If a crossover is designed with the speaker in an anechoic chamber...what would it sound like in a real room?

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it is absoultely impossible to predict how the speakers would sound without off axis- or power responses.
+1 In this case you know only the on axis response. That's all you would hear in an anechoic chamber. But in a room you'll hear the power response, which you do not know.
My guess is that the design will not lack treble in a normal room. Room gain is likely to push up the lower ranges, but that's what we expect to hear.
 
The lower mid and bass frequencies will be affected the most I'd think. Typical polar responses of loudspeakers show a narrowing directivity as frequencies rise. So, less room interaction. Loudspeaker position to walls, room modes will have more influence on lower frequencies.
 
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Perhaps your concern/confusion rests in the belief that a well designed speaker should measure flat in a room. Which is false. As has been mentioned, the standard for a well designed speaker is to measure flat anechoically but that doesn't mean the same is going to be true for the in-room measurement. Instead, the goal for the in-room measurement typically will be a smooth descending response (from lows to highs) of about 5 to 10dB.

Because, when you put an anechoically flat speaker into a room, very generally you will get:

1 - an increase in LF response due to room gain,
2 - an increase in LF response due to boundary re-enforcement, or the fact certain LF's are no longer spreading out into 4pi space but moving back closer to 2pi, and
3 - as has also been mentioned previously, because we hear more than just the direct on-axis sound, and because typically the higher frequencies have less energy off-axis than the lower frequencies, you will get a decrease in the in-room HF's.

And this is how it should be. Again as a generalization, this is what will be perceived as correct. Because remember, when that piece of music you are listening to was recorded/mixed in a studio, those speakers that were used there should also have measured flat anechoically (hopefully) but would therefore have also measured descendingly (if that's a word?) in the actual studio and so that's the same type of thing that you want to recreate in your own listening space. Granted this is an over simplification but maybe it will help a little as an overview of what's going on.
 
It is understood that all rooms are different. But typical rooms will be more like each other than they are like the anechoic chamber. There are no black-and-white answers.
room size matters a lot, and room construction too, so two room of different sizes and of different construction will sound a lot different, a wise man once said: rules of thumb are dumb rules
 
Perhaps your concern/confusion rests in the belief that a well designed speaker should measure flat in a room. Which is false. As has been mentioned, the standard for a well designed speaker is to measure flat anechoically but that doesn't mean the same is going to be true for the in-room measurement. Instead, the goal for the in-room measurement typically will be a smooth descending response (from lows to highs) of about 5 to 10dB.

Because, when you put an anechoically flat speaker into a room, very generally you will get:

1 - an increase in LF response due to room gain,
2 - an increase in LF response due to boundary re-enforcement, or the fact certain LF's are no longer spreading out into 4pi space but moving back closer to 2pi, and
3 - as has also been mentioned previously, because we hear more than just the direct on-axis sound, and because typically the higher frequencies have less energy off-axis than the lower frequencies, you will get a decrease in the in-room HF's.

And this is how it should be. Again as a generalization, this is what will be perceived as correct. Because remember, when that piece of music you are listening to was recorded/mixed in a studio, those speakers that were used there should also have measured flat anechoically (hopefully) but would therefore have also measured descendingly (if that's a word?) in the actual studio and so that's the same type of thing that you want to recreate in your own listening space. Granted this is an over simplification but maybe it will help a little as an overview of what's going on.

(y) Good explanation, it's precisely what I'm living with right now.
 
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FWIW, when I had a hypo-echoic listening room (lava cave) the on-axis response needed to be near flat. The typical descending house curve target sounded too dull. I have also found that to be true in many large spaces with a more normal RT-60; near flat sounds most balanced for palyback of recorded music. In more typical domestic settings I prefer the the 1dB/octave fall-off, or something similar. Rooms and speakers - they are complicated! :D
 
Hypothetically, let's say a crossover is designed with the speaker in an anechoic chamber. The resulting frequency response is flat, or whatever you prefer and design it to be. How would that frequency response change when that speaker is used in a real room?
I think you have to remember that, the job of a loudspeaker, is to represent the original sound on the recording - not tamper or fiddle with it. So all designs that differ too much from pretty much flat - are in my view - out! Why bother? Unless you really want a specific driver, and are somehow forced to go into compromise, to maybe achieve higher efficiency with a driver that has worse FR, when mated with another specific driver. But again.... it is actually not a needed compromise in many cases.
Let's stay roughly above 500Hz - since this is where the sound(wavelength) shift from being dominated by the room or the loudspeaker.
Like Toole said.... If you clearly know my voice in one room - you'll easily know it in any other room too. So if reproduction is done right - the room matters less. But.... it is much more comfortable and nice to listen to a voice, when it is not obstructed with lots of reflections(distortion) - so this is of course also the case with loudspeakers. But anechoic rooms or bathrooms - for listening to anything - are just two extremes. Again, why bother with these, when you really think it through.

The main reason - IMO - for using anechoic rooms and quasi/gated measurements(less precise but a good compromise) - is to know exactly what to do with the sound coming from the loudspeaker - how to work with the output. Simply put - a given FR from a given speaker driver, will change when placed in a given baffle and box. Many little things like positioning according to distance from the edge, type of edge, center-to-center from the other drivers and type of enclosure and so on. All this changes the original nice looking graph that we feel in love with on the data sheet to begin with. And we seriously need to isolate the loudspeaker from anything else, to know exactly what made these changes. So away with floor bounce, ceiling, walls and what have you. This is where anechoic rooms are perfect. We can now measure exactly what each driver does in a given box and baffle... how they interact, and where and how to place a crossover, so that the drivers work in coherence.
A driver can move 2 directions - in/out - producing soundwaves. This can be "framed" with a filter, to keep the driver within an area - lets say a midrange that operate between 500 and 3000Hz. And then we EQ it to become flat - within reason, power handling and limitations of radiation pattern - which we only truly can measure, when the room is taken completely out of the equation. Because any type of EQ or filter, can only truly correct a linear distortion. As soon as the soundwaves have left the driver and start to curve around baffle edges and reflect into the room... we now transition from a 2D EQ correctable soundwave( in/out, up/down 2-way problem) to a 3D sound field, which is complex and none correctable, at least above around the 500Hz mark or so. You could maybe dampen an area a bit to smooth things out. But again - the anechoic measurement shows you where the trouble started - because we want to fix things at the root - not just gloss it all over with makeup :p
When using EQ on measurements done in rooms - especially at the listening position... we simply move the issue around. We actually never really know what the issue was with the speaker to begin with.
We need fine detail, and details to analyze where to search and correct. Perhaps the speaker is simply badly build and not fit for the situation, and you can't force better sound out from a speaker, which is designed to try and cross an 8" to a tweeter at 3kHz... it will beam, it will lack energy here, have too much there, and you will not truly have the correct picture of the poor design, before several measurements at different angles are taken and compared directly.
All the little details of interference, resonances and off-axis issues, will show themselves clearly, when measured anechoicly. And then you are much better off, knowing what to do.

Now that you have a great loudspeaker... you put it in a room, measure again... and the difference... is your room. Because now you have two data-sets to compare. This can help you set up the system and figure out how to dampen the room, so that you get most of that original neutral sound from the loudspeaker.

Low frequencies are different. A huge anechoic room could give you some fine detail, but since we are less annoyed from distortion in bass, we can work around much more with EQ here... also because the wave length are long, so that everything is a practically a reflection anyway, before we actually hear it as bass sound. Of course, damping will help to counteract many of the sharp reflections that produce more pronounced peaks and dips - which leaves us with a FR that is much easier to EQ.

So anechoic rooms are a tool to figure out how to make the best possible loudspeaker. What you measure in a room afterwards, only matter at lower frequencies. But... if you use a speaker that is designed to a listening distance of 1m and you listen at 4 - or the other way around. Then it is not the loudspeakers fault... but simply wrong usage/choice of equipment. EQ can sometimes balance this out.... but again, this requires firm and accurate data, before using EQ to get there.

I use gated measurements to build my speakers. Up in the middle of the room, away with all furniture and on a silent calm day, with minimum of disturbance. It helps! When I put the speakers back in place, the sound is neutral in a way, that makes me forget the speakers in most cases.
At lower frequencies, I make a mix of spatial or averaged measurements, to find trends in my room at bass frequencies, so that EQ is only used to generally smooth out the response - trying not to ruin the original sound. And that's just a bit of fiddle, because a lot of music is not neutral and most rooms are absolutely not perfect for bass. So a balance is needed. And I am very happy for having 4 fully programmable presets on my DSP/pre-amp, to switch between movies and some types of music - and to test for listening fatigue, when trying out different ways to improve on my multiple sub-system.
 
Hypothetically, let's say a crossover is designed with the speaker in an anechoic chamber. The resulting frequency response is flat, or whatever you prefer and design it to be. How would that frequency response change when that speaker is used in a real room?
It depends on the room, the speakers, the positioning of the speakers and listening position in the room. It's fairly complicated to predict accurately, but maybe not impossible.
Almost every commercial loudspeaker is created in an anechoic chamber or by quasi-anechoic measrement datas.
 
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Hypothetically, let's say a crossover is designed with the speaker in an anechoic chamber. The resulting frequency response is flat, or whatever you prefer and design it to be. How would that frequency response change when that speaker is used in a real room?
The short answer is... They already are, properly designed ones anyways.

Fitting the speaker to your room is your problem.
 
FWIW, when I had a hypo-echoic listening room (lava cave) the on-axis response needed to be near flat. The typical descending house curve target sounded too dull. I have also found that to be true in many large spaces with a more normal RT-60; near flat sounds most balanced for palyback of recorded music. In more typical domestic settings I prefer the the 1dB/octave fall-off, or something similar. Rooms and speakers - they are complicated! :D


Pano, on a different topic, I want to acknowledge and thank you for this:
https://www.diyaudio.com/community/threads/what-is-gain-structure.186018/
I came across this many years back and it remains the best, clearest explanation of gain structure that I am aware of. That is saying something as I was a live sound engineer from the mid-1990's, where knowledge of gain structure is of much greater significance than it is in home audio. I had already read most the literature available on this topic prior to reading your thread.
 
And this is how it should be. Again as a generalization, this is what will be perceived as correct. Because remember, when that piece of music you are listening to was recorded/mixed in a studio, those speakers that were used there should also have measured flat anechoically (hopefully) but would therefore have also measured descendingly (if that's a word?) in the actual studio and so that's the same type of thing that you want to recreate in your own listening space. Granted this is an over simplification but maybe it will help a little as an overview of what's going on.
Umm, I think you are on the money here. Music is recorded and mixed to be played on monopole loudspeaker systems that mimic the system it was engineered with, namely monopoles - doh! The spaciousness and imaging that comes from omni, bi-pole or di-pole systems is an aberration, maybe fun or engaging for some listeners, but not what was 'in the mix' let alone 'in the room' that the recording engineer was mixing from and trying to emulate.