See harmonic series (mathematics) for the (related) mathematical concept. See Harmonic series (music for the (related musical concept In Mathematics, the harmonic series is the Infinite series

Harmonic series of a string.

Pitched musical instruments are usually based on a harmonic oscillator such as a string or a column of air. A musical instrument is a device constructed or modified for the purpose of making Music. This article is about the harmonic oscillator in classical mechanics Both can and do oscillate at numerous frequencies simultaneously. These oscillations are called 'standing waves' as the wave in the string or air column oscillates to and fro but does not travel along it. Interaction with the surrounding air causes sound waves - travelling waves which allow us to hear the instrument. Because of the self-filtering nature of resonance, these frequencies are mostly limited to integer multiples, or harmonics, of the lowest possible frequency, and such multiples form the harmonic series. In Physics, resonance is the tendency of a system to Oscillate at maximum Amplitude at certain frequencies, known as the system's In Acoustics and Telecommunication, the harmonic of a Wave is a component Frequency of the signal that is an Integer This frequency determines the musical pitch or note that is created by vibration over the full length of the string or air column. Pitch represents the perceived Fundamental frequency of a sound Vibration refers to mechanical Oscillations about an equilibrium point. The simplest case to visualise is a vibrating string, as in the illustration. Similar arguments apply to vibrating air columns in wind instruments. In most pitched musical instruments, the fundamental note (first harmonic) is accompanied by other, higher-frequency tones that are generally called overtones. An overtone is a natural resonance or vibration frequency of a system These shorter wavelength, higher frequency waves occur with varying prominence and give each instrument its characteristic tone quality. A wave is a disturbance that propagates through Space and Time, usually with transference of Energy. The fact that a string is fixed at each end means that the longest allowed wavelength (giving the fundamental tone) is twice the length of the string. Other allowed wavelengths are 1/2, 1/3, 1/4, 1/5, 1/6, etc. times that of the fundamental. To better understand this, see node. A node is a point along a Standing wave where the wave has minimal Amplitude. Theoretically, these shorter wavelengths produce vibrations at frequencies that are 2, 3, 4, 5, 6, etc. Vibration refers to mechanical Oscillations about an equilibrium point. times the fundamental frequency. Physical characteristics of the vibrating medium and/or the resonator against which it vibrates often alter these frequencies. (See inharmonicity and stretched tuning for alterations specific to wire-stringed instruments and certain electric pianos. In music inharmonicity is the degree to which the frequencies of Overtones (known as partials partial tones or Harmonics depart from whole Stretched tuning is a detail of Musical tuning, applied to wire-stringed Musical instruments, older non-digital Electric pianos (such as the Fender ) However, those alterations are small, and except for precise, highly specialized tuning, it is reasonable to think of the frequencies of the harmonic series as integer multiples of the fundamental frequency.

The harmonic series is an arithmetic series (1×f, 2×f, 3×f, 4×f, 5×f, . In Mathematics, an arithmetic progression or arithmetic sequence is a Sequence of Numbers such that the difference of any two successive members . . ). In terms of frequency (measured in cycles per second, or hertz (Hz) where f is the fundamental frequency), the difference between consecutive harmonics is therefore constant. The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. But because our ears respond to sound logarithmically (frequency ratios, not differences, determine musical intervals), we perceive higher harmonics as "closer together" than lower ones. In Mathematics, the logarithm of a number to a given base is the power or Exponent to which the base must be raised in order to produce On the other hand, the octave series is a geometric progression (2×f, 4×f, 8×f, 16×f, . In Music, an octave ( is the the use of which is "common in most musical systems In Mathematics, a geometric progression, also known as a geometric sequence, is a Sequence of Numbers where each term after the first is found . . ), and we hear these distances as "the same" in all ranges. In terms of what we hear, each octave in the harmonic series is divided into increasingly "smaller" and more numerous intervals.

The second harmonic, twice the frequency of the fundamental, sounds an octave higher; the third harmonic, three times the frequency of the fundamental, sounds a perfect fifth above the second. In Music, an octave ( is the the use of which is "common in most musical systems The perfect fifth ( is the Musical interval between a note and the note seven Semitones above it on the musical scale The fourth harmonic vibrates at four times the frequency of the fundamental and sounds a perfect fourth above the third (two octaves above the fundamental). The perfect fourth () is a Musical interval which spans four scale degrees Double the harmonic number means double the frequency (which sounds an octave higher). The combined oscillation of a string with several of its lowest harmonics can be seen clearly in an interactive animation at Edward Zobel's "Zona Land".

An illustration of the harmonic series as musical notation. The numbers above the harmonic indicate the number of cents it deviates from tempered tuning. The cent is a logarithmic unit of measure used for musical intervals. Red notes are sharp. Blue notes are flat.

For a fundamental of C1, the first 20 harmonics are notated as shown. You can listen to A2 (110 Hz) and 15 of its partials if you have a media player capable of playing Vorbis files. Vorbis is a free and open source, lossy audio Codec project headed by the Xiph You can also hear a sweep of the first 20 harmonics of A1 (55 Hz) in Quicktime format by clicking here.

Terminology

Harmonic vs. partial

Harmonics are all partial waves ("partials") within a sound that are integer multiples of the fundamental frequency. In music, and especially among tuning professionals, the words "harmonic" and "partial" are generally interchangeable.

In digital signal processing, a "partial" frequency can also refer to a constituent frequency of a sound which might not be harmonically related to the actual harmonics contained in the original sound that is being examined. Digital signal processing ( DSP) is concerned with the representation of the signals by a sequence of numbers or symbols and the processing of these signals According to the Fourier theorem, tonally complex signals can be constructed by adding sinewaves of different frequencies, amplitudes and phases. In Mathematics, the Fourier theorem is a Theorem stating that a Periodic function f ( x) which is reasonably continuous, In the case of a discrete time, discrete frequency Fourier transform it requires N/2+1 such partial frequencies to re-construct a given digital signal of N samples. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and

Likewise, many musicians use the term overtones as a synonym for harmonics, though not all overtones are necessarily harmonic: some are inharmonic or non-harmonic. That is, an overtone may be any frequency that sounds along with the fundamental tone, regardless of its relationship to the fundamental frequency. An overtone is a natural resonance or vibration frequency of a system The fundamental tone, often referred to simply as the fundamental and abbreviated fo, is the lowest frequency in a harmonic series. The sound of a cymbal or tam-tam includes overtones that are not harmonics; that's why the gong's sound doesn't seem to have a very definite pitch compared to the same fundamental note played on a piano. Barbershop quartets use overtone colloquially in reference to the psychoacoustic phenomenon of close harmony. Psychoacoustics is the study of subjective human Perception of Sounds Alternatively it can be described as the study of the Psychological correlates Scientists take the word harmonic very seriously and define it as integer multiples of the fundamental frequency, thereby separating the concept of harmonics from overtones. That is why the first harmonic is the fundamental frequency multiplied by one, and thus are the same frequency.

Harmonic numbering

In most contexts, the fundamental vibration of an oscillating body represents its first harmonic. However, some musicians, tuners, and even developers of piano tuning software do not consider the fundamental to be a harmonic; it is just the fundamental. For them, the harmonic one octave above the fundamental (the second mode of vibration) is the first harmonic or first partial. There are logical arguments for both approaches to numbering, but in this article, the fundamental vibration is referred to as the first harmonic for simplicity and consistency with the important notion of odd and even harmonics.

Harmonics and tuning

Image showing the relationship between notes of the diatonic scale and frequency (in Hz), starting with C1 and ending with C5 ("middle C" is C4). C or Do is the first Note of the fixed-Do Solfege. In Western Music, the expression " Middle C " refers to the note Horizontal grid lines correspond to the harmonic series for C1.

If the harmonics are transposed into the span of one octave, they approximate some of the notes in what the West has adopted as the chromatic scale based on the fundamental tone. In Music transposition refers to the process of moving a collection of notes ( pitches) up or down in pitch by a constant interval. In Music, an octave ( is the the use of which is "common in most musical systems The Western chromatic scale has been modified into twelve equal semitones, which is slightly out of tune with many of the harmonics, especially the 7th, 11th, and 13th harmonics. A semitone, also called a half step or a half tone, is the smallest Musical interval commonly used in Western tonal music and it is considered the In the late 1930s, composer Paul Hindemith ranked musical intervals according to their relative dissonance based on these and similar harmonic relationships. Paul Hindemith (16 November 1895 &ndash 28 December 1963 was a German Composer, Violist, violinist teacher music theorist and conductor.

Below is a comparison between the most of the first 31 harmonics and their closest frequencies in the 12-tone equal-tempered scale. Tinted fields highlight differences greater than 5 cents, which is the "just noticeable difference" for the human ear. The cent is a logarithmic unit of measure used for musical intervals. In Psychophysics, a just noticeable difference, customarily abbreviated with lowercase letters as jnd, is the smallest difference in a specified modality of sensory (Because physical characteristics of musical instruments cause significant variations from these theoretical values, they should not be used for tuning without adjusting for those variations. )

The frequencies of the overtone series, being a range of integral multiples of the fundamental frequency, are naturally related to each other by small whole number ratios and it is these small whole number ratios that are the basis of the consonance of musical intervals, for example, a perfect fifth, say 200 and 300 Hz (cycles per second), produces a combination tone of 100 Hz (the difference between 300 Hz and 200 Hz) that is, an octave below the lower note. The cent is a logarithmic unit of measure used for musical intervals. This 100 Hz first order combination tone then interacts with both notes of the interval to produce second order combination tones of 200 (300-100) and 100 (200-100) Hz and, of course, all further nth order combination tones are all the same, being formed from various subtraction of 100, 200, and 300. When we contrast this with a dissonant interval such as a tritone (not tempered) with a frequency ratio of 7:5 we get, for example, 700-500=200 (1st order combination tone)and 500-200=300 (2nd order). The rest of the combination tones are octaves of 100 Hz so the 7:5 interval actually contains 4 notes: 100 Hz (and its octaves), 300 Hz, 500 Hz and 700 Hz. All the intervals succumb to similar analysis as been demonstrated by Paul Hindemith in his book, The Craft of Musical Composition. Paul Hindemith (16 November 1895 &ndash 28 December 1963 was a German Composer, Violist, violinist teacher music theorist and conductor.

Timbre of musical instruments

The relative amplitudes of the various harmonics primarily determine the timbre of different instruments and sounds, though formants also have a role. Amplitude is the magnitude of change in the oscillating variable with each Oscillation, within an oscillating system In Music, timbre (ˈtæm-bər' like timber, or, from Fr timbre tɛ̃bʁ is the quality of a Musical note or sound that distinguishes different A formant is a peak in the Frequency spectrum of a sound caused by acoustic Resonance. For example, the clarinet and saxophone have similar mouthpieces and reeds, and both produce sound through resonance of air inside a chamber whose mouthpiece end is considered closed. The clarinet is a Musical instrument in the Woodwind family The name derives from adding the suffix -et meaning little to the Italian word The saxophone (commonly referred to simply as sax) is a conical- bored transposing Musical instrument considered a member of the Woodwind The mouthpiece of a Woodwind instrument is that part of the instrument which is placed partly in the player's mouth A reed is a thin strip of material which vibrates to produce a sound on a Musical instrument. In Physics, resonance is the tendency of a system to Oscillate at maximum Amplitude at certain frequencies, known as the system's Because the clarinet's resonator is cylindrical, the even-numbered harmonics are suppressed, which produces a purer tone. The saxophone's resonator is conical, which allows the even-numbered harmonics to sound more strongly and thus produces a more complex tone. Of course, the differences in resonance between the wood of the clarinet and the brass of the saxophone also affect their tones. In Physics, resonance is the tendency of a system to Oscillate at maximum Amplitude at certain frequencies, known as the system's The inharmonic ringing of the instrument's metal resonator is even more prominent in the sounds of brass instruments. In music inharmonicity is the degree to which the frequencies of Overtones (known as partials partial tones or Harmonics depart from whole

Our ears tend to resolve harmonically-related frequency components into a single sensation. Rather than perceiving the individual harmonics of a musical tone, we perceive them together as a tone color or timbre, and we hear the overall pitch as the fundamental of the harmonic series being experienced. Pitch represents the perceived Fundamental frequency of a sound If we hear a sound that is made up of even just a few simultaneous tones, and if the intervals among those tones form part of a harmonic series, our brains tend to resolve this input into a sensation of the pitch of the fundamental of that series, even if the fundamental is not sounding. This phenomenon is used to advantage in music recording, especially with low bass tones that will be reproduced on small speakers.

Variations in the frequency of harmonics can also affect the perceived fundamental pitch. These variations, most clearly documented in the piano and other stringed instruments but also apparent in brass instruments, are caused by a combination of metal stiffness and the interaction of the vibrating air or string with the resonating body of the instrument. The piano is a Musical instrument played by means of a keyboard that produces sound by striking steel strings with Felt covered hammers A brass instrument is a Musical instrument whose tone is produced by vibration of the lips as the player blows into a tubular Resonator. The complex splash of strong, high overtones and metallic ringing sounds from a cymbal almost completely hide its fundamental tone. An overtone is a natural resonance or vibration frequency of a system

Click to enlarge, Table of Harmonics up to the 7th harmonic. The colored dots indicate which positions have similar overtones.

See also

• Stretched tuning
• Inharmonicity
• Piano acoustics

External links

• Interaction of reflected waves on a string is illustrated in a simplified animation
• A Web-based Multimedia Approach to the Harmonic Series
• How to tune the normally inharmonic overtones of bells to the harmonic series
• Importance of prime harmonics in music theory
• Discrete Fourier Transform explained - splitting a sound into its partial frequencies
• Calculations of harmonics from fundamental frequency