## Averaging RT 60 values for room-acoustics

A primer on room ratios and other matters related to acoustic analysis

### Averaging RT 60 values for room-acoustics

STILL WORKING ON THIS ONE (I just save in-between stuff in order not to loose the already done work)

Original HTML text Jan. 19, 2002, formatting and layout adjusted to forum BBCode July 21 2009. This page will possibly be extended (future) with a more in-depth analysis. The text will also be adjusted to info available to the Author in 2009 rather than 2002 (as e.g. ASTM and other standards).
Any reference to  http://www.acoustics-noise.com on this page is old history which was the former site of the author.

How can one fill a complete page with a simple well known concept as a Reverberation Time.

As one knows Reverberation Time is inverse related with acoustic absorption expressed as an absorption coefficient or α. The lower the reverberation time (rooms sound dryer) the higher the average sound absorption present and vice versa.

and Blah, blah, .... rest summary folows in order to get a feel for the implications related to this page.

It does not intend to prove that, even when the acoustic math is, or isn't nicely solved, it should or shouldn't solve all related room-acoustic problems.
Therefore one can wonder, how important it is to solve those mathematical problems, taking into account that room-acoustics in its entirety covers much more, involving an enormous amount of uncertainties and question marks. Calculating/defining the real life a (not α Sabine with his known and accepted limitations) is one of the most complex matters in room-acoustics.
At the other hand the author experiences it as disturbing when the traditional basic acoustic mathematics should contribute to these physical question marks.
There really are enough real physical problems left to solve without needing to bother about the basic math itself.

This page discusses the possible theoretical approaches to average narrow band reverberation time values into broadband values. This assumes that the narrow band RT60 decay curves represent a correct statistical energetic decay based on a random (non-coherent) noise in a (pseudo) diffuse field.
Averaging can never exceed the accuracy of the data to be averaged.
Just to visualize the different approaches some arbitrary examples are chosen (don't bother about the sense of the used values).

1. USING THE CENTER 1/3 OCTAVE RT60 to represent the 1/1 octave band (NO averaging involved):

The center-frequency (f) of any 1/1 octave band equals the center-frequency of the second (center) 1/3 octave band within the same octave band.
With this approach one interprets this RT60 value of this second (center) 1/3 octave band as being representative for the corresponding entire 1/1 octave band (typical US method).

• Easy to apply.
• Basically it corresponds to a common commercial practice (noticed this on US sites) of using the Sabine αS value of this second 1/3 octave band as being representative for the aS value of the corresponding 1/1 octave band. Both RT60 and aS are inverse linear related.
• When the entire αS (Sabine absorption coefficient) curve shows a smooth logical behavior, any possible inaccuracy will remain relatively limited.
• There is neither a mathematical, nor any acoustic backing for this approach.
It's relative illogical that an approach mathematically even ignores the very existence of the adjacent frequency bands, based on the assumption that those values will behave smoothly.
• It only gives a rough approximation which can either be close to the physical truth or not.
• Since there is no physical or mathematical backing for this approach, subsequent acoustic calculations based on the input of this approach can be questionable.
SUMMARY:
One should avoid this approach, and if used one should be aware that this is only an approximation giving a rough picture of the acoustic environment.

It's neither correct from an energetic point of view, nor does it relate in any way to traditional subsequent room-acoustic calculations (as e.g. diffuse field).
As far as the designer is aware, this approach is never used by acoustic laboratories, nor did he ever find this approach in the standards he is familiar with (which, humbly admitting, indeed is only minor fraction of all existing standards).

2. ARITHMETIC AVERAGE:

By using this approach one applies an arithmetic average on the RT60 values of the 1/3 octave bands in order to obtain an averaged RT60 value that should be representative for the RT60 of the entire corresponding 1/1 octave bandwidth.
This method is a preferred method by laboratories and in Standards and Regulations, while fitting all subsequent traditional room-acoustic mathematics.

WHILE LOOKING NICE, SIMPLE and OK (why bother any other approach at all?) there is a TRAP.

From an energetic point of view this approach, while maybe better than the first one, is WRONG (which, for a significant part, is what room-acoustics is all about).

• Easy to apply.
• Commonly practiced method by laboratories and official acoustic standards/regulations.
• It nicely fits any traditional subsequent acoustic diffuse field and other room-acoustic calculations.
• While probably better than the previous approach (which ignores any possible influence and/or existence of the adjacent bands) this arithmetic averaging also IGNORES any energetic approach.
• Strictly from an acoustic and energetic point of view this approach is mathematically WRONG.
• AT THE OTHER HAND THERE IS A DISCREPANCY SINCE IT DOES FIT NICELY WITH TRADITIONAL SUBSEQUENT ROOM-ACOUSTIC CALCULATIONS.
SUMMARY:
As far as the author can see this approach will continue to be standard approach, since correct energetic averaging described further down isn't easy to apply.

Furthermore the question marks related to the mathematical and/or physical discrepancies for both Methods remain.

An example of a standard using this arithmetic averaging approach:
ISO 3382: Measurements of the reverberation time of rooms with reference to other acoustical parameters:

• Paragraph 7a): Related to spatial averaging
Arithmetic averaging of the reverberation times. The spatial average is given by taking the mean of the individual source and microphone positions.
Comment author: Even when not an averaging covering multiple frequency bands, the basic principle remains the same.
Basically this is how laboratories will average multiple measurements.

• Paragraph 8.1: Statement of results /Tables and curves.
A single figure reverberation time 'T30,mid' can be calculated by averaging T30 in the 500 Hz and 1000 Hz octave bands. (T20 may also be used). Alternatively take averages over the six one-third-octave bands from 400 Hz to 1250 Hz.
Comment author: This is a typical example where an official standard defines arithmetic averaging to represent a broader bandwidth in a single-number rating.

In Countries where lots of regulations and standards are still based on 1/1 octave band approaches (mostly related to long historical acoustic background), nowadays one will execute laboratory absorption measurements in 1/3 octave band. In order to prevent deviations and optimize repeatability, one standard publishes the corresponding 1/1 octave aS values by an arithmetic average of the respective 1/3 octave band values.
The Netherlands is a typical example of this approach.

Comments author: Since Absorption values as per laboratory measurements are inverse linear related with the reverberation time, the basic principle is equivalent to the arithmetic averaging of the reverberation times itself.
However the same Country recognizes the fact that mathematical averaging 1/3 octave band RT-values into 1/1 octave band values isn't that simple in order to obtain valid numbers when it matters.
Their Standard NEN 5077 : Noise control in buildings, related to sound proofing determination methods by field measurements, while explaining the mathematical formulas to convert measurement results obtained from 1/3 octave filters into 1/1 octave band values, makes an explicit exception for reverberation time measurements, not allowing direct mathematical conversion.

3. USING THE HIGHEST 1/3 OCTAVE BAND RT60 to represent the 1/1 octave band (NO averaging involved):

By using this approach one tries to approximate the RT60 as a real Energetic Decay Time, thereby trying to avoid the shortcomings of the 'arithmetic averaging'.
HOWEVER THIS METHOD IGNORES THE INITIAL ENERGY (noise, sound) DECAY.
While the arithmetic averaging method almost systematically will lead to RT60 values which are TOO LOW, this highest value approach will ALWAYS systematically lead to RT60 values which are TOO HIGH.
As such it solves one problem while simultaneously introducing another one.

• Easy to apply.
• It's meant to approximate the energetic decay, which is ignored by the arithmetic average approach.
• It will provide some safety when 'to be added absorptive materials' have to be calculated in function of redevelopment or upgrading projects.
• It completely ignores the initial noise (energy) decay causing systematically TOO HIGH reverberation times to be calculated.
• The return values of this approach do NOT fit in subsequent acoustic diffuse field calculations, which can or can't be a problem depending of the goal of the obtained RT60 figures.
SUMMARY:
While the 'Center 1/3 octave' approach is arbitrary, the 'Arithmetic Average' will mostly (or often) underestimate the energetically averaged RT60, this method will systematically overestimate the RT60 value.
4. Calculating a MATHEMATICAL EXACT Energetically Averaged RT60 value. (sorry author is a bit childish)

• This procedure calls for more complicated formulas.
• The author designed some acoustic algorithm's allowing to calculate a correct energetic averaged reverberation time.
• However this a a much more complicated procedure than the methods described above. There is no analytic solution to solve this averaging.
• As such it calls for iterative procedures. With spreadsheet programs as MS Excel it can be solved with the Goal Seek function.
It is possible to program such function as a standard Excel users function, making it accessible/applicable in the same manner as any other MS Excel worksheet function.
• Returns a theoretical mathematical exact averaged RT60 value
• Can also calculate weighted averages.
• The advantage is extreme: It allows to calculate weighted Single-number RT60 ratings for any arbitrary noise source or room purpose.
• In other words: one can calculate the weighted reverberation time for a complete spectrum of any sound or noise source in one single RT60 value.
While common practice for acoustic insulation single-number rating values, it is rarely used for room-acoustic purposes.
• Much more complicated procedure than the previous discussed approaches.
• It calls for some background in acoustics and logarithmic mathematics. Notions as EDT T10, T20, T30, T60 should be understood.
• There is still room for question marks in function of the 'to be used' integration time in order to evaluated the averaged decay curve.
• While theoretically and energetically correct, it does not fit into the traditional applied subsequent acoustic diffuse field calculations (room-acoustic calculations).
• Here the author was confronted with some strange discrepancies which call for much more in-depth analysis and investigation.
The method will be explained in much more detail further down on this page.
The idea of this page is to give a better feel for, and insight in the influencing factors.

The related algorithms (formulas) are completely shown here.
If a visitor wants to use them please respect the source, by clear reference to the author/designer and this site and page

What does this all means expressed in numbers?

Don't mind the respective RT values used in those example. They are just pragmatic arbitrary values for the purpose of these examples.
The calculations per Room (columns C:D) are executed as well as per the most common (simple) Sabine approach, as well the Eyring approach combined with the Room Constant for the diffuse field calculations. The used formulas are noted in the last section of this page.
As such any visitor (even without the related acoustic/mathematical background) can rebuild this model in any spreadsheet program.

The RT60 values shown in the orange cells (ranges A1:E11) always equal one another (which speaks in favor of the arithmetic average approach).
The differences related to T10, T20, T30, T60 (rows 10a to 10d) will become clear later on. Note: you can see all used formulas used in those tables at the bottom of this page (with reference in which cells they are used) + some additional explanation.

A few important notes related to the above tables:

1. The first approach, hence taking the center band of the 1/3 octave bands as being representative for the corresponding 1/1 octave band is NO good approach.
Not applying any way of averaging, thereby ignoring the existence of the adjacent 1/3 octave bands, is a mathematical very questionable approach.
This approach can be satisfying when the RT60 values of all bands are close together, within a smooth and logical relationship to one another etc.
When applying a preferred principle one should not count on luck or unnecessary assumptions. Any used principle should be backed by some logical arguments.
Also take into account that this approach does not give any reasonable possibility for subsequent acoustic calculations.
Basically this 'to be avoided' argument is also valid for the publication of absorption values which is inverse linear related to the RT60 values.

2. The second approach, hence applying an arithmetic average (commonly used), while energetically WRONG, results in exact subsequent calculations (at least from a mathematical point of view, from an acoustical point of view this entire page is cause for MORE questions than answers).

Look at the orange cells in the above tables:
The cells E11 (averaged Diffuse Field = LD = Level Diffuse) are not copied from, or linked to the cells E8, but are a real energetic (logarithmic) average from the diffuse fields levels calculated directly from the individual frequency bands shown in cells E4:E6 (green cells are related with green total beneath them).
This means that if one accepts that the traditional acoustic formulas commonly used to calculate the diffuse field are correct, the value in the cell E11 MUST ALSO BE CORRECT, representing an energetic averaged diffuse field.
However when we then calculate backwards (starting from the energetic averaged diffuse field dB values towards the RT60 value) one notices that the resulting RT60 values exactly equal the arithmetical averaged RT60 values.
The only conclusion should be that the arithmetical averaged RT60 values CORRECTLY represents an accurate energetic approach.
HOWEVER THIS IS NOT TRUE!

It's not difficult to prove that this way of averaging RT60 values is a WRONG energetic representation.
When one should measure the same RT in 1/3 octave and in 1/1 octave, one will notice that this arithmetical average is almost systematically TOO LOW.

3. The Third approach, hence using the highest 1/3 octave band RT60 value to represent the RT60 of the corresponding 1/1 octave band in order solve this energetic inaccuracy of the second approach will lead to (almost) systematically to TOO HIGH RT60 values.
One can imagine that acousticians will use this approach, in order to compensate (partly neutralize) for other subsequent inaccuracies, when upgrading projects are involved. Indeed the traditional calculation methods will often cause the number of 'to be added m² absorptive material' in function of a target RT60 to be too low. By starting from a too high RT60 this shortage will be (partly) compensated.
However allowing inaccuracies to compensate for other subsequent inaccuracies, while possibly having a practical empirical advantage, from an acoustical, mathematical point of view it isn't a clean path to walk.
Still this approach tries to solve the shortcoming of the previous approach by using this highest value as an approximation.
Note that also here (as also is with the first and fourth approach) there is a problem with the subsequent math.

4. The fourth approach is based on algorithms designed by the author in order to obtain a more accurate approach of such an Energetic Averaging (see below).
The problem is that there is no analytic solution to solve this averaging.
This means that one must use iterative procedures to get a calculation result (at least the author didn't found the analytic solution yet - if existing).
One can solve this by entering the function in a spreadsheet program and making use of the Goal Seek (is an iterative procedure) functionality of the program.
Easier of course is that such a function is programmed, meaning that the iterative procedure is out of sight for the user.
NOW, ONE IS CONFRONTED WITH A STRANGE DISCREPANCY:
WHILE THE ALGORITHM'S CALCULATE ENERGETICALLY CORRECT THEY DO NOT FIT INTO THE TRADITIONAL ROOM-ACOUSTIC MATHEMATICS.

As such, while the author is rather proud on the design of those algorithms (childish side of the author), knowing that they return correct energetically averaged RT 60 values, the author humbly admits that he DOES NOT KNOW exactly how to fit this in the traditional acoustic calculation methods.

Some Background: What is RT, RT60, T, T10,T20, T30, T60, Tdyn, ..., EDT ?
It is important to understand those title notions in order to understand how and why the newly designed energetic averaging algorithm works.

Reverberation Time :

Definition:
The time required, within an enclosed space (partly or not), for the initial sound pressure level to decay with 60 dB (= 1/1000 of the initial RMS sound pressure).

In practice, regulations and standards:
The reverberation time may be either the 60-dB extent(ion) of the straight-line fitting of the actual sound-decay curve (most common method), or if the decay is multisloped, it may be the 60-dB extension of a portion of the actual decay curve (which then should be defined or labeled as such).

The 60-dB extent:
The basic idea is to define this 60-dB decay in the diffuse field (in so far the field is diffuse) with exclusion of the direct noise, first reflections and possible influence of the cumulative effect with the ambient background noise.
In order to obtain the exclusion of those, in function of the reverberation measurement, initial and background polluting phenomena, standards will define the boundaries of the range in which the decay slope may be evaluated.

ISO 354: Measurement of sound absorption in a reverberation room.

• Beginning decay curve:
Exclusion of the initial ca 0.1 sec after the sound source has been switched off, or starting from a sound pressure level a few decibels lower than the initial level.

• Bottom of the range:
Shall be at least 15 dB above the combined background noise level of the reverberation and the recording equipment for each 1/3 octave band.
ASTM C423: Measurement if material absorption in a reverberation room (info provided by Angelo Campanella).

• Beginning decay curve:
Exclusion of the initial ca 0.1 sec after the sound source has been switched off.
Measurement range to ca -32 dB versus initial pressure level

• Bottom of range:
Author doesn't know exact description, but the basic principles must be comparable to ISO 354 (exclusion of background influence).
ISO 3382: Measurement of the reverberation time of rooms with reference to other acoustical parameters:

• Beginning decay curve:
Standardized at initial sound pressure level -5 dB

This common initial level -5 dB will be found in most literature, standards and regulations.
• Bottom of the range indirectly defined via minimum sound source level:
Shall be at least 10 dB above the background noise.
This common background +10 dB will be found in most literature, standards and regulations.
This all can deviate a bit in function of regional standards, regulations, etc.
Basic idea
When viewing those numbers one can state that in order to measure a valid reverberation time the available dynamic range must be at least 15 dB higher than the range available for the evaluation of the decay curves.
A direct RT60 measurement should assume a total dynamic range (between source level and background noise) of 60 + 15 = 75 dB, and this for any individual frequency band.
Such acoustic environment and circumstances will rarely occur in real life.

Initial level -5 dB to -35 dB
Therefore it is generally accepted that one evaluates the decaying noise pressure between the initial level -5 dB to -35 dB.
This result then is doubled to obtain the RT60.
When official Reverberation Time values or Sabine values (absorption coefficients) are published the visitor can be 99,5 % sure that they are based on this
RT60 = (-5 dB to -35 dB)·2 dynamic range approach.

CONFUSING NOTIONS/CONCEPTS: RT, RT60, T, T10,T20, T30, T60, Tdyn, ..., EDT ?

ISO 3382 defines the symbol for Reverberation Time as T (not T60, RT or RT60).
Other literature, courses, standards will refer to T60 or RT60

Note that the above mentioned notions: RT, RT60, T, T10,T20, T30, T60, Tdyn, ..., EDT ALL refer to the Reverberation Time answering to the SAME definition: The time required by the sound pressure level to decay by 60 dB.

So what's the difference?
First one should aware that those notions are not worldwide standardized.

RT, RT60, T60 is often used since the symbol T outside a clear context can be confused with other concepts. As such one can use (and find) T to express Temperature as well. This means that T can only be used within a clear context related to reverberation time.
RT and RT60 is often used for convenience since it is an easy to relate abbreviation for Reverberation Time, while the 60 relates to the definition representing the 60 dB decay.
T60 is also used for convenience since it relates this symbol T to the 60 dB decay definition.
Let's have a look at the way those notions are used by ISO 3382 (valid for most Europeans - the author doesn't have a copy of the corresponding US standards).

T: is the general symbol for Reverberation Time answering to the general definition. This is also used when measurement are obtained in the standard -5 dB to -35 dB dynamic range. As such one can state that a T symbol in fact equals T30.
T+suffix: these notions still answer to the general definition (e.g. a T20 is as well a decay of 60 dB), the difference is an additional definition related to the dynamic range from which this 60 dB decay is obtained.

This is important:
As described above most Reverberation Times are obtained by defining the decay in the range from -5 dB to -35 dB, which assumes an available dynamic measurement range of 30 + 15 = 45 dB. However in real live circumstances this dynamic range of 45 dB will often not be available (related to power sound source versus volume and level of background noise)
In that case the standards and day by day practice, will allow to measure this decay with a smaller dynamic range as long as the respective initial -5 dB to background +10 dB remains respected. For standardization purposes in official publications or studies one mostly uses rounded numbers as T20 in order to preserve compatibility, which then means that the T20 is evaluated over the dynamic range of -5 dB to -25 dB which then are multiplied with 3 to convert to a 60 dB decay..
Why mentioning this suffix at all, when they are recalculated anyhow to a 60 dB decay, answering to the same definition?
Well, that's easy: While all representing the decay expressed in seconds over a 60 dB range, a T10, T20, T30 etc. can return significant deviating RT values.
When a real life sound decay curve should behave as a perfect straight decaying line all those different values should indeed be equal. However lots of factors can influence this behavior. Mostly towards the lower frequencies those curves don't look very nice anymore, but also in the higher frequencies things as multisloped curves or convex or concave shapes can occur.
To make thing easy comprehensible in books and courses one often uses nice decay curves as examples, in order to explain the reverberation time principles. Practicing acousticians however know that reality isn't always that easy.

What this means is extremely important:
While one names a reverberation time value the noise decay of an initial level with 60 dB, which by definition represents a corresponding decay slope, this slope IS NOT EVALUATED on this 60 dB decay but on the range as stated in the suffix behind the T symbol.
Since most RT measurements are executed over this 30 dB range (reason that ISO 3382, interprets T30 as T and vice versa, only calling for the additional suffix 30, in case of ambiguity), ONLY FEW PEOPLE, INCLUDING PRACTICING ACOUSTICIANS, WILL HAVE EVER SEEN A REAL T60.

Is a range deviating from this standard 30 dB range always related to, or caused by a lack of dynamic measurement range?
A T20 mostly is, but one can define the initial covered range also in function of a specific purpose.
Some standards related to on site (in the field) measurements, in function of transmission loss (insulation) and/or related reverberation time measurements, will describe the reverberation time measurements standard as a T20 measurement. This is done since in a huge amount of cases, in the field measurements will show a shortage in dynamic range, not allowing a T30 procedure. In order to preserve compatibility the standard directly defines the T20 procedure.

T10 also called EDT (Early Decay Time):
In function of speech intelligibility and perceived (subjective) reverberance the Early Decay Time is a useful parameter. So here the shorter dynamic range isn't related to a lack of dynamic measurement range but to extract useful data.
While strictly theoretically (analogy with T20 etc.) the range should be defined between -5 dB to -15 dB, the EDT is often defined less strictly, leaving it to the responsible acoustician, to find the earliest possible representative -10 dB decay. So the initial -5 dB rule is applied less strictly in order to make sure that indeed the 'Early Decay Time' slope is grasped.

Tdyn abbreviated from T dynamic:
This is often used by practicing acousticians but also in acoustic laboratories.
Basically the idea is to measure the reverberation time by evaluating the decay curve over the maximum available dynamic range, respecting the rules as described by ISO 354 (for laboratories) and/or the initial level -5 dB to the background noise + 10 dB rule (depending on regional standards this can deviate a minor bit).

A laboratory will often measure more parameters than published in the official measurement reports. This then is done for scientific purposes, but it also gives a good check on the stability of the executed measurements (can also give info related to the modal behavior of the room), or other material related specifics.
The visitor should notice that even in such special designed laboratory environment a real T60 is seldom (if ever) measured.

A practicing acoustician will often apply an arbitrary dynamic range in order to get as close as possible to a T30 or for any other investigative purpose.

SUMMARY:

• Notions as RT, RT60 and T60 are used for convenience. T60 is also to be found in scientific books and literature.
• The suffix behind the symbol T is important and influences the returned RT value, unless the decay curve is a perfect straight line.
• As per ISO 3382 (most important European related standard) T is automatically interpreted as T30.
• As per ISO 3382 everything deviating from T30 should be labeled as such. T30 must only be labeled in ambiguous circumstances (e.g. different frequency bands measured over deviating dynamic ranges, if not all bands allow a T30)
• As per NEN 5077 (The Netherlands) T equals T20 when applying the ISO reasoning (but standard defines exactly how T is must be obtained).
• A real T60 will seldom occur. The author assumes that only minor number of people, including practicing acousticians, have ever seen a measured T60.
• And last but not least, but meant as a JOKE ;)
When a T60 almost NEVER (if ever ...) occurs, and the worldwide accepted approach (unless lacking dynamic range forcing to go lower) equals the T30 definition, why not change the definition of the reverberation time in order to represent what is supposed to do, namely representing the reverberation time for a decay of 30 dB (of course one should skip the current multiplication by 2).
Don't take this last point seriously, just understand what this acoustically means: Basically an RT60 mathematically almost NEVER (or coincidentally) represents the slope for a decay of 60 dB. That this is important will become clear further down. The author (for now) can't remember exactly where the original 60 dB decay originates from (should check that one again).
THE AUTHOR FEELS THERE REALLY IS A NEED FOR STANDARDIZATION OF THOSE REVERBERATION NOTIONS. (For now there is too much room for interpretation).

A Graphical approach (based on the numbers of the above left-hand table = room 632 m³)

Lets analyze the different approaches graphically:
Note: All further comments relate to this one example (room 632 m³ in table). Those Graphs are meant to get a better feel and insight in the numbers mentioned in the left-hand table. The numbers of the other tables should lead to other graphs.
So when interpreting the graphs, don't make general conclusions which could be different with other examples and numbers.
By definition a standard averaging assumes that all 'to be averaged' frequency bands have an equal initial steady state noise pressure level.

• Approach 1: center frequency equals the 250 Hz RT value of 0.70 sec.
• Approach 2: arithmetic average 0.73 sec.
• Approach 3: highest frequency band equals the 200 Hz band (also shown as maximum value approach) of 0.97 sec.
• Approach 4: Exact energetic average, will be explained further down.

GRAPH 1a What happens?

Reverberation Time measurements and calculations are based on the assumption of a (pseudo) diffuse field. The frequency bands are no discrete frequencies but are the center frequencies of the bandwidth they represent. Assumed is that the noise behaves as a non-coherent pink noise (equal energy content per bandwidth).

While during measurement one will notice that the original 200 Hz, 250 Hz and 315 Hz bands did not behave nicely as a straight line, any Reverberation Time calculation will attempt to find a straight line fitting through this original curve in order to define the statistical noise decay. In practice lots of measurements are averaged (or moving microphones) to define this RT value.
Any further calculation can only be executed accepting the assumption that those resulting respective straight line fittings represent the statistical noise decay per frequency band.

If one accepts that the individual RT straight line fittings represent the energetic decay (noise pressure in dB) then the total level in function of the elapsed time is nothing more than the logarithmic addition of those 3 individual bands.
Since this noise decay in function of reverberation time values are not interesting from an absolute point of view but relative to the initial steady state level (the level at point 0 sec) one can put all those graphs on top of one another to compare this relative decay of the diverse curves.
Mathematically it means that one doesn't add the levels per frequency band but one applies a logarithmic average. So we can set all curves to an initial 0 dB and directly read the respective RT60 value from the x-axis (since minimum y-axis is set to - 60 dB)

Look now at 'exactly the same' Graph 1b where the initial level is shifted to 0 dB:

GRAPH 1b

Note :
This graph equals the previous one. The main difference is that all frequency band curves are shifted to intersect the 0 dB point, since the energetic decay (RT60) is not absolute but relative to the initial level. This basically means that as shown on the graph the 'Energetic Total' isn't really an 'Energetic Total' anymore but an 'Energetic Average'. The author didn't change the name of the series to show that the visitor is looking to the same curve as in the previous graph.
So to prevent any possible confusion: As shown here, mathematically the 'Energetic Total' should read now 'Energetic Average' but the shape remains exactly the same. Lets analyze those results:
Let's assume that this energetic total (which is in fact now an energetic average on this GRAPH 1b) indeed graphically represents the relative noise decay of the three other bands combined.
What's then the real RT60? The answer LOOKS clear but ISN'T!

If one should interpret the Reverberation Time definition literally, then the answer is really clear and could easily be solved: Just read the x-axis value where the energetic decay intersects this x-axis. Since the initial level was set to 0 (zero) and the x-axis crosses the y-axis at exactly -60 dB, the answer seems to be shown there, with the emphasis on 'seems'.
As one easily can see this purpel energetic decay curve is NO straight line. One still must find a straight line fitting for this curve. This will be explained more in-depth in the next graph.

Let's limit ourselves here to some major conclusions:

1. Approach 1: using the center band.
Basically there is little to tell about this approach since the conclusion should differ in function of the used example. In this case this center band is a bit lower than the Arithmetic Average, but in other examples it could equal as well the highest or lowest frequency band. As said before: this approach has no mathematical backing at all and should be interpreted as a very rough approach, which can or can't be about correct, but be as well completely wrong.
One can expect trouble when one takes 1 value out of three to be representative for the combination of those three, while mathematically ignoring the impact or even the very existence of those other 2 values.
It's the strong opinion of the author that this approach should be avoided, as well for RT values as well for absorption α Sabine values (which are inverse related).

2. Approach 2: the commonly used arithmetic averaging
As explained before (and easy to check in the Tables) this approach fits into the traditional room-acoustic formulas. It's not yet clear to the author how this arithmetic averaged RT, as only method, fits into subsequent energetic room-acoustic calculations, while at the same time clearly wrong from an energetic point of view.
However here one can see that ALMOST systematically this arithmetic average will result in a TOO LOW energetic decay. The author did emphasize 'almost' because it is still depending how one defines the correct averaged RT. IN OTHER WORDS: WHICH VALUE DO WE USE TO COMPARE THE 'ARITHMETIC AVERAGED RT' TO?
So, what is the correct averaged RT value?

3. Approach 3: using the highest frequency band
As one can clearly can notice in GRAPH 1a at a certain point in time the sound/noise decay is indeed almost completely defined by the highest frequency band. This highest frequency band and the energetic total (almost) completely hide one another, being equal in slope and in level (there are minor differences since in the logarithmic addition the 2 other bands will still influence this total, but this concerns extreme minor fractions of a dB, so practically they can be interpreted as equal).
This method basically introduces this energetic phenomenon ignored by the arithmetic averaging method.

HOWEVER THERE IS A TRAP:
GRAPH 1a also shows that this approach completely ignores the initial noise/sound decay. Indeed while at a certain point in time the highest band will define the slope and level of the decay, with extremely minor impact of the other 2 bands, THIS IS NOT THE CASE IN THE FIRST PART OF THE ENERGETIC DECAY WHERE THE OTHER 2 FREQUENCY BANDS INDEED STILL HAVE A SIGNIFICANT IMPACT ON THE ENERGETIC DECAY.
The decay of this first ignored part will significantly define the subsequent calculation of the energetic averaged RT value.

BASICALLY THIS METHOD LEADS ALWAYS to TOO HIGH RT VALUES, by completely ignoring the initial energy content of the two other frequency bands. Further more (and this is probably the most defining question): On which part of the energetic decay curve (dynamic range) should the RT value be calculated?
While (specific in this example) calculated over a 60 dB range, based on the real energetic decay (here exactly 0.89 sec) and the highest value approach (0.97 sec), there is an overestimation of 0.08 sec, the real overestimation will be larger.
WHY? Here it becomes clear why the author used a complete section to explain concepts as the T20, T30, T60 etc. So if you skipped that section assuming it was only indirect or hardly related please reread it.
An RT value is (almost) NEVER calculated on a decay of -60 dB but (as far as the author knows) worldwide standard accepted as the slope of the ca first -30 dB (exact measurement methods can differ a bit) So basically the original 1/3 octave bands one averages are as well defined on this -30 dB slope.

Basically this is why the author made this joke in the previous section related to the definition of the reverberation time: The definition relates to a 60 dB decay, while ALL related measurement and calculation methods (also as defined by standards) ignore the slope of the second half 30 dB, defining the RT60 based on the slope of the FIRST HALF 30 dB.
(note that the author discusses basic principles here. It's not the idea of going in-depth for any possible occurrence, circumstance or exception)

And here question marks arise. What is in fact correct?

4. Approach 4: using an exact calculated energetic average This then should be the correct answer! Or doesn't it ??? • Indeed this approach gives a theoretical and mathematical correct calculation method based on the assumption that the individual 'to be averaged' respective RT60 values do represent their respective energetic decay.

• Still a pragmatic decision should be made which dynamic range (the previous described suffix for T) should be used to obtain a correct energetic answer.
The author assumes (read this clear: not assures) that a T30 approach is a logical approach, since most standards will define this -5 dB to -35 dB as being representative for the RT60. This means that it ,at least, sounds logical that the averaging procedure is executed over the same dynamic range as the to be averaged RT values.

Pragmatically the author assumes that, in normal circumstances, one should ignore those question marks, reason being:
• This -5 dB is not meant to exclude the first -5 dB slope but to exclude measurement uncertainties (direct field, first reflections etc.,), in order to make sure that one measures the decaying (pseudo) diffuse field.

• This T20 is basically meant to solve a shortage in dynamic range preventing to measure a valid T30. Therefor this RT must be labeled as T20, which is more meant to point to a possible inaccuracy, or the measurement limitations.
So in common situation the T20 allows an alternative method, to accommodate acoustic boundary limits, in order to obtain T (which stands for T30)
However, the author can imagine that there are circumstances that one wittingly should prefer to measure a T20 for specific acoustic, scientific or compatibility reasons.
Basically the author doesn't have a conclusive answer. The provided algorithms allow to custom define the dynamic range used for the energetic averaging.
While theoretically those algorithms should equal a direct measured broadband RT60, only empirical testing can confirm or contradict that assumption.
Basically it should be interesting to do a study on those algorithms by comparing the approach with real life RT measurements in 1/3 and 1/1 octave bands (on same signal), or start from the original noise pressure multispectra in 1/3 octave, which can also correctly be added into 1/1 noise pressure spectra.
This also should give the possibility to check which dynamic range setting most suits real live measurements.

GRAPH 2

Let's analyze the effect of this dynamic range (the suffix of T), or, in other words, how to define a straight-line fitting for the Energetic decay? This graph doesn't need too much explanation. So comments are limited to a few major points:

• It is clear that the maximum value approach always exceeds the real live RT60 value, whatever dynamic range one should accept as being representative for the correct slope of the energetic noise/sound decay.

• While the T60 basically answers to the definition of the Reverberation Time, namely the time expressed in seconds for a decay of 60 dB, it's clear that this contradicts all practiced and described Reverberation Time measurements and calculations.

• The T10 is only included as information. For now it's not clear to the author how this first steep decay corresponds with real life EDT (Early Decay Time) broadband measurements.

• The T20 is shown as information, more to show the impact of the different dynamic ranges. However when a user should have measured the narrowband RT values to extract specific energetic decay information, it could be advisable to apply the averaging on the same dynamic range.

• The T30, for now, looks as the logical approach for standard applications (even when the included subdata should have been based on a T20, Tdyn or others to accommodate a shortage in dynamic range). Reason being that the original 1/3 octave bands most likely are also derived from a 30 dB decay slope or meant to be derived from.

GRAPH 4 A combined overview based on an Energetic Total
While easy to recognize and to get the feel how this energetic total behaves relative to the respective frequency bands it's easier (but with less clear oversight) to bring all initial noise pressure levels to a 0 (zero) dB level allowing to directly read the noise decay versus time.
One should take into account that reverberation time expressing the noise decay in function of time is a relative, not absolute concept.

GRAPH 5
A combined overview based on an Energetic Average
In this overview ALL decay slopes are set relative to 0 dB which translated in arithmetics to the factor 1 or 100%.
This is basically how the mathematical formulas are buildup. Here you can find the most significant approaches all together:
The center band equals the 250 Hz band - blue line (don't compare this with anything else, since that's entirely dependent of the used example).
The maximum value approach equals the 200 Hz band (green line).
All the others are mentioned in the Legend.

FINAL SUMMARY:

1. This problem is not conclusively solved and calls for a more in-depth study.
1. It should be good to do a statistical analysis of lots of reverberation time measurements, directly measured in 1/3 and 1/1 octave, or as alternative:
2. To start from the original noise pressure multi-spectra measured in 1/3 octave. One can easily recalculate them to corresponding 1/1 octave pressure measurements.
Based on this data one can calculate the respective RT values for as well the 1/1 as 1/3 octave measurements.

2. Temporary conclusion:
1. For standard use one can continue using the arithmetic averaging method, which is often, directly or indirectly, defined by lots of standards, and probably the most practiced method.
It also fits in the traditional room-acoustic calculations, which can not be said for any other approach.
However one should be aware that the resulting RT60 value is most likely underrated, and does NOT represent the energetic averaged decay, which is what any RT value basically is supposed to do.
2. When a more accurate energetic approach is needed the energetic averaging method as applied and explained on this page, with the Dynamic range (Tx) set to 30 dB seems to be the most logical and accurate approach for common situations, until proven otherwise.

GRAPH 6
A combined overview based on a WEIGHTED Energetic Average
This graph equals the previous one with this exception that the initial noise pressure levels are made asymmetric:

Those values are arbitrary chosen and adjusted to a level causing the energetic average to be 0 dB (this is how the function calculates)

1. 200 Hz: -6.72 dB

2. 250 Hz:   2.74 dB

3.  315 Hz: -0.42 dB
While maintaining THE SAME RT values for the respective frequency bands the visitor can notice the significant difference for the WEIGHTED energetic averaged RT values versus the other Graphs and Table with the NON-WEIGHTED averaged RT values. Visitors less familiar with acoustic calculations maybe wonder what this nonsense and chaotic graph is all about!

Until now this page only discussed how one should average three 1/3 octave bands, with equal initial energy content, into the corresponding 1/1 octave band.
However it gives an enormous amount of useful information when one can express the reverberation time as a Single-number-rating of a room in relation to the function of this room, a specific noise/sound source, a specific weighting curve, and so on.

In transmission loss (insulation) calculations most acousticians are rather familiar with single-number ratings. As such the ISO 717-1 RW, the ASTM STC and OITC are such single-number ratings, but there are many more. As such one can express the transmission loss of a window or wall in one single dB value for air-traffic, road-traffic, railway-traffic, for different types of music, technical equipment, machinery and so on.

In order to do that one applies a sound source (or any other) related weighting in the acoustic calculation.

In principle one could do the same for room-acoustic parameters.
However this is very little practiced in reverberation time calculations, sound absorption, and basically for about all room-acoustic parameters.
As the visitor knows the reverberation time is related to the acoustic properties of the room, of which the averaged absorption in function of frequency is very important.

As such the reverberation time for disco music, transfos, pumps in function of the total spectrum of those sources will most likely be much longer than the reverberation time for music where the vocals are dominant or grinding tools etc.
This means that one can express a reverberation time T, as well as the a, A and LD values as single-number ratings allowing significant more exact and helpful measures to compare room-acoustic parameters or to set target values.
Such a single-number rating can as well be applied on the total spectrum (or whatever weighting curve), as on published 1/1 octave bands, when such data is based on 1/3 octave band data. In the latter case the 1/1 octave band RT60 value represents a single-number rating of the corresponding 1/3 octave bands.

For now, even in official standards as ISO 3382 and others (400 to 1250 Hz arithmetic averaged) , RT values for a room are mostly expressed as a value around 800 to 1250 Hz, without any mathematical relationship to the purpose (read weighting spectrum) of the room, sound source and so on.

So when being able to average energetically those respective frequency related RT values, weighted in function of a noise source or any other thinkable weighting curve this provides a POWERFUL ROOM-ACOUSTIC TOOL. The included algorithms are designed to do just that! Algorithms designed by author (please do respect source)

The author designed some formulas allowing to calculate the energetic average and weighted average for 3 of ANY number of adjacent Reverberation Times, no matter the bandwidth and number of frequencies.
A common approach is averaging 1/3 octave bands into octave bands, but it is as well applicable to show the masking effect of e.g. Reverberation Room measurements, when one wants to investigate peaky behavior of resonant narrow band absorbers with high Q. For instance in critical music listening circumstances as e.g. a home theater, a studio or a tracking room, a linear absorption behavior can be a desirable goal. While dedicated narrow band absorption can be useful to attack resonant phenomena as room modes, in most cases absorbers should not show a peaky but a very smooth, rather linear behavior.

Still the problem remains, since applying the results of this calculation for subsequent diffuse field calculations causes a discrepancy between the total diffuse level and the energetic sum of the diffuse field levels of the individual covered frequency bands.
This all means that some additional work is needed to find the cause and logic of those discrepancies.
The author suggests (for now) to set the Tx: Dynamic Range variable to 30 for standard applications. The formulas used in the above tables

Note: the © New Formula ® in cells B10 refers to the algorithms explained in the above formulas

These formulas are based on a metric approach:
V: Volume = ; A: Equivalent absorption = ; S: Room boundary Surface = ; LD: Level Diffuse field in dB  The Sabine Method:
Note that the formulas in Rows 1 to 10 calculate from Left to Right. Basically the a value is only mentioned for info.
This method neither needs the α value, nor the direct related Room boundary surface.
Row 11 calculates backwards, starting from the energetic average (E11) of the diffuse field in cells E4 to E6.
This then allows to check which direct averaging method applied on the RT60 values themselves corresponds with this backward calculated RT60.

The Eyring method:
Note that this is a bit a poetic license. The diffuse field field is calculated based on the Room Constant (R) which uses correction terms comparable to the nonlinear Eyring approach.
R = A/(1-α). So the formula 9 looks in fact as 10·log(4/R) which translates into the formula shown above.
The basic principle is equal. Only in the Eyring method the Room boundary surface (S) and the α values make an integral part of the approach.

Have fun with it (As such I get some indirect reward for the energy involved)
Kind regards from Antwerp The Author has still to recheck this topic , currently based on an old 2002 HTML page (© the author himself).
Eric.Desart
Moderator

Posts: 2461
Joined: Tue Feb 10, 2004 4:29 am
Location: Antwerp, Belgium 