Musical acoustics

(′myü·zə·kəl ə′kü·stiks)

(acoustics) That part of acoustics which is relevant to the composition, performance, and appreciation of music, including the physical characteristics of sounds that may be heard as music, laws governing the action, design, and construction of musical instruments, and the effects of musical sounds upon listeners.

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Sci-Tech Encyclopedia: Musical acoustics

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The branch of acoustics that deals with the generation of sound by musical instruments, the transmission of sound to the listener, and the perception of musical sound. A main research activity in musical acoustics is the study of the way in which musical instruments vibrate and produce sound. The most common way of classifying musical instruments is according to the nature of the primary vibrator, into string instruments, wind instruments, and percussion instruments. The vibrations of a plucked string, a struck membrane, or a blown pipe can be described in terms of normal modes of vibration. Determining the normal modes of a complex vibrator is often termed modal analysis. Much of the progress in understanding how musical instruments generate sound is due to new methods of modal analysis, such as holographic interferometry and experimental modal testing. See also Cavity resonator; Interferometry; Mode of vibration; Vibration.

In the case of most percussion, plucked string, and struck string instruments, the player delivers energy to the primary vibrator (string, membrane, bar, or plate) and thereafter has little control over the way it vibrates. In the case of wind and bowed string instruments, however, the continuing flow of energy is controlled by feedback from the vibrating system. In brass and reed woodwinds, pressure feedback opens or closes the input valve. In flutes or flue organ pipes, however, the input valve is flow-controlled. In bowed string instruments, pulses on the string control the stick-slip action of the bow on the string.

Four attributes are frequently used to describe musical sound: loudness, pitch, timbre, and duration. Each of the subjective qualities depends on one or more physical parameters that can be measured. Loudness, for example, depends mainly on sound pressure but also on the spectrum of the partials and the physical duration. Pitch depends mainly on frequency, but also shows lesser dependence on sound pressure and envelope. Timbre includes all the attributes by which sounds with the same pitch and loudness are distinguished. Relating the subjective qualities of sound to the physical parameters is a central problem in psychoacoustics, and musical acousticians are concerned with this same problem as it applies to musical sound. See also Psychoacoustics.

Sound pressure level is measured with a sound level meter and is generally expressed on a logarithmic scale of decibels (dB) using an appropriate reference level and weighting network. From measurements of the sound pressure level at different frequencies, it is possible to calculate a subjective loudness, expressed in sones, which describes the sensation of loudness heard by an average listener. Musicians prefer to use dynamic markings ranging from ppp (very soft) to fff (very loud). See also Decibel; Loudness; Sound; Sound pressure.

Pitch is defined as that attribute of auditory sensation in terms of which sounds may be ordered on a scale extending from low to high. Pitch is generally related to a musical scale where the octave, rather than the critical bandwidth, is the “natural” pitch interval. See also Pitch; Scale (music).

Timbre is defined as that attribute of auditory sensation in terms of which a listener can judge two sounds similarly presented and having the same loudness and pitch as dissimilar. Timbre depends primarily on the spectrum of the sound, but it also depends upon the waveform, the sound pressure, the frequency location of the spectrum, and the temporal characteristics of the sound. It has been found impossible to construct a single subjective scale of timbre (such as the sone scale of loudness); multidimensional scales have been constructed. The term “tone color” is often used to refer to that part of timbre that is attributable to the steady-state part of the tone, but the time envelope (and especially the attack) has been found to be very important in determining timbre as well.

Another subject relating to the perception of music is combination tones. When two tones that are close together in frequency are sounded at the same time, beats generally are heard, at a rate that is equal to their frequency difference. When the frequency difference Δf exceeds 15 Hz or so, the beat sensation disappears, and a roughness appears. As Δf increases still further, a point is reached at which the “fused” tone at the average frequency gives way to two tones, still with roughness. The respective resonance regions on the basilar membrane are now separated sufficiently to give two distinct pitches, but the excitations overlap to give a sense of roughness. When the separation Δf exceeds the width of the critical band, the roughness disappears, and the two tones begin to blend. See also Ear (vertebrate).

Pythagoras of ancient Greece is considered to have discovered that the tones produced by a string vibrating in two parts with simple ratios such as 2:1, 3:2, or 4:3 sound harmonious. These ratios define the so-called perfect intervals of music, which are considered to have the greatest consonance. Other consonant intervals in music are the major sixth (f₂/f₁ = 5/3), the major third (f₂/f₁ = 5/4), the major sixth (f₂/f₁ = 8/5), and the minor third (f₂/f₁ = 6/5). Why are some intervals more consonant than others? H. Helmholtz concluded that dissonance (the opposite of consonance) is greatest when partials of the two tones produce 30 to 40 beats per second (which are not heard as beats but produce roughness). The more the partials of one tone coincide in frequency with the partials of the other, the less chance of roughness. This explains why simple frequency ratios define the most consonant intervals. More recent research has concluded that consonance is related to the critical band. If the frequency difference between two pure tones is greater than a critical band, they sound consonant; if it is less than a critical band, they sound dissonant. The maximum dissonance occurs when Δf is approximately 1/4 of a critical band, which agrees reasonably well with Helmholtz's criterion for tones around 500 Hz.

Wikipedia: Musical acoustics

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Musical acoustics or music acoustics is the branch of acoustics concerned with researching and describing the physics of music — how sounds employed as music work. Examples of areas of study are the function of musical instruments, the human voice (the physics of speech and singing), computer analysis of melody.

1 Methods and fields of study
2 Physical aspects
3 Subjective aspects
4 Frequency Range of Music
5 Harmonics, partials, and overtones
6 Harmonics and non-linearities
7 Harmony
8 Scales
9 Further reading
10 See also
11 External links

Methods and fields of study

Frequency Range of Music
Frequency analysis
Computer analysis of musical structure
Synthesis of musical sounds
Music cognition, based on physics (also known as psychoacoustics)

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. As a result, any given sound wave which is more complicated than a sine wave can be modelled 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 harmonic 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 in order 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 air pressure 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 (called a definite pitch), and we refer to the transmission mechanism as a "sound wave". In a very simple case, the sound of a sine wave, which is considered to be 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 varies governs is the frequency of the tone, which is measured in oscillations per second, called hertz. Frequency is a primary determinate of the perceived pitch. Frequency can change with Altitude due to changes in air pressure. This is called the Adiabatic Lapse Rate

Frequency Range of Music

*This chart only displays to a C0, though the Octocontrabass clarinet extends down the B♭ below that C. Also some pipe organs, such as the Boardwalk Hall Auditorium Organ, extends down to C-1 (one octave below C0).

A spectrogram of violin playing. The bright lines along the bottom are the fundamentals of each note, and the other bright lines are (nearly) harmonic overtones; collectively, they are spectra.

Harmonics, partials, and overtones

Scale of harmonics

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 which 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 half-wave symmetric and asymmetric waveform. The red contains only the fundamental and odd harmonics, the green contains the fundamental, odd, and even harmonics.

200 and 300 Hz waves and their sum, showing the periods of each.

A spectrogram of a violin playing a note and then a perfect fifth above it. The shared partials are highlighted by the white dashes.

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 will be asymmetrical; the top half will not be a mirror image of the bottom.

The opposite is also true. A system which 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 will only be odd, if asymmetrically, at least one even harmonic will be produced (and probably also odd).

Harmony

Main article: Harmony

If two notes are simultaneously played, with frequency ratios that are simple fractions (e.g. 2/1, 3/2 or 5/4), then the composite wave will still be periodic with a short period, and the combination will sound consonant. For instance, a note vibrating at 200 Hz and a note vibrating at 300 Hz (a perfect fifth, or 3/2 ratio, above 200 Hz) will add together to make a wave that repeats at 100 Hz: every 1/100 of a second, the 300 Hz wave will repeat thrice and the 200 Hz wave will repeat twice. Note that the total wave repeats at 100 Hz, but there is not actually a 100 Hz sinusoidal component present.

Additionally, the two notes will have many of the same partials. For instance, a note with a fundamental frequency of 200 Hz will have harmonics at:

(200,) 400, 600, 800, 1000, 1200, …

A note with fundamental frequency of 300 Hz will have harmonics at:

(300,) 600, 900, 1200, 1500, …

The two notes have the harmonics 600 and 1200 in common, and more will 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 to be 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 theory.

The difference between consonance and dissonance is not clearly defined, but the higher the beat frequency, the more likely the interval to be dissonant. Helmholtz proposed that maximum dissonance would arise between two pure tones when the beat rate is roughly 35 Hz. [1]

Scales

Main article: Musical scale

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).

Main article: Just intonation

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 which do not adhere to just intonation, and instead have their intervals adjusted to meet other needs are known as temperaments, of which equal temperament is the most used. Temperaments, though they obscure the acoustical purity of just intervals often have other desirable properties, such as a closed circle of fifths.

External links

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	beat tone (engineering acoustics)
	mean-tone scale (acoustics)
	Pythagorean scale (acoustics)

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