Quizzical Question about Q

  • Thread starter Thread starter dobro
  • Start date Start date
dobro

dobro

Well-known member
I'd like to know more about Q. One definition I've heard is that Q is frequency divided by bandwidth. Okay, so if your parametric is centered on 1000 Hz and your bandwidth is 750 Hz, with the lower end on 500 and the upper limit at 1250, then the Q is 2.

First question: so, that means that if you center your parametric on 2K, a Q of 2 means that you're covering twice as much bandwidth (2000 divided by 1000 = 2). Have I got that right? Q is a shifting value depending on the center frequency?

I'd appreciate an answer to the above, but the reason I'm posting this is because I want to know how much of an octave I'm affecting when I set the Q at 2 or 3 or 4 or...

Does a Q of 2 affect a similar octave range no matter what the center frequency is?
 
Here is a definition from the Waves Ltd. help file for their EQ (Q10)

Q is a way of expressing the frequency width of a filter in relation to the center frequency of the filter band and
may be represented by the mathematical relationship: Fc/Fw (center frequency divided by frequency width at
the traditional ‘-3dB point’, both in Hertz). For example, a Q of 2.0 at 1,000Hz gives a bandwidth of 500Hz
(1000Hz divided by 2).
Click to expand...

You are correct, Q is a "relative" setting dependant on the frequency. If I recall correctly, a Q setting of 1.1 is equal to an octave.
 
That's right... or in short sentences, you can say "Q" is the sharpness of the peak response in an equalization circuit. ;)
 
James, I didn't understand that at all. :D

And by the way, in my original post, I made a mistake - I should have said the bandwidth was 500. Duh.

Dachay - so, if a Q of 1.1 covers a range of an octave, then 2.2 would deal with half an octave, right? And 4.4 would nail a quarter of an octave, and so finally (and this is what I'm trying to get at), 8.8 would affect a frequency range of only about a single note in an octave scale. So what would the Q for a semitone bandwidth be? Halfway between 8.8 and 17.6?
 
Yo Dobro:

I thought "Q" was a character on Star Trek....

Nice alliterative phrasing in your question though.

Green Hornet :D :D
 
Nessbass - yeah, Lionel's article is good, but it still doesn't explain the relation between Q and octaves. Lionel says 'work in octaves, not Q', but all my gear and software operates in Q.

What I want to know is this: what Q should I use to cover an octave of sound? A third of an octave? One note? A semitone?
 
That article by Lionel explains it all.

A Q of 1.5 is one octave.

The Q is = center frequency/ (high freq - low freq)

One problem with all of this is that the Q actually changes somewhat according to the amplitude of the center frequency. If you want to figure out the Q of a single note than you need to just plug in the right fequencies in the formula.

For instance to EQ out the fundamental tone of a 440hz A

440/441-439 = Q of 220

Now you just need to find an EQ with a Q that high.
 
The concept that the "number of Hz" in the range gets bigger as the center frequency gets bigger sort of sounds strange, but it's not. What we perceive as "pitch" is also logarithmic. We (I think) perceive a semitone interval (or a third, or a fifth, or whatever) to sound just as "big" at different pitches -- but the "number of Hz" in a particular interval gets bigger as the frequency gets bigger.

If you want to go up a semitone above A (440 Hz), you would go up by 26.16 Hz, to 466.16. If you want to go up a semitone above a C at 1,046.5 Hz, you would go up by 62.23 Hz, to 1,108.73 Hz. If you want to go up a semitone above an E at 41.2 Hz, you would go up by just 2.45 Hz to 43.65.*

Actually, I think the calculation of Q is a bit more complicated than it might at first seem, for just this reason. Where the "center frequency" is is also affected by the logarithmic nature of the scale. Take a fairly obvious example: "Pitch 2" is two octaves above "Pitch 1." The "center pitch" is (I think) one octave above "Pitch 1." So, say Pitch 1=100 and Pitch 2=400 (two octaves above. The center pitch is 200, not 250. So far as I understand it, anyway.

More generally, the center pitch is = P1 * (P2/P1)^1/2. To put it in words: the lower pitch times the square root of the ratio of the higher pitch divided by the lower pitch.

To get even more general, the pitch that's 1/nth of the way from P1 to P2 is = P1 * (P2/P1)^1/n. One-twelfth of an octave is a semitone, right? The pitch that's 1/12th of an octave above P1 is = P1 * 2^1/12 (P2 is on octave higher than P1, so P2/P1 = 2).

That's how I understand it, anyway. On the other hand, various sources do say things like 1 octave is a Q of 1.5, which I don't think is right.

dobro said:
What I want to know is this: what Q should I use to cover an octave of sound?
Click to expand...

1.41 (the square root of 2).

A third of an octave?
Click to expand...


4.32

One note?
Click to expand...


Assuming you mean two semitones (a "full step"): 8.65.

A semitone?
Click to expand...

17.31.


_______
*Actually, it's a bit more complicated: these examples use "even" intonation, in which the space between every pair of semitones is the same (multiply by the twelfth root of 2), and it doesn't matter what key you're in. In "just" intonation, the space between different semitones depends on where they are in the scale, so it varies ... a little bit.
 
Your formula makes complete sense to me.

Only thing is, how are you using it to calculate Q? For the octave, it's easy: Q = 2^0.5 = Sqrt (2) ~1.41. But what about the rest of them?
 
Hugh Jazz said:
Your formula makes complete sense to me.

Only thing is, how are you using it to calculate Q? For the octave, it's easy: Q = 2^0.5 = Sqrt (2) ~1.41. But what about the rest of them?
Click to expand...

This would be easier to follow if I could write exponents in real superscript and use "square-root" symbols, but:

by definition, Q = m / (h - l), where
h = high frequency, in Hz
l = low frequency, in Hz
m = "mid" frequency, in Hz

let n = the interval from l to h, in octaves
by definition, h = l * 2^n
and m = (h/l) ^ 1/2 (as discussed in my previous post)

say l = 1 to simplify formulas (doesn't matter)

h = 2^n
m = h^1/2
so m = (2^n)^1/2 = 2^n/2
and Q = (2^n/2) / (2^n - 1)

So, for an interval of 1 octave
Q = (2^1/2) / (2^1 - 1) = 2^1/2

So, for an interval of 1/3 of on octave (aka 4 semitones):
Q = (2^1/6) / (2^1/3 - 1) = 4.32

for an interval of a whole step = two semitones = 1/6 of an octave
Q = (2^1/12) / (2^1/6 - 1) = 8.65

for an interval of a semitone = 1/12 of an octave
Q = (2^1/24) / (2^1/12 - 1) = 17.31

(I did the arithmetic at the end with Excel, but you could use a calculator, or a slide rule, if you're so inclined)
 
tex, mr johnson - thank you both - I've got a much clearer idea of what's involved now. I can handle it better now - I don't need double decimal precision here - I just want to know how much of an octave I'm affecting when I choose this or that Q, and I think I have a clearer idea of that now. Thanks.
 
Awesome! Thanks a ton!

Being a math geek, I love this kind of stuff. :D
 
You must log in or register to reply here.
Back
Top