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Given BW in octaves to find Q and given Q to find BW in octaves.
BW = Δf = f0 / Q Q = f0 / BW f0= BW × Q = √ (f1 × f2)
BW = f2 − f1 f1 = f02 / f2 = f2 − BW f2 = f02 / f1 = f1 + BW
Conversion formula: 'octave bandwidth' N to quality factor Q:
Conversion formula: Quality factor Q to 'octave bandwidth' N:
Also known is this longer formula with 4 Qs; see its development at:
Bandwidth in octaves versus Q in bandpass filters − RaneNote 170
Frequency ratio of an octave:
Formula to convert quality factor Q to 'bandwidth in octaves' N,
but with logarithmus naturalis:
And the very short formula to convert quality factor Q
to 'bandwidth in octaves' N, but with sinh-1 :
Conversion chart or table
'bandwidth in octaves' N to quality factor Q
|1/60||86.6||1 1/4||1.12||4 1/4||0.242||7 1/4||0.082|
|1/50||72.1||1 1/3||1.04||4 1/3||0.234||7 1/3||0.079|
|1/40||57.7||1 1/2||0.92||4 1/2||0.220||7 1/2||0.075|
|1/30||43.3||1 2/3||0.82||4 2/3||0.207||7 2/3||0.071|
|1/25||36.1||1 3/4||0.78||4 3/4||0.200||7 3/4||0.068|
|1/16||23.1||2 1/4||0.58||5 1/4||0.166||8 1/4||0.058|
|1/12||17.3||2 1/3||0.56||5 1/3||0.161||8 1/3||0.056|
|1/10||14.4||2 1/2||0.51||5 1/2||0.152||8 1/2||0.053|
|1/8||11.5||2 2/3||0.47||5 2/3||0.143||8 2/3||0.050|
|1/6||8.65||2 3/4||0.45||5 3/4||0.139||8 3/4||0.048|
|1/4||5.76||3 1/4||0.36||6 1/4||0.116||9 1/4||0.041|
|1/3||4.32||3 1/3||0.35||6 1/3||0.113||9 1/3||0.039|
|1/2||2.87||3 1/2||0.33||6 1/2||0.106||9 1/2||0.037|
|2/3||2.14||3 2/3||0.30||6 2/3||0.100||9 2/3||0.035|
|3/4||1.90||3 3/4||0.29||6 3/4||0.097||9 3/4||0.034|
Q factor as a function of the bandwidth in octaves N
|3.0 wide||0.404 low|
|1/12 small||17.310 high|
A low Q factor gives a broad band (wide) bandwidth or
a high Q factor gives a narrow band (small) bandwidth.
| A high filter quality means narrow-band filtering (notch), with a large Q factor.
This results in steep filter flanks with a small bandwidth.
A low filter quality means broad-band filtering, with a small Q factor.
This results in flat filter flanks with a large bandwidth.
The larger the Q the more narrow the resonance peak.
The smaller the Q the more broad the resonance peak.
Notch filters have a large quality factor (Q), corresponding to a small bandwidth.
|The Q factor or the bandwidth does
not tell the "steepness" in dB/oct.
|Slope in dB/oct = steepness of the filter flanks
● Only with high pass and low pass filters − not with bell curves ●
|Note: The Q factor (quality factor) or the bandwidth is not convertable to the "slope" as dB/oct.
There are mastering equalizers with false information regarding the filter setting as
"Slope in dB/octave" and not Q factor (width), see:
Filter slope or steepness (dB/oct) is not bandwidth = Slope in dB/oct or steepness of filter slopes is not the bandwidth.
|Given Q factor and center frequency − Find the 3 dB cut-off frequencies
Interrelationship of 'octave bandwidth' N and the quality factor Q
Formulas for conversion of bandwidth in octaves to quality factor
Questions on "Parametric filter adjustment"
Conversion table Q to N and N to Q for parametric filters
Excel conversion − quality factor Q to bandwidth in octaves N
Filter Slope or steepness (dB/oct) is not Bandwidth
Adding decibels of one-third octave bands to level of one octave band
|With "quality" is not meant how valuable the signal is. It is meant the quality of the filter.
If the filter has flat slopes many frequencies are influenced around the cutoff frequency.
The filter has therefore a larger bandwidth and the so-called quality factor Q is specified as a low number.
If the filter has steep slopes, its bandwidth is smaller. Here a few frequencies below and above its cutoff frequency are affected and the quality factor Q is specified as a high number.
|Why is the bandwidth and the cutoff frequency found at a level of "−3 dB"?
Full width at half maximum (FWHM).
That is the point where the energy (power) is fallen to the 1/2 value or 0.5 = 50 percent of the initial energy quantity.
There the voltage is fallen to the value of √(1/2) = 1/√2 or 0.71 = 70.1 percent of the initial voltage as field quantity. A 3 dB voltage drop is a decrease of 29.29 % to 70.71 %.