## Sinusodial Creation and Simple Harmonic Motion

The first premise to understand is that simple harmonic motion through time generates sinusoidal motion.

The following diagram will display the amplitude of the harmonic motion and for this we need to use the term A in our formula. We will also be using θ.

I have used the equation A sin θ where θ completes one cycle (degrees).
The axis displays values based on a unit circle with being interpreted as amplitude.
The x axis denotes degrees (θ)

It then follows that:

When the angle θ is 0° or 180° then y = 0
sin 0° and sin 180° = y/A = 0

When the angle θ is 90° then y = 1
sin 90° = y/A = 1

When the angle θ is 270° then y = −1
sin 270° = y/A = −1

When constructing and working with sinusoids we need to plot our graph and define the axis.

I have chosen the y-axis for amplitude and the x-axis for time with phase expressed in degrees. However, I will later define the formulae that define the variables when we come to expressing the processes.

For now, a simple sine waveform, using each axis and defining them, will be enough.

I will create the y-axis as amplitude with a range that is set from -1 to +1.
y: amplitude

Now to create the x-axis and define its variables to display across the axis.

The range will be from -90 deg to 360 deg
x: time/phase/deg

The following diagram displays the axis plus the waveform and the simplest formula to create a sinusoid is y-sinx

The diagram shows one cycle of the waveform starting at 0, peaking at +1 (positive), dropping to the 0 axis and then down to -1 (negative).

The phase values are expressed in degrees and lie on the x-axis. A cycle, sometimes referred to as a period, of a sine wave is a total motion across all the phase values.

I will now copy the same sine wave and phase offset (phase shift and phase angle) so you can see the phase values and to do this we need another simple formula and that is:
y=sin(x-t) where t (time/phase value) being a constant will, for now, have a value set to 0. This allows me to shift by any number of degrees to display the phase relationships between the two sine waves.

The shift value is set at 90 which denotes a phase shift of 90 degrees. In essence, the two waveforms are now 90 degrees out of phase.

The next step is to phase shift by 180 deg and this will result in total phase cancelation. The two waveforms together, when played and summed, will produce silence as each peak cancels out each trough.

Relevant content:

Frequency and Period of Sound

Total and Partial Phase cancellation

Digital Audio – Understanding and Processing

## Low End

Whenever I have been called into a studio to assist a producer in managing frequencies for pre-mastering I have always been surprised at the fact that people seem to want to attribute a frequency range for the low end of a track. Every track has its own qualities and criteria that need addressing based on the entire frequency content of the track before a range can be attributed to the low end.

I have come across producers affording insights into some interesting low-end frequency ranges and these ranges are relevant only to the context that the track resides in. If we are talking about a heavy Hip Hop track that uses 808 kicks supplemented with sine waves then the low end of that track will vary dramatically to that of a mainstream EDM (electronic dance music) that will incorporate stronger kicks supplemented with ducked bass tones.

So, working on the premise of a frequency range will not help you at all. What is far more important is to understand both the frequencies required for the low end of a specific track and the interaction of these frequencies within themselves and the other elements/frequencies that share this particular range. This might sound strange: ‘within themselves’ but this is the exact area of the physics of mixing and managing low end that we need to explore. When we come to the chapters that pertain to both the harmonic content of a specific frequency range and the manipulation of those frequencies using advanced techniques then all will become clearer.

To fully understand how to manage low-end frequencies we need to look at frequencies, some of the problems encountered with manipulating frequencies, and some of the terminology related to it, in far more detail.

Timbre

We use the term Timbre to describe the tonal characteristics of a sound. It is simply a phrase to distinguish the differences between different sounds and is not reliant on pitch or volume. In other words, two different sounds at the same frequency and amplitude do not signify that they are the same. It is the timbre that distinguishes the tonal differences between the two sounds. This is how we are able to distinguish a violin from a guitar.

Sinusoids

However, to help you in understanding what this has to do with the low end it’s best to explain the first thing about sound, any sound, and that it is made up of sine waves at different frequencies and amplitudes. If you understand this basic concept then you will understand why some sounds are tonal and others are atonal, why a sampled kick drum might exhibit ‘noise’ as opposed to a discernible pitch and why a pure sine wave has no harmonic content.

To explain the diagrams below: I have drawn a simple sine wave that starts at 0, rises to +1 which we call the positive, drops to 0 and then drops below 0 to -1 which we call the negative. From 0 to +1 to 0 then to -1 and finally back to 0 is considered one complete cycle.

The phase values are expressed in degrees and lie on the x-axis. A cycle, sometimes referred to as a period, of a sine wave is a total motion across all the phase values.

This cycle is measured in Hertz (Hz) over 1 second and represents frequency. A good example of this is the note A4, which you have come across so many times. A4 is 440 Hz: this means that the waveform cycles 440 times per second (repeats itself) and this frequency represents pitch. If I jump to A5, which is one octave higher, I double the frequency 880 Hz. If I halve the A4 I get A3 (220 Hz) which is one octave lower.

Partial and total phase cancellations are critical to understand as I will be showing you how to use some very specific techniques to create new sonic textures using these concepts. Understanding that a sound has a timbre and that timbre can be expressed by partials which form, apart from the fundamental, both overtones and undertones is equally important as we will cover techniques in managing low frequencies without having to use the fundamental frequency of the sound. Additionally, when we come to managing shared frequencies (bass and drums) then the concept of harmonics is very useful as we are continually fighting the battle of clashing frequencies, frequency smearing, gain summing and so on. For example, sine waves have no harmonic content and therefore some dynamic processes yield no useful results and more specialised techniques are required. Whereas saw waveforms are rich in harmonics and therefore we are able to use pretty standard techniques to accent the sweet spots and eradicate artifacts.

I will now copy the same sine wave and phase offset (phase shift and phase angle) so you can see the phase values:

The shift value is set at 90 which denotes a phase shift of 90 degrees. In essence, the two waveforms are now 90 degrees out of phase.

The next step is to phase shift by 180 deg and this will result in total phase cancellation. The two waveforms together, when played and summed, will produce silence as each peak cancels out each trough.

Summing

When two shared (the same) frequencies (from different layers) of the same gain value are layered you invariably get a gain boost at that particular frequency. This form of summing can be good if intended or it can imbalance a layer and make certain frequencies stand out that were not intended to be prominent. A good way around this problem is to leave ample headroom in each waveform file so that when two or more files are summed they do not exceed the ceiling and clip.

If you take two sine waves of the same frequency and amplitude and sum them one on top of the other you will get a resultant gain value of 6dB.

Summing is important when dealing with the low end as any form of layering will have to take into account summed values.

When two shared frequencies are layered and one has a higher gain value than the other then it can ‘hide’ or ‘mask’ the lower gain value frequency. How many times have you used a sound that on its own sounds excellent, but gets swallowed up when placed alongside another sound? This happens because the two sounds have very similar frequencies and one is at a higher gain; hence one ‘masks’, or hides, the other sound. This results in the masked sound sounding dull, or just simply unheard. As we are dealing with low end this problem is actually very common because we are layering, in one form or another, similar frequencies.

Partials

The individual sinusoids that collectively form an instrument’s Timbre are called Partials also referred to as Components. Partials contain Frequencies and Amplitudes and, more critically, Time (please refer to my book on the subject of EQ – EQ Uncovered). How we perceive the relationships between all three determines the Timbre of a sound.

Fundamental

The Fundamental is determined by the lowest pitched partial. This can be the root note of a sound or what our ears perceive as the ‘primary pitch’ of a sound (the pitch you hear when a note is struck).

Overtones/Undertones

Using the fundamental as our root note, partials pitched above the fundamental are called overtones and partials pitched beneath the fundamental are called undertones, also referred to as Sub Harmonics. These partials are referred to, collectively, as Harmonics. This can be easily represented with a simple formula using positive integers:

f, 2f, 3f, 4f etc..

f denotes the fundamental and is the first harmonic. 2f is the second harmonic and so on.
If we take A4 = 440 Hz then f = 440 Hz (first harmonic and fundamental).
The second harmonic (overtone) would be 2 x 440 Hz (2f) = 880 Hz.

Sub Harmonics are represented by the formula: 1/n x f where n is a positive integer. Using the 440 Hz frequency as our example we can deduce the 2nd subharmonic (undertone) to be ½ x 440 Hz = 220 Hz and so on.

An area that can be very confusing is that of harmonics being overtones. They are not. Even-numbered harmonics are odd-numbered overtones and vice versa. The easiest way of looking at this, or rather, counting is to think of it as follows:

Let’s take the A4 440 Hz example:
If A4 is the fundamental tone then it is also regarded as the 1st Harmonic.
The 1st Overtone would then be the 2nd Harmonic.
The 2nd Overtone would be the 3rd Harmonic and so on…

Inharmonic/Inharmonicity

Most musical sounds consist of a series of closely related harmonics that are simple multiples of each other, but some (such as bells and drums for instance) do contain partials at more unusual frequencies, as well as some partials that may initially seem to bear no relation to the fundamental tone, but we can go into more detail about these later on.

It is important to understand this concept as the area of tuning drum sounds and marrying and complimenting the frequencies with tonal basses, is an area that troubles most producers.

When managing low-end frequencies the phase relationships and harmonic content are more important than any other concept because of the limited frequency range we have to process, the nature of the sounds we are dealing with and the types of processing we need to apply.

I have often found frequency charts excellent for ‘normal’ acoustic instruments but a little hit and miss when it comes to synthetic sounds as these sounds will invariably contain a combination of waveforms and associated attributes that will vary dramatically from the standard pre-defined acoustical frequencies. However, ranges of this type can help as a starting point and some of the following might be helpful to you:

Sub Bass

This is the one frequency range that causes most of the problems when mixing low-end elements and for a number of reasons:

We tend to attribute a range starting from (about) 12 Hz to 60 Hz for this vital area. Although our hearing range has a ballpark figure 20 Hz – 20 kHz we can ‘feel’ energies well below 20 Hz. In fact, you can ‘hear’ the same energies by running a sine wave at high amplitude, but I don’t recommend that at all. In fact, we use these sub frequencies at high amplitudes to test audio systems. It is often said that cutting low end frequencies will brighten your mix. Yes, this is true. It is said that too much low-end energy will muffle and muddy up a track. Yes, this is also true. In fact, I cut out any redundant frequencies before I even start to mix a track. However, this is not the only reason we cut certain frequencies below the frequency we are trying to isolate and enhance and it has to do with the impact the lower end of this range has on using processors like compressors (more on this in later chapters).

Bass

I have seen some wild figures for this range as bass can encompass a huge range of frequencies depending on whether it is acoustic or synthetic. But the ‘going rate’ seems to be anywhere between 60 Hz all the way to 300 Hz. The reason this range is so critical is that most sounds, relevant to this low end, in your mix, will carry fundamentals and undertones in this range and will form the ‘boom’ of a track. This frequency range presents us with some of the most common problems that we will try to resolve in later chapters as so many frequencies reside in this range that their summed amplitudes alone will create metering nightmares.

We will deal with frequencies above these ranges when we come to working through the exercises otherwise it is simply a case of me writing another frequency chart and attributing descriptions for each range. I am only concerned with the relevance of these frequencies in relation to the low end and not for anything else.

Kick Drum

I find kick drum frequency ranges almost useless because in today’s music or the genres this book is concerned with, EDM and Urban, kick drums are multi-layered and in most cases samples as opposed to tuned acoustic kicks. So, remarks like ‘boost between 60 Hz – 100 Hz to add low end’, although a guide, is both misleading and unhelpful. We have a general rule in audio engineering/production: you cannot boost frequencies that are not there. Although most sounds will have a sensible frequency range we can use as a guide the kick drum is an entity on its own, simply because of the move away from using only acoustically tuned drum kits to sample-based content. Tonal synthetic kick drums are a different story entirely as the tone will have a pitch but layer that with other drum sounds and they can amass into one big mess if not handled sensibly. The TR 808, through design, behaves tonally but in quite a specific manner thanks to its clever oscillator Bridged T-network, Triggering and Accent.

To help you, purely as a guide, here is a basic chart outlining fundamental and harmonic ranges.

I have included some of the higher frequency ‘instruments’ like the Soprano voice so you can get an idea of the range of frequencies that we have to consider when mixing one frequency range in a track with another. As I said at the start of this chapter, low-end frequency ranges can only be assigned when the entire frequency content of a track is known otherwise it will be a process in isolation and when it comes to mixing that one frequency range with the rest of the track you will encounter problems in ‘fitting it in’.

I have covered the above in most of my eBooks and on my website www.samplecraze.com as part of my ongoing free tutorials. So, if you find the above a little overwhelming please feel free to explore my other books or head on over to my site and read at leisure.

Extract taken from the eBook Low End.

Relevant content:

Low End – what is Low End and how to Analyse it

Sinusodial Creation and Simple Harmonic Motion

Frequency and Period of Sound

Total and Partial Phase cancellation