an interval known as the “perfect fifth”. The perfect fifth brings us to the first note in the series that is different from the fundamental. The fifth tone on a 7-tone scale is a perfect fifth away from the first tone. This first tone is called the “tonic” and the fifth is called the “dominant”. Due to their proximity on the harmonic series, the tonic and the dominant have a very strong relationship, which is fundamental to Western music and instruments.
The interval after the perfect fifth is the perfect fourth, which describes a ratio of 4:3 between two frequencies. Similar to the perfect fifth, the fourth note on most 7-note scales is a perfect fourth away from the tonic, and it is called the “subdominant”. The most common chord progressions involve the tonic transitioning into the subdominant, which develops into the dominant, which ultimately resolves back to the tonic, or, alternatively, a similar progression with the subdominant and dominant switched around. The simple ratios of the perfect fifth and perfect fourth with the tonic mean that these types of chord progressions are found all over musical genres and eras. If you see a video or an article with a title to the effect of “You can learn thousands of songs with these four easy chords!”, this progression is behind it.
The following two intervals are the major third and the minor third, described by the ratios 5:4 and 6:5 respectively. As the names suggest, the major third is used in major keys, while the minor third is used in minor keys. In fact, the position of the third is what differentiates a major triad from a minor triad. While we’re here, why do major chords sound happy or fulfilled while minor chords sound sad or unsettling? One reason could be that the smaller interval between the tonic and the minor third makes for a more dissonant sound. However, the more important reason has to do with the harmonic series. The note that is a major third away from a given tonic note in our 12-note system (e.g. E is a major third away from C) happens to be the fourth overtone of that tonic note on the harmonic series, and the second distinct overtone. Meanwhile, the minor third (e.g. Eb is a minor third from C) is the eighteenth overtone on the harmonic series, and therefore much higher and much quieter. So whenever we hear a given tonic note played on an instrument, we are hearing the major third as a fundamental part of that note. When we hear a minor chord, the minor third that is being played conflicts with the major third that is implicit in the tonic (e.g. Eb conflicts with the ‘E’ pitch that occurs when the C is played in the C minor chord).
The next interval I want to highlight is the semitone, represented by a frequency ratio of 17:16. Given how complicated a pattern of 16 beats for 17 beats is to our ears, the semitone is the most dissonant interval on the keyboard/The semitone is represented by the distance between any two notes on a piano, so all the other intervals that we have mentioned can all be defined in terms of the semitone - the minor third is 3 semitones, major third is 4 semitones, perfect fourth is 5 semitones, perfect fifth is 7 semitones, and the octave, of course, is 12 semitones, spanning the twelve different common notes and the twelve distinct keys on a piano. The collection of all twelve of these notes arranged in order is called the “chromatic scale”.
The Chromatic Scale
There are other intervals on the chromatic scale which we haven’t mentioned, such as the major and minor sixths and sevenths, but they can all be accounted for if you look far enough on the harmonic series. These are not the only intervals used in music, though. Musical cultures throughout the world use intervals, and therefore notes, that are completely absent from most Western music and which are impossible to play on many common instruments. For example, the sixth overtone on the harmonic series has a 7:6 frequency ratio, which is equivalent to about 2.7 semitones. This means that it cannot be played on a standard piano, but it appears in some Hindu ragas and is represented in temperament systems other than the Western 12-tone one. It is also possible to have intervals smaller than one semitone, known as microtones. The blues, and musical genres influenced by it, use microtones, known as ‘blue notes,’ fairly frequently. Experimental classical composers like Charles Ives and Ivan Wyschnegradsky have implemented microtones in their music by tuning two pianos exactly half a semitone apart. The bağlama, a string instrument used in Turkish folk, Kurdish, Azberbaijani, and other types of music, naturally plays notes at microtonal intervals.
So why are there twelve notes in the chromatic scale, and why these twelve notes in particular? The specific twelve pitches represented on most Western instruments were set in place with the development of clavichords and pianos (a clavichord is a keyboard instrument predating the piano), which have strings of fixed length. These instruments were designed so that important intervals such as the major and minor thirds and the perfect fourth and fifth were represented or nearly represented in every key, and so that each of the twelve notes between a fundamental note and its octave were equidistant, with that initial fundamental note being chosen somewhat arbitrarily. But why twelve notes in particular? The answer lies in that unbreakable bond between the tonic and the fifth dominant. You can travel up the chromatic scale with the perfect fifth interval, moving from tonic to dominant, then again with the last dominant acting as the next tonic. Traveling this way, you will reach every distinct note on a keyboard, and after 12 repetitions, you will wind up back at the same note you started with. This is called the “Circle of Fifths.”
Harmonic Series - Table