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Old 8th October 2011, 10:58 AM   #1
forr is offline forr  France
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Default The 400000 Hz² rule

The 400000 Hz² rule states that a sound reproduction system must have the product of its lowpass limit frequency by its high pass limit frequency to be around 400000 to be "auditively well balanced".

I would like to read any references and comments on it, if it has some pertinence apart from the 20*20000 Hz case, and particurlarly if somebody (particularly sound engineers) has ever used it at whatever stage of sound reproduction.

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Old 8th October 2011, 11:08 AM   #2
triode3 is offline triode3  United States
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Nyquist rate - Wikipedia, the free encyclopedia

Note that Nyquist does not take into account harmonics, and that he was dealing with the telegraph... which is why as you increase bandwidth, in some instances, sound gets "better" (e.g. increase sampling rate and quantization bits for better sound)...
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Old 8th October 2011, 11:09 AM   #3
sreten is offline sreten  United Kingdom
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IMO its a complete myth. Take a typical domestic lower limit of 50Hz,
this implies an upper limit of 8KHz is needed, its simply not true, and
no-one makes loudspeakers with such a curtailed top end.

Tonal balance and frequency extension are not the same thing.

rgds, sreten.

For telephony "speech" you need 300Hz to 4kHz, anything less
and quality really suffers, music think of a small AM radio.

Things improve massively with 150Hz to 8KHz, think of a
quality radio, but hifi IMO starts around <100Hz to >10KHz.
There is one octave left at the top, two at the bottom.

Last edited by sreten; 8th October 2011 at 11:21 AM.
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Old 8th October 2011, 11:58 AM   #4
DF96 is online now DF96  England
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This sort of guideline is just that, a guideline. A factor of 2 either way doesn't matter too much. Also, it is about balance, not fidelity. It tells you, roughly, whether a bandwidth-limited system will sound 'toppy' or 'boomy'. The telephone line example implies a product of 900k rather than 400k. People who follow it slavishly, and people who dismiss it as a myth, are both showing that they don't understand it.
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Old 9th October 2011, 03:22 PM   #5
forr is offline forr  France
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Thanks for your replies.

Many of us have heard of this "rule" but it is not much documented.
I thought it came from the beginnings of voice and sound transmission by electronic means so I've been looking at its historical background without real success. This is the current state of my research :

From the ASA, nothing.

From the AES, one pertinent reference :
Functionalized Listening Facility for the Multilingual Auditorium
Author: Jones, D. D.
JAES Volume 12 Issue 3 pp. 245-247, 250, 252, 254, 256; July 1964

From the SAE, a paper by Stroud :

I also had a reference in a book published in 1947, I think, but currently can't find it

There were some short discussions about it at DiyAudio :


Human Hearing.................

post #22 Curmudgeon 14th January 2006
the 400,000 "rule" was an observation by J. Gordon Holt, and was not intended to be an ironclad rule. The 640,000 was touted by a manufacturer whose system was 32-20,000. That was late "Golden Era" and tweeters (and other equipment, like cartridges) were soon to improve enough that they became less obvious and needed less bass to counterbalance any inappropriate prominence. With the better of today's smooth "sweet" tweeters, I'm not sure the rule has much relevance any more.

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Old 9th October 2011, 06:08 PM   #6
DF96 is online now DF96  England
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The rule is mentioned in "High Quality Sound Reproduction", by James Moir. I have the second edition, 1961. He says 400k has been suggested, but then offers 750k himself. He says that the product should be between these two values.
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Old 9th October 2011, 06:17 PM   #7
Conrad Hoffman is offline Conrad Hoffman  United States
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That "rule" is useless when applied to a high fidelity reproduction system. Do you believe there's much audible difference between systems that cut off at 5 Hz vs 10 Hz vs 20 Hz? How old are you and what's your high frequency hearing cutoff? IMO, it only applied, if it ever did, to systems where both the high and low frequency cutoffs were well within the audible band.
I may be barking up the wrong tree, but at least I'm barking!
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Old 9th October 2011, 06:30 PM   #8
AndrewT is offline AndrewT  Scotland
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The rule, if there is one, that I have heard is that the bandwidth either side of 1kHz should be roughly equal.

300Hz to 3300Hz, 200Hz to 5kHz, 100Hz to 10kHz, 50Hz to 20kHz.
The first would be good for speech intelligibility, the second for a small portable AM radio, the third for our common low quality hifi, and finally the fourth for High Fidelity. Some might even add a fifth @ 20Hz to 50kHz.
regards Andrew T.
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Old 9th October 2011, 06:35 PM   #9
ghg is offline ghg  Austria
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I remember that.

Stumbled across this rule , when I was really that young.
On the green grass, starring at the stars,
listening to Al Steward's "Year of the cat" via a cheap ghettoblaster.
No true bass, no good highs, but perfect.

... but maybe the feeling was caused by the girl near me ...
I hate "sounding amps", except these are Marshalls, driven by Eric C. or Jeff B., period.
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Old 9th October 2011, 06:43 PM   #10
DF96 is online now DF96  England
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Moir's 750k rule comes from measured data on perceived changes to quality as the bandwidth is changed separately at either end of the spectrum. The balance point (for equal goodness/badness at both ends) turns out to be around 860-870Hz, not 1kHz.

It would be interesting to repeat these measurements with modern equipment.
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