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I'm writing this because someone sent me a note, which I looked at, but then it seemed to disappear before I could reply. So, I'm replying here, and hope the person who sent the note gets to read this.
All "equalizers" have the potential to either boost or cut or do both, depending on their design. Some, by virtue of their design, only do the one. So, for instance, Fender and Marshall amps have a "tonestack" that includes a treble, midrange, and bass control, but none of those controls actually do any boosting at all. All they can do is arrange to bleed off different ranges to ground, or preserve whatever bass, mid or treble content is already there.
Other equalizer circuits have the ability to not just passively bleed off content, but to apply gain selectively to different ranges of the spectrum. So, they can provide boost as well as cut.
In some instances, it is possible to alter where the boost and cut is applied by simply changing a few component values. Where this is easy and cheap to do, and where there is room on the chassis to do it, you will see a variety of secondary controls that determine how the primary ones work. So, for instance, on my stereo at home, there is a 2-position switch for the treble and bass controls that shifts their "zone of influence" so that the treble control covers either only treble, or treble and some upper midrange too, while the bass control can cover only bass or bass and some lower midrange as well.
The size of that zone or "selectivity" or "focus" varies in other ways too. Trebe and bass controls generally affect everything above a certain frequency range or below it. What is called "resonant equalization" will provide boost and cut to a zone or range of frequencies that has both an upper and lower limit. It is important to note that resonant equalization is not a precise thing. Even though a control might be legended on the panel as "1khz", that same control is also affecting adjacent frequencies above and below it, and for quite a ways above and below. The frequency it is labelled with is simply that zone where it has the most effect, and not where it has the only effect. So, if I were to push the 1khz slider to full boost, and the 2khz slider to full cut, the 1khz resonant circuit would be "fighting" the 2khz resonant circuit with respect to the frequency content in between them. In view of this, it is quite likely that a slider for 200hz would not be "fighting" a slider for 16khz, since they are far enough apart that each sub-circuit has only minimal influence on frequencies substantially higher or lower. However, as the range of frequencies where the sub-circuit has maximal impact starts to get closer to another sub-circuit, they will tend to counteract each other, such that you might have to apply a little more boost than you think if you have applied some cut in an adjacent band.
Those resonant sub-circuits can have varying degrees of focus or selectivity, however. For resonant circuits, this is referred to by the technical term "Q". Typically, in graphic EQ devices, the Q that is designed into the resonant circuit is a function of how many bands it has. So, if you have a 3-band graphic EQ, typically each band will be made to be fairly unselective such that there is no "dead space" between where the controls have maximal effect. The circuits of a 31-band EQ will be designed to be more selective than those of a 15-band, which will in turn be more selective than an 11-band, and so on until we get down to a 3-band. So, any resonant circuit can have several "parameters" that describe what it does, including how much boost or cut it can provide, where that boost/cut is centred, and how selectively that boost/cut is applied. If the resonant circuit offers "parametric control" over those 3 aspects, we call it a "parametric equalizer" (surprise!!).
Although being able to adjust the selectivity/focus/Q of a resonant sub-circuit permits one to do things like zero in on a specific feedback frequency without affecting nearby frequencies, sometimes, you don't really want or need all that control. For instance, maybe all you really need is the ability to provide some boost or cut in one or two places to change the "character" of the sound, and you don't want to be bothered by the cost and size of a 31-band device. So, 2 or 3 boost/cut controls that permit you to vary where the boost/cut is applied may do everything you want. Such an EQ is referred to as "semi-parametric" because it provides more control than a simple graphic EQ with sliders, but less than an EQ with full parametric control over boost/cut, center-frequency, and Q.
In general, the kind of equalization circuit/device one uses, or needs to use, is a function of what you're trying to do. People who work the house P.A. in a concert hall or regular venue will prefer a 31-band graphic , and maybe even a 4-band parametric on top of that, since there may be multiple bumps and dips in the frequency response of the room/hall that need to be adjusted to "equalize" the live sound and overcome room resonances and feedback at specific frequencies. In the case of musicians simply trying to change the character of their instrument, often a few bands will be sufficient. That could be 5 bands, each covering an octave, or perhaps a few resonant bands with semi-parametric control. Many distortion pedals, for example, come with bass, treble, midrange, and a fourth control for moving where the midrange boost/cut is applied. I've been saying for years that one could probably have more control with a pedal that include a standard bass shelving control, a variable treble cut filter, and two semi-parametric resonant controls, than most folks are able to achieve with 7-band, and sometimes even 15-band EQ.
The key is to always think about what it is you need to accomplish, and how fast you have to do it. Full parametric control allows more precision, but takes more time to use. 31-band graphic is simple to use, but permits for exerting more complex control. Seven-band is dead simple to use but limits what you can control.
I hope this answers your concerns.
All "equalizers" have the potential to either boost or cut or do both, depending on their design. Some, by virtue of their design, only do the one. So, for instance, Fender and Marshall amps have a "tonestack" that includes a treble, midrange, and bass control, but none of those controls actually do any boosting at all. All they can do is arrange to bleed off different ranges to ground, or preserve whatever bass, mid or treble content is already there.
Other equalizer circuits have the ability to not just passively bleed off content, but to apply gain selectively to different ranges of the spectrum. So, they can provide boost as well as cut.
In some instances, it is possible to alter where the boost and cut is applied by simply changing a few component values. Where this is easy and cheap to do, and where there is room on the chassis to do it, you will see a variety of secondary controls that determine how the primary ones work. So, for instance, on my stereo at home, there is a 2-position switch for the treble and bass controls that shifts their "zone of influence" so that the treble control covers either only treble, or treble and some upper midrange too, while the bass control can cover only bass or bass and some lower midrange as well.
The size of that zone or "selectivity" or "focus" varies in other ways too. Trebe and bass controls generally affect everything above a certain frequency range or below it. What is called "resonant equalization" will provide boost and cut to a zone or range of frequencies that has both an upper and lower limit. It is important to note that resonant equalization is not a precise thing. Even though a control might be legended on the panel as "1khz", that same control is also affecting adjacent frequencies above and below it, and for quite a ways above and below. The frequency it is labelled with is simply that zone where it has the most effect, and not where it has the only effect. So, if I were to push the 1khz slider to full boost, and the 2khz slider to full cut, the 1khz resonant circuit would be "fighting" the 2khz resonant circuit with respect to the frequency content in between them. In view of this, it is quite likely that a slider for 200hz would not be "fighting" a slider for 16khz, since they are far enough apart that each sub-circuit has only minimal influence on frequencies substantially higher or lower. However, as the range of frequencies where the sub-circuit has maximal impact starts to get closer to another sub-circuit, they will tend to counteract each other, such that you might have to apply a little more boost than you think if you have applied some cut in an adjacent band.
Those resonant sub-circuits can have varying degrees of focus or selectivity, however. For resonant circuits, this is referred to by the technical term "Q". Typically, in graphic EQ devices, the Q that is designed into the resonant circuit is a function of how many bands it has. So, if you have a 3-band graphic EQ, typically each band will be made to be fairly unselective such that there is no "dead space" between where the controls have maximal effect. The circuits of a 31-band EQ will be designed to be more selective than those of a 15-band, which will in turn be more selective than an 11-band, and so on until we get down to a 3-band. So, any resonant circuit can have several "parameters" that describe what it does, including how much boost or cut it can provide, where that boost/cut is centred, and how selectively that boost/cut is applied. If the resonant circuit offers "parametric control" over those 3 aspects, we call it a "parametric equalizer" (surprise!!).
Although being able to adjust the selectivity/focus/Q of a resonant sub-circuit permits one to do things like zero in on a specific feedback frequency without affecting nearby frequencies, sometimes, you don't really want or need all that control. For instance, maybe all you really need is the ability to provide some boost or cut in one or two places to change the "character" of the sound, and you don't want to be bothered by the cost and size of a 31-band device. So, 2 or 3 boost/cut controls that permit you to vary where the boost/cut is applied may do everything you want. Such an EQ is referred to as "semi-parametric" because it provides more control than a simple graphic EQ with sliders, but less than an EQ with full parametric control over boost/cut, center-frequency, and Q.
In general, the kind of equalization circuit/device one uses, or needs to use, is a function of what you're trying to do. People who work the house P.A. in a concert hall or regular venue will prefer a 31-band graphic , and maybe even a 4-band parametric on top of that, since there may be multiple bumps and dips in the frequency response of the room/hall that need to be adjusted to "equalize" the live sound and overcome room resonances and feedback at specific frequencies. In the case of musicians simply trying to change the character of their instrument, often a few bands will be sufficient. That could be 5 bands, each covering an octave, or perhaps a few resonant bands with semi-parametric control. Many distortion pedals, for example, come with bass, treble, midrange, and a fourth control for moving where the midrange boost/cut is applied. I've been saying for years that one could probably have more control with a pedal that include a standard bass shelving control, a variable treble cut filter, and two semi-parametric resonant controls, than most folks are able to achieve with 7-band, and sometimes even 15-band EQ.
The key is to always think about what it is you need to accomplish, and how fast you have to do it. Full parametric control allows more precision, but takes more time to use. 31-band graphic is simple to use, but permits for exerting more complex control. Seven-band is dead simple to use but limits what you can control.
I hope this answers your concerns.