Musical acoustics or music acoustics is a multidisciplinary field that combines knowledge from

physics Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which ...

psychophysics Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they produce. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, ...

, organology (classification of the instruments),

physiology Physiology (; ) is the scientific study of functions and mechanisms in a living system. As a sub-discipline of biology, physiology focuses on how organisms, organ systems, individual organs, cells, and biomolecules carry out the chemic ...

music theory Music theory is the study of the practices and possibilities of music. ''The Oxford Companion to Music'' describes three interrelated uses of the term "music theory". The first is the " rudiments", that are needed to understand music notation (k ...

ethnomusicology Ethnomusicology is the study of music from the cultural and social aspects of the people who make it. It encompasses distinct theoretical and methodical approaches that emphasize cultural, social, material, cognitive, biological, and other dim ...

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

and instrument building, among other disciplines. As a branch of

acoustics Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acousticia ...

, it is concerned with researching and describing the physics of

music Music is generally defined as the art of arranging sound to create some combination of form, harmony, melody, rhythm or otherwise expressive content. Exact definitions of music vary considerably around the world, though it is an aspe ...

– how

sound In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by ...

s are employed to make music. Examples of areas of study are the function of musical instruments, the

human voice The human voice consists of sound made by a human being using the vocal tract, including talking, singing, laughing, crying, screaming, shouting, humming or yelling. The human voice frequency is specifically a part of human sound producti ...

(the physics of

speech Speech is a human vocal communication using language. Each language uses phonetic combinations of vowel and consonant sounds that form the sound of its words (that is, all English words sound different from all French words, even if they are th ...

and

singing Singing is the act of creating musical sounds with the voice. A person who sings is called a singer, artist or vocalist (in jazz and/or popular music). Singers perform music ( arias, recitatives, songs, etc.) that can be sung with or ...

), computer analysis of

melody A melody (from Greek μελῳδία, ''melōidía'', "singing, chanting"), also tune, voice or line, is a linear succession of musical tones that the listener perceives as a single entity. In its most literal sense, a melody is a combina ...

, and in the clinical use of music in

music therapy Music therapy, an allied health profession, "is the clinical and evidence-based use of music interventions to accomplish individualized goals within a therapeutic relationship by a credentialed professional who has completed an approved music t ...

. The pioneer of music acoustics was

Hermann von Helmholtz Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associat ...

, a German polymath of the 19th century who was an influential

physician A physician (American English), medical practitioner (Commonwealth English), medical doctor, or simply doctor, is a health professional who practices medicine, which is concerned with promoting, maintaining or restoring health through th ...

physicist A physicist is a scientist who specializes in the field of physics, which encompasses the interactions of matter and energy at all length and time scales in the physical universe. Physicists generally are interested in the root or ultimate cau ...

, physiologist, musician, mathematician and philosopher. His book '' On the Sensations of Tone as a Physiological Basis for the Theory of Music'' is a revolutionary compendium of several studies and approaches that provided a complete new perspective to

, musical performance, music psychology and the physical behaviour of musical instruments.

Methods and fields of study

*The

of musical instruments * Frequency range of music *

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph ...

*Computer

analysis Analysis ( : analyses) is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (3 ...

of musical structure *

Synthesis Synthesis or synthesize may refer to: Science Chemistry and biochemistry * Chemical synthesis, the execution of chemical reactions to form a more complex molecule from chemical precursors **Organic synthesis, the chemical synthesis of organ ...

of musical sounds * Music cognition, based on physics (also known as

psychoacoustics Psychoacoustics is the branch of psychophysics involving the scientific study of sound perception and audiology—how humans perceive various sounds. More specifically, it is the branch of science studying the psychological responses associated wi ...

)

Physical aspects

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

human Humans (''Homo sapiens'') are the most abundant and widespread species of primate, characterized by bipedalism and exceptional cognitive skills due to a large and complex brain. This has enabled the development of advanced tools, cultu ...

s the hearing apparatus (composed of the ears and

brain A brain is an organ (biology), organ that serves as the center of the nervous system in all vertebrate and most invertebrate animals. It is located in the head, usually close to the sensory organs for senses such as Visual perception, vision. I ...

) 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

harmonic A harmonic is a wave with a frequency that is a positive integer multiple of the ''fundamental frequency'', the frequency of the original periodic signal, such as a sinusoidal wave. The original signal is also called the ''1st harmonic'', t ...

partial Partial may refer to: Mathematics *Partial derivative, derivative with respect to one of several variables of a function, with the other variables held constant ** ∂, a symbol that can denote a partial derivative, sometimes pronounced "partial d ...

s, or overtones. The sounds have

frequency 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.

Subjective aspects

Variations in air

pressure Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...

against the ear drum, and the subsequent physical and neurological processing and interpretation, give rise to the subjective experience called ''

''. Most sound that people recognize as

al 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 physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (re ...

. In a very simple case, the sound of a

sine wave A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...

, 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 Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...

of the tone, which is measured in oscillations per second, called

hertz The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...

. 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

Harmonics, partials, and overtones

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

s. Together they form the harmonic series. Overtones that are perfect integer multiples of the fundamental are called

s. 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

When a periodic wave is composed of a fundamental and only odd harmonics (f, 3f, 5f, 7f, ...), 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 (0f, 2f, 4f, 6f, ...), it is asymmetrical; the top half is not a mirror image of 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 In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathematicians, and many other ...

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).

Harmony

If two notes are simultaneously played, with frequency

ratio In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...

s 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 In articulatory phonetics, a consonant is a speech sound that is articulated with complete or partial closure of the vocal tract. Examples are and pronounced with the lips; and pronounced with the front of the tongue; and pronounced w ...

. For instance, a note vibrating at 200 Hz and a note vibrating at 300 Hz (a

perfect fifth In music theory, a perfect fifth is the musical interval corresponding to a pair of pitches with a frequency ratio of 3:2, or very nearly so. In classical music from Western culture, a fifth is the interval from the first to the last of five ...

, or 3/2 ratio, above 200 Hz) add together to make a wave that repeats at 100 Hz: every 1/100 of a second, the 300 Hz wave repeats three times and the 200 Hz wave repeats twice. Note that the total wave repeats at 100 Hz, but there is no actual 100 Hz sinusoidal component. Additionally, the two notes have many of the same partials. For instance, a note with a fundamental frequency of 200 Hz has harmonics at: :(200,) 400, 600, 800, 1000, 1200, … A note with fundamental frequency of 300 Hz has harmonics at: :(300,) 600, 900, 1200, 1500, … The two notes share harmonics at 600 and 1200 Hz, 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, , 200 Hz - 300 Hz, = 100 Hz. As another example, a combination of 3425 Hz and 3426 Hz would beat once per second (, 3425 Hz - 3426 Hz, = 1 Hz). This follows from

modulation In electronics and telecommunications, modulation is the process of varying one or more properties of a periodic waveform, called the '' carrier signal'', with a separate signal called the ''modulation signal'' that typically contains informat ...

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 Hermann Ludwig Ferdinand von Helmholtz (31 August 1821 – 8 September 1894) was a German physicist and physician who made significant contributions in several scientific fields, particularly hydrodynamic stability. The Helmholtz Associatio ...

proposed that maximum dissonance would arise between two pure tones when the beat rate is roughly 35 Hz

Scales

The material of a musical composition is usually taken from a collection of pitches known as a Musical scale, scale. Because most people cannot adequately determine

absolute Absolute may refer to: Companies * Absolute Entertainment, a video game publisher * Absolute Radio, (formerly Virgin Radio), independent national radio station in the UK * Absolute Software Corporation, specializes in security and data risk manag ...

frequencies, the identity of a scale lies in the ratios of frequencies between its tones (known as intervals). The

diatonic scale In music theory, a diatonic scale is any heptatonic scale that includes five whole steps (whole tones) and two half steps (semitones) in each octave, in which the two half steps are separated from each other by either two or three whole st ...

appears in writing throughout history, consisting of seven tones in each

octave In music, an octave ( la, octavus: eighth) or perfect octave (sometimes called the diapason) is the interval between one musical pitch and another with double its frequency. The octave relationship is a natural phenomenon that has been refer ...

. In

just intonation In music, just intonation or pure intonation is the tuning of musical intervals as whole number ratios (such as 3:2 or 4:3) of frequencies. An interval tuned in this way is said to be pure, and is called a just interval. Just intervals (and ...

the diatonic scale may be easily constructed using the three simplest intervals within the octave, the

(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 The major scale (or Ionian mode) is one of the most commonly used musical scales, especially in Western music. It is one of the diatonic scales. Like many musical scales, it is made up of seven notes: the eighth duplicates the first at doub ...

and the fixed frequency of the first note of the scale. 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 An equal temperament is a musical temperament or tuning system, which approximates just intervals by dividing an octave (or other interval) into equal steps. This means the ratio of the frequencies of any adjacent pair of notes is the same, ...

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.

References

External links

Music acoustics - sound files, animations and illustrations - University of New South WalesAcoustics collection - descriptions, photos, and video clips of the apparatus for research in musical acoustics by Prof.
Dayton Miller
The Technical Committee on Musical Acoustics (TCMU) of the Acoustical Society of America (ASA)The Musical Acoustics Research Library (MARL)Acoustics Group/Acoustics and Music Technology courses - University of EdinburghAcoustics Research Group - Open UniversitySavart Journal - The open access online journal of science and technology of stringed musical instrumentsInterference and Consonance
fro
PhysclipsCurso de Acústica Musical
(Spanish) {{DEFAULTSORT:Musical Acoustics Acoustics Musical terminology