The physics of music.

Physics and Music - one would think that they have nothing in common! Well the truth is, they are so profoundly related and entertwined that many terms used are common.

The most important is harmonics, commomly referred to in music as "overtones".

There is a mathematical relationship between musical notes, for example octaves are multiples of two in frequency, e.g. A=440 Hz (cycles per second), the upper A is 440 x 2 = 880 Hz.

Understanding how we hear is quite a complex subject. Psychology, physiology and acoustics can explain this. The ability of the ear to do this allows us to perceive the pitch of sounds by detection of the frequency of the sound wave, the loudness of sound wave (amplitude) and the timbre of the sound determines the quality. The ear consists of three basic parts - the outer, the middle and the inner ear. The outer ear focusses sound to the middle ear which transforms the energy of the sound waves to the bone structure of the middle ear. The inner ear then transforms the energy of the compression waves within the fluid within the inner ear fluid into nerve impulses which are then transmitted to the brain.
The ear can detect sounds caused by vibrations from 20 cycles per second (or Hertz, Hz) to over almost 20,000 Hz. Sound waves can be measured by their frequency, which is the number of compression pulses that go past a fixed point for one second.
The frequency of audible sound is measured in Hertz, (cycles per second). Wavelength is the physical distance between compression pulses, like waves on the sea. The distance from crest to crest is the wavelength . The shorter the wavelength is the higher the frequency. Below 20 Hz they are more felt than heard, and then at lower frequencies they are heard as separate pulses. Above 20,000 Hz, they are inaudible but can still affect us. Animals can hear outside this range, e.g. whales can hear very low frequencies, while dolphins and bats can easily hear well over 20,000 Hz and use this for sonar detection or echo location of objects.
The source of the sound pulses pushes against the air around it: these compression pulses cause the air to “spring out” past the pressure zone where it began, creating a zone of less pressure or rarefaction. Sound waves are longitudinal, that is they consist of alternating compression and rarefaction pulses of air in the direction of the sound. Of course sound cannot travel in the vacuum of outer space, but travel much more quickly in solids and fluids such as water. In the ear, compression forces the eardrum inward and a rarefaction forces the eardrum outward, so the eardrum vibrates at the same frequency of the sound wave. Inside the ear the vibrations of the eardrum will set the hammer, anvil and stirrup into motion at the same frequency of the sound wave. When the frequency of the compression wave resonates with the natural frequency of the nerve cell, the increased amplitude induces the cell to release electrical impulses which transmits via the auditory nerve towards the brain.
High frequencies are interpreted by the ear as high notes, lower frequencies as low notes. Higher notes are produced in stringed instruments by lighter strings under higher tension, e.g. guitars. The length, tension and mass of the strings affect the pitch of the note. High tension and short strings give rise toi high frequencies; low tension and long strings can generate low frequencies. When a string is first plucked, it vibrates at many frequencies. All of these except the harmonics are quickly filtered out. The harmonics make up the tone we hear and it is called the “timbre” of the note. Wind instruments use longer lengths of vibrating air columns for low notes, and shorter lengths for high notes, e.g. the trombone, where you can see the player extending the sliding portion and changing the length of the air column and therefore the pitch of the sound. Other wind instruments use valves or stops to change the effective length, e.g. in a recorder covering or releasing a hole changes the effective resonant length of the vibrating column to change the pitch and thus the resonance of the column of air.
Resonance
Strike tuning fork on your knee and hold it right next to your ear: the sound is very soft. This is because the small surface area of the tuning fork cannot transmit much energy to the surrounding air. Now take the same tuning fork, whack it on your knee again, and touch the handle to a tabletop: the sound will be a lot louder. This is because the vibrating tuning fork causes the larger area of tabletop to vibrate. This principle is used in the guitar, violin, cello and other stringed instruments. If the dimensions of the box are a multiple of the wavelength of the sound, the sound waves will tend to reinforce each other and thus create more volume. This phenomenon is called resonance.

4) Resonance, octaves, fifths
5) Universality of resonance in mechanics, bridges, electronics
6) Harmony -thirds, fifths and octaves
7) Mathematical relationship.

Resonance as applied to an acoustic piano.
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 (upper) wave contains only the fundamental and odd harmonics; the green (lower) wave contains the fundamental, odd, and even harmonics.
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
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. Vibration


Overtones
The vibrating and resonating parts of musical instruments don't produce sound waves of just one frequency. This is because the vibrating body (e.g. string or air column) does not just vibrate as a whole; smaller sections vibrate as well. In the case of musical instruments, these additional frequencies are usually even multiples of the vibration frequency of the whole string, air column, bar, etc. The vibrating reed will generate sound waves with a frequency of 440 Hz. (cycles per second), which happens to correspond to the A above Middle C. Because of other physical properties of the reed and the accordion, the instrument will also generate waves with a frequency of 880 Hz. (2 x 440), 1,320 Hz (3 x 440), 1,760 Hz. (4 x 440), etc. These extra frequencies are called overtones. Amazingly enough, when the overtones are close to even multiples of the fundamental frequency, our brains interpret the whole conglomeration of frequencies as a single pitch. Different instruments differ in the relative strengths of the various overtones, and that is what gives the instruments different timbres. This is also what makes your voice sound different from someone else's, even when you sing the exact same pitch. In the case of cymbals, gongs, snare drums, and the other indefinite-pitch percussion instruments, there are so many frequencies and overtones all at the same time that our brains don't pick out a definite pitch. You might notice, though, that the sound of a drum or woodblock can still be "higher" or "lower" than the sound of another.
Since a pressure-time plot shows the fluctuations in pressure over time, the period of the sound wave can be found by measuring the time between successive high pressure points (corresponding to the compressions) or the time between successive low pressure points (corresponding to the rarefactions). As discussed in an earlier unit, the frequency is simply the reciprocal of the period. For this reason, a sound wave with a high frequency would correspond to a pressure time plot with a small period - that is, a plot corresponding to a small amount of time between successive high pressure points. Conversely, a sound wave with a low frequency would correspond to a pressure time plot with a large period - that is, a plot corresponding to a large amount of time between successive high pressure points. The diagram below shows two pressure-time plots, one corresponding to a high frequency and the other to a low frequency.

The ears of a human (and other animals) are sensitive detectors capable of detecting the fluctuations in air pressure that impinge upon the eardrum. The mechanics of the ear's detection ability will be discussed later in this lesson. For now, it is sufficient to say that the human ear is capable of detecting sound waves with a wide range of frequencies, ranging between approximately 20 Hz to 20 000 Hz. Any sound with a frequency below the audible range of hearing (i.e., less than 20 Hz) is known as an infrasound and any sound with a frequency above the audible range of hearing (i.e., more than 20 000 Hz) is known as an ultrasound. Humans are not alone in their ability to detect a wide range of frequencies. Dogs can detect frequencies as low as approximately 50 Hz and as high as 45 000 Hz. Cats can detect frequencies as low as approximately 45 Hz and as high as 85 000 Hz. Bats, being nocturnal creature, must rely on sound echolocation for navigation and hunting. Bats can detect frequencies as high as 120 000 Hz. Dolphins can detect frequencies as high as 200 000 Hz. While dogs, cats, bats, and dolphins have an unusual ability to detect ultrasound, an elephant possesses the unusual ability to detect infrasound, having an audible range from approximately 5 Hz to approximately 10 000 Hz.
The sensation of a frequency is commonly referred to as the pitch of a sound. A high pitch sound corresponds to a high frequency sound wave and a low pitch sound corresponds to a low frequency sound wave. Amazingly, many people, especially those who have been musically trained, are capable of detecting a difference in frequency between two separate sounds that is as little as 2 Hz. When two sounds with a frequency difference of greater than 7 Hz are played simultaneously, most people are capable of detecting the presence of a complex wave pattern resulting from the interference and superposition of the two sound waves. Certain sound waves when played (and heard) simultaneously will produce a particularly pleasant sensation when heard, are said to be consonant. Such sound waves form the basis of intervals in music. For example, any two sounds whose frequencies make a 2:1 ratio are said to be separated by an octave and result in a particularly pleasing sensation when heard. That is, two sound waves sound good when played together if one sound has twice the frequency of the other. Similarly two sounds with a frequency ratio of 5:4 are said to be separated by an interval of a third; such sound waves also sound good when played together. Examples of other sound wave intervals and their respective frequency ratios are listed in the table below.
Interval Frequency Ratio Examples
Octave 2:1 512 Hz and 256 Hz
Third 5:4 320 Hz and 256 Hz
Fourth 4:3 342 Hz and 256 Hz
Fifth 3:2 384 Hz and 256 Hz

The ability of humans to perceive pitch is associated with the frequency of the sound wave that impinges upon the ear. Because sound waves traveling through air are longitudinal waves that produce high- and low-pressure disturbances of the particles of the air at a given frequency, the ear has an ability to detect such frequencies and associate them with the pitch of the sound. But pitch is not the only property of a sound wave detectable by the human ear. In the next part of Lesson 2, we will investigate the ability of the ear to perceive the intensity of a sound wave.

String and wind instruments are good examples of standing waves on strings and pipes.
One way to describe standing waves is to count nodes. Recall that a node is a point on a string that does not move as the wave changes. The anti-nodes are the highest and lowest points on the wave. There is a node at each end of a fixed string. There is also a node at the closed end of a pipe. But an open end of a pipe has an anti-node.
What causes a standing wave? There are incident and reflected waves traveling back and forth on our string or pipe. For some frequencies, these waves combine in just the right way so that the whole wave appears to be standing still. These special cases are called harmonic frequencies, or harmonics. They depend on the length and material of the medium.
Definition 1: Harmonic
A harmonic frequency is a frequency at which standing waves can be made in a particular object or on a particular instrument.
Standing Waves in String Instruments
Let us look at a basic "instrument": a string pulled tight and fixed at both ends. When you pluck the string, you hear a certain pitch. This pitch is made by a certain frequency. What causes the string to emit sounds at this pitch?
You have learned that the frequency of a standing wave depends on the length of the wave. The wavelength depends on the nodes and anti-nodes. The longest wave that can "fit" on the string is shown in Figure 1. This is called the fundamental or natural frequency of the string. The string has nodes at both ends. The wavelength of the fundamental is twice the length of the string.
Now put your finger on the center of the string. Hold it down gently and pluck it. The standing wave now has a node in the middle of the string. There are three nodes. We can fit a whole wave between the ends of the string. This means the wavelength is equal to the length of the string. This wave is called the first harmonic. As we add more nodes, we find the second harmonic, third harmonic, and so on. We must keep the nodes equally spaced or we will lose our standing wave.
Guitars use strings with high tension. The length, tension and mass of the strings affect the pitches you hear. High tension and short strings make high frequencies; low tension and long strings make low frequencies. When a string is first plucked, it vibrates at many frequencies. All of these except the harmonics are quickly filtered out. The harmonics make up the tone we hear.
The body of a guitar acts as a large wooden soundboard. Here is how a soundboard works: the body picks up the vibrations of the strings. It then passes these vibrations to the air. A sound hole allows the soundboard of the guitar to vibrate more freely. It also helps sound waves to get out of the body.
The neck of the guitar has thin metal bumps on it called frets. Pressing a string against a fret shortens the length of that string. This raises the natural frequency and the pitch of that string.
Most guitars use an "equal tempered" tuning of 12 notes per octave. A 6 string guitar has a range of 4
Piano
Let us look at another stringed instrument: the piano. The piano has strings that you cannot see. When a key is pressed, a felt-tipped hammer hits a string inside the piano. The pitch depends on the length, tension and mass of the string. But there are many more strings than keys on a piano. This is because the short and thin strings are not as loud as the long and heavy strings. To make up for this, the higher keys have groups of two to four strings each.
The soundboard in a piano is a large cast iron plate. It picks up vibrations from the strings. This heavy plate can withstand over 200 tons of pressure from string tension! Its mass also allows the piano to sustain notes for long periods of time.
The piano has a wide frequency range, from 27,5 Hz (low A) to 4186,0 Hz (upper C). But these are just the fundamental frequencies. A piano plays complex, rich tones with over 20 harmonics per note. Some of these are out of the range of human hearing. Very low piano notes can be heard mostly because of their higher harmonics.
Traveling Waves
Sound is produced when something vibrates. The vibrating body causes the medium (water, air, etc.) around it to vibrate. Vibrations in air are called traveling longitudinal waves, which we can hear. Sound waves consist of areas of high and low pressure called compressions and rarefactions, respectively. Shown in the diagram below is a traveling wave. The shaded bar above it represents the varying pressure of the wave. Lighter areas are low pressure (rarefactions) and darker areas are high pressure (compressions). One wavelength of the wave is highlighted in red. This pattern repeats indefinitely. The wavelength of voice is about one meter long. The wavelength and the speed of the wave determine the pitch, or frequency of the sound. Wavelength, frequency, and speed are related by the equation speed = frequency * wavelength. Since sound travels at 343 meters per second at standard temperature and pressure (STP), speed is a constant. Thus, frequency is determined by speed / wavelength. The longer the wavelength, the lower the pitch. The 'height' of the wave is its amplitude. The amplitude determines how loud a sound will be. Greater amplitude means the sound will be louder.
Standing Waves
Vibration inside a tube forms a standing wave. A standing wave is the result of the wave reflecting off the end of the tube (whether closed or open) and interfering with itself. When sound is produced in an instrument by blowing it, only the waves that will fit in the tube resonate, while other frequencies are lost. The longest wave that can fit in the tube is the fundamental, while other waves that fit are overtones . Overtones are multiples of the fundamental. The areas of highest vibration are called antinodes (labeled 'A' on the diagram), while the areas of least vibration are called nodes (labeled 'N' in the diagram). In an open pipe, the ends are antinodes. However, in a pipe closed at one end, the closed end is a node, while the blown end is an antinode. Thus, closed pipes yield only half the harmonics.
How a guitar works
A typical guitar has six strings. These are all of the same length, and all under about the same tension, so why do they put out sound of different frequency? If you look at the different strings, they're of different sizes, so the mass/length of all the strings is different. The one at the bottom has the smallest mass/length, so it has the highest frequency. The strings increase in mass/length as you move up, so the top string, the heaviest, has the lowest frequency.
Tuning a guitar simply means setting the fundamental frequency of each string to the correct value. This is done by adjusting the tension in each string. If the tension is increased, the fundamental frequency increases; if the tension is reduced the frequency will decrease.
To obtain different notes (i.e., different frequencies) from a string, the string's length is changed by pressing the string down until it touches a fret. This shortens a string, and the frequency will be increased.
Beats
When two waves which are of slightly different frequency interfere, the interference cycles from constructive to destructive and back again. This is known as beats; two sound waves producing beats will generate a sound with an intensity that continually cycles from loud to soft and back again. The frequency of the sound you hear will be the average of the frequency of the two waves; the intensity will vary with a frequency (known as the beat frequency) that is the difference between the frequencies of the two waves.
The Octave
The octave is simply the doubling of the frequency, or the creation of a sound with an interval ratio of 2 to 1.
The Fifth
The interval of a fifth is the most common and has a calming effect. It is obtained from the interval ratio of 3 to 2. This interval is created by simply taking a lower octave of the third harmonic. So, if we had a frequency of 100 Hz, the third harmonic is 300 Hz. One octave below 300 Hz is 150 Hz. So, the interval of the fifth would be created by playing a sound at 100 Hz and another at 150 Hz simultaneously. Note: It's easier to multiply the fundamental frequency by the ratio, 3/2 in this case, to obtain the second frequency.
you measure these waves?

The

Constructive Interference
Unlike matter, two or more waves and Problems). When the crest of one interference.

Destructive Interference
One the other hand, if the crest of one wave meets the trough of the other, the displacement from equilibrium will be lees, or possibly zero, this is known as Destructive Interference. It should be noted that after the two waves have completely passed through each other, they will be unchanged from their original form.
So loud is it?

More accurately, the question should read how intense is the sound. While loudness
The layout of the notes in the pan contributes to the ergonomics of performance. Positioning of the notes is critical - when a note is struck, the segments which are adjacent will be forced to vibrate or”resonate”
To understand the theoretical foundations for a good note layout it is useful to know the musical relations of the partials of a harmonic tone, see the chapter about partials above. The intervals that have the most harmonic relationship are the octave, the fifth, the fourth and the third. If notes at these intervals ring together with the struck note, they will support its harmonic spectrum. Therefore, a favourable design is to put these notes close to each other.
The general idea for a good note layout in steel pans is to position notes with a harmonic relationship as close to each other as possible, while placing notes with a non-harmonic relationship as far apart as possible. A design notion that is valid for all steel pans is that the octave counterparts always are placed close together. Fifths, fourths or thirds are also consequently placed close to each other in some pans, as in the fourths-and-fifths tenor and the quadrophonic pan with their ingenious designs.
Notes with a non-harmonic relationship – as the minor or the major second – are usually placed as far apart as possible, preferably in separate drums. If the notes with semitone intervals are spread consistently over different drums, pans with different numbers of drums will have correlated minimum intervals that have to be placed in the same drum: Two drums – a major second, three drums – a minor third, four drums – a major third.
If notes with a dissonant harmonic interval – one or two semitones apart – have to be placed in the same drum, they are usually placed on opposite sides of the drum. On the other hand, if the smallest interval between the notes that have to be placed in the same drum is harmonic, the notes are put close to each other to support each other’s harmonic spectra.
The acoustical implications of a good layout are that it will make the pan sound better and make it easier to tune. A pan with more octave notes will be harder to tune due to the interaction of the notes. But it will also be easier to get a good tone in the end, because the notes in the upper octave will support the lower ones with higher partials. This is very easy to demonstrate; just put a finger on a high note while playing on its lower octave counterpart – this will usually make the brilliance of the lower tone disappear.
“Roads” between notes
The acoustic function of the “roads” – the space between the notes – is to damp the acoustic waves coming from the vibrating note before they reach the surrounding notes. This means that an increased distance (or a double groove) reduces the interaction between two adjacent notes.
The more dissonant the relation between two notes, the more they need to be separated. On the contrary, the better the acoustic separation between the notes, the less the need will be to keep the dissonant tones apart.
Sometimes, ergonomic or construction considerations are judged to be more important than the acoustical ones. The double tenor is an example of this. A double groove has been introduced to make it possible to put dissonant notes close together and still have a well-sounding instrument with many notes in it.
Pans that are designed with harmonically sounding notes close to each other, as the fifths-and-fourths tenor, may have adjacent notes put close together, with just a single groove between them. Octave counterparts should always be put as close together as possible, to enable positive feedback.
Two of the most important contributors to pan evolution are Ellie Mannette and Anthony Williams. It must be recognised that there are other pan configurations being developed but they are outside the scope of this book.

Physics of Music

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 (upper) wave contains only the fundamental and odd harmonics; the green (lower) wave contains the fundamental, odd, and even harmonics.
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.
The pan layouts are illustrative only – actual areas of notes and thus final positioning will depend on the manufacturer/tuner so will vary from the diagrams.