Re: Re: Re: Re: Golden Section and


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Posted by Eric Bamberg on March 01, 2004 at 01:03:05:

In Reply to: Re: Re: Re: Golden Section and posted by Rick Denney on February 29, 2004 at 22:59:13:

In speaker design, the goal is typically accurate sound reproduction. To keep the sound output of the speaker as close to linear over the range of frequencies it reproduces, it is necessary to design the enclosure so that the inevitable resonances that occur inside are spread as evenly as possible so that they don't reinforce each other and cause peaks or dips in the frequency response of the speaker. In the case of speaker enclosures, the internal acoustic resonances occur at harmonics of the fundamental resonance frequency C/2/W where C is the speed of sound and W is the width in inches. The concept of using a number similar to the golden mean as the ratio of dimensions began in the 1940's when the boxes were empty and quite resonant whereas today speakers are stuffed with fiberglass to absorb all the internal resonances. With these resonances most of the energy is in the lower harmonics, so the lower the harmonic number the farther it has to be from other harmonics compared to the weaker higher frequencies. The ratio is not actually the golden mean, but it's close enough for jazz.

The resonances occur at integer multiples of the fundamental frequency C/2/L where C is the speed of sound and L is the internal dimension so the frequency of resonance is inversely proportional to the box dimension i.e double the length and the frequency cuts in half. To evenly spread the resonances apart, ideally you want the harmonics of one resonance to go exactly halfway between the harmonics from another resonance, so you start with a ratio of 1.5 to put the fundamental of the 2nd resonance in between harmonics 1 and 2 of the first resonance. This causes the the 3rd harmonic of the first resonance to line up with the 2nd harmonic of the 2nd resonance though, so to fix that you change the ratio so that the second 2nd harmonic of the 2nd resonance falls halfway in between harmonics 3 and 4 of the 1st resonance, a ratio of 1.75. But that ratio causes a problem at the 7th harmonic of the 1st resonance, so you adjust again to put the harmonics between each other, and this time adjust downward, and you get a ratio of 1.625. This ratio has its first coincidence at the 13th harmonic while still keeping lower harmonics as far apart as possible.

Anyway, in complex terms that was the rationale used to decide speaker enclosure internal dimensions way back when, and although it is obsolete for that purpose now because speakers are stuffed with sound absorbent materials, it is still used to dictate the external shape of speakers and the placement of particular speaker drivers on the baffle. Diffraction effects from the edge of a speaker enclosure have a harmonic signature and by using the 1.62 ratio (or thereabouts) to dictate the off center placement of a driver on front of the speaker, the diffraction effects tend to even out rather than concentrating on one particular frequency. Human ears are very sensitive to a resonance at one particular frequency from our use of vowels in speech.

This, of course, has nothing to do with tuba playing.

-Eric


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