Acoustic Calculators

These calculators are written in Javascript (saves our bandwidth). Find the calculation you want, and enter the various parameters required then click the 'Calculate' button. If you don't like 'em or think you can improve 'em, the code for the calculators is here. For the equations used in each calculator follow the Notes and Equations Used link.

1. Musical Notes - Table of notes in the range of human hearing to selectable decimal places and with user defined A4 frequency.
2. Center Frequencies - center and band frequencies for octave filters (1 - 1/48).

Serious Health Warning: In all cases the equations used can be found by following the link in the calculator. Standard algorithms are used where they are known and their source documented. Unfortunately the google search engine picked up the Center Frequency calculator before it was either finished or published. The calculator is now complete and provides its full range of target features. Some of them may even be correct. Please read the Notes and Equations Used section for this calculator before use.

<OUCH>The calculator did not always display the full range of results between the user supplied upper and lower limits when using anything other than 1000Hz as a base frequency. It now does. Many thanks to Adam Liberman for taking the time to point out this problem.</OUCH>

Musical Notes

Use this calculator to generate the standard western musical notes to a user defined number of places for the 11 Octave range C0 to B10 (Additional Information). The table is produced using the American Standard Pitch value of A4 = 440 Hz but this can be changed. Enter the number of required places in the Decimal Place box (defaults to 2 if no valu supplied). The A4 Pitch is 440 Hz by default (normal International value) but enter any other value in the range 430 to 450 Hz if using different tunings. The Number of Harmonics may be any value in the range 2 to 5. Click Calculate when ready. Reset clears all values to their default settings. The calculator displays the Results as the Note,Fundamental (1st Harmonic), 2nd Harmonic (up to the number requested). The comma separted format is ugly but has the overriding merit that it may be copied (or a selected subset), pasted into a text file and imported into a spreadsheet or any other application that supports comma separated format files. Are we helpful or what? Calculations and equations used. Parameters   Decimal Places:  (range 1 to 5) A4 Pitch:  (range 430 to 450) Harmonics:  (range 2 to 5) Result

 

Fractional Octave Centers

Generates the Preferred and Calculated centers together with the Upper and Lower band edges for a range of Fractional Octaves and covering a user defined frequency range. These values are typically used in the context of Equalization and filtering. Depending on the operational parameters being used the results will be ISO'ish. However, as an example, if you change the base frequency then the results will clearly not be ISO compatible, even if you are going downhill with a following wind. Use the Calculations and equations used link below for all the gruesome details. Enter the Decimal Places: for the upper and lower band edge (default is 2). The Calculated center is always shown to 5 decimal places. Select the required Octave Fraction: from the drop down box. Select the required Method: from the drop down list. Preferred Centers - Contiguous Bands will calculate the Upper and Lower band edges using the geometric mean of adjacent band Preferred centers. For each band displayed the Lower band edge is the geometric mean of the Preferred center of the band and the next lowest band and the Upper band is the geometric mean of the Preferred center of the band and the next highest band . This ensures no band gaps or overlaps. Preferred Centers - from Band Center will calculate both Upper and Lower band edges from the Preferred center using standard algorithms. Overlaps or gaps will result from using this method. Calculated Centers - Contiguous Bands and Calculated Centers - from Band Center provide the same functionality but all centers are computed using the Calculated center not the Preferred center. Set the Lower Limit: and Upper Limit: to any required values - they are defaulted to the audible range (20 Hz and 20000 Hz). The calculator will stop on the Octave boundary after the limits have been exceeded. Click the Calculate button to see the results. Clicking Reset will set all fields to their default values. The first line of the Results field displays the parameters used. The second and subsequent lines show a comma separated list of values: Preferred Center frequency, Calculated Center frequency, Lower band limit, Upper band limit (depending on the Method selected the Upper and Lower limits may be computed from the Preferred or Calculated center). The comma separated format is ugly but has the overriding merit that the results may be copied (or a selected subset), pasted into a text editor and imported into a spreadsheet or any other application that supports comma separated format files. Notes and equations used. Parameters   Decimal Places:  (range 0 to 5) Fractional Octave: 1 Octave Intervals1/3 Octave Intervals1/2 Octave Intervals2/3 Octave Intervals1/6 Octave Intervals1/12 Octave Intervals1/24 Octave Intervals1/48 Octave Intervals Method: Preferred Centers - Contiguous BandsPreferred Centers - from Band CenterCalculated Centers - Contiguous BandsCalculated Centers - from Band Center Lower Limit:   Hz (range 1 to 100000) Upper Limit:   Hz (range 1 to 100000) Base Frequency:   Hz (range 10 to 20000) Result

Musical Notes Calculations

1. Each octave is comprised of 12 semi-tones which are displayed as C, C#, D, D#, E, F, F#, G, G#, A, A#, B. Where C# (C sharp) = D^b (D flat), D# (D sharp) = E^b (E flat), F# (F sharp) = G^b (G flat), G# (G sharp) = A^b (A flat) , A# (A sharp) = B^b (B flat). We show the # version in all cases for brevity which probably has already sent real musicians into a paroxysm of teeth-gnashing.

2. All notes are assumed to be even tempered (equal intervals) using 1.05946309 as the interval.

3. All notes are derived from an A4 pitch base using a user defined value (defaulted to 440 Hz representing American Standard Pitch). The range C0 to G3# is calculated as lower than the pitch base, A4# to B10 are calculated as higher than the pitch base.

4. Higher notes are calculated as: this Note * 1.05946309 = next highest note

5. Lower notes are calculated as: this Note / 1.05946309 = next lowest note

6. Harmonics are calculated as integer multiples. The 2nd harmonic is 2 x the fundamental (1st harmonic), the 3rd harmonic as 3 x the fundamental etc. to the user defined number (maximum allowed is 5).

Fractional Octave Calculator

1. The base frequency (default is ISO 1000 Hz value) is always assumed to be an Octave boundary.

2. When calculating 2/3 Octaves the calculator normalizes all centers to 1/3 from the base frequency and then calculates the edge bands based on the selected set (every 2nd value). This can lead to minor rounding errors.

3. When calculating 1/2 Octaves the calculator normalizes all centers to 1/6 from the base frequency and then calculates the edge bands based on the selected set (every 3rd value). This can lead to minor rounding errors.

4. Centers are calculated using a base 10 equation:

   next lowest center = center / (10 ^ (3 / (10 x interval)))

   next highest center = center * 10 ^ (3 / (10 x interval))

   Where interval is 1 for an Octave, 2 for 1/2 Octave, 3 for 1/3 Octave, 6 for 1/6 Octave etc.

5. Calculations use either the Preferred frequency or the Calculated frequency as defined by the selected Method. Both appear permissible under the ISO standard.

6. The Preferred frequency is taken from the preferred number series R10 for 1, 1/3 and 2/3 Octave values, R20 for 1/6 and 1/2 Octave values, R40 for 1/12 Octave values, R80 for 1/24 and R160 for 1/48 Octave values. Note: The R160 series for 1/48 Octaves does not exist as a standard but is computed by generating a Renard series with an interval of the 160th root of 10 (1.0145) with some manual tweaking which is the same technique apparently (by observation) used by ISO and therefore may be said to be an ISO like Preferred Frequency. The R160 series has a problem in that it can require 3 decimal places to differentiate between adjacent values reliably which may be why ISO has not, in fact, generated such a series. The R160 series uses the R80 values and then 3 decimal places only for those items not covered by R80 and which require the additional resolution.

7. The calculator uses the following formulas to calculate upper and lower edge/crossover frequencies from a single center:

   Upper = Center * 2 ^ (number / (2 x interval))

   Lower = Center / 2 ^ (number / (2 x interval))

   Where interval is 1 for an Octave, 3 for 1/3 Octave, 6 for 1/6 Octave etc. and number is the number of required fractions such as 2 for 2/3 Octave, and 1 for 1/6 octaves etc..

   The above formulas yield two results for each band and a problem. The same calculations for adjacent bands never coincide. Thus lower band::high edge != higher band::low edge - there is almost always a gap or an overlap. Due to the nature of analog equalization circuits this problem is probably not terribly significant. In a digital filter this is much more serious. The calculator provides two methods for generating edge frequencies the first (labeled from Band Center in the Method selector) uses the above equations to generate both the Upper and Lower cutoffs from a single center. The second method (labeled Contiguous Bands in the Method selector) uses the geometric mean of two adjacent centers as the cutoff frequency using the following equation (assuming band 1 > band 2):

   band 1 to band 2 cutoff = Sqrt (band 1 center * band 2 center)

8. While not relevant to the calculator the problem with a single crossover frequency is: which band does it belong to? In the abscence of any other guidance, we use the algorithm:

   If target frequency >= this-band::low edge and < this-band::high edge = this-band.

9. The number of octaves in a frequency range f1 to f2 is given by:

   no. of octave = (log(f2) - log(f1))/log(2)

10. The center of a band, given the upper and lower bands, is:

    Center = (fl x fh)^ 1/2

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