The Science of Synthesis - Part 6 - Voltage Controlled Filters

A voltage-controlled filter (VCF) is a processor, a filter whose operating characteristics (primarily cutoff frequency) can be controlled by means of a control voltage applied to control inputs. It can be considered to be a frequency-dependent amplifier. Although popularly known for their use in analog music synthesizers, in general, they have other applications in military and industrial electronics.

Following the oscillator's mixer section are the filters for sculpting the previously created signal. In the synthesizer world, if the oscillator's signal is thought of as a piece of wood that is yet to be carved, the filters are the hammer and chisels that are used to shape it. Filters are used to chip away pieces of the original signal until a rough image of the required sound remains.

This makes filters the most vital element of any subtractive synthesizer because if the available filters, are of poor quality, few sound sculpting options will be available and it will be impossible to create the sound you require. Indeed the choice of filters combined with the oscillators waveforms is often the reason why specific synthesizers must be used to recreate certain 'classic' dance timbres.

The most common filter used in basic subtractive synthesizers is a low-pass filter. This is used to remove frequencies above a defined cut-off point. The effect is progressive, meaning that more frequencies are removed from a sound, the further the control is reduced, starting with the higher harmonics and gradually moving to the lowest. If this filter cut-off point is recused far enough, all harmonics above the fundamental can be removed leaving just the fundamental frequency. While it may appear senseless to create a bright sound with oscillators only to remove them later with a filter, there are several reasons why you may wish to do this.

• Using a variable filter on a bright sounds allows you to determine the color of the sound much more precisely than if you tried to create the same effect using oscillators alone.
• This method enables you to employ real-time movement of a sound.

This latter movement is an essential aspect of sound design because we naturally expect dynamic movement of sound throughout the length of the note.

Using our previous example of a piano string being struck, the initial sound is very bright, becoming duller as it dies away. This effect can be simulated by opening the filter as the note starts and then gradually sweeping the cut-off frequency down to create the effect of the note dying away.

Notably, when using this effect, frequencies that lie above the cut-off point are not attenuated at right angles to the cut-off frequency; therefore, the rate at which they die away will depend on the transition period. This is why different filters that essentially perform the same function can make beautiful sweeps, whist others can produce quite uneventful results.

Action of a low-pass filter

When a cut-off point is designated, small quantities of the harmonics that lie above this point are not removed completely and are instead attenuated by a certain degree. The degree of attenuation is dependent on the transition band of the filter being used. The gradient of this transition is important because it defines the sound of any one particular filter. If the slope is steep, the filter is said to be 'sharp0 and if the slope is more gradual the filter is said to be 'soft.' To fully understand the action of this transition, some prior knowledge of the electronics involved in analogue synthesizer is required.

When the first analogue-synthesizers appeared in the 1960s, different voltages were used to control both the oscillators and the filters. Any harmonics produced by the oscillators could be removed gradually by physically manipulating the electrical current. This was achieved using a resistor (to reduce the voltage) and a capacitor (RC) circuit. Because a single RC circuit produces a 6dB transition, the attenuation increases by 6dB every time a frequency is doubled.

One RC element creates a 6dB per octave 1-pole filter that is very similar to the gentle slope created by a mixing desks EQ. Consequently, manufacturers soon implemented additional RC elements into their designs to create 2-pole filters, which attenuated 12dB per octave, and 4-pole filters, to provide 24dB per octave attenuation. Because 4-poles filters attenuate 24dB per octave making substantial changes to the sound, they tend to sound more synthesized than sounds create by a 2-pole filter; so it's important to decide which transition period is best suited for the sound. For example, if a 24 dB filter is used to sweep a pad, it will result in strong attenuation throughout the sweep, while a 12 dB will create a more natural flowing movement.

The difference between 12dB and 24dB slopes

Although low-pass filters are the most commonly used type, there are numerous variations including high pass, band pass, notch and comb. These utilize the same transition periods as the low-pass filter, but each has a wide different effect on the sound.

Action of the high pass filter

A high-pass filter has the opposite effect to a low-pass filter, first removing the low frequencies from the sound and gradually moving towards the highest. This is less useful than the low-pass filter because it effectively removes the fundamental frequency of the sound, leaving only the fizzy harmonic overtones. Because of this, high-pass filters are rarely used in the create of instruments and are predominantly used to create effervescent sound effects or bright timbres that can be laid over the top of another lowness sound to increase the harmonic content.

Action of the Band select filter

The typical euphoric trance leads are a good example of this, as they are often created from a tone with the fundamental overlaid with numerous other tones that have been created using a high-pass filter. This prevents the timbre from becoming too muddy as a consequence of stacking together fundamental frequencies. In both remixing and dance music, it's commonplace to run a high.pass filter over an entire mix to eliminate the lower frequencies, creating an effect similar to a transistor radio or a telephone. By reducing the cut-off control, gradually or immediately, the track morphs from a thin sound to a fatter one, which can produce a dramatic effect in the right context.

If high- and low-pass filters are connected in series, then it's possible to create a band-pass, or band-select filter. These permit a set of frequencies to pass unaltered through the filter while the frequencies either side of the two filters are attenuated. The frequencies that pass through unaltered are known as the 'bandwidth' or the 'band pass' of the filter, and clearly, if the low pass is set to attenuate a range of frequencies that are above the current high-pass setting, no frequencies will pass through and no sound is produced.

Band-pass filters, like high-pass filters, are often used to create timbres consisting of fizzy harmonics (Figure 1.15). They can also be used to determine the frequency content of a waveform, as by sweeping through the frequencies each individual harmonic can be heard. Because this type of filter frequently removes the fundamental, it is often used as the basis of sound effects or lo-fi and trip-hop timbres or to create very thin sounds that will form the basis of sound effects.

Although band-pass filters can be used to thin a sound, they should not be confused with band-reset filters, which can be used for a similar purpose. Band-reject filters, often referred to as notch filters, attenuate a selected range of frequencies effectively creating a notch in the sound - hence the name - and usually leave the fundamentals unaffected. This type of filter is handy for scooping out frequencies, thinning out a sound while leaving the fundamentals intact, making them useful for creating timbres that contain a discernible pitch but do not have a high level of harmonic content.

Action of the notch filter

One final form of the filter is the comb filter. With these, some of the samples entering the filter are delayed in time and the output is the fed back in the filter to be reprocessed to produces the results, effectively creating a comb appearance, hence the name. Using this method, sounds can be tuned, to amplify or reduce harmonics based on the length of the delay and the sample rate, making it useful for creating complex sounding timbres that cannot be accomplished any other way. Because of the way they operate, however is rare to find these featured on a synthesizer and are usually available only as a third-party effect.

Action of the comb filter

As an example, if a 1kHZ signal is put through the filter with a 1ms delay, the signal will result in phase because 1ms is coincident with the inputted signal, equalling one. However, if a 500 Hz signal with a 1ms delay were used instead it would be half of the period length and so it would be shifted out of phase by 180°, resulting in a zero. It's this constructive and deconstructive period that creates the continual bump then dip in harmonics, resulting in a comb like appearance when represented graphically, as in Figure 1.17. This method applies to all frequencies, with integer multiples of 1 kHZ producing ones and odd multiples of 500 Hz (1.5, 2.5, 3.5 kHz etc.) producing zeros. The effect of using this filter can at best be described as highly resonant, and forms the basis of flanger effects; therefore, it's use is commonly limited to sound design rather than the more basic sound sculpting.

One final element of sound manipulation in a synthesizer's filter section is the resonance control. Also referred to as peak, this refers to the amount of the output of the filter that is fed back directly into the input, emphasizing any frequencies that are situated around the cut-off frequency. This has a similar effect to employing a band-pass filter at the cut-off point, effectively creating a peak. Although this also affects the filter's transition period, it is more noticeable at the actual cut-off frequency than anywhere else. Indeed, as you sweep through the cut-off range, the resonance follows the curve, continually peaking at the cut-off point. In terms of the final sound, increasing the resonance makes the filter sound more dramatic and is particularly effective when used in conjunction with low-pass filter sweeps.

The effects of resonance

On many analogue and DSP-analogue-modelled synthesizers, if the resonance is turned up high enough it will fed back on itself. As more and more of the signal is fed back, the signal is exaggerated until the filter breaks into self-oscillation. This produces a sine wave with a frequency equal to that of the set cut-off point and is often a purer sine wave than that produced by the oscillators. Because of this, self-oscillating filters are commonly used to create deep, powerful sub-basses that are particularly suited to the drum 'n' bass and rap genres.

Notably, some filters may also feature a saturation parameter which essentially overdrives the filters. If applied heavily, this can be used to create distortion effects, but more often it's used to thicken out timbres and add even more harmonics and partials to the signal to create rich sounding leads or basses.

The keyboard's pitch can also be closely related to the action of the filters, using a method known as pitch tracking, keyboard scaling or more frequently 'key follow'. On many synthesizers the depth of this parameter is adjustable, allowing you to determine how much or how little the filter should follow the pitch.

The effect of filter key follow

When this parameter is set to it's neutral state (neither negative nor positive), as a note is played on the keyboard the cut-off frequency tracks the pitch and each note is subjected to the same level of filtering. If this is used on a low-pass filter, for example, the filter setting remains fixed, so as progressively higher notes are played fewer and fewer harmonics will be present in the sound, making the timbre of the higher notes mellower than that of the lower note. If they key follow parameter is set to positive, the higher notes will have a higher cut-off hand, the key follow parameter is set to negative, the higher notes will lower the cut-off frequency, making the high notes even mellower than when key follow is set to it's neutral state. Key follow is useful for recreating real instruments such as brass, where the higher notes are often mellower than the lower notes, and is also useful on complex bass lines that jump over an octave, adding further variation to a rhythm.

Here concludes the sixth part of this post, if you want to know more about acoustic science please read. Rick Snoman's Dance Music Manual (Second Edition) Tools, Toys and Techniques.

The science of Synthesis - Part 5 Waveshapes and it's properties

Oscillators generate a consistent, repeating signal. Signals from oscillators and other sources are used to control the movement of the cones in our speakers, which make real sound waves which travel to our ears.

Oscillators generate a consistent, repeating signal. Signals from oscillators and other sources are used to control the movement of the cones in our speakers, which make real sound waves which travel to our ears. If you tie one end of a rope to a doorknob, stand back a few feet, and wiggle the other end of the rope up and down really fast, you're doing roughly the same thing as an oscillator. The difference is that you're wiggling a rope, whereas the oscillator is wiggling an audio signal.

Audio signals are often represented on a graph where the horizontal x-axis represents time and the vertical y-axis represents the pressure of the signal. This is called a time domain representation of audio.

The Sine Wave

A sine wave is the simplest wave shape and is based on the mathematical sine function. A sine wave consists of the fundamental frequency alone and does not contain harmonics. This means that they are not suitable for sole use in a subtractive sense, because if the fundamental is removed no sound is produced (and there are no harmonics upon which the modifiers could act). Consequently, the sine wave is used independently to created sub-basses or whistling timbres or is mixed with other waveforms to add extra body or bottom end to a sound.

The Square Wave 

A square wave is the simplest waveform for an electrical circuit to generate because it exists only in two states: high and low (Figure 1.7). This wave produces only odd harmonics resulting in a mellow, hollow sound. This makes it particularly suitable for emulating wind instruments, adding width to strings and pads, or for the creation of deep, wide bass sounds.

The Pulse Wave

Although pulse waves are often confused with square waves, there is a significant difference between the two. Unlike a square wave, a pulse wave allows the width of the high and low states to be adjusted, thereby varying the harmonic content of the sound.

Today it is unusual to see both square and pulse waves featured in a synthesizer. Rather the square wave offers an additional control allowing you to vary the width of the pulses.

The benefit of this is that reductions in the width allow you to produce thin reed-like timbres along with the wide, hollow sounds created by a square wave.

The Sawtooth Wave

A sawtooth wave produces even and odd harmonics in series and therefore produces a bright sounds that is an excellent starting point for brassy, raspy sounds. It's also suitable for creating the gritty, bright sounds needed for leads and raspy basses. Because of it's harmonic richness, it is often employed in sounds that will be filter swept.

The Triangle Wave

The triangle wave shape features two linear slopes and is not as harmonically rich as a sawtooth wave since it only contains odd harmonics (partials) ideally this time of waveform is mixed with a sine square or pulse wave to add a sparkling or bright effect to a sound and is often emptied on pads to give them a glittery feel.

The Noise Wave

Noise waveforms are unlike the other five waveforms because they create a random mixture of frequencies rather than actual tones (Figure 1.11). Noise waveforms can be 'pink' or 'white' depending on the energy of the mixed frequencies they contain. White noise contains equal amounts of energy at every frequency and is comparable to radio, static, while pink noise contains equal amounts of energy in every musical octave and therefore we perceive it to produce a heavier, deeper hiss.

Noise is useful for generating percussive sounds and was commonly used in early drum machines to create snares and hand claps. Although this remains it's main use, it can also be used for simulating wind or sea effects, for producing breath effects in wind instrument timbres or for producing the typical trance leads.

Creating more complex waveforms

Whether oscillators are created by analogue or DSP circuitry, listening to individual oscillators in isolation can be a mind numbing experience. To create interesting sounds, a number of oscillators should be mixed together and used with the available modulation options.

This is achieved by first mixing different oscillator waveforms together and then detuning them all or just those that share the same waveforms so that they are out of phase from one another, resulting in a beating effect. Detuning is accomplished using the detune parameter on the synthesizer, usually by odd rather than even numbers. This is because detuning by an even number introduces further harmonic content that may mirror the harmonics already provided by the oscillators, causing the already present harmonics to be summed together.

It should be noted here that there is a limit to the level that oscillators can be detuned from one another. As previously discussed, oscillators should be detuned so that they beat, but if the speed of these beats is increased by any more than 20 Hz the oscillators separate, resulting in two noticeably different sounds. This can be sometimes be used to good effect if the two oscillators are to be mixed with a timbre from another synthesizer because the additional timbre can help to fuse the two separate oscillators. As a general rule of thumb, it is unusual to detune an oscillator by more than an octave.

Additional frequencies can also be added into a signal using ring modulation and sync controls. Oscillator sync, usually found within the oscillator section of a synthesizer, allows a number of oscillators' cycles to be synced to one another. Usually all oscillators are synced to the first oscillators' cycle; hence, no matter where in the cycle any other oscillator is, when the first starts it's cycle again the others are forced to begin again too.

For example, if the two oscillators are used, with both set to a sawtooth wave and detuned by -5 cents (one-hundredth of a tone), every time the first oscillator restarts it's cycle so too will the second, regardless of the position in it's own cycle. This tends to produce a timbre with no harmonics and can be ideal for creating big, bold leads. Furthermore, if the first oscillator is unchanged and pitch bend is applied to the second to speed up or slow it's cycle, screaming lead sounds typical of the Chemical Brothers are created as a consequence of the second oscillator fighting against the syncing with the first.

After the signals have left the oscillators, they enter the mixer section where the volume of each oscillator can be adjusted and features such as ring modulation can be applied to introduce further harmonics. (The ring modulation feature can sometimes be found within the oscillator section but is more commonly located in the mixer section, directly after the oscillators). Ring modulation works by providing a signal that is the sum and difference compound of two signals (while also removing the original tones). Essentially, this means that both signals from a two-oscillator synthesizer enter the ring modulator and come out from the other end as one combined signal with no evidence of the original timbre remaining.

As an example, if one oscillator produces a signal frequency of 440 Hz (A4 on a keyboard) and the second produces a frequency of 660Hz (E5 on a keyboard), the frequency of the first oscillator is subtracted from the second.

660Hz - 440Hz = 220Hz(A3)

Then the first oscillator's frequency is added to that of the second.

660Hz + 440Hz = 11000Hz(C#6)

Based on this example, the difference of 220Hz provides the fundamental frequency while the sum of the two signals, 11000Hz, results in a fifth harmonic overtone. When working with synthesizers, though, this calculation is rarely performed. This result is commonly achieved by ring modulating the oscillators together at any frequency and then tuning the oscillator. Ring modulation is typically used in the production of metallic-type effect (ring modulators were used to create the Dalek voice from Dr Who) and bell-like sounds. If ring modulation is used to create actual pitched sounds, a large number of in-harmonic overtones are introduced into the signal creating dissonant, unpitched results.

The option to add noise may also be included in the oscillator's mix section to introduce additional harmonics, making the signal leaving the oscillator/mix section full of frequencies that can then be shaped further using the options available.

Here concludes the fifth part of this post, if you want to know more about acoustic science, please read Rick Snoman's Dance Music Manual (Second Edition) Toys, Tools and Techniques.

The science of Synthesis - Part 4 Subtractive Synthesizers

Subtractive synthesis is the basis of many forms of synthesizers and is commonly related to analogue synthesis. It is achieved by combining a number of sounds or 'oscillators' together to create a timbre that is very rich in harmonics.

Having looked into the theory of sound, we can look at how this relates to a synthesizer. Subtractive synthesis is the basis of many forms of synthesizers and is commonly related to analogue synthesis. it is achieved by combining a number of sounds or 'oscillators' together to create a timbre that is very rich in harmonics.

Layout of a basic synthesizer

This rich sound can then be sculpted using a series of 'modifiers'. The number of modifiers available on a synthesizer is entirely dependent on the model, but all synthesizers offer a way of filtering out certain harmonics and of shaping the overall volume of the timbre.

The next part of this series of posts looks at how a real analogue synthesizer operates, although any synthesizer that emulates analogue synthesis (i.e. digital signal processing (DSP) analogue) will operate in essentially the same way, with the only difference being that the original analogue synthesizer voltages do not apply to their DSP equivalents.

An analogue synthesizer can be said to consist of three components.

• An oscillator to make the initial sound.
• A filter to remove frequencies within the sound.
• An amplifier to define to overall level of the sound.

Each o these components and their role in synthesis are discussed in the sections below. 

Voltage controlled Oscillator (VCO)

When a key on a keyboard is pressed, a signal is sent to the oscillator to activate it, followed by a specific control voltage (CV) to determine the pitch. The CV that is sent is unique to the key that is pressed, allowing the oscillator to determine the pitch it should reproduce. For this approach to work correctly, the circuitry in the keyboard and the oscillator must be incredibly precise in order to prevent the tuning from drifting, so the synthesizer must be serviced regularly. In addition, changes in external temperature and fluctuations in the power supply may also cause the oscillator's tuning to drift.

This instability gives analogue synthesizers their charm and is the reason why many purists will invest small fortunes in second-hand models rather than using the latest DSP-based analogue emulations. Although, that said, if too much detuning is present, it will be immediately evident and could become a major problem! There is still an ongoing argument over whether is possible for DSP oscillators to faithfully reproduce analogue-based synthesizers, but the argument in favor of DSP synthesizers is that they offer more waveforms and do not drift too widely, therefore prove more reliable in the long run.

In most early subtractive synthesizers the oscillator generated only three types of waveforms: square, sawtooth and triangle waveforms. Today this number has increased and many synthesizers now offer additional sine, noise, tai-saw, pulse and numerous variable wave shapes as well.

Although these additional waveforms produce different sounds, they are all based around the three basic wave shapes and are often introduced into synthesizers to prevent mixing of numerous basic waveforms together, a task that would reduce the number of oscillators.

For example, a tri-saw wave is commonly a sample of three sawtooth waves blended together to produce a sound that is rich in harmonics, with the advantage that the whole sound is contained in one oscillator. Without this waveform it would take three oscillators to recreate this sound, which could be beyond the capabilities of the synthesizer. Even if the synthesizer could utilize three oscillators to produce this one sound, the number of available oscillators would be reduced. 

Here concludes the fourth part of this post, if you want to know more about acoustic science, please read Rick Snoman's Dance Music Manual (Second Edition) Toys, Tools and Techniques.

The science of Synthesis Part 3

So far we've looked at how both the pitch and the timbre are determined. The final characteristic to consider is volume.

So far we've looked at how both the pitch and the timbre are determined. The final characteristic to consider is volume. Changes in volume are caused by the amount of air molecules an oscillating object displaces. The more air an object displaces, the louder the perceived sound. This volume also called 'amplitude', is measured by the degree of motion of the air molecules within the sound waves corresponding to the extent of rarefaction and compression that accompanies a wave. The problem, however, is that many simple vibrating objects produce a sound that is inaudible to the human ear because so little air is displaced; therefore for the sound wave to be heard most musical instruments must amplify the sound that's created. To do this, acoustic instruments use the principle of forced vibration that utilizes either a sounding board, as in a piano or similar stringed instruments, or a hollow tube, as in the case of wind instruments.

When a piano string is struck, it's vibrations not only set other strings in motion but also vibrate a board located underneath the strings. Because this sounding board does not share the same frequency as the vibrating wires, the reaction is not very sympathetic and the board is forced to resonate. This resonance moves a large number of air particles than the original sound alone, in effect amplifying the sound. Similarly, when a tuning fork is struck and placed on a tabletop, the table's frequency is forced to match that of the tuning fork and the sound is amplified.

Of course, neither of these methods of amplification offers any physical control over the amplitude. If the level of amplification can be adjusted, then the ration between the original and the changed amplitude is called the 'gain'.

It shoyud be noted, however that loudness itself is difficult to quantify because it's entirely subjective to the listener. Generally speaking, the human ear can detect frequencies from as low as 20Hz up to 20kHz; however, this depends on a number of factors. Indeed, while most of us are capable of hearing (or more accurately feeling) frequencies as low as 20 Hz, the perception of higher frequencies changes with age. Most teenagers are capable of hearing frequencies as high as 18 kHz, while the middle-aged tend not to hear frequencies above 14kHz. A person's level of hearing may also have been damaged, for example by overexposure to loud noise or music. Whether is possible for us to perceive sounds higher than 18kHz with the presence of other sounds is a subject of debate that has yet to be proven. However, it is important to remember that sounds that are between 3 and 5kHz appear perceivably louder than frequencies that are out of this range.

Here concludes the third part of this post, if you want to know more about acoustic science please read. Rick Snoman's Dance Music Manual (Second Edition) Tools, Toys and Techniques.

The Science of Synthesis - Part 2

A single wave produces a single tone known as the fundamental frequency, which in effect determines the pitch of the note. When further sine waves that are out of phase from the original are introduced, if they are integer multiples of the fundamental frequency they are known as 'harmonics'. Harmonics make the sound appear more complex.

A single wave produces a single tone known as the fundamental frequency, which in effect determines the pitch of the note. When further sine waves that are out of phase from the original are introduced, if they are integer multiples of the fundamental frequency they are known as 'harmonics'. Harmonics make the sound appear more complex.

Otherwise if they are not integer multiples of the fundamental they are called 'partials', which also contribute to the complexity of the sound. Through the introduction and relationship of these harmonics and partials an infinite number of sounds can be created.

The harmonic content or 'timbre' of a sound determines the shape of the resulting waveform. It should be noted that the diagrams shown below are simple representations, since the waveforms generated by an instrument are incredibly complex which makes it impossible to accurately reproduce it on paper.

In an attempt to overcome this, Joseph Fourier, a French scientist, discovered that now matter how complex any sound is, it could be broken down into it's frequency components and, using a given set of harmonics, it was possible to reproduce it in a simple form.

To use his words, 'Every periodic wave can be seen as the sum of sine waves with certain lengths and amplitudes, the wave lengths of which have harmonic relations. This is based around the principle that the content of any sound is determined by the relationship between the level of the fundamental frequency and its harmonics and their evolution over a period of time. From this theory, known as the Fourier theorem, the waveforms that are common to most synthesizers are derived.

Addition of sine waves to create a square wave

Addition of sine waves to create a Sawtooth wave

Addition of sine waves to create a triangle wave

Here concludes the second part of this post, if you want to know more about acoustic science, please read Rick Snoman's Dance Music Manual (Second Edition) Toys, Tools and Techniques.

The Science of Synthesis - Part 1

Today's recording techniques would have been regarded as science fiction forty years ago.

Today's dance . and club-based music relies just as heavily on the technology as it does on the musicality; therefore, to be proficient at creating this genre of music it is first necessary to fully comprehend the technology behind its creation. Indeed, before we can even begin to look at how to produce to music, thorough understanding of both the science and the technology behind the music is paramount. You wouldn't attempt to repair a car without some knowledge of what you were tweaking, and the same applies for dance - and club - based music.

Therefore, we should start at the very beginning and where better to start than the instrument that encapsulated the genre - the analogue synthesizer. Without a doubt, the analogue synthesizers are becoming increasing difficult to source today, nearly all synthesizers in production, whether hardware or software, follow the same path laid down by their predecessors. However to make sense of the various knobs and buttons that adorn a typical synthesizer and observe the effects that each has on a sound, we need to start by examining some basic acoustic science.

Acoustic Science

When an object vibrates, air molecules surrounding it begin to vibrate sympathetically in all directions creating a series of sound waves. These sound waves then create vibrations in the ear drum that the brain perceives as sound.

The movement of sound waves is analogous to the way that waves spread when a stone is thrown into a pool of water. The moment the stone hits the water, the reaction is immediately visible as a series of small waves spread outwards in every direction. This is almost identical to the way in which sound behaves with each wave of water being similar to the vibrations of air particles.

Difference between low and high frequencies

For instance, when a tuning fork is struck, the forks first move towards one another compressing the air molecules before moving in the opposite direction. In this movement from 'compression' to 'rarefaction' there is a moment where there are less air molecules filling the space between the forks. When this occurs, the surrounding air molecules crowd into this space and are then compressed when the forks return on their next cycle. As the fork continues to vibrate, the previously compressed air molecules are pushed further outwards by the next cycle of the fork and a series of alternating compressions and rarefactions pass through the air.

The number of rarefactions and compressions or 'cycles', that are completed every second is referred to as the operating frequency and is measured in Hertz (Hz). Any vibrating object that completes, say, 300 cycles/s has a frequency of 300Hz while an object that completes 3000 cycles/s has a frequency of 3kHz.

The frequency of a vibrating object determines its perceived pitch, with faster frequencies producing sounds at a higher pitch than slower frequencies. From this we can determine that the faster an object vibrates, or 'oscillates' , the shorter the cycle between compression and rarefaction. An example of this is shown in the previous picture.

Any object that vibrates must repeatedly pass through the same position as it moves back and forth through its cycle. Any particular point during this movement is referred to as the 'phase' of the cycle and is measured in degrees, similar to the measurement of a geometric circle. As shown in Figure 1.2, each cycle starts at position zero, passes back through this position, known as the 'zero crossing', and returns to zero.

The zero crossing of a waveform

Consequently, if two objects vibrate at different speeds and the resulting waveforms are mixed together, both waveforms will start at the same zero point but the higher frequency waveform will overtake the phase of the lower frequency. Provided that these waveforms continue to oscillate, they will eventually catch up with one other and then repeat the process all over again. This produces an effect known as 'beating'.

The speed at which waveforms 'beat' together depends on the difference in the frequency between them. It's important to note that if two waves have the same frequency and are 180º out of phase with one another, one waveform reaches its peak while the second is at its through, and no sound is produced.

This effect, where two waves cancel one another out and nos sound is produced, is known as 'phase cancellation' and will be shown in the next picture.

As long as waveforms are not 180º out of phase with one another, the interference between the two can be used to created more complex waveforms than the simple sine wave. In fact, every waveform is made up of a series of sine waves, each slightly out of phase with one another. The more complex the waveform this produces the more complex the resulting sound. This is because as an increasing number of waves are combined a greater number of harmonics are introduced. This can be better understood by examining how an everyday piano produces its sound.

The strings in a piano are adjusted so that each oscillates at an exact frequency. When a key is struck, a hammer strikes the corresponding string forcing it to oscillate. This produces the fundamental pitch of the note and also, if the vibrations from this string are the same as any of the other strings natural vibrations rates, sets these into motion too. These are called 'sympathetic vibrations' and are important to understand because most musical instruments are based around this principle. The piano is turned so that the strings that vibrate sympathetically with the originally struck string create a series of waves that are slightly out of phase with one another producing a complex sound.

Two waves out of phase

Any frequencies that are an integer multiple of the lowest frequency (i.e. the fundamental) will in harmony with one another, a phenomenon that was first realized by Pythagoras, from which he derived the following three rules.

• If a note's frequency is multiplied or divided by two, the same note is created but in a different octave.
• If a note's frequency is multiplied or divided by three, the strongest harmonic relation is created. This is the basis of the western musical scale. If we look at the first rule, the ration 2:3 is known as a perfect fifth and is used as the basis of the scale.
• If a note's frequency is multiplied or divided by five, this also creates a strong harmonic relation. Again, if we look at the first rule, the ration 5:4 gives the same harmonic relation but this interval is known as the major third.

Here concludes the second part of this post, if you want to know more about acoustic science please read Rick Snowman's Dance Music Manual (Second Edition) Tools, Toys and Techniques.

Music Production with Ableton Live & Push 2

I made this video tutorial using ableton live 9.5 and Push 2 I hope you enjoy it.