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Given BW in octaves to find Q and given Q to find BW in octaves.
BW = Δf = f2 − f1 = f0/Q f1 = f02/f2 = f2 − BW f2 = f02/f1 = f2 + BW Q = f0/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.
|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.
|Calculating the 3 dB cut-off frequencies; given center frequency f0 and the Q factor
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.
At a filter with flat slopes many frequencies are influenced around the cutoff frequency.
The filter has therefore a larger bandwidth. The so-called quality factor is given with a low number specified. 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 %.