Invest in better measuring speakers or room equalisation ? When does one or the other make a bigger difference in your experience ?

[levimax]

, as far as I know they don’t address room echo or reverberation.

I have not tried it yet, I will as soon as it is available for PC, but it appears DIRAC ART does a good job with echo and reverberation. Check out some of the before and after REW measurements on this long thread. Hard not to be impressed https://www.audiosciencereview.com/forum/index.php?threads/active-room-treatment-art-by-dirac.40743/

 

[kemmler3D]

Room EQ can be done free (if you are already using a PC for a source) or very cheap (WiiM Mini can do it for $80-90) and so I'd argue it's worth doing in any speaker system that costs more than ~$200 total, excluding the room EQ gear.

Adding subs is a different question, but if you do it ,it makes the room EQ even more important.

Now, rebutting an unrelated comment:

we should surely not need to resort to deliberately spoiling of our amp's qualities, just because we have poorly chosen speakers, or rooms where acoustic difficulties are ignored, or we haven’t got our speakers properly set up.

What specific qualities of an amp are you spoiling by applying EQ before the amp?

I accept I’m in a minority here, but get the basics properly sorted and there should be no need to add an extra processor that the precious signal will have to endure.

You're in the minority, rightly, because in order to "get the basics sorted" (i.e. correct low frequency room modes) without DSP, you need to install very extensive acoustic treatment. Speaker placement can help, but modes always exist regardless, and do typically have a major detrimental effect at the listening position.

I've seen large (like 1 x 2m or larger) membrane absorbers improve modes by 2-3dB at the listening position at low frequencies (30hz IIRC?), but I have also RARELY seen anyone's uncorrected room without at least one +/- 10dB swing from modes. So you need several very large, heavy, decor-unfriendly, expensive absorbers to "get the basics sorted properly".

Or you can just spend $100-200 for a mic and something capable of doing DSP, and at least get the amplitude under control.

I think acoustic treatment is still worth it even if you EQ, to get decay times under control, but that's not an option everyone has.

If you are concerned about phase distortion against the "precious signal" then you can look into FIR and have it both ways.

 

[PGansz]

If your speakers have rough off-axis, compared to ‘on’ they are always going to sound coloured.
Keith

What are the qualifications for "rough" in your view? +/- what db and in what frequency ranges?

 

[dlaloum]

Old habits die hard I guess.

I do have a dedicated but these days we're finishing the top floor which is basically a living room.
Fairly minimal as you can imagine by my comments all this time, imagine a big T shape room which creates essentially two boxes at the front ending up to what I can describe as a crystal wall (which can be opened, completely, high view is nice) .

So no where to place speakers as "rules" dictate other than in the middle of this opening (imagine the upper section of the T) which is out of the question of course.
Still...

You can guess that the dedicated 20A lines are in place at the side of one "box" and most likely it well end up in a thing having a wall on one side and a crystal on the other
I guess something minimal will work there , at the 5 x 6 meters that create this virtual box I don't dream of 200 lit. mains monitors. Also treatment is out of the question, curtains as well.

Will that stop me setting up something half-decent so the place can be alive and show that actual people living in? Nope!
So, it does make sense to get a decent speaker and correct as much as I can down low with DSP. Speaker must be able to help with this though.
And if it has the meat at mid-bass that I value, even at lower SPL even better!

If we are talking mid and high frequencies, and their reflections, then curtains are one option, but there are also various sound absorbent treatments that can be used, or sound absorbent panels (which can look like artworks) - bookshelves full of books can work very well too... but yeah, the barebones "modern scandinavian" look (as shown in most B&O brochures...) is a mess for reflections!!

If the space is large enough, and that means the reflections are sufficiently delayed, then it will merely end up being an ample soundstage... our ears and brain automatically filter out reflections of the primary direct sound if it is sufficiently delayed - so as long as the main speakers are far enough from the side walls, it may not be an issue!

Bass frequencies are not susceptible to furnishings and standard absorbers/diffusers, they need bass traps which are LARGE...

Having said that, the construction material used in the space will have a major impact - if it is "solid" such as brick or concrete - then it will reflect bass - the room will pressurise (making the bass sound substantially louder) and the room will have long decay/reverb - if the room boundaries are "lightweight" (drywall on studs) - then they are effectively transparent, and the bass will pass through to either the outside of the building or to the boundary where it meets something solid.
The bass is very amenable to treatment via DSP with Dirac ART - where the electro-acoustic wizardry is used to control/eliminate the reverb/decay.

The key thing you need to avoid, is a speaker right next to a reflecting surface - think of sound as a light, and surfaces as mirrors... a surface that breaks up the audio, disperses it in different directions, is like a matt painted wall - a light you shine on it, sort of illuminates the entire room, but you don't get the beam from your torch reflecting directly back or around the room as you do in a mirror. (Books on bookshelves both absorb certain frequencies, but the varied shapes of the spines also disperse it, breaking up the soundwave... very handy!)

Keep the speakers 2m away from a reflective sidewall if possible ( you can get much closer if the wall has dispersing "treatments")

Another consideration is how the speaker disperses sound - narrow/controlled dispersion speakers will minimise the sound outside of their narrow cone ... wide dispersion or omni speakers will spread sound in very direction... controlled dispersion speakers are more easily placed...
A good room will provide quite a bit of reflected sound which will fill out the soundstage, but the reflections are sufficiently delayed so as not to smear the imaging effect. - if the room is large enough, you can even use omni's (or bipole / dipoles) and place them well out away from walls - the resulting effect when listening to stereo is fabulous, and typically there is much less of a "sweet spot" effect.... Controlled dispersion speakers are the opposite and at the extreme, can result in the "head in a vice" effect, with only a very small sweet spot.

At 5 x 6 meters, you could place speakers on the 5m side 1m away from each wall, 3m appart - a little furnishing on the side walls might break up reflections close to the speakers - if one of the 6m side walls is all glass (and curtains not an option), you probably should aim for controlled dispersion speakers - and place them toed inwards towards the listening position.

Make sure you purchase a Receiver/Integrated that supports Dirac ART - as that can be used to control the bass reverb/decay.

Good luck!

 

[dlaloum]

Evidence of this claim would be interesting. Science seems to show that "phase correction" is not or only barely audible (simplified statement), and I am skeptical that Dirac is able to correct the anechoic phase response of the speakers when measured in room with only one microphone (direct sound and reflections cannot be separated).

Impulse alignment of speakers is measurable from a single measurement point and adjustable via DSP.

However the net effect of Dirac Live RC (the basic EQ part of the Dirac suite) has been quite noticeable in terms of the enhanced clarity, even when intentionally opting to constrain any frequency-amplitude adjustment - by using the speakers native response in the room as the "EQ target". (my comments with regards to how that is achieved are my own musings and may or may not be right)

 

[dlaloum]

What are the qualifications for "rough" in your view? +/- what db and in what frequency ranges?

Although Keith can respond for himself - the dispersion pattern of the speaker should be timbrally even in a perfect world - that is to say, the frequency response of the speaker at any angle from the speaker should be flat... even though amplitude might vary...

Omni's have both frequency and amplitude equal all around... directed speakers in a perfect world attempt to have amplitude lobing, but try to avoid varying the frequency response at various angles and of course inherently fail, due to the variance in directivity of differing frequencies - so then it comes down to managing the reflections with their varying F/R to ensure that the timbral variation in the reflections doesn't mess with the overall sound. - But even with omni's once the sound hits a room barrier, different materials absorb/reflect/refract sound in different frequency dependent ways... (hence the seperation conceptually in the way we treat sounds below and above a rooms schroeder frequency)

 

[dlaloum]

I don’t think DSP or speakers alone can control the naturally occurring echo and reverberation in an untreated room. I use both Dirac and Neumann DSP processors (not at the same time), and while they’re effective at managing frequency response, as far as I know they don’t address room echo or reverberation.

Perhaps the reports and tests you’ve seen refer to less effective room treatments, such as foam or thin panels with a low NRC (Noise Reduction Coefficient).

In room decay, is exactly that: echo and reverb....

And people who by design "pressurise" the bass in the room, are leveraging echo/reverb (and resulting standing waves, etc.... with modes and nodes) - to achieve the result.

Do a basic REW measurement and take a look at the waterfall chart... it very clearly shows the echo/reverb

In the 20Hz to 150Hz region, Dirac ART very effectively controls the decay - ie: the echo and reverb - and measurably so!

Above 150Hz - we need to continue working "old school" to optimise via delaying, absorbing, and diffusing reflections - and yes as you say it is not (yet?) amenable to DSP treatment.

 

[PGansz]

Although Keith can respond for himself - the dispersion pattern of the speaker should be timbrally even in a perfect world - that is to say, the frequency response of the speaker at any angle from the speaker should be flat... even though amplitude might vary...

Omni's have both frequency and amplitude equal all around... directed speakers in a perfect world attempt to have amplitude lobing, but try to avoid varying the frequency response at various angles and of course inherently fail, due to the variance in directivity of differing frequencies - so then it comes down to managing the reflections with their varying F/R to ensure that the timbral variation in the reflections doesn't mess with the overall sound. - But even with omni's once the sound hits a room barrier, different materials absorb/reflect/refract sound in different frequency dependent ways... (hence the seperation conceptually in the way we treat sounds below and above a rooms schroeder frequency)

Off-axis meaning both vertically and horozontally? How many degrees?

Hard to find any implementations other than very well designed coaxial that can do this...there will be lobing and cancellation somewhere...

 

[dlaloum]

Off-axis meaning both vertically and horozontally? How many degrees?

Hard to find any implementations other than very well designed coaxial that can do this...there will be lobing and cancellation somewhere...

In a perfect world - all angles, vertical and horizontal - that way the timbral profile of both direct and reflected sound would match...

Hence the rather academic ideal objective of the "pulsing sphere" being the theoretically perfect speaker - hence Amar Boses early research (take a look at his early research speaker prototypes!), and the 901 speakers, also the Quad ESL63 and its electrostatic descendants with built in delay lines in the electrostatic panel to simulate a pulsing sphere about a meter behind the speaker...

One thing is the ideal perfect world... the other thing is achieving an optimal practical result in a room.... which involves accepting the flaws of what the speakers we can get can actually do, and the room and furnishings within which we need them to "live".

 

[Mnyb]

The signal purity argument does not hold water when using digital PEQ systems as they are transparent to us humans unless broken.

If you use a PC it’s very good basically perfect . Some of the built in ones like mini DSP can present some measurable artifacts when in use , but they are not audible.

Especially when not in use doing nothing, the argument that they impact signal fidelity by just being there is bunk . Then they really does nothing.

But you are trading some dB of signal to noise ratio when it’s in use , but use good DAC’s who are more than quiet enough and you’re fine .

As said be careful to not overcook the correction . And basically not use any above the transition frequency of the room where your brain can separate direct sound from reflected. If used at higher frequencies it’s to correct the speaker rather than the room .


DSP gets some bad rep due it being misunderstood it can’t magically fix bad speakers and the automated things HT receivers does is sometimes rather brutal trying to correct everything better nowadays than in the past .

But it can do stuff totally impractical with treatment, very few has the room filled with half meter thick traps needed for bass control.

But it can not replace good speakers with good of axis behavior.
In fact with really good speakers it might not be needed to control the acoustic as much as with speakers with bad directivity you only need to fix bass modes which are unavoidable. But psychoacoustic make you hear trough the room above the transition frequency and hear the speakers. Then you only migth need some discrete treatment for flutter echoes and to much reverb ( or don’t live in a minimalist glass and concrete cave , have furniture sofas rugs drapes curtains book cases with actual books etc )

The perfect storm for a bad DSP experience seems to bee speakers with bad off axis behavior when you use to much correction
”Bad” speakers also seems more finicky with placement and offers a very small sweetspot . So constant fidgeting with placement is not that the speaker are so high end that they are very sensitive to placement it’s because they are crap get new speakers.

 

[OldTimer]

$1200 speakers sound great. Then $600 (half of speakers’ cost) for DDC/DAC/Amp. For player/streamer can’t be defined.

 

[Miniyouuuu]

The signal purity argument does not hold water when using digital PEQ systems as they are transparent to us humans unless broken.

If you use a PC it’s very good basically perfect . Some of the built in ones like mini DSP can present some measurable artifacts when in use , but they are not audible.

Especially when not in use doing nothing, the argument that they impact signal fidelity by just being there is bunk . Then they really does nothing.

But you are trading some dB of signal to noise ratio when it’s in use , but use good DAC’s who are more than quiet enough and you’re fine .

As said be careful to not overcook the correction . And basically not use any above the transition frequency of the room where your brain can separate direct sound from reflected. If used at higher frequencies it’s to correct the speaker rather than the room .


DSP gets some bad rep due it being misunderstood it can’t magically fix bad speakers and the automated things HT receivers does is sometimes rather brutal trying to correct everything better nowadays than in the past .

But it can do stuff totally impractical with treatment, very few has the room filled with half meter thick traps needed for bass control.

But it can not replace good speakers with good of axis behavior.
In fact with really good speakers it might not be needed to control the acoustic as much as with speakers with bad directivity you only need to fix bass modes which are unavoidable. But psychoacoustic make you hear trough the room above the transition frequency and hear the speakers. Then you only migth need some discrete treatment for flutter echoes and to much reverb ( or don’t live in a minimalist glass and concrete cave , have furniture sofas rugs drapes curtains book cases with actual books etc )

The perfect storm for a bad DSP experience seems to bee speakers with bad off axis behavior when you use to much correction
”Bad” speakers also seems more finicky with placement and offers a very small sweetspot . So constant fidgeting with placement is not that the speaker are so high end that they are very sensitive to placement it’s because they are crap get new speakers.

Additionally without any double check commercial room correction software defined “nice to see” frequency graphs but don’t tell the whole story of the room.

My initial Dirac Live tests defined a nice frequency graph but REW waterfall was awfull (worst than no DSP).

Room correction is difficult to automate. You have to know the target you like, the range you want to affect…

 

[Sokel]

If we are talking mid and high frequencies, and their reflections, then curtains are one option, but there are also various sound absorbent treatments that can be used, or sound absorbent panels (which can look like artworks) - bookshelves full of books can work very well too... but yeah, the barebones "modern scandinavian" look (as shown in most B&O brochures...) is a mess for reflections!!

If the space is large enough, and that means the reflections are sufficiently delayed, then it will merely end up being an ample soundstage... our ears and brain automatically filter out reflections of the primary direct sound if it is sufficiently delayed - so as long as the main speakers are far enough from the side walls, it may not be an issue!

Bass frequencies are not susceptible to furnishings and standard absorbers/diffusers, they need bass traps which are LARGE...

Having said that, the construction material used in the space will have a major impact - if it is "solid" such as brick or concrete - then it will reflect bass - the room will pressurise (making the bass sound substantially louder) and the room will have long decay/reverb - if the room boundaries are "lightweight" (drywall on studs) - then they are effectively transparent, and the bass will pass through to either the outside of the building or to the boundary where it meets something solid.
The bass is very amenable to treatment via DSP with Dirac ART - where the electro-acoustic wizardry is used to control/eliminate the reverb/decay.

The key thing you need to avoid, is a speaker right next to a reflecting surface - think of sound as a light, and surfaces as mirrors... a surface that breaks up the audio, disperses it in different directions, is like a matt painted wall - a light you shine on it, sort of illuminates the entire room, but you don't get the beam from your torch reflecting directly back or around the room as you do in a mirror. (Books on bookshelves both absorb certain frequencies, but the varied shapes of the spines also disperse it, breaking up the soundwave... very handy!)

Keep the speakers 2m away from a reflective sidewall if possible ( you can get much closer if the wall has dispersing "treatments")

Another consideration is how the speaker disperses sound - narrow/controlled dispersion speakers will minimise the sound outside of their narrow cone ... wide dispersion or omni speakers will spread sound in very direction... controlled dispersion speakers are more easily placed...
A good room will provide quite a bit of reflected sound which will fill out the soundstage, but the reflections are sufficiently delayed so as not to smear the imaging effect. - if the room is large enough, you can even use omni's (or bipole / dipoles) and place them well out away from walls - the resulting effect when listening to stereo is fabulous, and typically there is much less of a "sweet spot" effect.... Controlled dispersion speakers are the opposite and at the extreme, can result in the "head in a vice" effect, with only a very small sweet spot.

At 5 x 6 meters, you could place speakers on the 5m side 1m away from each wall, 3m appart - a little furnishing on the side walls might break up reflections close to the speakers - if one of the 6m side walls is all glass (and curtains not an option), you probably should aim for controlled dispersion speakers - and place them toed inwards towards the listening position.

Make sure you purchase a Receiver/Integrated that supports Dirac ART - as that can be used to control the bass reverb/decay.

Good luck!

Thank you very much for suggestions.

I don't expect miracles though.
Set up will be pretty much as you suggest, but I know that my brains will always compare the results with that of my dedicated room setup-up.

The main two differences (or one as it's dependent) I expect is that the small one will not get physical (no chest punch ) as this demands SPL+big cabinets+strong midbass driver up to 400-500Hz straight.

Spatial qualities are important (classical music here mainly) but at a highly reflective, concrete and glass room you're right, dispersion better be at the narrow side.

Mic will tell some of the story at the end

 

[FrantzM]

On the subject of room treatments...I won't discuss Room Treatments for Bass. The reality is that ROom Treatments in the bass is quasi impossible to implement in most home, even in a medium say 8 x 5 x 3 meters Dedicated room.
Now concerning Room Treatments for the non-bass part of the spectrum:

Are there clear and well research guidelines on the matter?
What are we treating?
What should you measure before and after?
Why and How?

I believe Acoustic/Room Treatment is one of the most poorly understood subject matter in audio. It seems at first evident .. You "just" put this thing here and other things, "there", and you will magically reduce reflection, perhaps your RT60 time ... I have put things on my wall and on my ceiling and .. perhaps they do something. I would have to remove them to be sure and even then ...
Not sure that even those selling Room Treatment wares, are very clear on the objectives, and goal to reach with their treatment. Sure the room may truly sound different but ... is that better? Or even, just confirmation bias?
Of course there are horrible rooms where the presence of a few absorbers will help but ... what are the guidelines?
or I could be the ignoramus here..
OTOH, I can clearly hear the benefits of DRC, especially in the bass but also in the medium Not simple EQ, the results are usually , not that good "up there" DRC on the other hand actions are immediately heard. No need for plated or pure gold ears, to hear the differences. Truly and blindfolded-proof "night and day" ...
Waiting to be corrected and educated on the matter.

 

[Keith_W]

On the subject of room treatments...I won't discuss Room Treatments for Bass. The reality is that ROom Treatments in the bass is quasi impossible to implement in most home, even in a medium say 8 x 5 x 3 meters Dedicated room.

Well there are other options:

- tuneable membrane absorbers. I think Artnovion makes one. The membrane absorber is tuned by hanging tuning weights on the membrane.
- VPR's ("Verbundpanel Resonator", or "Compound Panel Resonator"). Essentially a large metal plate with a dampening material like foam adhered to the other side. This is said to be more broadband than a membrane absorber. It sounds pretty simple to construct but it might be pretty ugly. There is precious little on the net about how to construct a VPR, let alone measure and tune it.

The problem with both is that very large surface areas need to be treated and they are inflexible as to where they can be deployed.

Now concerning Room Treatments for the non-bass part of the spectrum:

Are there clear and well research guidelines on the matter?
What are we treating?
What should you measure before and after?
Why and How?

Some acousticians I have spoken to say that it is 50% measurement and 50% rules of thumb and experience. Re: the measurements, they use the RT60 and Clarity.

People say that small listening rooms do not have an RT60 because reverberant fields do not exist. Toole calls it "early reflection T60" to emphasize the point that we are measuring specular reflections, and not reverberant fields. What this means is that the "reverberant field" will change depending on where in the room it is measured.

However I think that above a certain frequency, depending on the size of your room, you do get reverberant fields, so I decided to test that. My eBook has a procedure for measuring the RT60. Essentially, you place your mic in different positions in the room and sweep, then compare the RT60's of all the measurements. If they do not diverge, then it must be a reverberant field. I have not seen anyone criticize it yet.

Re: guidelines. The book also has a discussion on EBU3276 and DIN 18041 guidelines, along with links to the original documents. The newer betas of REW has a RT60 calculator. You guessed it, also in the book. Essentially, "not too little, not too much". There is an optimum range that depends on your room size, application, and preference.

OTOH, I can clearly hear the benefits of DRC, especially in the bass but also in the medium Not simple EQ, the results are usually , not that good "up there" DRC on the other hand actions are immediately heard. No need for plated or pure gold ears, to hear the differences. Truly and blindfolded-proof "night and day" ...

I think that an optimum listening room needs both adequate room treatment and DSP. But Toole says that the target RT60 can be achieved with ordinary room furnishings. He is right, my RT60 sits at the upper end of the target. I do have some foam absorbers, which I should ideally hang from the ceiling or place behind the listening position, but I don't do that for aesthetic reasons. So I chuck them behind the speakers where they probably don't do much. I say "probably" because I haven't taken before-after measurements to compare.

This thread needs some @Tim Link and @Bjorn.

 

[Bjorn]

However I think that above a certain frequency, depending on the size of your room, you do get reverberant fields, so I decided to test that. My eBook has a procedure for measuring the RT60. Essentially, you place your mic in different positions in the room and sweep, then compare the RT60's of all the measurements. If they do not diverge, then it must be a reverberant field. I have not seen anyone criticize it yet.

A reverberant sound field means you don't have specular or discrete reflections or room modes. I can assure you that you have that unless the room is highly treated.

I see little reason to contribute regarding acoustics on this forum anymore. It's not science based unfortunately and is highly misleading.

 

[Tim Link]

As someone who sells acoustic products I can say that one big area we are focusing on is upper bass, which is where a lot of audible problems in small rooms happen. Traps that work in the upper bass and lower midrange to help improve clarity are not overly burdensome to add to a room and can definitely work to improve articulation in those frequency ranges. We use a modulated sweep, have people record their room with that signal playing and then see how much the dips and peaks in the modulation are getting obscured by the room.

In the picture below you can see the level read out over time. The faint blue is the actual electronic signal rapidly bouncing up and down over 40 dB, the fruity colored line superimposed over it is a room measurement of that same signal at the listening position. The red areas indicate frequencies where the sound bursts are difficult to audibly distinguish. We can make that a lot better with room treatments.

For lower bass it's hard to put something in the room that will adequately absorb it. A better approach is to let the room leak bass. Make the walls flexible and include constrained layer damping in the walls so they don't get out of control. They can act like very large membrane absorbers.

Our hearing isn't very timing sensitive below 100 Hz, which means it's difficult for us to tell the difference if there is too much bass because it is lingering in the room for too long, or if it's just too loud. That means you can just EQ down some irritating loud notes and it perceptually works well. This kind of EQing down bass peaks in the room is the only kind of DSP I've found useful in the long run. It can be a different story if you have a surround sound system and a room DSP system that allows the speakers to work as active absorbers for other speakers in the room. I've never owned a system like that but I've seen in-room measurements that show not just flatting of response but also a faster response at troublesome frequencies. I've seen some DSP treatments notably improve the tone burst clarity shown in the chart. The best looking rooms are always well treated. Rooms that have DSP treatment will show much more evenness on peak volume of each tone burst. The depth of the dropoffs still varies widely, but if it goes down at least 10 dB it's in the green zone and is very clear.

 

[Keith_W]

A reverberant sound field means you don't have specular or discrete reflections or room modes. I can assure you that you have that unless the room is highly treated.

I see little reason to contribute regarding acoustics on this forum anymore. It's not science based unfortunately and is highly misleading.

I am not out to spread misinformation. If you think I am wrong, say so and point me to some resources. I will happily change my view. If I seem stubborn, it's only because I think about it and examine evidence before I change. I'm not stubborn at all, I like to think that I am intellectually humble and willing to learn.

BTW you have made that point about specular reflections a few times now. I have wondered how you would measure it, and I am starting to think I need a Trinnov-style microphone and an upgraded REW license that enables multi-input capture. Just because I can't measure it right now doesn't mean it doesn't exist, or that you're not correct. It might mean that I am not taking the right measurement.

 

[Bjorn]

I am not out to spread misinformation. If you think I am wrong, say so and point me to some resources. I will happily change my view. If I seem stubborn, it's only because I think about it and examine evidence before I change. I'm not stubborn at all, I like to think that I am intellectually humble and willing to learn.

BTW you have made that point about specular reflections a few times now. I have wondered how you would measure it, and I am starting to think I need a Trinnov-style microphone and an upgraded REW license that enables multi-input capture. Just because I can't measure it right now doesn't mean it doesn't exist, or that you're not correct. It might mean that I am not taking the right measurement.

My post wasn't directed at your personally but generally.

So how does your ETC of each front speaker look like only at only one single point? Post them.

 

[Keith_W]

My post wasn't directed at your personally but generally.

So how does your ETC of each front speaker look like only at only one single point? Post them.

That is an excellent point, thank you. I have looked at those reflections dozens of times and I forgot about them I even made the point in the book that you can work out where they came from with some basic math.