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


Postby Eric.Desart » Fri Jun 18, 2004 1:21 pm


If something should be wrong, or interesting data can be added, please send an Email or PM to the author, who will take any well meant comments certainly into account.
I restored this post at Nov 5, 2007. These picture were lost for a relative long time already. Also added the Louden map + accompanying text. In order not to loose overview an additional separate topic was added with a M.M. Louden ratios analysis

The pictures below give an oversight of the Room Ratios as Published by world-wide known and respected acousticians over many decades.

The author DOES NOT ALLOW to copy those pictures and distribute them in any direct or indirect way.
The COMPLETE content of this topic belongs exclusively here in the Studiotips Acoustics Forum
If a visitor wants to help others, nothing is easier than linking to this topic, as such respecting the work and study done by others.
Link to this topic:

For comparison, independent of the room ratios (read shape)  ALL Rooms are fixed at 84 m³ [± 2966 cft].  
By scaling the volume of those rooms the relative modal distribution remains equal.

This room volume is arbitrary defined as:
1) An acceptable common relative small room.
2) This size guaranteed a good common scale for the different graphs (just a matter of presentation/layout).


Tangential and Oblique Modes
    Some NON-acousticians on the net claim the UNimportance of the Tangential and Oblique modes.  

    This is based on either:
    a) A lack of knowledge and understanding and a denial of physics and studies executed by the most respected worldwide known acousticians.
    b) The assumption that Tangential and even more Oblique modes are weaker than axial modes, thereby completely ignoring the exponential increasing number of those Tangential and Oblique modes, and the importance of the remaining individual strength.
Let's analyze the number of modes
Taking a fixed order [n] of modes then (e.g.: n=10):
Axial modes: 3 x n = 30
Tangential modes: 3 x n^2 = 300
Oblique modes: n^3 = 1000
More follows I'm preparing some statistics in function of real distributions in function of frequency boundaries.

Philip M Morse & Richard H Bolt:
    Sound Waves in Rooms
    Reviews of Modern Physics.V16 No2. 1944
Morse & Bolt state that tangential modes require twice the power to produce the same sound pressure as a similar axial one, while oblique modes require four times as much.
It is therefore necessary to divide the sound pressure so calculated by sqrt(2) for  each non-zero nx, ny, or nz over one.
Having computed the properties of each mode, it is then necessary to integrate them together to produce the power response curve.
Bolt himself also integrated the Tangential and Oblique modes in the by him published formulas.

Richard. H. Bolt
    Note on the normal frequency statistics in rectangular rooms.
    J.Acoust.Soc.Am. 18(1) 130-133. (1946)
Bolt studied the behavior of sound in enclosures where the enclosure was of dimensions comparable to the wavelength, and suggested a family of ratios of height to length to width such that the modal frequencies would be well distributed.
Bolt gives a pair of formulae which, for medium sized rooms, should give a response which is "good enough", without defining any criteria for "good enough".

Note: In function of optimum or acceptable room ratios, much more is published than represented here by those predefined ratios.

Richard. H. Bolt: Published a mathematical approach
Note on the normal frequency statistics in rectangular rooms. J.Acoust.Soc.Am. 18(1) 130-133. (1946)

Newman, Bolt & Beranek 1957: Publication of a Pictogram with acceptable ratios.

IEC 60268-13, BS6840 13 (1998) publishes formulas.

European Broadcasting union
EBU Technical Recommendations R22-1998.  Based on the J. Walker, BBC approach

Dr. Trevor J. Cox and Dr. Peter D'Antonio
Published a mathematical approach (integrated in the related RPG software tools).

Oscar Juan Bonello
A new criterion for the distribution of normal room modes. J.Audio.Eng.Soc. 29(9) 597-605 (1981). Note erratum given in 29(12) 905 (1981).

Very interesting to note is that world-wide respected acousticians USED the positioning of Tangential and Oblique modes in order to come to their respective world-wide accepted room ratios.

L. W. Sepmeyer: 1965 - 1:1.14:1.39.
This is one of world's most common used ratio.  When one judges those room ratios based on the Axial modes alone one can only conclude that the modal distribution of this ratio looks terrible.
However L.W. Sepmeyer counted on the NON-Axial modes in order to optimize the modal energetic distribution.
The same is valid for his other ratios.

M. M. Louden: 1971: Best ratio as per Louden - 1:1.4:1.9
Another of world's most common used ratios.
When studying lots of Louden's room ratios, even the best ranked (he published 125 ratios in decaying order), one will notice that without the NON-Axial modes, only judged on the Axial modes, lots of those common used  and worldwide accepted ratios should be rejected resulting from an UGLY looking distribution.

A strong advice:

    If a visitor wants to judge a room in function of its ratios,
    NEVER use a Room Modes Calculator which DOES NOT take the
    Tangential and Oblique modes into account

    Only taking the Axial modes into account can be alright but can as well lead to WRONG conclusions.
    One just doesn't know!

Understanding the graphs (from bottom to top):

  • 3 bottom lines:  The axial modes on their respective x, y and z-axis.
  • The bold Axial line:  All Axial modes shown together (gives easier picture on distribution). The total number of axial modes per 1/3 oct. is mentioned on top of this line.
  • 3 higher lines with light blue markers: The Tangential modes in their respective plains.
  • The bold Tangential line:  All Tangential modes shown together (gives easier picture on distribution).
    Here the numbers on top of this line are split as xx [xx]. The numbers between the square brackets are the real number of modes. The number preceding the brackets is the same number divided by 2 (50%). This relates to the - 3 dB (10log(0.5)) per mode as described by Morse and Bolt and gives a better energetic representation.
  • The bold Axial + Tan. line. Here all Axial and Tangential modes are shown together. The numbers between and preceding the square brackets are described above. In the related summation lines the number of Axial modes are always included in both the numbers between and preceding the square brackets (since they are counted at 100%).
  • The higher oblique modes line with the light green markers:  Here the numbers preceding the brackets are the number of modes divided by 4 (25%). This relates to the - 6 dB (10log(0.25)) per mode as described by Morse and Bolt and gives a better energetic representation.
  • The Top Line:  All modes together.  The numbers preceding the square brackets is the sum of the Axial modes at 100%, the Tangential modes at 50% and the Oblique modes at 25%.
    As per Oscar Juan Bonello the numbers between brackets (all modes at 100%) should be either equal or higher than the preceding 1/3 octave band.
    Adjusted to the Morse and Bolt energetic approach the numbers preceding the brackets can be judged in a comparable manner as the Bonello approach. The author feels (personal opinion) that there is much in favor of preferring the energetic approach over the Bonello approach.
  • Warning: One should be somewhat careful interpreting a Bonello like approach.  When one scales a room maintaining the same ratios, the relative modal distribution remains equal. One will notice a frequency shift.
    It is possible that the Bonello approach will qualify such a room with volume x as OK while rejecting a room with volume y. It should be clear that this can't be true. Acoustic quality in function of the modal distribution can not depend on the accidental positioning of the same relative distribution in a mathematical frequency scale, but on the relative distribution itself.
  • The light gray numbers at the left bottom: xxx/xxxx represent the total number of modes shown on the graph, followed by the number of markers shown on the graph (modes are repeated in the summation lines).[/i]
An overview picture:
The 3 areas enclosed by the green border lines (ABCD: left-top, EFG: right-top, HIJ: bottom) are the graphical representation of the mathematical approach of the accepted room ratios published by the European Broadcasting Union in the EBU R22-1998 recommendation.
The gray area is the Pictogram as Published by Newman, Bolt & Beranek in 1957.  Important is to note that this pictogram does not take the coinciding modes in function of the z versus x-axis into consideration.  The author must still analyze this, at a first glance, seemingly strange approach.

The marker ID numbers in this EBU picture matches the Graph numbers In the "Room modes" pictures below.
The Bolt area equals the one to be found In Alton Everest's "Master Handbook of Acoustics". This however diverts somewhat from the original Bolt paper (will be adjusted in some later stage).


One can easily use those pictures to locate the closest well-known ratios that matches the users needs.
The left bottom corner = the z (height) ratio equaling 1.
The top scale is the y (width) ratio.
The right-hand scale is the x (length) ratio.

If one wants to prevent Axial modes to become to closely spaced as one can see in the Room Ratio Graphs 5, 6, 7, 10 etc. (using the Tangential and Oblique modes to assure a good modal distribution), one can adjust the 5% safety rule as described in EBU R22-1988.  One can adjust this to any %, e.g. this can result in the second Graph. In this Graph the author increased the EBU 5 % rule, to 100/11 = 9.0909....%. This strange looking number is rather logical when analyzing the rest of the EBU approach.
To prevent misunderstanding: The first EBU graph above is the official EBU approach, the second EBU graph a poetic license of the author, causing the emphasis to be shifted more to a better axial distribution for the low order axial modes. This however does not necessarily guarantees an overall improved modal distribution.
While the EBU approach (based on the J. Walker BBC approach) gives a rather flexible tool to handle Room Ratios by excluding bad ranges, it STILL is strongly advisable to check the final selected ratios in a Room Modes Calculator.

Don't look for nr 1). The numbers are used to match the Picture Id. numbers.

2)  L. W. Sepmeyer: 1965 - 1:1.14:1.39
Computed Frequency and Angular Distribution of the Normal Modes of Vibration in Rectangular Rooms.
Journal Acoustic Society America Volume 37 No.3, March 1965, pp 413-423.


3)  L. W. Sepmeyer: 1965 - 1:1.28:1.54

4)  L. W. Sepmeyer: 1965 - 1:1.6:2.33

5 to 10)  M. M. Louden: 1971: 125 ratios
M. M. Louden listed 125 dimension ratios arranged in descending order of room acoustic quality
Link to more details & graphs: ROOM RATIOS: M.M. LOUDEN ANALYSIS
    Note that this does NOT represent 125 acceptable room ratios but a list resulting from a grid with 1 decimal resolution.
    In fact he had 153 Matrix points of which 28 were double with  reversed x/y ratios.
    These doubles are masked by the hatched triangle in the graph.  
    Some of the published ratios are located in that hatched triangle, and the corresponding ones in the main area lacking.
    That seems to justify the assumption that M. M. Louden's publication in 1971 was influenced by much less powerful and flexible calculation means than available nowadays.

    Therefore, unlike ratios of other authors ALL Louden ratios yield standard 1 decimal on the x & y ratios.
    With modern means it's possible to extract the acoustic logic from this huge list of ratios ranging from best to really ugly, to be avoided ratios, reorganize and map them.
    One can see that within the by M. M. Louden published ratios and used method, better ratio points/ranges can be extracted when 2 (rather than 1) decimals are used for the Length/Width values.
    Several in-between ranges are better than the ones published by M. M. Louden himself.
    This graphical approach is easier and user friendlier than the original published data, giving additional insight.
    This results in the following stylized graphical analysis (Copyright ©: Eric Desart)


    The darker the area the better, with the center of the darkest areas being best (note: based on the Louden approach).
    The numbered markers are the as 5 best published ratios in order of quality.
    What you can see that there is not such a thing as a perfect ratio. The darkest green starts from 0.38
    Reminder: More details on these numbers and graphs in the above referred link (also part of the FAQ)

    Also note that the whole room ratio approach as per the different Authors is a topic still under discussion.

5)  M. M. Louden: 1971: Best ratio as per Louden - 1:1.4:1.9
Dimension-ratios of Rectangular Rooms With Good Distribution of Eigentones.
Louden listed 125 dimension ratios arranged in descending order of room acoustical quality.
Acustica, Volume 24, 1971, pp 101-103/104.


6)  M. M. Louden: 1971: 2nd best ratio - 1:1.3:1.9

7)  M. M. Louden: 1971: 3rd best ratio - 1:1.5:2.1

8)  M. M. Louden: 1971: 4th best ratio - 1:1.5:2.2

9)  M. M. Louden: 1971: 5th best ratio - 1:1.2:1.5

10)  M. M. Louden: 1971: 6th best ratio - 1:1.4:2.1

11)  J. E. Volkmann: 1942 (later discussed by H. Bolt) - 2:3:5
Polycylindrical Diffusers in Room Acoustical Design.
Journal Acoustic Society America Volume 13 , 1942, pp 234-243.


12)  C. P. Boner: 1942 - 1: 2^(1/3):4^(1/3) - This equals Id. O - 1:1.26:1.587
Performance of Broadcast Studios Designed With Convex Surfaces of Plywood.
Journal Acoustic Society America Volume 13 , 1942, pp 244-247.

Note: this is a mathematical very interesting ratio to study.
This ratio is exclusively calculated on the axial modes guaranteeing the mathematical maximum ratio between x, y and z axis (22.62%).
This however causes some less desirable coincidences with 3 Tangential an one Oblique mode.


13)  Rounded C. P. Boner ratio - 1:1.26:1.59

14)  Derived from C. P. Boner (stylized ratio) - 1:1.25:1.6

15)  Golden rule ratio: 1968 - 1.236:2:3.236

16)  IEC 268-13: Recommendation for listening room: 1987 - 2.8:4.2:6.7

17)  IEC 60268-13: Recommendation for listening room:1998 - 2.7:5.3:7

18)  Dolby's optimum ratios for Film & Music Room - 0.67:1:1.55

19)  Origin unknown: equals ± 1:1.6:2.56 - 0.625:1:1.6

20)  Origin unknown: Resembles Id. 19 - 1:1.618:2.588

21)  Origin unknown: meant for small room - 1:1.5:1.6

22)  Origin unknown: normal - often used ratio - 1:1.6:2.5

23)  Origin unknown: meant for long rooms. - 1:1.25:3.2

24)  A worst case scenario calculated by RPG - 1:1.075:1.868

NOTE: The sometimes non-standard looking ratios, e.g. Volkmann's 2:3:5 rather than 1:1.5:2.5 is given in respect to the original published ratio.
The IEC ratios are room measures (in meter = metric) rather than ratios.
However the pictures themselves show also the exact traditional shown ratios (z : y : x, where z = 1).

Some interesting calculation files on the Net:

Room mode calculators can be found via the Studiotips section: Calculation Tools:
It includes calculators by Jeff D. Szymanski, Scott Foster and Eric Desart.

This Excel File by Chris Whealy is a VERY good one, calculating more than Room Modes alone.

Yet another good calculator by Dr. Floyd E. Toole - Harman International.
Look also at the other articles and papers via this page. They are incredibly valuable.

There are more files on the net. Just search via Google.
Lot's of them are good. One is more extended than the other.
Just play with them, to check which one feels most comfortable and suits one's needs and purpose.
Just don't forget:  Room Mode calculators should calculate ALL modes (Axial, Tangential and Oblique)!

This topic will be further extended by the author.
Additionally some more ratios and theories will be discussuded based on recent Dr. Trevor J. Cox and Dr. Peter D'Antonio papers and room mode calculations as well as contributions by Jeff D. Szymanski from Auralex.

Enjoy !
Eric Desart

All the above pictures are Auto generated from within an MS Excel file (without the aid of a graphic tool), not yet available for the public.
This will come in the future (matter of having time)!
Sometimes one will find references to "not shown" ratios.  In the referred MS Excel file over 50 Room Ratios are included.
However it has little sense to shown them here out of their proper calculation context.

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