BS: Lisa Null's Question at FSGW Getaway

Subject: Lisa Null's Question at FSGW Getaway
From: Bruce O.
Date: 15 Oct 99 - 05:54 PM

A query asked by Lisa Null in a workshop under Charle Baum's direction at the FSGW Getaway last Sunday went unanswered at the time. I squeezed the question in under a somewhat different subject on rec.music. folk and got the following informative reply. [Crude HTML added]

[How small a frequency difference can an ear hear?]

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From: Peter Hughes
Newsgroups: rec.music.folk,uk.music.folk

I got interested in this a few years ago and ended up buying 'The Physics and Psychophysics of Music - an Introduction' by Juan G. Roederer. Publisher. Springer-Verlag. I found it very useful and I think it will answer most questions on this topic.

Its a complicated topic, but a brief precis of his figures is:

* For a single pure sine wave at about 2kHz, most people can detect a change of frequency of about 0.5% if its a smooth transition, and much smaller if its a step change. It's worse at lower frequencies - 3% at 100Hz. To put this into somthing more tangible, an even tempered semitone is an increase of 5.95%, and a 'cent' is 0.05778%.

* For a pair of tones, if you have two steady equal amplitude sine waves of frequency F, and F+deltaF, for small values of deltaF (up to 10-15Hz) what you actually hear is a single frequency equal to F + deltaF/2, beating in amplitude at a frequency of deltaF [see note @ below] We don't actually hear two clear individual signals until deltaF gets much larger. Depending on the value of F, this can be as much as a *whole tone* difference. I.E. over a 2nd, which is a surprizingly large interval.

Note that these results are for pure sine waves, not musical instruments, whose harmonic content gives our hearing more infomation to work with.

I hope this helps. Oh, and by the way, I make no claim whatsoever to be an expert in this field, just an interested engineer and occasional musician.

Peter.

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@ [My interpretation, so blame me alone if you don't like it. I think must this must be an eror and should be changed from deltaF to deltaF/2 because of the trigonometrical identity :

sin A + sin B = 2*sin {1/2(A + B)}*cos {1/2(A - B) ]

where A-B is deltaF and 1/2(A+B) = F + deltaF/2

It looks like the ear for small deltaF/2 is on the right hand side of the equation, so what we actually hear as beat notes of low frequency is the modulation by that cosine term of the 1/2(A+B) high frequency. We know full well we can't hear anything close to a 0.1 cycles per second pure single tone, but its not hard to tune an instrument to that accuracy by the very low beat frequency that we hear (and even feel) when two strings (or note and tuning fork) of higher frequency are sounded simultaneously. - WBO]