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A one-pole highpass filter with cutoff at 880Hz (6dB per octave). A two-pole highpass filter with cutoff at 880Hz. (12dB per octave) A three-pole highpass filter with cutoff at 880Hz. (18dB per octave) A four-pole highpass filter with cutoff at 880Hz. (24dB per octave) A one-pole lowpass filter with cutoff at 320Hz (6dB per octave). A four-pole lowpass filter with cutoff at 320Hz (24dB per octave). High- and low-pass filters are combined to create a bandpass. A bandpass filter whose cutoff frequencies are 100 and 300 Hz has a center frequency of  173 Hz, not 200. The center of a bandpass filter with cutoffs at 80 and 12,000 Hz is 980, not 6,040 Hz. A low-shelf filter with a Q of .74 has no resonance peaks. A low-shelf filter with a Q of 2 has noticable resonance peaks. A low-shelf filter with a gain of 9dB. A peaknotch filter with a bandwidth of 200 Hz (Q of 60) at 12,000 Hz affects a very small range. A peaknotch filter with a bandwidth of 200 Hz (Q of 1.5) at 300 Hz affects a larger range. A peaknotch filter with a Q of 60 at 300 Hz has a bandwidth of only 5 Hz, but is equivalent in musical terms to the example of a 200 Hz bandwidth at 12,000 Hz. A 30-band, 1/3-octave graphical eq in Cubase as used to modify the snare tone in TCRM14_18.wav SSL XR625 Channel Equalizer Module API 550A Discrete 3 Band EQ Tube-Tech PE 1C based on the classic Pultec Program EQ A plug-in version of a Pultec Program EQ by Waves Manley Massive Passive EQ for mastering A plug-in version of the Manley Massive Passive EQ for mastering by Universal Audio The dbx 131, a 1/3-octave graphic EQ with 31 bands
A one-pole highpass filter with cutoff at 880Hz (6dB per octave).
A two-pole highpass filter with cutoff at 880Hz. (12dB per octave)
A three-pole highpass filter with cutoff at 880Hz. (18dB per octave)
A four-pole highpass filter with cutoff at 880Hz. (24dB per octave)
A one-pole lowpass filter with cutoff at 320Hz (6dB per octave).
A four-pole lowpass filter with cutoff at 320Hz (24dB per octave).
High- and low-pass filters are combined to create a bandpass.
A bandpass filter whose cutoff frequencies are 100 and 300 Hz has a center frequency of 173 Hz, not 200.
The center of a bandpass filter with cutoffs at 80 and 12,000 Hz is 980, not 6,040 Hz.
A low-shelf filter with a Q of .74 has no resonance peaks.
A low-shelf filter with a Q of 2 has noticable resonance peaks.
A low-shelf filter with a gain of 9dB.
A peaknotch filter with a bandwidth of 200 Hz (Q of 60) at 12,000 Hz affects a very small range.
A peaknotch filter with a bandwidth of 200 Hz (Q of 1.5) at 300 Hz affects a larger range.
A peaknotch filter with a Q of 60 at 300 Hz has a bandwidth of only 5 Hz, but is equivalent in musical terms to the example of a 200 Hz bandwidth at 12,000 Hz.
A 30-band, 1/3-octave graphical eq in Cubase as used to modify the snare tone in TCRM14_18.wav
SSL XR625 Channel Equalizer Module
API 550A Discrete 3 Band EQ
Tube-Tech PE 1C based on the classic Pultec Program EQ
A plug-in version of a Pultec Program EQ by Waves
Manley Massive Passive EQ for mastering
A plug-in version of the Manley Massive Passive EQ for mastering by Universal Audio
The dbx 131, a 1/3-octave graphic EQ with 31 bands

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The Compleat Recording Musician - Part 14
Equalization, part 1 -- Filter Types and Parameters
By John Shirley

For the last 11 installments, this series has been delving into microphones and miking techniques, tracking acoustics, preamps, line-level signals, DIs, cabling, connectors, gain structure, digital audio conversion, and audio interfaces. These are the basic elements needed for getting the best version of any sound to your recorder.

Once the sounds are recorded, editing and mixing begins. Editing is the process of creating a definitive version of a basic track by removing unwanted parts (silence, coughing, blunders) and assembling the best bits from multiple takes or repetitions. Mixing is the process of modifying the individual tracks so that they achieve the desired sound, form an appropriate balance between them, create a specific sense of stereo or surround space, and elicit the intended response in the listener.

For the next seven articles, we will explore ways of treating recorded sounds to modify their timbre, sense of space, and placement in the overall mix. Tools such as equalization, dynamics processing, effects, bussing, and panning will be examined extensively. The theory and parameters of each will be outlined along with both technical and creative uses. So without further ado… let’s start with eq!

Equalization

Equalization (eq) is the boosting or cutting (attenuating) of specific frequency areas. The simplest version of eq on consumer equipment is called tone controls – usually just labeled bass and treble. On recording and mixing equipment we have infinitely more subtle ways to affect the frequency responses of our sound sources.

Eq earned its name during the early years of radio and telephone when eq was needed to make up for the shortcomings of the technology. Its use was mostly technical in nature, intended to flatten-out the frequency response of these systems, making the frequency ranges “equal”. 

In the modern recording studio, the goals of eq can be much more creative in nature. Often, it is intentionally used to make frequency areas unequal. But before delving into specific uses let’s look at what equalizers do and how they do it.

Band-Pass Filters

A band-pass filter (BPF) allows a specified range (band) of frequencies to pass through unaltered while cutting (reducing or attenuating) anything outside that range. The most common types are highpass and lowpass. A high-pass filter (HPF) attenuates audio information below a specified cutoff frequency, allowing the higher frequencies “to pass.” A low-pass filter (LPF) does exactly the opposite: it attenuates audio information above a specified cutoff frequency and passes the frequencies below it.

While these two filter types technically fall within the definition of bandpass filtering, you’ll more commonly find the term “bandpass filter” applied to a filter that passes audio in a particular frequency range and attenuates audio both above and below that range. We’ll discuss this filter type more below.

The cutoff frequency is commonly defined as being the frequency at which the filter has attenuated the signal by 3 dB.

It is important to note that all frequencies below the cutoff may not be attenuated equally as there is a slope to the amplitude reduction. This slope is best described in decibels per octave (dB/oct), but is sometimes also noted by order number (or, especially in synthesizer design, the number of poles), where each increasing order number represents a change of -6 dB/oct. For example, a first order (1-pole) filter has a slope of -6 dB/oct while a fourth order (4 pole) filter has a slope of (-6 * 4) = -24 dB/oct.

I have been careful here to use appropriate negative numbers to show that these filters only attenuate frequency areas, rather than boosting them. If additional gain is needed, it is often accomplished after the actual filtering stage. This combination of filtering with makeup gain can make a bandpass filter appear to boost a frequency range – but, as we’ll discover later, it’s also why boosting in this manner adds more noise than cutting.

Slope

The slope of the filter is one of the most important elements of its design, yet is often overlooked by users. This is generally because most consoles, microphones and preamps fix the slope so that it cannot be changed. Even when the slope is fixed, it is important to know its exact nature to determine just how much of a specific unwanted “problem” frequency is being removed.

When you want to figure out how a filter will affect your sound, remember this: ascending octaves are frequency doubles of each other (the note A above middle C is 440 Hz, and the A an octave higher is 880 Hz), and the frequencies of descending octaves are halved from one to the next (the A below middle C is 220 Hz).

Now let’s look at real-life examples. A first order HPF with a cutoff at 880 Hz would attenuate a 110 Hz signal (three octaves lower than the cutoff frequency) by -18 dB. In practice, that means that you could still hear the low A (an octave and a third below middle C) faintly, even though the filter cutoff is set almost two octaves above middle C. By contrast, a fourth order filter with the same cutoff at 880 Hz would reduce that same 100 Hz bass note by -72 dB, making it disappear.

Digital audio software often offers a lot of eq flexibility to address audio problem frequencies. For example, if faced with a 60 Hz hum on a male vocal track where the lowest note is around 110 Hz, a first order filter would offer little hum relief without undue coloration of the vocals. A fourth order filter, however, could bring the hum down by around 24 dB with little effect on the vocals.

This brings us to an often-asked question: why not always use as steep a slope as possible? There are a couple main reasons not to. For musical reasons you may not want to be that heavy handed. Steep slopes can be a bit too obvious, giving a track an unnatural quality. In part, this is due to excessive removal of room modes and instrumental formant frequencies. Furthermore, a HPF could be used to balance one or two strong lower harmonics with the higher ones; a steep slop would remove them, not balance them….

There are also compelling technical reasons. Generally, the more aggressive the slope, the lower the audio fidelity. Though the design of filters can vary significantly, it is common for frequency dependant phase shifts and distortion to occur. In digital designs there can be other noticeable artifacting and distortion as the slope becomes steeper. So the rule of thumb is to use only as steep a slope as the music requires.

Bandpass versus band-reject

When both high and low-pass functions are combined, leaving an unaltered frequency band in the middle, the less-common band-pass filter is created. Here, a range of frequencies is allowed to pass unaltered, while anything outside of that range (both above and below) is attenuated. A band-reject filter creates the opposite situation, in which a range of frequencies is attenuated while those around it are not.

In most cases, this form of filter incorporates more extreme slopes (fourth order and above). Because of their sonically aggressive nature, dedicated band-pass filters of this type are not generally found on consoles, or even in DAW software. Their primary musical use is in analog synthesizers and effects processors. Of course, as noted above, a similar EQ function can be achieved by combing a HPF and a LPF manually, but this may not give you as clinically accurate results.

Integrated band-pass filters usually have controls for setting a center frequency and a range to let pass. Sometimes, slope is also user-assignable. The center frequency is the point (in Hz) that is exactly in the middle of the passed range. The range itself is best described as a frequency bandwidth, stating the portion of an octave (or octaves) in which the sound is attenuated by no more than -3 dB.

Bandwidth and center frequency

Before going on to other types of filters it’s prudent to go a bit deeper into concepts of bandwidth. A frequency range can be described either in musical terms (relative to octaves) or by the difference between highest and lowest frequency (in Hz). For this reason, bandwidth can also be expressed using either method. This can be a cause for great confusion, however, as musical intervals are based on an exponential function, simulating the way we hear frequency relationships, while the actual difference in Hertz is simply a linear function.

Note that the center frequency of a bandpass setting is not the “obvious” midway point between the low cutoff and the high cutoff. That’s because of the logarithmic nature of the frequencies (as seen in the doubling of frequencies of ascending octaves, where it takes 60 Hz to get from 60 Hz to the next highest octave of 120 Hz, but it takes 120 Hz to get from 120 Hz to the next highest octave of 240 Hz, etc.).

So the center frequency of a bandpass setting with cutoffs at 100 Hz and 300 Hz is not at the obvious midway point of 200 Hz, but at 173 Hz. (For the curious: multiply the low and high cutoff frequencies (100*300), then determine the square root (173). Or, for a much wider bandpass range, the center frequency of a bandpass with cutoffs at 80 Hz and 12 kHz is not at the obvious halfway point of 6040 Hz, but at 980 Hz ((SQRT(80*12000)) = 980 Hz)

To illustrate: 200 Hz is a small bandwidth when it is centered at 12,000 Hz. It is a very large bandwidth when centered at 283 Hz. At 12,000 Hz, a 200 Hz band spans around a third of a semitone (roughly from a slightly sharp f#9 to a slightly sharper f#9). But centered at 283 Hz, that same range is exactly one octave wide (200 Hz to 400 Hz is from around G3 to G4).

Musical expressions are often the easiest to handle, as they are consistent to our way of hearing (throughout the audible range) and musicians are already familiar with the terminology and pitch relationships. For this reason, bandwidth is sometimes expressed in musical terms. Frequency differences, however, are still a common means of noting bandwidth as they are more specific, and often more accurate. When selecting a bandwidth in Hz, be keenly aware of their relative nature; a given bandwidth in the lower frequency ranges is much larger than the same bandwidth, in Hertz, turns out to be in the higher frequencies.

Bandwidth and Q

There is another way of stating the relationship between bandwidth and center frequency, with the letter Q and a number. (Q stands for quality factor.) Q is the ratio of filter center frequency (f) to bandwidth (BW), and can be worked out with the formula:

Q = f/BW

Let’s work out the Q of our previous bandwidth examples.

We had a bandwidth of 200 Hz centered around 12 kHz. The Q here is 60 (12,000/200). We also had a bandwidth of 200 Hz centered around 283 Hz. Here the Q is 1.415 (283/200).

We established earlier that a bandwidth of 200 Hz centered around 12 kHz is tiny (a fraction of a tone in musical terms), while the same 200 Hz centered around 283 Hz covers an entire octave.

Conclusion: The larger the Q number, the smaller the bandwidth.

Q numbers are nice in that, like musical intervals, they retain the same perceptual size regardless of octave range. A Q of 4 will always be around a major third wide, in both high and low registers.

Approximate equivalents from Q to musical interval:

A Q of 1 = ca. an octave and a fifth. A Q of 2 = ca. a major sixth A Q of 3 = ca. an augmented fourth A Q of 4 = ca. a major third

Many digital consoles and DAWs use graphical displays to show exactly how your eq settings are affecting your audio. These are very useful to see how various bands interact, especially as it pertains to their overlapping slopes. These graphs display frequency on a logarithmic scale so that the x-axis spread for a given Q setting remains constant in size regardless of what octave a filter is used in.

Peaking Filters

Peaking filters can either boost or cut a frequency area according to a bell-curve function, with the peak of the curve set at a specific center frequency. Peaking filters are now quite commonly referred to as notch filters, though that term was originally coined to describe a band-reject filter. The basic parameters for a peaking filter include frequency (center), gain (amount of boost or cut in dB), and bandwidth (or Q). If all of these are user selectable, the equalizer is said to be parametric (or fully-parametric).

In many circumstances, such as the mid frequencies on many inexpensive mixers, only controls for frequency and gain are available. In this case, the equalizer is semi- or quasi- parametric. They are sometimes also referred to as sweepable, though this does not really differentiate them enough from fully-parametric. On lots of lower-end equipment, especially for the high and low frequencies, the filtering has preset frequency and bandwidth, with only gain controls. These bands are referred to as fixed or non-parametric.

Graphic equalizers are collections of peaking filters whose gain is controlled by vertical faders. Most often, these are non-parametric, but there are some semi- and fully-parametric models (though these are quite rare). Graphic EQ is popular on home and car stereos as well as in live sound applications due to its ease of use. Often, in professional audio applications, there are 31 bands spanning the audible spectrum in 1/3-octave widths (Q = ca. 4.3).

Shelving Filters

As with high and low-pass filters, shelving filters affect frequencies above or below a specified cutoff, and there are both high and low shelves. Unlike their band-pass brethren, they can either boost or cut the signal by a specified amount (usually labeled gain or amount). They do this by creating a sort of plateau, rather than a constantly descending slope. This is not to say that the concept of slope does not apply to the design of shelving filters….

A high frequency shelf really has three frequency areas: lower unaffected, higher completely affected, transitional. The transitional area describes a slope between the lower and higher. Audio within this area is altered by the equalizer, but not to the full gain setting specified by the user. It’s common to have high-shelving or low-shelving filters as tonal “tilt controls” to broadly and subtly alter the high treble or deep bass in a signal without cutting it off entirely.

Resonance

Most EQ designs are based on varying feedback, where some of the output of the filter is sent back through the input. This causes peaks in the frequency response of the system, due to circuit resonance. These are dependent on the exact design incorporated, gain and Q. Resonance can be seen (heard) as part of the particular character of that EQ. In some cases, EQ models offer a resonance control, especially on synthesizers’ filters. This controls an extra boost at the cutoff frequency. When the resonance is turned up high enough, the filter can actually begin to self-oscillate and “squeal” at that frequency. This is the same sort of regenerative loop oscillation that you hear when an electric guitar or open mic/PA create feedback.  

Sonic Character

As with a lot of audio gear, the price of similarly equipped equalizers can vary greatly by make and model. These price differentials (sometimes thousands of dollars per channel) can be due to the quality of the components, design characteristics, or even the “sexiness factor.” But, if all features are otherwise equal, what’s the difference between EQ designs?

The bottom line, as mentioned above, is “sonic character.” Different designs, components, distortion, noise, phase, and resonance factors (along with some other engineering voodoo) make the sound of one EQ distinct from another. This is even true of digital EQ based on such things as the algorithm used, bit depth, sample rate, processor, and whether the number crunching is done in fixed or floating point.

Some eqs simply have a pleasing sound, even though not very accurate. This accuracy versus character idea is an important element in EQ, mic preamps, dynamics processors, converters… and even multitrack digital audio software. These factors, and their relative worth ($$$), are subjective to say the least. You must be the judge of what sounds are pleasing and when to add which particular “flavor” to your mix.

Once again, the mantra is… “recording is an art.”

Next time, now armed with a thorough understanding of the types and workings of various filters, we’ll look at specific uses for each.

John Shirley is a recording engineer, composer, programmer and producer. He’s also a Professor in the Sound Recording Technology program at the University of Massachusetts Lowell and chairman of their music department. Check out his wacky electronic music CD, Sonic Ninjutsu, at http://cycling74.com/products/c74music/c74009/

Supplemental Media Examples

Here a synth bass note is put through a highpass filter with a fixed cutoff at 880 Hz and the slope is varied. Note how the tone changes. First, the original recording: TCRM14_1.wav

Now through a one-pole highpass filter with cutoff at 880Hz (6dB per octave). TCRM14_2.wav

Now through a two-pole highpass filter with cutoff at 880Hz (12dB per octave). TCRM14_3.wav

Now through a three-pole highpass filter with cutoff at 880Hz (18dB per octave). TCRM14_4.wav

Now through a four-pole highpass filter with cutoff at 880Hz (24dB per octave). TCRM14_5.wav

High- and low-pass filters are combined to create a bandpass with cutoffs at 596 Hz and 2,110 Hz. This is a common method for producing a lo-fi, band-limited effect. TCRM14_6.wav

Now, a band-reject filter is placed on the bass at 440 Hz, one of it's overtones. TCRM14_7.wav

The following examples use a snare recording to demonstrate various filters. First the original snare: TCRM14_8.wav

Now the snare is put through a one-pole lowpass filter with cutoff at 320Hz (6dB per octave). TCRM14_9.wav

The snare is put through a two-pole lowpass filter with cutoff at 320Hz (12dB per octave). TCRM14_10.wav

The snare through a three-pole lowpass filter with cutoff at 320Hz (18dB per octave). TCRM14_11.wav

The snare through a four-pole lowpass filter with cutoff at 320Hz (24dB per octave). TCRM14_12.wav

Now the snare is used to demonstrate shelving filters. First, a low-shelf set at 234Hz is used to boost the lows by 9dB. TCRM14_13.wav

The same low-shelf is now used to reduce the lows by 9dB. TCRM14_14.wav

If the Q on this low-shelf is increased, the resonance peaks begin to appear. TCRM14_15.wav

A high-shelf at 1.8 kHz with a gain of 9 dB. TCRM14_16.wav

A high-shelf at 1.8 kHz with a gain reduction of 9 dB. TCRM14_17.wav

Finally, a graphic eq is used to adjust the tone of the snare. TCRM14_18.wav

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