Sound spectrography of infrasound recording 30301A spectrogram of a violin playing a note and then a perfect fifth above it. The shared partials are highlighted by the white dashes.
Whenever two different pitches are played at the same time, their sound waves interact with each other – the highs and lows in the air pressure reinforce each other to produce a different sound wave. Any repeating sound wave that is not a sine wave can be modeled by many different sine waves of the appropriate frequencies and amplitudes (a frequency spectrum). In humans the hearing apparatus (composed of the ears and brain) can usually isolate these tones and hear them distinctly. When two or more tones are played at once, a variation of air pressure at the ear "contains" the pitches of each, and the ear and/or brain isolate and decode them into distinct tones.
When the original sound sources are perfectly periodic, the note consists of several related sine waves (which mathematically add to each other) called the fundamental and the harmonics, partials, or overtones. The sounds have harmonicfrequency spectra. The lowest frequency present is the fundamental, and is the frequency at which the entire wave vibrates. The overtones vibrate faster than the fundamental, but must vibrate at integer multiples of the fundamental frequency for the total wave to be exactly the same each cycle. Real instruments are close to periodic, but the frequencies of the overtones are slightly imperfect, so the shape of the wave changes slightly over time.[citation needed]
Subjective aspects
Variations in airpressure against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the subjective experience called sound. Most sound that people recognize as musical is dominated by periodic or regular vibrations rather than non-periodic ones; that is, musical sounds typically have a definite pitch. The transmission of these variations through air is via a sound wave. In a very simple case, the sound of a sine wave, which is considered the most basic model of a sound waveform, causes the air pressure to increase and decrease in a regular fashion, and is heard as a very pure tone. Pure tones can be produced by tuning forks or whistling. The rate at which the air pressure oscillates is the frequency of the tone, which is measured in oscillations per second, called hertz. Frequency is the primary determinant of the perceived pitch. Frequency of musical instruments can change with altitude due to changes in air pressure.
Pitch ranges of musical instruments
*This chart only displays down to C0, though some pipe organs, such as the Boardwalk Hall Auditorium Organ, extend down to C−1 (one octave below C0). Also, the fundamental frequency of the subcontrabass tuba is B♭−1.
The fundamental is the frequency at which the entire wave vibrates. Overtones are other sinusoidal components present at frequencies above the fundamental. All of the frequency components that make up the total waveform, including the fundamental and the overtones, are called partials. Together they form the harmonic series.
Overtones that are perfect integer multiples of the fundamental are called harmonics. When an overtone is near to being harmonic, but not exact, it is sometimes called a harmonic partial, although they are often referred to simply as harmonics. Sometimes overtones are created that are not anywhere near a harmonic, and are just called partials or inharmonic overtones.
The fundamental frequency is considered the first harmonic and the first partial. The numbering of the partials and harmonics is then usually the same; the second partial is the second harmonic, etc. But if there are inharmonic partials, the numbering no longer coincides. Overtones are numbered as they appear above the fundamental. So strictly speaking, the first overtone is the second partial (and usually the second harmonic). As this can result in confusion, only harmonics are usually referred to by their numbers, and overtones and partials are described by their relationships to those harmonics.
Harmonics and non-linearities
A symmetric and asymmetric waveform. The red (upper) wave contains only the fundamental and odd harmonics; the green (lower) wave contains the fundamental and even harmonics.
When a periodic wave is composed of a fundamental and only odd harmonics (f, 3 f, 5 f, 7 f, ...), the summed wave is half-wave symmetric; it can be inverted and phase shifted and be exactly the same. If the wave has any even harmonics (2 f, 4 f, 5 f, ...), it is asymmetrical; the top half of the plotted wave form does not mirror image the bottom.
Conversely, a system that changes the shape of the wave (beyond simple scaling or shifting) creates additional harmonics (harmonic distortion). This is called a non-linear system. If it affects the wave symmetrically, the harmonics produced are all odd. If it affects the harmonics asymmetrically, at least one even harmonic is produced (and probably also odd harmonics).
If two notes are simultaneously played, with frequency ratios that are simple fractions (e.g. 2/1, 3/2 or 5/4), the composite wave is still periodic, with a short period—and the combination sounds consonant. For instance, a note vibrating at 200Hz and a note vibrating at 300Hz (a perfect fifth, or 3/2 ratio, above 200Hz) add together to make a wave that repeats at 100Hz: every 1/100 of a second, the 300Hz wave repeats three times and the 200Hz wave repeats twice. Note that the total wave repeats at 100Hz, but there is no actual 100Hz sinusoidal component.
Additionally, the two notes have many of the same partials. For instance, a note with a fundamental frequency of 200Hz has harmonics at::(200,) 400, 600, 800, 1000, 1200, ...
A note with fundamental frequency of 300Hz has harmonics at::(300,) 600, 900, 1200, 1500, ... The two notes share harmonics at 600 and 1200Hz, and more coincide further up the series.
The combination of composite waves with short fundamental frequencies and shared or closely related partials is what causes the sensation of harmony. When two frequencies are near to a simple fraction, but not exact, the composite wave cycles slowly enough to hear the cancellation of the waves as a steady pulsing instead of a tone. This is called beating, and is considered unpleasant, or dissonant.
The frequency of beating is calculated as the difference between the frequencies of the two notes. For the example above, |200Hz - 300Hz| = 100Hz. As another example, a combination of 3425Hz and 3426Hz would beat once per second (|3425Hz - 3426Hz| = 1Hz). This follows from modulation theory.
The difference between consonance and dissonance is not clearly defined, but the higher the beat frequency, the more likely the interval is dissonant. Helmholtz proposed that maximum dissonance would arise between two pure tones when the beat rate is roughly 35Hz.
The material of a musical composition is usually taken from a collection of pitches known as a scale. Because most people cannot adequately determine absolute frequencies, the identity of a scale lies in the ratios of frequencies between its tones (known as intervals).
The diatonic scale appears in writing throughout history, consisting of seven tones in each octave. In just intonation the diatonic scale may be easily constructed using the three simplest intervals within the octave, the perfect fifth (3/2), perfect fourth (4/3), and the major third (5/4). As forms of the fifth and third are naturally present in the overtone series of harmonic resonators, this is a very simple process.
The following table shows the ratios between the frequencies of all the notes of the just major scale and the fixed frequency of the first note of the scale.
C
D
E
F
G
A
B
C
1
9/8
5/4
4/3
3/2
5/3
15/8
2
There are other scales available through just intonation, for example the minor scale. Scales that do not adhere to just intonation, and instead have their intervals adjusted to meet other needs are called temperaments, of which equal temperament is the most used. Temperaments, though they obscure the acoustical purity of just intervals, often have desirable properties, such as a closed circle of fifths.
Additive synthesis is a sound synthesis technique that creates timbre by adding sine waves together.
The fundamental frequency, often referred to simply as the fundamental, is defined as the lowest frequency of a periodic waveform. In music, the fundamental is the musical pitch of a note that is perceived as the lowest partial present. In terms of a superposition of sinusoids, the fundamental frequency is the lowest frequency sinusoidal in the sum of harmonically related frequencies, or the frequency of the difference between adjacent frequencies. In some contexts, the fundamental is usually abbreviated as f0, indicating the lowest frequency counting from zero. In other contexts, it is more common to abbreviate it as f1, the first harmonic.
A harmonic series is the sequence of harmonics, musical tones, or pure tones whose frequency is an integer multiple of a fundamental frequency.
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.
An overtone is any resonant frequency above the fundamental frequency of a sound. 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.
In music, timbre, also known as tone color or tone quality, is the perceived sound quality of a musical note, sound or tone. Timbre distinguishes different types of sound production, such as choir voices and musical instruments. It also enables listeners to distinguish different instruments in the same category.
Pitch is a perceptual property that allows sounds to be ordered on a frequency-related scale, or more commonly, pitch is the quality that makes it possible to judge sounds as "higher" and "lower" in the sense associated with musical melodies. Pitch is a major auditory attribute of musical tones, along with duration, loudness, and timbre.
In music, inharmonicity is the degree to which the frequencies of overtones depart from whole multiples of the fundamental frequency.
A combination tone is a psychoacoustic phenomenon of an additional tone or tones that are artificially perceived when two real tones are sounded at the same time. Their discovery is credited to the violinist Giuseppe Tartini and so they are also called Tartini tones.
A pseudo-octave, pseudooctave, or paradoxical octave in music is an interval whose frequency ratio is not 2:1, that of the octave, but is perceived or treated as equivalent to this ratio, and whose pitches are considered equivalent to each other as with octave equivalency.
Piano acoustics is the set of physical properties of the piano that affect its sound. It is an area of study within musical acoustics.
Piano tuning is the act of adjusting the tension of the strings of an acoustic piano so that the musical intervals between strings are in tune. The meaning of the term 'in tune', in the context of piano tuning, is not simply a particular fixed set of pitches. Fine piano tuning requires an assessment of the vibration interaction among notes, which is different for every piano, thus in practice requiring slightly different pitches from any theoretical standard. Pianos are usually tuned to a modified version of the system called equal temperament.
A multiphonic is an extended technique on a monophonic musical instrument in which several notes are produced at once. This includes wind, reed, and brass instruments, as well as the human voice. Multiphonic-like sounds on string instruments, both bowed and hammered, have also been called multiphonics, for lack of better terminology and scarcity of research.
In music, consonance and dissonance are categorizations of simultaneous or successive sounds. Within the Western tradition, some listeners associate consonance with sweetness, pleasantness, and acceptability, and dissonance with harshness, unpleasantness, or unacceptability, although there is broad acknowledgement that this depends also on familiarity and musical expertise. The terms form a structural dichotomy in which they define each other by mutual exclusion: a consonance is what is not dissonant, and a dissonance is what is not consonant. However, a finer consideration shows that the distinction forms a gradation, from the most consonant to the most dissonant. In casual discourse, as German composer and music theorist Paul Hindemith stressed, "The two concepts have never been completely explained, and for a thousand years the definitions have varied". The term sonance has been proposed to encompass or refer indistinctly to the terms consonance and dissonance.
In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, perceived as a periodic variation in volume whose rate is the difference of the two frequencies.
Stretched tuning is a detail of musical tuning, applied to wire-stringed musical instruments, older, non-digital electric pianos, and some sample-based synthesizers based on these instruments, to accommodate the natural inharmonicity of their vibrating elements. In stretched tuning, two notes an octave apart, whose fundamental frequencies theoretically have an exact 2:1 ratio, are tuned slightly farther apart. "For a stretched tuning the octave is greater than a factor of 2; for a compressed tuning the octave is smaller than a factor of 2."
Acoustic resonance is a phenomenon in which an acoustic system amplifies sound waves whose frequency matches one of its own natural frequencies of vibration.
In music, the undertone series or subharmonic series is a sequence of notes that results from inverting the intervals of the overtone series. While overtones naturally occur with the physical production of music on instruments, undertones must be produced in unusual ways. While the overtone series is based upon arithmetic multiplication of frequencies, resulting in a harmonic series, the undertone series is based on arithmetic division.
Violin acoustics is an area of study within musical acoustics concerned with how the sound of a violin is created as the result of interactions between its many parts. These acoustic qualities are similar to those of other members of the violin family, such as the viola.
Playing a string harmonic is a string instrument technique that uses the nodes of natural harmonics of a musical string to isolate overtones. Playing string harmonics produces high pitched tones, often compared in timbre to a whistle or flute. Overtones can be isolated "by lightly touching the string with the finger instead of pressing it down" against the fingerboard. For some instruments this is a fundamental technique, such as the Chinese guqin, where it is known as fan yin, and the Vietnamese đàn bầu.
↑ Watson, Lanham, Alan H. D., ML (2009). The Biology of Musical Performance and Performance-Related Injury. Cambridge: Scarecrow Press. ISBN9780810863590.{{cite book}}: CS1 maint: multiple names: authors list (link)
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