Hi,
   I think the best book you could read on the perception of loudness 
in both pure and complex tones is:- 

'An Introduction to the Psychology of Hearing' by Brian C.J. Moore 
(no relation?), a new edition of the book has just come out (Acedemic 
Press) 

    The whole book is a comprehensive introduction to the 
psychophysics of hearing, with Chapters 2 and 3 being particularly 
relevant. Chapter 3 deal with 'Frequency Analysis, masking, and the 
critical band'. If you read about auditory filters and the critical 
band concept this will point in the right direction for 
approximate scaling of loudness for complex tones derived from 
additive synthesis. Basically component tones close enough in 
frequency to fall within the 'critical band' will interact. From 
the above book 2nd Ed. p.84:-

'The increase in loudness with increasing bandwidth can be understood 
if we assume that when the bandwidth of a sound is sufficient to 
occupy more than one critical band, the loudness in adjacent, but not 
overlapping, bands is summed to give total loudness.'


    Critical Bands are approximately 10 % - 15 % of therir centre 
frequency.

 (However, Moore (Brian C.J !) notes that the whole area is one of 
contention).

On page 85 he uses Steven's Power Law that says


        Loudness (aprox.) = kI^0.3
        
k = constant (depending on units and subject !)
I = Physical Intensity

So, if you have two pure tones of the same intensity, that are close 
enough to occupy the same 'critical band' and you keep the same 
intensities, but slowly increase the frequency separation what is the 
effect on loudness perception ? When the two tones occupy different 
frequency bands then in effect you now have two bands with half the 
original intensity. Halving the intensity is equivalent to 0.81 the 
original loudness (by Steven's). So the total loudness in the two 
bands will now be

            2 * 0.81 = 1.62 times the original !
            
 
 It's a woolly area though. The other thing I would suggest would be 
to get hold of some specifications for sophisticated environmental 
noise meters. The good ones use algorithms that take psychophisical 
data into consideration. 

 Don't despair just yet !
 
                             Pete Kearton


> certainly convinced to abandon all hope at this time. in the meantime i
> suppose i will gain whatever perceptive sense there is to gain for
> manipulating timbre just working with plain old decibels. as my primary
> objective is to improve judgement for tweaking relative strengths of
> partials within a musical tone constructed via additive synthesis, rather
> than the overall perceived volume of the resulting musical tone, let alone
> its presence in the overall mix, i may just as easily struggle, as i go, to
> develop a sense for adjusting volume of sine waves using different numbers
> of decibels at different frequencies. moreover any such decibel perception
> may prove useful in other situations and would certainly not be gained
> through reliance on a mathematical model programmed by others.
 *************************************************
Peter Kearton,
University of Surrey Psychology Department,
Guildford,
United Kingdom.
GU2 5XH
Tel. UK (01483) 300800 Ext.3346
e-mail - P.kearton@surrey.ac.uk
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