Vertical reflections, symmetric crossover and tilt -experiment

For sure, the forum is nice outlet, a tribe. Writing helps imagination a lot, also works as a memo so mostly selfish motives :) Lots of fun time with the hobby, try and think things through, how visual and written concepts translate into perceived sound and there is engineering layer in between. Enthusiasm toward music, source for real joy and enjoyment. Hearing system being the interface, room the medium and loudspeaker system the messenger unless a live event.
 
wesayso, have you changed the Haas kicker since 2015, I mean have you tuned the system since and how? You seemed to be very happy with it on your thread back then. I remember you have new drivers to be installed, any other advancements?

I've got from all kinds of studies and discussions that small rooms ar tough ones (regarding early reflections) as they are small and represent them selves sooner than whats on the records and you seem to have solved most of it, to my eye. Clean up reflections by reducing them with directivity, positioning and acoustic treatment and then introduce some late reverberation back in with the Haas kicker to essentially create reflection signature of a bigger room. Nice one, need to think about it more. But thats about it what we can do I think if searching better and better sound. I dont want to abandon the topic yet though, it would be cool to gain some more info on the first reflections and what we can do with them, how to make best of them. Perhaps you are right with your system and there is no other way than reduce them when room is small.

Ps. as you start swapping drivers would you have possibility to listen the speakers with only few drivers in? to listen how much difference there is to full line array? I suspect the difference would mostly be with reducing vertical reflections, first and all the rest.
 
The basics are still the same as back then, however I have experimented a lot with those ambience speakers. And will continue to do so
to try and learn more. For example, I've used those speakers using IR's as measured in real spaces as a back drop (so an IR recording of a real
room's response). It wasn't nearly as flexible as using the Haas kicker with some virtual room values added, like the Random Hall reverb can do.

One can hide a small room, I'd imagine it would be much more difficult to make a large room sound small ;).

I already swapped out all drivers and have no opportunity (nor do I feel the need, to be honest) to listen to smaller arrays. They simply don't function
the same and as such I have no specific interest in them. I'd think about CBT with shorter (finite) arrays, but not with drivers the size of a TC9 or 10F.
To me it's the infinite array that is of interest. Or better said; something large enough that it can get away with mimicking the interesting stuf of the
infinite part. The arrays have filters in them these days and are another experiment of their own. I still need to redo all of it, as simply too many things
have changed. It is a heck of a lot of work though... And it still sounds pretty awesome even with the wrong DSP (lol).

When @nc535 started simulating these type of arrays I wasn't sure of the value in that, until we added the floor reflection and I started recognizing
my measurements in his results. That's when I jumped on the Vituixcad train to see what it could bring. for me that resulted in the added filters.

Maybe a useful tip:

Try and simulate the speaker of interest with real models as reflections for floor and ceiling. We've done that for these arrays and it brings an opportunity
to view the resulting IR as if one were measuring it in the room. That itself can teach us more of what the speaker does and what part is caused by the
room. As you can play with the time factor and see when reflections hit and what they do to the frequency curve. It was a heavy load within Vituixcad,
having 75 drivers, 25 playing the floor mirror part, 25 being the array and 25 playing the ceiling part. But it's one of the things that could teach me
a thing or two about my room and speakers. I had researched my IR's as measured in the room in a same way, so it's a great confirmation of what each
part in the room does to the frequency curve, over time... The prediction from the check marks in the "room" tab is static in comparison. One can not go
back in time, filter it, to see what happens when. You can if you model the reflections like a (copy of the) real speaker.
 
^VituixCAD has the room tab that displays some single reflections nicely. Did you do additional filtering to the "manual" reflections, simulating acoustic treatment or something, or why would you do it? Wasn't the room -tab available back then? I remember kimmosto added the room tab by nc535 request.

What were you looking at from the simulated reflections? Combined sound at listening position, in other words interference, or something else?
 
Like I said, that room tab shows the results just fine. But it creates a static frequency prediction without time info.
By creating the mirror images manually, we can actually save the impulse. The damping factor is just a reduction in SPL for the mirror images.
I can create the same output as the room tab, by reducing the SPL of the mirror images accordingly.

Indeed we started it before the room tab was giving us the answers we needed. But the main attraction is being able to save the IR.

Export the IR to REW and dissect it in time. You have many tools in REW to look at it, and learn from it.
 
Ok here is something I've not read, or thought about, earlier than yesterday after watching Dave Rat on PA systems, where he uses decorrelation to help with comb filtering perceived within live audience, for better sound for all. Take a look

What Dave kind of says, reading between the lines almost, is that when there is two sound sources outputting the exact same sound we perveive a phantom image at best, if we are listening roughly equidistant to both sources. As soon as we move the correlation and phantom image breaks down, it was a construct by hearing system. If we were to mic the sound it would be very wooshy sounding because of changing comb filter effect, very hard to be equidistant from both sources to make good sound. To reduce the comb filtering (especially for a mic), to remove the phantom center for hearing system, he could utilize decorrelation, make either of the two sound sources vary in level / timing / spectrum and the comb filtering reduces which results better sound to any location in the audience. I speculate this is compromise between "imaging" and "spaciousness". One can try these out with headphones, Haas effect for example, look for listening tests, decorrelation makes spacious spread out sound while high correlation makes mono, focus.

I thinks the image, envelopment and stuff, various qualities of stereo, are highly dependent on the first reflections and their correlation / decorrelation to direct sound and each other. Here is what I mean, a simplified thought experiment: Think direct sound of a single loudspeaker and only its lateral first reflections, left and right sidewall. One wall is closer to speaker, the other is further away and both reside at different angle relative to direct sound path, either side of listening axis from loudspeakers perspective. Because the walls are highly reflective, not too much diffusion or attenuation, these could be thought of as secondary sound sources, we have now three sound sources at different locations relative to listening spot. One could have same though experiment with floor and ceiling reflections.

Alright, same "input signal" is effectively making through all three different locations, a direct sound and reflected sound from two boundaries. Now, there is some options how to affect resulting perceived qualities, how much the reflections correlate to the direct sound and perhaps to each other. Note, I'm speculating here just out of interest and not sure how much all this plays out, precedence effect and all, assuming it would work along the lines as it works with PA (outdoors) and in headphones, with multiple sound sources.

Ok, assuming we have ideal direct sound, just the way you like it, perfect frequency response you like. What if the side wall reflections have the exact same frequency response? They have delay due to path length difference with direct sound path and also some attenuation due to the longer path length introducing some decorrelation to direct sound. What would this sound like? I would assume this would sound like some kind of focus, relatively focused sound, correlation and resulting focus wouldn't be perfect as there is delay and attenuation. Now if we take some old-school loudspeaker that never got its off-axis response considered or inspected there is probably some difference between its frequency spectrum to direct sound making the side wall reflections more decorrelated to direct sound because the spectrums would also differ. They would probably decorrelate with each other as well ( but not sure how much this is audible, decorrelation to direct sound but correlation between the reflections). Now we probably perceive relatively less focused sound, brain is not able to construct a phantom image and instead we have more spacious sound. Then, if we have much attenuation to the wall reflections perhaps all that is left is the direct sound, single sound source, maximum focus. Or some combination of.

What if we had the close wall reflection correlate more than the far side wall reflection? They already have delay and level difference supporting this, of the two the close wall differs less from direct sound. What if we introduce deliberately asymmetric response to the speaker and position it so that spectrum of the close wall reflection is as much same as direct sound as possible and the far wall reflection as much different as possible? Now the speaker and close wall would correlate as well as possible perhaps extending the perceived sound to be coming between the two (would possibly extend sound stage beyond speakers in stereo situation) and decorrelation on the far side which would add spaciousness. Perhaps focus and spaciousness would be in balance as I believe both belong to good sound.

Ok lets think about a room without acoustic treatment: how much various first reflections correlate with direct sound? left and right sidewall are dependent on the horizontal response of the speaker and toe-in, but in general near sidewall could be more correlated as delay and level difference would be less. Front wall reflection would be relatively decorrelated as there is usually quite long path length difference and spectrum is very different due to bafflestep. Vertical reflections are also somewhat decorrelated by the spectrum on a traditional multiway speaker with stacked drivers and inherent lobing. Back wall reflection is quite correlated to direct sound, also depending on toe in and speaker directivity.

Reasoning how to get more spacious sound in room, wider deeper and higher sound stage, we have to reduce correlation of first reflections to direct sound. This makes near sidewall and back wall most important as they were the most correlated to begin with, add diffusion, utilize toe in, maybe attenuation. Decorrelation makes less focus so if one wants to get more focused sound then increase correlation, make more uniform spectrum to first reflections so that hearing system is able to construct the focus, phantom images. To aid how to balance focus and spaciousness, what reflections benefit from correlation or decorrelation, we need to rely on studies about this.

This is all just reasoning and speculation by me, so, I don't know what everyone prefers or would it work like so cannot give any guidelines what to do and what not to do. But it kind of makes sense to think through correlation and decorrelation, helps imagining the stuff. What do you think?
So, I'll bite on this. This guy is putting a bandaid on the discussed problem that doesnt really apply to hifi home audio reproduction. He understands the problem but his so called solution is to create a false self interpreted facsimile of the original source - at home we would usually try to do the exact opposite ie an accurate holografic stereo image. He discredits and questions someone's valid theory on sound reinforcement, elevating himself and turning it into a self promoting plug along with some narcissistic chatter in efforts to save face and look like the smarter person. Reality is that most pro audio sound guys create their own eclectic approaches in making something sound different, then label it as a cure all to a specific sound reinforcement problem. Fact is most sound guys have fragile egos, mainly because alot of them wish they could be the lead guitarist on stage instead of being stuck behind the console in back. Having been a musician and sound guy, on both sides of the stage, I understand the dynamic. Sound guys often get upset at you for not accepting their ways and methods, for which in most csses there isn't much if any plausible reasoning behind.

The point I'm making is in sound reinforcement your doing the exact opposite of creating an accurate stereo image, mainly because you're trying to give everyone in the room the same experience to some degree and accepting that you're willing to sacrifice fidelity for even audience coverage and SPL. In home audio its usually the reverse situation. The environment is more controlled and a smaller sweet spot is typically accepted in return for the smoothest FR and best three dimensional sound stage. At least that's how it is for me.

So sound reinforcement is a completely different animal and poses other sets of problems compared to hifi home audio. Some of the issues are the same between both, but the accepted compromises are completely different. With home hifi we usually believe point soirces are the correct approach to the least amount of combing and destructive interference. In reality HF combing isn't as much of an issue because its something the ear is used to. Even with a mono point source you have HF combing from room reflections. You can treat the room, but you'd have to turn the room into an anechoic chamber to fully get rid of it. You then end up with an over-dampened room that sounds dead and sterile.

My approach to combating speaker baffle diffraction related HF combing is using dampening foam and/or thicker felt around the baffle perimeter, mainly around the tweeter. It works very well, but isn't the most visually attractive solution. Dunlavy, Lipinsky and a few other companies employ foam dampening on the front baffle to great success. Limiting vertical dispersion is another effective way to deal with vertical combing and ceiling bounce. It depends on how important a wide veritcal listening window is to you. Its nice to hear a decent stereo image standing up but I'd personally rather have less combing and a better listening experience sitting down. People who have a lot of parties and entertain with music playing in the background will feel different about it, which sort of borders on a sound reinforcement approach. Pick your poison...
 
So, I'll bite on this. This guy is putting a bandaid on the discussed problem that doesnt really apply to hifi home audio reproduction. He understands the problem but his so called solution is to create a false self interpreted facsimile of the original source - at home we would usually try to do the exact opposite ie an accurate holografic stereo image. He discredits and questions someone's valid theory on sound reinforcement, elevating himself and turning it into a self promoting plug along with some narcissistic chatter in efforts to save face and look like the smarter person. Reality is that most pro audio sound guys create their own eclectic approaches in making something sound different, then label it as a cure all to a specific sound reinforcement problem. Fact is most sound guys have fragile egos, mainly because alot of them wish they could be the lead guitarist on stage instead of being stuck behind the console in back. Having been a musician and sound guy, on both sides of the stage, I understand the dynamic. Sound guys often get upset at you for not accepting their ways and methods, for which in most csses there isn't much if any plausible reasoning behind.

The point I'm making is in sound reinforcement your doing the exact opposite of creating an accurate stereo image, mainly because you're trying to give everyone in the room the same experience to some degree and accepting that you're willing to sacrifice fidelity for even audience coverage and SPL. In home audio its usually the reverse situation. The environment is more controlled and a smaller sweet spot is typically accepted in return for the smoothest FR and best three dimensional sound stage. At least that's how it is for me.

So sound reinforcement is a completely different animal and poses other sets of problems compared to hifi home audio. Some of the issues are the same between both, but the accepted compromises are completely different. With home hifi we usually believe point soirces are the correct approach to the least amount of combing and destructive interference. In reality HF combing isn't as much of an issue because its something the ear is used to. Even with a mono point source you have HF combing from room reflections. You can treat the room, but you'd have to turn the room into an anechoic chamber to fully get rid of it. You then end up with an over-dampened room that sounds dead and sterile.

My approach to combating speaker baffle diffraction related HF combing is using dampening foam and/or thicker felt around the baffle perimeter, mainly around the tweeter. It works very well, but isn't the most visually attractive solution. Dunlavy, Lipinsky and a few other companies employ foam dampening on the front baffle to great success. Limiting vertical dispersion is another effective way to deal with vertical combing and ceiling bounce. It depends on how important a wide veritcal listening window is to you. Its nice to hear a decent stereo image standing up but I'd personally rather have less combing and a better listening experience sitting down. People who have a lot of parties and entertain with music playing in the background will feel different about it, which sort of borders on a sound reinforcement approach. Pick your poison...
Don't judge Dave before checking out more of his stuff, he is very down to earth guy sharing knowledge openly, opposite what you describe. The channel is full of demonstrations made with simple equipment and procedures so that anyone can repeat them at their own with their own gear and draw their own conclusions what would be good or bad in their own application (in PA), exactly opposite to promoting his own ego.

You are right on comb filtering, in PA world the distances are so long that for example bass alley or wind are real problems. Also PA application is to give good sound to crowd not just for single person, very different application than at home you are right and on point. There is some point in this at home too, when delay (between real source the speaker, and secondary source a reflection) is very short the resulting comb filter starts at high frequencies where we might have enough directivity in the speaker so that the comb filtering is still reduced. Even though it wasn't too audible to start with its now even less. Something to utilize if it seems to be worthwhile perceptually, something trying to find out here in this thread.

I'm fully aware the thing is completely different for home stereo than to PA and application he explains in the video, he can choose what sources to put on what loudspeakers and utilize correlation and decorrelation there to have good sound to all. It doesnt have to be compromised to everyone, how about having own mono stack just for vocals for example, pristine center sound for everyone in audience without any combfiltering.

I took only the concept of correlation / decorrelation he explains and tried to apply the idea to reflections in a loudspeaker listening setup. Reflections happen our home, not so much on PA world with huge venues so its a different thing, but similar sound event in a way that there is multiple sound sources due to boundaries and reflections on them. Reread my posts on this if you have time, think single loudspeaker as one source and then two adjacent boundaries as secondary sources of the same. We cannot select what each of the three outputs because the reflections are made by our speaker, cannot put vocals on the speakers and guitars on the walls. Instead we can manipulate the correlation by changing position and various toe-in/tilt stuff, utilize the polar pattern to create correlation and decorrelation to where we want, at best. These are separate to direct sound in a way that brain tries to suppress the reflections, still they are there and we could try and help brain to make out the perception by manipulating correlation and decorrelation of the reflections.
 
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Limiting vertical dispersion is another effective way to deal with vertical combing and ceiling bounce. It depends on how important a wide veritcal listening window is to you. Its nice to hear a decent stereo image standing up but I'd personally rather have less combing and a better listening experience sitting down. People who have a lot of parties and entertain with music playing in the background will feel different about it, which sort of borders on a sound reinfor
Yes limiting the vertical coverage angle, like a line array, would be fine, taking them out, or narrow pattern multiple entry horn, but about anything else lays their sound to ceiling and floor. Angles towards the reflections in terms of loudspeaker off-axis angle change with elevation and tilt, also path lengths could change and crossover region changes with the slopes. This is the initial subject on the thread. Have you tried the tilt yet? :) Just trying to scope if there is anything useful we could do with the vertical reflections with our traditional loudspeakers, just by reorientating them.

ps. one could design a speaker that works both seated and standing up. For example traditional 90cm elevated speaker with LR slopes and symmetric vertical response has kind of ideal vertical reflections when listening standing up, roughly same path length difference through both ceiling and floor assuming ~2.5m high room and ~1.8m height person. Similar situation if speaker was elevated to ~170cm or so and listening around 90cm high, like on a sofa. Anyway, still might be perceptually different as sound through floor has perhaps different meaning than through ceiling. What happens to stereo image if you stand up or sit down? What if both were mid height, speaker and listener? "perfect" symmetry regarding height and vertical reflections. Sitting on a bar stool might get close.
 
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Have to get back on that tonight, got other stuff getting in the way right now :).
I just typed in a very long reply, only to delete it myself... pretty inefficient, must be the heat we have currently :D.

To show a little préqual:
25x TC9 FR Shaded Groundplane Six-pack.png

A modeled result of the floor and ceiling mirrors (mirrors at -6 dB).

When viewed in REW:
example1.gif

So in short: browse trough the Window (ms) value to travel trough time and see the wave form itself. Keep Rise Time low, but play with it.
It makes the floor and ceiling reflections visible over time. Making it easier to find it in other graphs.

I used this tool, and the spectrogram extensively to figure out what windowing to use in FIR processing for my arrays. While measuring a lot in my room, armed with
fiberglass damping panels to figure out what did what and making lots of measurements with those panels in every place you can imagine:
ceiling.jpg

Afterwards vacuuming the house before the rest of the family came home ;).

It resulted after quite a while in knowing where every bump in a graph like this comes from:
stereo.jpg


And more important: learning to make the right choices for my EQ in DSP processing using DRC-FIR, making it about speaker control, way more than room
correction! Except maybe for the low end, as that is where room correction actually makes sense. But first I had to learn to be able to separate the two.

If I would have had modeling like Vituix presents us, I would have have been able to see the arrays by themselves, saved as an IR trough View _>
Impulse Resonse -> Total SPL. I could have added the floor and ceiling reflections separately, while comparing it with actual results. But it would provide me
with way more information than the end frequency response, as you're able to scroll trough it. Both in Spectogram and Waterfall tabs. Not to mention the
filtered IR's that we have at our disposal within REW. Disecting a room + speaker is harder than it may look, these programs make it more accessible, as
we can separate the results and "see" it happen.

Hope this helps... not typing it again :D.
 
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Yeah it helps, thanks, appreciating your time!

That seems a lot like what I might have done as well, try figure out whats what, diving the deeps :) Sound is tough to visualize as there is one more dimension than the three + time we can see easily, frequency. But yes, looking from multiple angles, all kinds of possible visuals, help. Real time adjustability in VituixCAD (and REW I see, hornresp..) is great teaching tool, fiddling around parameters and see what happens.
 
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Found interesting read to help thinking about the stuff, coherence mostly. Gotta leave it here so I remember to read it again some day :D
https://www.merlijnvanveen.nl/en/study-hall/154-coherence-and-reverberation

There is passage for critical distance, closer which direct sound dominates and further out reverberation dominates direct sound. There is formula which includes directivity along with some other variables like damping in the room which also varies. Anyway, point is the critical distance varies with frequency.
"... This means that for real loudspeakers, at any given distance, for some frequencies you'll end up in the direct field with little to no ripple and high coherence whereas for other frequencies you might already find yourself in the reverberant field with lots of ripple and low coherence."
 
Yeah, also Griesinger is mostly on concert halls. He talks about proximity, which is kind of the similar thing as I have understood it. There is nice illustration about it in Youtube
The video contains audio samples where he has separated direct sound, (single) first reflection and reverberance and plays them back in various mixes. First reflection there don't seem to do any good, only bad. Reverberance dominates quite much when they all are combined, direct sound is mostly responsible for the "proximity", or intimacy, grabbing the attention. This is for concert hall again though, and for live sound not for stereo playback other than stereo playback is used to illustrate proximity.

I don't know how relevant this is in homes, interesting stuff nevertheless, as you say :)

From these two the question is how low of a frequency we'd want to control directivity and get proximity at home (or what ever, more quality to perceived sound), if its a thing. I suspect there is no harm drawing conclusion that some directivity as low as possible and as smooth as possible would be a good thing :D There is point of diminishing returns somewhere, perhaps around schroeder frequency, perhaps higher up what Geddes often mentions, or lower down like dipole people tout.
 
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Yeah I guess so that its very much in play at home. Critical distance, or the zone where direct sound and reverberation is in good balance (whats that?) changes with positioning and speaker directivity, furnishing / acoustic treatment. The narrower directivity speakers the further from speakers this point moves. Test, take chair with wheels and go closer / further from speakers and try to find distance where the balance is nice (try to ignore changing low end due to room modes). Need to try this after I get stereo speakers playing. I suspect the distance is different for speakers with waveguides (and arrays and fullrangers etc.) than it is for direct radiating tweeters. Nice balance could be very close to speaker due to very early reflections in small room? Perhaps speaker directivity (smoothness of DI, ERDI) has a lot to do what the perceived quality is at this point, at some nice distance with good direct / reflected sound balance. Perhaps there is no good balance if off-axis sound (reverberation) is not good. Any idea how all this would relate to stereo setup, phantom images?
 
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Hello,

Long story short I suspect tilting back speakers about measly 10 degrees give or take would positively affect imaging.

This is a thread where it would be nice to collect some observations and experiences on tilting back loudspeakers, is it worth it.

Without further ado I'll explain an observation I had recently from VituixCAD simulation and I'd like to know better what you have experienced in terms of sound stage height and / or stability with and without tilt. If you have any comments regarding vertical early reflections and speaker tilt and height, crossovers, how we perceive the vertical reflections, what ever related.

There are attachments after the longish text explanation, scroll back and worth, it was bit hard to condense.

Observation from simulations:
Multiway speaker will have narrowing vertical response around crossover frequency on most typical two(+) way speakers between a tweeter and a woofer due to the drivers being not coincident but stacked one above the other, lobing. With symmetric crossover this narrowing is symmetric to vertical angles relative to listening axis meaning that above and below listening axis the frequency response is roughly similar 10 and -10, 20 and -20 etc.

On a typical listening situation a loudspeaker is positioned so that listening axis is about at ear height which is 90cm from floor in this example. Usually rooms (here in Finland at least) are about 2.5m tall and listening distance is 3m. Due to speaker and listeners ears being closer to floor than to ceiling angles towards first vertical specular reflections differ between floor and ceiling. On this example scenario the angles are roughly 30 degrees towards floor specular reflection and about 50 degrees towards ceiling. This means that with symmetric crossover frequency response towards these reflections are different. Also, floor being closer the path length through the floor is shorter than through the ceiling and this means less attenuation through floor reflection, which is louder of the two and arrives first to ear, right after direct sound making it probably dominant of the two.

Now the hypothesis from this observation regards to "vertical imaging" is that due to differing frequency responses that arrive to ear from high up and down below, and particularly nasty zigzag move right around crossover point which usually is right at the important vocal range and where ear is sensitive, affects perceived vertical image height/soundstage clarity/how would you call it. This is something I think I heard with quick listening test and would like to know what you think and hear.

Tilting speaker back some 10-15 degrees on this example scenario will make the response towards the first reflection points similar, no zig zag move and highs towards ceiling get boost and respectively attenuation towards floor. Same thing happens with my prototype speaker. I assume but don't know if its true, that if there is more sound through floor than through ceiling, and especially since the floor reflection arrives earlier, the perceived sound image height stays low. The zig zag move perhaps makes important vocal range image little bit hazy because there is more sound towards floor in general except around the crossover there is suddenly more sound to ceiling which perhaps pulls up this frequency zone a bit in perceived image, a confusing thing. When the two reflections are roughly balanced I assume sound image is stable at the speaker height, less confusing perceived vertical image. In addition if there was more highs through ceiling and more lows through floor would possibly elongate perceived vertical image some. If you know better or have links to studies, please comment!

This makes a lot of sense to me logically and I think I heard it but not so sure if its just bias :D Keep in mind direct sound frequency response also changes as speaker is tilted, but this could be adjusted if the scenario is planned in design phase. I'd be glad if you can share thoughts on it and especially if you have tried it. did vertical image or image in general get somehow sharper and perhaps even taller?

Here are some attachments for the text above.
This is made up sim with ideal drivers, its a two way box with 5" woofer and 1" tweeter, 20cm wide and 31cm tall, rather typical bookshelf speaker. The situation would be about the same for any loudspeaker that has a small tweeter and a woofer who are not coincident but stacked one above the other, perhaps 90% of all speakers out there, could be two or more ways, bookshelf or tower. Speaker with coincident tweeter and woofer (or a fullrange driver, or an array etc.) would have pretty much similar frequency response towards these reflections and perhaps there is not much difference. Perhaps this is partly the reason people like such speakers?

Drivers and baffle, responses made with mic at center for simplicity.
View attachment 1073049 View attachment 1073050

Here is the basic simple simulation setup. Arbitrary crossover point at 2500Hz, could be more or less doesn't matter, this is just some plausible crossover for such system.
View attachment 1073052

Angle calculations, the room setup and simple right angle triangle calculator to get how much I need to increase tweeter Z coordinate with the Tilt angle in simulator. On the calculator B was set to driver c-c distance of 10cm and desired angle to alpha. This is not too accurate do it like so but accurate enough to zone in the tilt for listening experiment.
View attachment 1073053 View attachment 1073054

Here is the angles visible illustrating zig zag. Bottom graph shows -30 angle towards floor, direct sound and +50 angle towards ceiling. The zig zag happens roughly between 2-5kHz in this example, when ceiling reflection is relatively louder than floor reflection ( I think these graphs don't include attenuation due to path length ). Same graphs are also in power and DI window as light blue and light brown, these are default in VituixCAD so one doesn't have to setup the user defined angle if you are checking the stuff out in your project.
View attachment 1073055

Now with small 10 degree tilt we can align the floor and ceiling reflection responses better so that dip due to lobing on both reflections happens roughly at same frequency. The hypothesis was that now both floor and ceiling reflections are about as loud through the whole frequency spectrum keeping the vertical image more stationary, stable. Alignment is not perfect but there is about no zig zag. 15 degrees would work fine as well but in the example listening axis response suffers already quite much. I don't know how bad this would be with your speaker, perhaps affects perceived sound or not.
View attachment 1073056

Here is also 15 and 20 degree tilts. 20 degrees is too much in this case, zig zag is worse.
View attachment 1073058 View attachment 1073057

And here is completely another kind of system a three way with waveguide based on real measurements, not sure what kind of crossover slopes are, symmetric enough at least to make the zig zag. See the ER lines, light blue and brown. The first image is system adjusted before checking out this stuff, for basically power and DI and that kind of stuff. The second image shows that if I was to sacrifice some of the other lines and just adjust delay of the system to get tweeter back some and steer the nulls toward reflection angles the zig zag is gone. Not much of a sacrifice because the response can be adjusted for this new delay. Third image also tilts the waveguide little, now there is relatively more sound through ceiling on high frequencies than through floor, similar thing than perhaps a thick rug does as well. As frequency goes up more from above and less sound below listening axis. Here the 10 degree tilt works just fine.
View attachment 1073060 View attachment 1073063 View attachment 1073068


I suspect your speakers and listening setup are suitable for the test. If you try tilting your speakers I suggest you to concentrate on vocals when listening for the effects of tilting but anything goes. What do you hear or is there any difference at all on sound image?

Here are some quick rough numbers to tilt your speaker roughly 10 degrees, put some Lego (a standard block is ~1cm high) or something else under front edge of your speaker. You could try more or less tilt if you wish. Careful not to tip over your speakers, I don't want to be held responsible for broken stuff! :) If you have DSP you could just delay the tweeter (driver closest to ceiling) some extra to tilt the nulls instead of physically tilting the speaker.

Box depth affects how high the front edge needs to be lifted to achieve 10 degree tilt. Here is few for starters:
20cm deep speaker box use ~5cm riser on the front, 5 lego blocks.
30cm deep speaker box use ~7cm riser.
40cm deep speaker box use ~10cm riser.
50cm deep speaker box use ~13cm riser.

ps. I'm sorry we have people coming in for visit so I might not be able to comment extensively too much until weekend. I hope the text is extensive enough to get an idea and comments flowing :)
"I suspect tilting back speakers about measly 10 degrees give or take would positively affect imaging."

Ha! I've had my speakers tilted back about 30 degrees for the past 25 years. Why? Because the upper mids and high end need to get into the room and be part of the reverberant field.

Sorry, I did not read your entire post, nor look at your graphs. No need to - it's well known in the industry that a high quality reverberant field is essential to high quality sound for indoor environments. It's fine that you're exploring this as an amateur, but this has all been researched and documented in the professional literature for many decades.
 
Yeah approaching the issue through some observations from simulations. Although I had kind of different things in mind, more accurate imaging, rather than highly reverberant sound which is kind of anti-imaging :) Relevant still

Do you have any references to studies on this (especially that would consider tilting speaker, or vertical reflections)? I think the reverberation / envelopment / spaciousness part is true, interested if there is some literature particularly for the early reflections and vertical early reflections especially. Thanks!