Q factor bandwidth in octaves db bandwidth frequency filters conversion converter octave 3 dB bandwidth calculator - sengpielaudio
 
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● Calculation  −  Equalization  −  Filter ●

Calculating the −3 dB  cut-off frequencies  f1 and f2
when center frequency  f0 and Q factor is given.
 
3 dB Bandwidth BW = f2 − f1 = f0/Q and quality factor is Q factor
Parametric peak equalizer and notch (dip) equalizer

 
Center frequency f0   Hz 
Q factor or quality factor Q    
     
          ↓          
     
Lower cutoff frequency f1   Hz
Upper cutoff frequency f2   Hz
 
Filter resonance

Formula for the lower cutoff frequency:
Formula f1
Formula for the upper cutoff frequency:
Formula f2
Formulas mailed by Daniel Fournier

Formula for the Q factor:
Formula f2
Formula for the bandwidth:
Formula f1

 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.
 
 
 Notice:
 A low Q factor gives a broad band (wide) bandwidth or
 a high Q factor gives a narrow band (small) bandwidth.
 

Q factor as a function of the bandwidth in octaves N (octave bandwidth)

Bandwidth in
octaves N		 Q factor
         3.0 wide          0.404 low
         2.0          0.667
         1.5          0.920
         1.0          1.414
         2/3          2.145
         1/2          2.871
         1/3          4.318
         1/6          8.651
       1/12 small        17.310 high

Conversion: 'bandwidth per octave' N to quality factor Q
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 (dB/oct) is not bandwidth
Calculating the center frequency from a given bandwidth
Finding the filter center frequency - geometric mean

3 dB implies 1/2 the power and since the power is proportional to the
square of voltage, the voltage will be 0.707 or 70.7 % of the passband voltage.
√ (0.5) = 0.707

Bandwidth for Yamaha Parametric Equalizer

For a Yamaha parametric equalizer EQ there is the filter
bandwidth of an octave divided in 60/60 (12 semitones).
One half tone step (semitone) is then 5/60 − 01V Digital Mixing Console.
Conversion:
N = "bandwidth in octaves" (semi tone or half tone distance). Q = Q factor
 
Filter EQ N Q Interval        Filter EQ N Q Interval
         5/60      0.083    17.31      Semitone step               95/60      1.583      0.867  
       10/60      0.167      8.651  Whole tone             100/60      1.667      0.819  
       15/60      0.25      5.764               105/60      1.75       0.776  
       20/60      0.333      4.318  1/3 octave             110/60      1.833      0.736  
       25/60      0.417      3.45               115/60      1.917      0.7  
       30/60      0.5      2.871  1/2 octave               120/60      2      0.667  2 octaves
       35/60      0.583      2.456  Fifth             125/60      2.083      0.,636       
       40/60      0.667      2.145               130/60      2.167      0.607  
       45/60      0.75      1.902               135/60      2.25      0.581  
       50/60      0.833      1.707               140/60      2.333      0.556  
       55/60      0.917      1.548               145/60      2.417      0.532  
       60/60      1      1.,414  1 octave             150/60      2.5      0.511  2.5 octaves
       65/60      1.083      1.301               155/60      2.583      0.49  
       70/60      1.67        1.204               160/60       2.667      0.471  
       75/60      1.25        1.119               165/60      2.,75      0.453  
       80/60      1.333      1.044               170/60      2.883      0.436  
       85/60      1.417      0.979               175/60      2.917      0.419  
       90/60       1.5      0.92  1.5 octaves             180/60       3      0.404  3 octaves
 
The "BW/60" control replicates the effect of the Behringer Pro DSP1124P - Feedback Destroyer bandwidth setting.
This control sets the bandwidth of the filter between the half-gain points with:
BW (Hz) = f0 × (BW / 60) × √2
For example, at a bandwidth setting of 60/60 a filter centred on 1 kHz with a gain of −6 dB will have a bandwidth of 1,414 Hz
between the points where its response crosses −3 dB. This bandwidth remains constant as the filter's gain is adjusted.
Note that the Behringer DSP1100 - 24 band parametric equalizer software package does NOT correctly reproduce the way
the bandwidth control actually operates, its bandwidths are too small by a factor of sqrt(2).
Defining filter bandwidth in this way is not uncommon (the TMREQ filters use a similar definition).
The relationship between Q and BW for the DSP1124P is:
Q = 60 / [(BW / 60) × √2]
So the bandwidth range of 1/60 to 120/60 gives a range from Q = 42.4 to 0.35.
 
Why is the bandwidth at "−3 dB"?
That is the point where the energy (power) is fallen to the 1/2 value. 0.5 = 50 precent (−6 dB) of the initial energy quantity.
There the voltage is fallen to the √(1/2) value. 0.71 = 70.1 percent (−3 dB) of the initial voltage as field quantity.
 
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