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If I take a note with frequency 101 Hz (roughly G2), then the critical bandwidth (CB) will be made up of 51 Hz on the low end and 151 Hz on the high end. This means that if I play a D3 with frequency of 152, it will fall outside the G2's CB and thus will not interfere with the G2.

However, if I analyze this with as a CB rate (CBR) (Fastl and Zwicker, Psychoacoustics, page 160, table 6.1), then both the 101 and 152 Hz notes fall within the 2nd CBR "bucket"... 0 to 100 Hz is the first CBR bucket and 100 to 200 Hz is the second. This means that the D3 will interfere with the G2.

So my confusion is that if I view this as CB, then that G2 and D3 are acoustically independent (linear) as they are outside each others CB... but if I view it as a CBR, then both notes are in the same CBR "bucket" and thus will be acoustically dependent (non-linear).

My proposed solution would be to consider the CBR as a "sliding scale" and thus create it using the G2 as a reference for the initial center frequency of the first bin and then create the remaining CBR from that. But even though there is mention on page 159 of this "slide", I'm am not at all sure this is viable or accurate or that I'm doing it right.

Ricardo J Rademacher
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2 Answers2

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This might explain your confusion, copied from the Wikipedia page quoting Zwicker:

It must be pointed out that the measurements taken so far indicate that the critical bands have a certain width, but that their position on the frequency scale is not fixed; rather, the position can be changed continuously, perhaps by the ear itself.

It seems, based on that quote, that you can't say that one note is always in band 1 and another is always in band 2, because the bands might move up or down to include both notes. You can say that two notes can't be in the same band no matter how far the bands move if they are farther apart than the widest bandwidth of the nearby critical bands.

Since G2 and D3 are less than 100 Hz apart, and band 2 is 100 Hz wide and nearby to both notes, they could reasonably be placed, psychoacoustically, in the same critical band by a listener's hearing apparatus/brain.

Todd Wilcox
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  • That quote is the "slide" I mentioned on page 159 of his book.

    But yeah, what you say is how I'm leaning towards thinking about it, not as a static scale between 0 and 100 for the first bin but a dynamic scale such that 100 would "create" a 50 to 150 bin.

    But remember that G2 and D3 need to be only 50 Hz apart for them to not interfere with each other... 100 Hz is the full bandwidth at this frequency and thus G2's CB would be from 50 to 150 Hz as I explain above. And the nature of my question is viewed as a CB, these notes don't interfere but viewed as a STATIC CBR they do.

     – Ricardo J Rademacher Oct 16 '15 at 18:23
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You have to be aware that all of those psychoacoustical rules are soft rules and you need to employ fuzzy logic (in the colloquial sense or even some of the more mathematically inclined versions) in order to apply them to in-and-out decisions.

One obvious factor is that frequency recognition itself is fuzzy: if you want to recognize the frequency of an event lasting for a tenth of a second before the next, the variance in perceived frequency will be at least 4Hz. It's sort of an "Unschärferelation" underlying frequency analysis itself.

So if you combine several decisions based on a threshold, your decisions will turn out better when the outcome of each decision is not strictly true/false and thus the operations for combining various criteria have to work with more than just two possible values for each variable. Even if you do want to arrive at a binary decision in the end, the "hard" binarization should usually occur at the end.

Admittedly, this answer is rather generic apart from mentioning the general duration-frequency Unschärferelation: it applies for a lot of diagnostic procedures. For example, if you have some diagnostic catalogue where you have rules like "If the spots are larger than 0.5in in diameter, go to page 35. If not, go to page 87." and the skin spots in question are 0.45in in diameter, are you never going to look at page 35 even when the other symptoms are so-so to inconclusive?

So if you are putting rules like that into code or other forms of absolute decision-making, going into true/false for single components of a composite decision is not likely to reflect realities well.

user24138
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  • Thanks for your contribution but your answer has no relevance to the question at hand and betrays a lack of understanding of psychoacoustics if you feel that experimentally verified relationships are "soft rules". Further, your talk of "in and out" and "code" have no relevance to the topic at hand. – Ricardo J Rademacher Oct 17 '15 at 14:42