Fundamental frequency in the context of Zero-based numbering

In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a harmonic series.

The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz.

Acoustic phonetics is a subfield of phonetics, which deals with acoustic aspects of speech sounds. Acoustic phonetics investigates features of waveforms as they pertain to the time domain (e.g. duration, amplitude, fundamental frequency), frequency domain (e.g. frequency spectrum), or combined spectrotemporal domains. Acoustic phonetics is also concerned with how these properties relate to other branches of phonetics (e.g. articulatory or auditory phonetics), as well as abstract linguistic concepts such as phonemes, phrases, or utterances.

The study of acoustic phonetics was greatly enhanced in the late 19th century by the invention of the Edison phonograph. The phonograph allowed the speech signal to be recorded and then later processed and analyzed. By replaying the same speech signal from the phonograph several times, filtering it each time with a different band-pass filter, a spectrogram of the speech utterance could be built up. A series of papers by Ludimar Hermann published in Pflügers Archiv in the last two decades of the 19th century investigated the spectral properties of vowels and consonants using the Edison phonograph, and it was in these papers that the term formant was first introduced. Hermann also played back vowel recordings made with the Edison phonograph at different speeds to distinguish between Willis' and Wheatstone's theories of vowel production.

In music theory, a scale is "any consecutive series of notes that form a progression between one note and its octave", typically by order of pitch or fundamental frequency.

The word scale originates from the Latin scala, which literally means "ladder". Therefore, any scale is distinguishable by its "step-pattern", or how its intervals interact with each other.

A vibration in a string is a wave. Initial disturbance (such as plucking or striking) causes a vibrating string to produce a sound with constant frequency, i.e., constant pitch. The nature of this frequency selection process occurs for a stretched string with a finite length, which means that only particular frequencies can survive on this string. If the length, tension, and linear density (e.g., the thickness or material choices) of the string are correctly specified, the sound produced is a musical tone. Vibrating strings are the basis of string instruments such as guitars, cellos, and pianos. For a homogeneous string, the motion is given by the wave equation.

In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input, and outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex valued function of frequency. The term Fourier transform refers to both the mathematical operation and to this complex-valued function. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches.

Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statistics as well as in the study of physical phenomena exhibiting normal distribution (e.g., diffusion). The Fourier transform of a Gaussian function is another Gaussian function. Joseph Fourier introduced sine and cosine transforms (which correspond to the imaginary and real components of the modern Fourier transform) in his study of heat transfer, where Gaussian functions appear as solutions of the heat equation.

A two-dimensional elastic membrane under tension can support transverse vibrations. The properties of an idealized drumhead can be modeled by the vibrations of a circular membrane of uniform thickness, attached to a rigid frame. Based on the applied boundary condition, at certain vibration frequencies, its natural frequencies, the surface moves in a characteristic pattern of standing waves. This is called a normal mode. A membrane has an infinite number of these normal modes, starting with a lowest frequency one called the fundamental frequency.

There exist infinitely many ways in which a membrane can vibrate, each depending on the shape of the membrane at some initial time, and the transverse velocity of each point on the membrane at that time. The vibrations of the membrane are given by the solutions of the two-dimensional wave equation with Dirichlet boundary conditions which represent the constraint of the frame. It can be shown that any arbitrarily complex vibration of the membrane can be decomposed into a possibly infinite series of the membrane's normal modes. This is analogous to the decomposition of a time signal into a Fourier series.

An overtone is any resonant frequency above the fundamental frequency of a sound (or of any oscillation). An overtone may or may not be a harmonic. In other words, overtones are all pitches higher than the lowest pitch within an individual sound; the fundamental is the lowest pitch. While the fundamental is usually heard most prominently, overtones are actually present in any pitch except a true sine wave. The relative volume or amplitude of various overtone partials is one of the key identifying features of timbre, or the individual characteristic of a sound.

Using the model of Fourier analysis, the fundamental and the overtones together are called partials. Harmonics, or more precisely, harmonic partials, are partials whose frequencies are numerical integer multiples of the fundamental (including the fundamental, which is 1 times itself). These overlapping terms are variously used when discussing the acoustic behavior of musical instruments. (See etymology below.) The model of Fourier analysis provides for the inclusion of inharmonic partials, which are partials whose frequencies are not whole-number ratios of the fundamental (such as 1.1 or 2.14179).

The harmonic series (also overtone series) is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency.

Pitched musical instruments are often based on an acoustic resonator such as a string or a column of air, which oscillates at numerous modes simultaneously. As waves travel in both directions along the string or air column, they reinforce and cancel one another to form standing waves. Interaction with the surrounding air produces audible sound waves, which travel away from the instrument. These frequencies are generally integer multiples, or harmonics, of the fundamental and such multiples form the harmonic series.

Sibilants (from Latin: sibilans 'hissing') are fricative and affricate consonants of higher amplitude and pitch, made by directing a stream of air with the tongue towards the teeth. Examples of sibilants are the consonants at the beginning of the English words sip, zip, ship, and genre. The symbols in the International Phonetic Alphabet used to denote the sibilant sounds in these words are, respectively, [s] [z] [ʃ] [ʒ]. Sibilants have a characteristically intense sound, which accounts for their paralinguistic use in getting one's attention (e.g. calling someone using "psst!" or quieting someone using "shhhh!").

The strike tone, strike note, or tap note, of a percussion instrument (e.g. bell, chime or gong) when struck, is the dominant note perceived immediately by the human ear. It is also known as the prime or fundamental note. However, an analysis of the bell's frequency spectrum reveals that the fundamental only exists weakly and its dominance is a human perception of a note built up by the complex series of harmonics that are generated. The correct and accurate harmonic tuning is therefore important in creating a good strike tone.

The blue whale (Balaenoptera musculus) is a marine mammal and a baleen whale. Reaching a maximum confirmed length of 29.9–30.5 m (98–100 ft) and weighing up to 190–200 t (190–200 long tons; 210–220 short tons), it is the largest animal known to have ever existed. The blue whale's long and slender body can be of various shades of greyish-blue on its upper surface and somewhat lighter underneath. Four subspecies are recognized: B. m. musculus in the North Atlantic and North Pacific, B. m. intermedia in the Southern Ocean, B. m. brevicauda (the pygmy blue whale) in the Indian Ocean and South Pacific Ocean, and B. m. indica in the Northern Indian Ocean. There is a population in the waters off Chile that may constitute a fifth subspecies.

In general, blue whale populations migrate between their summer feeding areas near the poles and their winter breeding grounds near the tropics. There is also evidence of year-round residencies, and partial or age- and sex-based migration. Blue whales are filter feeders; their diet consists almost exclusively of krill. They are generally solitary or gather in small groups, and have no well-defined social structure other than mother–calf bonds. Blue whales vocalize, with a fundamental frequency ranging from 8 to 25 Hz; their vocalizations may vary by region, season, behavior, and time of day. Orcas are their only natural predators.

In music, just intonation or pure intonation is a tuning system in which the space between notes' frequencies (called intervals) is a whole number ratio. Intervals spaced in this way are said to be pure, and are called just intervals. Just intervals (and chords created by combining them) consist of tones from a single harmonic series of an implied fundamental. For example, in the diagram, if the notes G₃ and C₄ (labelled 3 and 4) are tuned as members of the harmonic series of the lowest C, their frequencies will be 3 and 4 times the fundamental frequency. The interval ratio between C₄ and G₃ is therefore 4:3, a just fourth.

In Western musical practice, bowed instruments such as violins, violas, cellos, and double basses are tuned using pure fifths or fourths. In contrast, keyboard instruments are rarely tuned using only pure intervals—the desire for different keys to have identical intervals in Western music makes this impractical. Some instruments of fixed pitch, such as electric pianos, are commonly tuned using equal temperament, in which all intervals other than octaves consist of irrational-number frequency ratios. Acoustic pianos are usually tuned with the octaves slightly widened, and thus with no pure intervals at all.

The pitch of a brass instrument corresponds to the lowest playable resonance frequency of the open instrument. The combined resonances resemble a harmonic series. The fundamental frequency of the harmonic series can be varied by adjusting the length of the tubing using the instrument's valve, slide, key or crook system, while the player's embouchure, lip tension and air flow serve to select a specific harmonic from the available series for playing. The fundamental is essentially missing from the resonances and is impractical to play on most brass instruments, but the overtones account for most pitches.

The following table provides the pitch of the second harmonic (the lowest playable resonance on most brass instruments, an octave above the fundamental frequency) and length for some common brass instruments in descending order of pitch. This pitch is notated transpositionally as middle C for many of these brass instruments.

A conch (US: /kɑːŋk, kɑːntʃ/ KONK, KONCH, UK: /kɒntʃ/ KONCH) or conque, also called a "seashell instrument" or "shell natural instruments", is a wind instrument that is made from a conch, the shell of several different kinds of sea snails. Their natural conical bore is used to produce a musical tone. Conch shell natural instruments have been played in many Pacific island countries, as well as South America and South Asia.

The shells of large marine gastropods are blown into as if they were natural instruments, as in blowing instrument. A completely unmodified conch may be used, or a mouth hole may be created. Wooden, bamboo, metal, or any kind of material used to make mouthpieces may be inserted into the end of the shell. Embouchure is used to produce notes from the harmonic series. A tone hole may be added to change the fundamental frequency but globally this is extremely rare.