PROCEEDINGS of the
23rd International Congress on Acoustics
9 to 13 September 2019 in Aachen, Germany

Multidimensional Visual Cluster Analysis of Room AcousticalParameter Values as Means to gain Scientific Insights
and Design / Consulting Tool
Klaus-Hendrik LORENZ-KIERAKIEWITZ1; Benjamin PFÄNDNER2;
Christoph REUTER3, Stefan OSTROWSKI4
1,2
2

Peutz Consult GmbH, Germany

Institute of Musicology, University of Vienna, Austria
3

admost Ostrowski, Germany

ABSTRACTAt the conference DAGA 2017, a method of multidimensional visual analysis of basic room-acousticalparameter values incorporating reverberation time RTMid, strength GMid and volume V was presented. Using
this method, it is possible to demonstrate that different room types such as e.g. concert halls, opera orchamber music halls form separated clusters in partial volumes of the acoustical parameter space spanned by
the acoustical parameters. For concert and opera halls rated as "good" regarding their acoustical quality it has
been shown that these are localized in narrow partial volumes within the centers of the room type clusters. At
the DAGA 2018 this approach was extended by investigating additional room types and by comparisons of
the spatial positions of several halls before and after renovations. In the present paper, this method will beshown to serve as a valuable tool obtaining new scientific insights as well as being a helpful planning tool:
For the room type "concert hall" the historical development is analyzed from the beginnings to the 21st
century yielding characteristic trajectories across the parameter spaces clarifying the evolutionary trends aswell as geographical differences. The immediate practical advantage of the method described here as aconsulting and design tool will be demonstrated by application to orchestral rehearsal rooms.
Keywords: Room-Acoustics, Acoustical Parameter Space, Acoustical Quality

1. INTRODUCTION
For a long time, the analysis and presentation of room acoustical parameter values of rooms rated
for their acoustic quality relied on tables with two-dimensional graphs based upon two variables. Bymeans of recent computer programs the available possibilities of data analysis and presentation now
allow for a multi-dimensional analysis of more than two room-acoustical parameters simultaneously.
In (1) a new method using multi-dimensional visualization was presented showing how new insights
can be derived with different examples of various room types plotted into a parameter space spannedby basic room-acoustical parameters (without loss of generality by this method in the first instance e.g.
reverberation time RTMid, strength GMid and room volume V). In such initially three-dimensionalparameter spaces the room examples are represented by characteristic parameter triples withcharacteristic positions within the parameter space. Insights into the nature of the room types can begained fast by visual analysis from the graphs by different angles. In (1) it was further shown, thatdifferent room types form clusters in separate parameter space sub-volumes in characteristic
parameter-value ranges with comparatively small overlap regions.
Moreover, in (1) and (2) was shown how this data analysis method can be expanded by anotherdimension to data quadruples, e.g. by adding a quality rating, and an adapted visual data representation,e.g. by means of marker color. It could further be demonstrated that the rooms rated as “good” with1

khl@peutz.de
bp@peutz.de3
christoph.reuter@univie.ac.at2

4

stefan@admost.de795

regard to their acoustical quality meant for their genuine use (here: concert halls and opera houses) arelocated in smaller sub-clusters within the certain room-type clusters and therefore displaying even aseparation of the different room types. In reverse, this analysis may lead to sets of requirements for theparameter-value ranges which are necessary in order to achieve acoustical properties rated as “good”
for the different room types.
In (2) this multi-dimensional method was extended by incorporating a larger number of rooms andadditional room types: churches, rehearsal rooms, lecture halls and theatres. This expansion of the databasis confirmed the result of cluster formation. Furthermore, it has been demonstrated in (2) how thiscluster analysis can be used for consulting purposes. A comparison of the location of a room
before/after a refurbishment or reconstruction in the parameter space can reveal if, and to what extent
construction measures are able to move the position of the room under examination in the parameter
space (with help of actual examples, among them the Tonhalle Düsseldorf, Staatsoper Berlin andKulturpalast Dresden) if possible even towards the sub-clusters with high quality ranking.
For adequate acoustical conditions in a room in regard to the main use, the first-order necessary
condition in general is a value of the reverberation time in the value ranges judged as appropriate forthe use of the room examined. However, for adequate acoustical conditions sufficient conditions
should be met as well. This means that also the values of the other room acoustical parameters ought tobe in the value ranges judged to be adequate for the use of the room type, e.g. early decay time EDT,
binaural quality index BQI, Clarity C80 etc.. Therefore, one-dimensional considerations are
insufficient for covering the sufficient conditions of the acoustical quality. In order to examine
first-order necessary and sufficient conditions simultaneously, considerations had to be given to
two-dimensional methods; for an adequate simultaneous consideration of more than two parameters
more-dimensional methods are necessary. Dependencies between more parameters are not easily
detectable in two-dimensional representations. Figure 2 in (1) exhibits an example from (3) of a
two-dimensional graph of the parameters reverberation time RT and strength G, where a thirdparameter - the room volume V - is added as a family of curves. Conform the revised diffuse field
theory after Kuttruff and Barron, in the diffuse sound field the three quantities V, RT and G are strongly
coupled by equation (1):
 V 
Gtheo , Diff = 45dB − 10dB ⋅ log10  
 T (1)

In (1) common analysis methods of existing data sets were described and the disadvantage of thesecommon methods discussed namely that many tables of two-dimensional graphs have to be taken into
consideration. Because of the a priori unknown interdependencies between the commonroom-acoustical parameters, the conventional analysis methods lead to limited informative values.
Therefore the conclusion in (1) and (2) was drawn that in order to understand and measure thesufficient conditions of acoustical quality in rooms regarding their use in addition to the first ordercondition - namely that the reverberation time shows values in the range adequate for the use - amulti-dimensional analysis method is required. At its best, such a multi-dimensional analysis method
is able to reveal the mutual interdependencies of the parameters.

2. DATA BASIS
The following quantities are at least in prinpiple capable of being controlled and therefore they
have to be taken into consideration in any acoustical design process: room volume V and main
geometry / room shape, absorption and especially the audience capacity (seating/persons/position ofaudience areas); consequently, reverberation time RT and strength G; room width (and maximum
distance from the stage), and last but not least structure and acoustical behavior (reflection, absorption
or diffusion) of bounding surfaces. As data basis for the comprehensive multidimensional analysis in
this context, per room data from the following categories were collected:- name, city, country, continent, opening year;- shape (cuboid, hexagon, fan, horseshoe, surround/vineyard, ellipsoid); room width w;- volume V, number of seats N (resulting in values of the specific volume (V/N));- quality rating (from (14));- spatial averages of the room acoustical parameters for seats with stage distance d > 2 r H:
reverberation time RTMid,seated ;Clarity C80,Mid,unseated ;Gtheo,Mid,unseated resulting from equation (1);strength G meas,Mid,unseated ;BQIMid,unseated ;ALcons,P88,seated.

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Data collections of room acoustical parameter values of culture buildings provide numerous
objective data (4-17) as well as correlations with subjective preference ratings (14-16), in which
especially the rating results in (14) are only one of the possible quality judgments and therefore
questionable to a certain degree. The collected data were sorted in sets of n-tuples per room in an
extensive table, see tables 1 in (1) and (2), the structure of which is given in table 1, showing the
averages of the values per room type, as well as of all values available in the actual study:

RTMid,s [s]Gmeas,Mid,u [dB]

Gtheo,Mid,s [dB]

BQIu

Width W [m]C80,Mid [dB]

1,8

5,8

6,3

0,56

27

-0,8

1933

Chamber Music

-

3.400

425

1,6

12,9

12,6

0,72

15

-0,1

1911

Opera Halls

(14)

9.100

1.325

1,4

4,7

7,8

0,57

26

+2,8

1885

Churches-

15.000

515

3,1

13,7

11,2

-

20

-2,4

1771

Lecture Rooms

-

3.900

390

1,5

11,1

13,7

-

17

+7,7

1953

Theatres

-

6.000

1.150

1,2

7,0

8,5

-

20

+4,3

1939

Rehearsal Rooms

-

4.000

190

1,4

12,9

11,9

-

17

+0,7

2000

All Room Types-

11.200

1.290

1,8

7,8

8,6

-

24

+1,0

1900

opening year

Number of Seats
2.000

Shape

Volume V [m3]16.000

Hall

(14)

City

Concert Halls

Room type

RatingTable 1 – Structure of the data sets, average values of the room types and average of all values (last line)

Note: for some rooms not all data has been available (lacking are e.g. Rating, BQI, C 80 and G meas),
but for all room data sets the parameter values of V, RT, G theo , W, shape, type and opening year arecollected and incorporated (for churches mostly the RTMid,unseated ).
Note: the values in table 1 are displaying the averages of the data sets per parameter. They are notmeant as a reference of how concert halls, opera halls and other buildings should be built.
2.1 Enlargement of the Data Basis
For the study in (1) a selection of 32 concert halls, 11 opera houses and 8 chamber music halls from
(14) served as data basis, in (2) this was extended by data sets taken from (4-12) to 150 rooms, among
them 59 concert halls, 19 chamber music halls, 20 opera houses and four new room types: churches (40examples), lecture halls (14 rooms), rehearsal rooms (8 rooms), theatres (4 additional rooms).
In the present study, this data basis was extended by all examples from (4-14) and (17) as well as byadditional data from the Peutz group to 403 room data sets, which are available for analysis, with 147
concert hall data sets, 39 chamber music halls, 118 opera halls, 50 churches, 15 lecture halls, 17
theatres and 17 rehearsal halls. The actual number of different rooms though is somewhat below 400 in
total, for multipurpose rooms like for instance the "Spiegel" in Zwolle the corresponding parametervalues for their uses as theatre, opera and concert hall has been added also in all applicable room types.

3. EXTENSION TO (3+)-DIMENSIONAL ANALYSIS: TOOL FOR NEW INSIGHTS
The analysis method presented here is a valuable tool to obtain new scientific insights. Computer
technology allows a comparatively simple (3+)-dimensional graphic representation of data. For thevisual analysis of the data n-tuples per room presented in (1) and (2), a MATLAB®-script was used.
For the present study the code was totally newly implemented within the open-source software GNU
octave, version 4.2.1 in order to plot the available parameter values of the data sets into(3+)-dimensional graphs on arbitrary axes within a Cartesian coordinate system.
A multitude of parameters in these graphs as viewing angle (azimuth, elevation), axis scaling(linear, logarithmic, range) as well as marker color and marker size for additional dimensions above797

three allow for the presentation of the input data n-tuples to be illustrated in comparatison to each other,see e.g. figure 1.
For the representation proposed here a parameter space can be assumed without loss of generality
as orthogonal in the first instances, as long as the inter-parameter dependencies are unknown, e.g.
RT mid,bes ; G meas,mid,u and V. In order to examine these 3-dimensional representation of triplets in avisual analysis, see e.g. figure 1 showing all rooms of this study; marker color indicates the room typeas additional dimension.

Figure 1 –RT - V - Gtheo plot of all data sets examined, color indicates room type.
3.1 (3+)-Dimensional Representation of Dependent Parameters
In such (3+)-dimensional graphs, a dependency between correlated quantities becomes directly
visible while rotating the graph by choosing an appropriate angle of view of the data sets:
Obviously, there is a strong correlation between the quantities RT, G and V, see figure 2:

Figure 2 – RT - V - Gmeas plot of all data sets with Gmeas, color indicates room type (see figure 1)
Figure 2 shows that the in the first instance w.l.o.g. chosen quantities RT mid,seated, G meas,mid,u and V tospan the parameter space obviously are correlated; if viewed in an appropriate viewing angle, the798

majority of the data sets are located along a straight line in the parameter space. Exceptions appear in
the data sets of opera houses and churches, where a correct estimation of the acoustically effective
volume is non-trivial due to coupled spaces as stage tower or naves etc.. Nevertheless, the value of
Pearson's correlation coefficient of G meas,u and G theo,s is 0,87 for the data sets examined here. The
conclusion may be drawn that in the majority of the rooms analyzed here, in source distances of d > 2
times the reverberation radius a nearly diffuse field behavior can be assumed. This allows in thefollowing sections for transferring findings from the smaller number of rooms with measured date of
G meas,u to that ones, where Gtheo,s is calculated by equation (1), which is valid in the diffuse field.
When viewed vertically, already in (1) and (2) was shown that the RT / G / V - triples of the rooms
are located in or at least near a plane in the parameter space, see figure 1. Moreover, the diverse room
types are located in partly disjoint subspaces. Within these subspaces, the majority of samples
cumulate in clusters in specific subspaces characteristic for the genuine room type. In the analysis herewith extended number of data sets, it can be observed that the separation in clusters according to theuse turns out to be slightly less distinctive than in the earlier data sets due to many rooms, which arebuilt as multipurpose halls or as a compromise between different applications desired without alteringof the acoustical parameters: these are located in between the subspaces characteristic for the room
type. Moreover, in the actual data samples of known acoustical appropriateness are included as well.
3.2 (3+)-Dimensional Representation of Independent Parameters
The visual representation of samples in a multi-dimensional parameter space can be assumed to be
especially significant if a parameter space is chosen in which the quantities to span this space are
independent. In order to identify independent parameters, in (1) the correlation of each pairs of room
acoustical parameters was tested.
As independent parameters turned out to be, among others e.g. RT mid,seated , G meas,unseated, and
BQIunseated . In figure 3 the actual data n-tuples are displayed in a parameter spaces spanned by thesethree little correlated quantities.

Figure 3 – RT - Gmeas - BQI plot of all data sets with BQI-values, color indicates room type, marker the shape
A visual analysis of figure 3 shows a clear separation of the room types chamber music halls (green),
concert halls (blue) and opera halls (magenta) in clusters in almost disjoint subspaces of the chosenparameter space RT - G - BQI with some data samples in between two adjacent room types.
3.3 Cluster Accumulation of Rooms with High Quality Rating Score
Within the clusters of the room types, sub-clusters can be identified with rating scores from (14):799Figure 4 – RT - Gtheo - BQI plot of concert halls with BQI-values, color indicates rating, marker the shape
In figure 4, it can be seen that in (14) higher rated concert halls have higher BQI values and higher
RT-values as well as high G-values, and form a sub-cluster within the concert hall cluster (cyan marker
color: top rated) with characteristic parameter values within a certain range and room shape (rectangle:
shoebox-style halls). Concert hall data sets with lower rating exhibit lower values of RT, G and BQI.
3.4 Historical Development of Room Types
Let us regard the development of for instance the room type concert hall with time: In figure 5 thehistorical development is analyzed from the first examples of this room type to the 21st century,
plotting number of seats against room volume and opening year.

Figure 5 – Time - Seats - Volume plot of concert halls, color indicates rating, see fig. 4, marker: shape
In figure 5 it is obvious, that characteristic trajectories across the parameter spaces through timeclarify the evolutionary trends of this room type: starting with a low number of small halls (see (12)),which would nowadays serve as chamber music halls, the room type over time has changed
dramatically by adding more and more seats (i.e. more and more sold tickets as motivation), requiring
higher and higher room volumes. Not visible in the figure: the resulting demand to enlarge theorchestra to compensate for the loss of strength. Within a certain slot of some 40 years around 1890,

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the halls mostly are rated (14) as excellent, followed by decades with rather mediocre ratings causedby even larger number of seats (a trend especially in North America) and high volumes (as well as hallwidths), resulting in low values of G and BQI, respectively.

Figure 6 – Time - Seats - Volume plot of opera halls, color indicates rating, see fig. 4, marker indicates shape
Figure 6 shows the same view as chosen in figure 5, but now displaying the historical developmentof opera halls. Whereas the quality rating is not that clear, it can be seen easily that the room type“opera hall” starts to exist earlier and spread out at first in Europe all over the continent, whereas theconcert hall room type was just at its beginning (the first data point in figure 4 indicates an old buildingin Vienna which has a hall inside, which has been used in the 18th century for concerts, see (12)).

4. BENEFIT AS CONSULTING TOOL
Objective of room acoustical consulting is the judgment whether an existing or prospective room isable to exhibit room-acoustical qualities adequate to the (main) intended use of the room.
The analysis method presented here offers a real benefit in terms of consulting purposes: already inthe very early states of consulting the room acoustical conditions of a newly to build room or before arefurbishment with the basic data V and RTMid (which is strongly linked to the number auf audience)and resulting value of G mid,theo (assuming a diffuse sound field in the room further away from thesource) can indicate the relative position of this new room in relation to the room type clusters, if
within a type cluster or even in the subspace with high rankings, inside the type-clusters, but outside
the subspaces with good ranking scores, or even beyond the type clusters. In discussions with clientsand architects, consequences of global decisions defining the acoustical quality can be visualized.

Figure 7 – RT - V - Gtheo plot of orchestra rehearsal halls, color indicates rating, marker indicate shape801

Figure 7 displays orchestra rehearsal rooms inclusive ratings by the users. A clear pattern is
identifiable: if the volume of a rehearsal room is large enough for the intended ensemble size there isthe chance that an acoustically successful rehearsal room can be built within the geometrical frame.
If room acoustical conditions in given rooms by the users are judged as suboptimal, measures
should be defined to improve the rating for the intended use. For this proposal the analysis method
presented here offers possibilities to judge the effect of alterations before going into the consulting
process with computer simulations or scale models: if the spatial position of a room in the chosenparameter space is outside the room type cluster or inside the type cluster, but still outside the subspacecluster with high rankings, planned measures can be pre-viewed within the same plot and analyzed forthe spatial position change in the parameter space, the planned or discussed measures would cause.
Effective proposed measures are visible as spatial position change in the parameter space, as wellas whether the aim can be achieved by the proposed measures: then the parameter n-tuple of the roomwith measures should meet the subspace clusters of room type or of the desired rating, see (2).
The Effect of refurbishment measures can be tested before building activities on a very basic level.

5. CONCLUSION
The method presented in this paper, multidimensional visual cluster analysis of room-acousticalparameter values in three- or moredimensional parameter spaces based upon numerous data-sets ofexisting halls, represented as parameter n-tuples with well-defined spatial positions offers
comprehensive possibilities as means to gain scientific insights into the nature of usage-dependent
room acoustics and can be very beneficial as a fast consulting tool for planning new rooms or
refurbishment measures of existing ones.
Further work still has to be done e.g. by adding acoustical quality rating scores to other room types,
especially chamber music halls, and to find the just noticeable difference (JND) for spatial positionchanges in the parameter spaces.

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