Show harmonics: 1-12 8-20 16-24

Fig. 1: The first 12 partials of an harmonic series for the pitch C2 (65.41 Hz.) expressed in traditional musical staff notation with frequency multiples indicated between the staves. The minus (-) symbol indicates that the indicated pitch is lower than the same pitch in 12TET. To see harmonics 8-20 or 16-24 for the pitch C2, choose the appropriate selection from the menu (top-right).

Overtone "One of the frequency components of a sound other than that of lowest frequency. Usually overtones are numbered consecutively in ascending order of frequency; they need not be harmonic."

Murray Campbell The Grove Dictionary of Music

T he term overtone series generally refers to a specific set of frequency components that appear above a musical tone. The related term harmonic series is a more precisely defined mathematical concept. Though musicians often use the two terms interchangeably, the term harmonic series specifically refers to a set of numbers related by whole number ratios. For example, the set of frequencies (in Hz.) 1000, 2000, 3000, 4000, 5000, 6000, etc., forms a harmonic series; So does 500, 1000, 1500, 2000, 2500, 3000, etc. The fundamental, or lowest component of the first series is 1000 Hz. The fundamental of the second series is 500 Hz. The other frequency components are called harmonics, overtones or partials. Expressing both of the series above as frequency multiples of their respective fundamentals reduces them to the same form:

f0, 2 f0, 3 f0, 4 f0, 5 f0, 6 f0, etc.

A harmonic series may alternatively be expressed using musical staff notation (see Fig. 1). Though this traditional way of notating the harmonic series is obviously less precise at conveying frequency than the mathematical sequence notation used above, it does provide a convenient way for musicians to memorize the series as a kind-of chord/scale of nature.

Fig. 2: Monochord diagram from Hindemith 1945 [1]

A simple explanation for why such a pattern of tones appears above the pitch one actually perceives may be found in the physical model of a plucked string. Fig. 2 shows an illustration of monochord, a simple device used by theorists since the Middle Ages to investigate the relationship between string length and pitch. A monochord has three main components: 1. a string fixed at both ends, 2. a moveable bridge, and 3. a resonating body. When plucked, the monochord's string vibrates at a rate directly proportional to its length.[2] However, this is not the only mode of vibration of the string. In addition to vibrating over its entire length, a string simultaneously vibrates over fractional divisions of its length (1/2, 1/3, 1/4, 1/5, 1/6, and so on; see Fig. 3a) producing ever higher frequencies (overtones) that are inversely proportional (2x, 3x, 4x, 5x, 6x, and so on) to those string length divisions.

Fig. 3a: String length division diagram from Hindemith 1945

Theoretically, an infinate number of these multiple modes of vibration exist, producing an infinate number of ever increasing frequency components that generally decrease in amplitude or loudness. Yet research has found that our brain tends to fuse these harmonically-related components into a single sensation we call pitch. Further, frequency components that are nearly harmonically-related also produce a sensation of pitch. The process by which the brain fuses all of these individual pure tone components together is called fusion. A stunning fact uncovered by psychologists doing research on fusion in the 1920s is that the ommision of the fundamental from a series of harmonically-related components does not change our sensation of pitch.[3]

The Path to Just Intervals

The discovery of a numerical relation between string length and musical interval is commonly attributed to Pythagoras (c. 500 BCE).[4] We say that the pitch produced by a string of length L/2 sounds an octave higher than the pitch produced by a string of length L. This relationship between string length and musical interval provides musicians with a precise way to express the size of a musical interval. For example, the octave can be defined as the musical interval whose frequency ratio is 2:1. All ratios should be expressed in their simplest form. For example, the interval whose frequency ratio is 6:4, should be expressed using the ratio 3:2.

Fig. 3b: A plucked string has many simultaneous modes of oscillation, vibrating over fractional divisions of its length to produce the pattern of overtones shown in Fig. 1. Fig. 3b is not drawn to scale. [5]

Fig. 1 can be used to look up which natural frequency ratios (intervals found in the harmonic series) correspond to which tonal interval names. Tonal interval names like perfect fifth, major third, minor second, and so on are defined by their relative position within in the major or minor scale. Now for practice, find two tones in Fig. 1 that form a perfect fifth. For example, the second and third tones of the harmonic series form a perfect fifth. So the natural interval frequency ratio for that interval can be determined by using the frequency multiple numbers between the staves, in this case 3:2. To gain a deeper understanding of the notion of a just interval, use Fig. 1 to look up the natural interval frequency ratios for the following tonal intervals:

Fig. 4a: Common tonal interval names.

a. perfect fifth, b. perfect fourth, c. major sixth, d. major third, e. minor third, f. minor sixth, g. minor seventh, h. major second, i. major seventh, j. minor second.

These intervals are equivalent to the following natural interval frequency ratios:

Fig. 4b: Natural interval frequency ratios corrosponding to the common tonal interval names in Fig. 4a.

a. 3:2, b. 4:3, c. 5:3, d. 5:4, e. 6:5, f. 8:5, g. 9:5, h. 9:8, i. 15:8, j. 16:15.

Natural interval frequency ratios are called just intervals. In his 1949 book Genesis of a Music, composer Harry Partch assigns interval names to these and many other natural interval frequency rations. It is definately worth the time for musicians to memorize the harmonic series on C2 up to the sixteenth partial. Once memorized for C2, the pattern may easily be transposed to any other pitch.

The Path to Scales and Tuning Systems

On the horns of dilema "On the whole of the historical period of instrumental music, Western music has based itself upon an acoutical lie. In our time this lie--that the normal musical ear hears twelve equal intervals within the span of an octave--has led to the impoverishment of pitch usage in our music."

Ben Johnston Dictionary of Contemporary Music

The octave, or interval whose frequency ratio is 2:1, is clearly too large an intervallic distance to be melodically expressive. Imagine a melody consisting of only octave-related pitches. Many cultures, including our own, have gone so far as to assume that pitches related by a 2:1 frequency ratio are to be considered equivalent in some sense. In music theory this important principle is known as octave equivalance. Scale creation is, of course, integrally related to both instrument design and tuning theory. To understand the nature of scale, one must learn a bit of tuning theory.

The word scale is dervied from the Greek word, Skala, meaning steps. Thus, it's convenient to think of scale creation as the process of dividing up the octave (or other basic interval) into discrete steps that are designed to achieve a desired musical result (e.g.: melodic, harmonic, etc.) Another way to think of scale creation is as a process of arranging interval patterns within the octave. One way to think of a diatonic scale is as a set of seven distinct tones arranged within the octave to form a particular interval succession pattern. For example, the interval succession could be expressed in semitones as: 2 2 1 2 2 2 1. But exactly what size are those semitones? That is where tuning theory comes in. Throughout history, the diatonic scale has been tuned in a variety of different ways to achieve different melodic and harmonic results. To get an idea of how all tuning systems are constructed, it is convenient to divide tuning theory into two basic approches and work through four historically significant tuning systems: Pythagorean, Just, Mean-tone, and 12TET.

J. Murray Barbor in his book Tuning and Temperament: A Historical Survey says there are two basic approaches:

1. Tuning - Steps are derived from the harmonic series (whole-number ratios

2. Temperement - Steps are derived by making adjustments to a tuning to achieve a desired musical result

On recognizble diatonic tunings "The structure of recognizable diatonic tunings is basically an array of intricate interconnections...which are the very foundation of what is perceived as tonal harmonic motion, are shaped by the short-term span of human memory, the tolerance range of the human ear, and the peculiar manner in which intervals are perceived."

Easley Blackwood The Structure of Recognizable Diatonic Tunings

For example, our current accepted standard for tuning 12TET was designed to do many things, however, its most important feature is that all intervals remain the same size in every possible key. The best way to learn how tuning theory works is to work through the math yourself for a few basic examples:

1. A Pythagorean Tuning of the Diatonic Scale: A tuning based on the line of fifths
2. A Just Tuning of the Diatonic Scale: A tuning based on the harmonic series
3. Meantone Temperement: Turning three different sized tones into two
4. 12-tone equal temperement (12TET) - An accepted standard for tuning since the time of J.S. Bach for fixed pitched instruments like the piano

The Path to Consonance and Dissonance

The 19th-Century physicist Herman von Helmholtz proposed a theory of consonance and dissonance based on the harmonic series, beating and roughness. Many cultures have noted that ratios involving relatively small integers (e.g: 1:1, 2:1, 3:2, 4:3, 5:3, 5:4, 6:5) produce the common "harmonious" intervals or consonances in the music of many cultures. Yet, why might this be so? In 1863, Helmholtz proposed that the degree of dissonance produced by an interval is related to the degree of roughness produced between the partials. For the middle register, a separation of one semintone produces the most roughness, with a separation of two semitons next. Fig. 99 shows the intervals encounteed in a just (5-limit) tuning the diatonic scale. The first six harmonics for each tone are shown.

Fig. 99: Degree of Dissonance Diagram after Cambell and Greated 1987.

For each interval, compare the number of semitone and tone collisions between the partials. Based on Helmholtz's model, how would you rank the following intervals in terms of degree of dissonance?

Fig. 10: The amplitude proressions of the partials of a trumpet tone as analyzed by Grey and Moorer. Click to see a larger version of this graph.

The Path to Timbre

In the 1970s, Grey, Moorer, Risset and others first used computers to analyze the tones produced by acoustic instruments. Using computers, researchers are able to isolate the individual partials of an acoustic instrument tone and show how each partial's amplitude progresses independantly through time. Fig. 10 shows such an analytical graph in three dimensions. The graph was created by Grey and Moorer. The three dimensions are: 1. Time (left to right), 2. Amplitude (down to up), and 3. Partial number (back to front). The graph displays the first .5 seconds of a trumpet tone. Notice how the lower partials (back) rise first and rise to be the loudest components. They are also the last to decay. Clearly the basic building blocks of this trumpet tone can be expressed as harmonically-related partials. Notice that only the first 12 partials are displayed of the trumpet tone are display. This type of analysis of an acoustic sound uses a very computationally intensive algorithm known as a fast-fourier transform (FFT). In 1822, the French mathematician Jean Baptist Fourier (1768-1830) came forth with a mathematical proof demonstrating that any waveform, no matter how complex, could be reduced to an infinate set of sine wave components. The modern FFT algorithms that run on personal computers are based on Fourier's work and are thus named after him. Though this algorithm is indeed computationally intensive, it can easly be performed by today's personal computers in real-time and is commonly exploited in technologies such as MP3 compression.

Further Study On-line

Levin, Theodore and Michael Edgerton. "The Throat Singers of Tuva." Scientific American (September 1999): Available on-line at: http://www.sciam.com/1999/0999issue/0999levin.html.

Learn more about just intonation and tuning theory in general at the Just Intonation Network Web site: http://www.dnai.com:80/~jinetwk/index.html