J. W. Head

RESEARCH DEPART1~NT

THE STATIS'I'ICAL AHALYSIS OF STUDIO QUALITY INTERIM REPORT

Heport No. ].057 Serial No. 1953/21

Report v~itten byg

J.W. Head

/1Cc('!o-'A-~ I~ /, ---------- ' (W. Proctor Wilson)

CONFIDENTIAL ____ .... _w ___ ~ .. ~ __

Resea.rch Depa.rtment July? 1953

Figs. Nos. B.057.1 - B.057~3

THE S_TATI~TICAL ANALYSIS OF STUDIO QUALITY INTERIM REPORT

Contents Page No.

1. 2. 3. 4.

5. 6.

Summary Introduction A Linear Regression Formula Quadratic Regression Formulae A Nomogram equivalent to one of the Quadratic Regression Formulae Conclusions Possible FUrther Work References Appendix 13 Comparison of Subjective Assessments of Various

Observers Appendix 2i The Somerville Criterion and the Derived

Counterparts G19 HI of the Observed Quality g

1 2

5 10

14 16 17 18

19

21

Fig. 1 Fig. 2 Fig. 3

Somerville's Criteriong The "Good Corridor" Quality Prediction in Greater Detail based upon Somerville's Criterion Nomogram for Quality Prediction by means of the Quadratic Regression Formula (19).

A formula is sought which shall enable the subjective quality of a studio or concert hall to be predicted with reasonable confidence from objective measurements of volume? reverberation time 9 a parameter D associated with the general behaviour of decay curves and a parameter R associated with variation of reverberation time with frequency. Data for twenty-five studios and concert halls are used~ subjective quality was assessed by two observers who knew all of these auditoria. A linear and a quadratic regression formula are obtained by statistical analysis of the d~t~9 and these are compared with a formula derived from Somerville's\l) criterion.

Figs. 1 and 2 show the way in which the parameters (D + O.7R) and Tm contribute to quality as predicted by means of Somerville's criterion. As the quadratic regression formula involves five variables? the best

- 2 ...

available substitute for the corresponding diagram is a nomogram (Fig. 3). The methods by which these formulae are derived and their limitations are fully discussed; further investigations are considered, by means of which these formulae could be improved.

1. . Introduction ----~---

Although many attempts have been made to obtain an objective parameter closely related to the subjective Quality of a studio or concert hall? none of the proposed criteria can be regarded as (1) generally satisfactory. These criteria are summdrized by Somerville 9

who has proposed a new tentative empirical criterion X involving a parameter D connected with the nature of the decay curves at certain frequencies? the mean reverberation time Tm? and a parameter R con­nected with the variation of reverberation-time with freQuency. For the twenty-five studios and concert halls for which values of D? Rand Tm are at present available? Somerville's criterion effec­tively separates the halls of high Quality from those of low Quality, but at present there does not seem to be any adeQuate theoretical explanation of its effectiveness? and a formula which could dis­tinguish more shades of quality may be required.

We here seek a formula for subjecti.ve quaE ty based upon a statistical analysis of the above-mentioned twenty-five studios and concert halls. The statistical technique used can be re-applied if further data later become available? or if further 'Quantities are thought likely to have a bearing upon Quality. The result of the statistical analysis may ultimately lead us towards a formula which has a theore­ticat justification. The parameters in terms of which the statisi;ical analysis is here carried out are D, Rand Tm already mentioned? vl /3 where V is the volume? and/a number ~ representing the subjective Quality. D? R9 Tm and VI 3 are all overall parameters of the hall or studio? and Somerville(l) points out that the first three are associated with what the ear detects.. For this analysis? .€i.. is an integer between I and 5 (1 for a bad hall? 5 for a very good one) written down after discussion between two experienced observers who Y~ew all the places analysed. It might be preferable to replace this ~ by a Quantity obtained by averaging the subjective asses.sments of many people. The difficulties of obtaining such an ave:rage at present are purely practical. In a recent investigation(2)? recordings of the same piece of music were made at four halls? and were played to a number of observers who were asked to place the halls concerned in order of preference. There was seldom statistically significant agreement between observers? even when they were divided into homogeneous €,TOUpS 9 such as performing musicians? or B.B.C. programme engineers. This suggests that only a few observers are competent to eive a subjective opinion Of studio or concert-hall Quality which has

- 3 -

meaning and signifioanoe for our present purpose. Even if a panel of oompetent observers were available, it would not be likely to know all the plaoes uLier oonsideration, whioh are soattered all over the British Isles, so that the task of obtaining a value of ~ based on any oonsepsus of opinion is formidable. We have here assumed that the values of .fi given in Table 1, based on the judgment of two experienoed observers, are the best available at present. Subjeotive assessments of other observers are tabulated in Appendix 1 (Table 5) and correlation ooeffioients among them or between them and the values used in the present oaloulations are inoluded where possible.

The values of 1£ given in the last, "average" oolumn of Table 5 (in Appendix 1), whioh were not available when the investigation was begun, might vuth advantage be taken as the observed values used in the oaloulations instead of the values given by observers A and B. At this stage, however, it seems best to leave any recalculation of this kind until more subjeotive opinions and measurements are available.

From the values given in Table 1, oorrelation ooeffioients between pairs of the variables D, R, Tm, Vl / 3 and Jr are first oaloulated and tabulated in Table 2. A linear regression for Jr ~n terms of the other variables is then obtained by the Esoalator method~3). The detail of this caloulation is given in Section 2. The derivation of a quadratic regression formula for .fi is considered in Seotion 3, and a' partioular quadratio regression formula is evaluated numerioallY9 for this formula the oaloulation has been greatly simplified at some expense in "goodness of fit". The linear and quadratio regression formulae are oompared in Table 4 with the observed values of E.: Somerville' s criterion number X is also tabulated there, and two possible formulae for Jr if Jr is assumed to be a funotion of X only are also included. The derivation of these formulae is disoussed in Appendix 2. Although formulae do in theory show the influence on predicted studio quality of the various parameters involved, a pictorial representation is much more convincing. This is easily obtained in the case of X and formulae based on X as only three variables (l9 D + 0.7R, and Tm)-are involved. Fig. 1 Treproduoed from report B.053 whioh is substantially equivalent to reference 1) shows how most of the good studios ocoupy positions within a "good corridor" for whioh 0.8 , X ~ 1.2, while most of the poor studios occupy positions outside this corridor. Fig. 2 shows a finer division of the (D + Oo7R, Tm) plane into regions within each of whioh a quantity ~4 = 10/(X4 + X-4) varies between specified limits. When we try to obtain a oorresponding pictorial representation of the quadratio regression formula (19) which involves five variables, the best available substitute is the nomogram shmvn in Fig. 3 9 its derivation and use are explained in Section 4. Section 5 gives oonclusions, and Seotion 6 disousses possible further work.

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Table 1 ---

Va,luesof D,9 R, _T~,' .. V!!.3 an~. ,g" f::r,twentY-five Studios and Concert Halls

Number and N'ameof Studio or Hall

1 Maida Vale 1 2 Concert Hall, B.H. 3 Swansea 1 4 Edinburgh I 5 Maida Vale 2 6 Maida Vale 3 7 Charles street, Cardiff 8 Criterion Restaurant 9 Paris Cinema

10 Manchester I 11 Birmingham 4 12 Bristol 1 13 Portland Place 3 14 Portland Place 5 15 Bristol 5 16 Portland Place 3 (modified) 17 Portland Place 5 (modified) 18 Glasgow 8 19 Belfast 2 20 Free Trade Hall, Manchester 21 Royal Festival Hall 22 st. Andrews Hall, Glasgow 23 Usher Hall, Edinburgh 24 Civic Hall, Wolverhampton 25 Belfast 2 (modified analysis)

D R

2.7 .3 2.7 4 J.4 3 303 3 3.4 2 3.1 3 3.4 1 3.3 3 2.8 4 3.1 12 2.8 6 3.1 6 4.2 8 3.1 8 3.6 -2 3.1 4 2.5 3 203 4 2.0 -1 207 6 3.4 8 3.4 6 4·1 3 4.0 0 2.0 2

1.66 1.38 1.23 1.2 1.22 1.24 103 1.64 1.14 1.16 0.98 0.96 0.61 0·56 0.62 0·57 0·51 0.42 0035 1·93 1.8 2.6 2.15 1~5 0.35

Vl/3

61.27 50.00 33.02 51.68 39.15 39.15 35·03 43.09 31.07 33.62 32.08 38.70 18.47 17.86 16.87 18.47 17.86 14.26 15·87 86.62 90.86 84.34 81.43 79037 15.87

g

5 5 4 5 4 3 4 5 4 2 2 2 1 1 1 3 4 5 3 4 2 5 4 2 3

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Table 2

Correlation Coefficients rij between the variables xl, x2,

x39 X4 9 x~! _~na..... X5,!6, Xl defined by ( 16)

~j_~ 1

-~.- -.-----.-2 3 4 5 6

1 2 0.08711 3 0.42521 0.16249 4 0-37688 0.14094 0.91117 5 -0.05152 -0.29165 -0.29542 -0.14461 6 0.12972 0.28337 -0.17025 -0.14350 0.06136 7 -0.02609 0.03877 0-37402 0.33387 0.29591 -0.09476 8

7

9 -0.28009 -0.23215 0.40936 0.26253 -0.16388 -0.57981 0.20813

2. ~_Linear Regression Formula

The values of D, R, Tm, Vl /3 and g for twenty-five studios and concert halls are tabulated in Table 1. Tl}e correlation coefficients rij between pairs of the variables D, R, Tm, Vl /3 and ~ are as tabulated in Table 2 (omitting the fifth, sixth and seventh rows and COlumns). We shall find it convenient to work with the variables

Xl = feD), x2 = feR), x3 = f(Tm), X4 = f(Vl/3), x9 = f(g) (1)

where -:- \"

f( ~ ) = ( ~ - ~ )/ [L ( ~ - ~ The bar denotes a m~an, and ~ a summation, for the twenty-five sets of observations involved. We then have

although for actual calculation of rij the simplest formulae are

(2 )

L (D - D)2 = ~ D2 - 25D2; L(D-D)(R-R) = L DR - 25DR (4)

and so on.

8 9

- 6 -

,,h It is possible that D~ R, Tm, VI/3 and g should be replaced by <p. (D), Y-'2.(R~, CP3(Tm), cfJy.(V) and CPs-(g) where the?> 's are any reasonable I

func.hons 9 as the tentative replacement of VI /3 by log V and of i£ by log i£ did not appear to have any marked effect on the matrix of corre­lation coefficients, it was decided only to use the variables (1).

The linear regression formula is of the form

where the Als are so chosen that, for the twenty-,five studios,

~ 2 2... (x9 - Alxl - A2x2 - A3x3 - A4X4 ) (6)

is a minimum.

The ~4)ivation and statistical significance of (5) is discussed by Kendall 9 the required quantities Ai satisfy the equations

Al + r l2 A2 + r13 A3 + r 14 A4 rl9 r 12 Al + A2 + r23 A3 + r 24 A4 r29

(7) r13 Al + r23 A2 + A3 + r 34 A4 r39 r 14 Al + r 24 A2 + r 34 A3 + A4 r

49

There are many well-known methods of SOlving(s~multaneous equations like (7). Here we use the Escalator method 3) because in the process of ' the calculation we obtain quantities X2' X39 X4~ Y3' Y4? Z4 such that the variables

~l = Xl

~2 x2 + X2xI

~3 x3 + X3xI + Y3x2 (8)

~ 4 x4 + X4xl + Y4x2 + Z4x3

are uncorrelated. When the A's satisfy (7), the correlation coeffi­cient £. between x9derived fr:>m (5) and the observed x9 is expected to have a known value R which is also closely related to a quantity which occurs in the course of the Escalator-method calculations.

When we wish to determine (5), the relevant correlation coeffi­cients are in the first, second, third, fourth and ninth rows and columns of Table 2. The computation proceeds by the step-by-step building up of Tables 3a, 3b, 3c and 3d as explained below.

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,Ta1?1e_2.

lIEsca1ator" Computation~ __ for Linear Regression 02

----.. la) Ho. of t 9 Row 1 2 3 4

-.~-~-.-

1 r1 t X2 -0.08711 -0.007588 -0.037040 -0.032830 0.024399

2 r 2t 0.08711 1 0.16249 0.14094 -0.23215

---- ---_.-

3 Sum 0 <12 "" s23 = s24 = s29 =

e 0·992412 0.125450 0.108110 -0.207751 -~--.--- '"---'---",---~--..-.-.. ---. .~----..

4 r1tX3 -0.414199 -0.036081 -0.176122 -0.156103 0.116013

5 r2tY3 -0.01l011 -0.126409 -0.020540 -0.077816 0.029346

6 r3t 0.42521 0.16249 1 0·91117 0.40936 ----~--

7 Sum 0 0 <13 = s34 = s39 =

0.803338 0.737251 0·554719 ._-_._. 8 r ltX4 0.012734 0.001109 0.005415 0.004799 -0.003567

9 r2tY4 0.000616 0.007074 0.001l49 0.000997 -0.001642

10 r3t Z4 -0.390230 -0.149123 -0.917734 -0.836212 -0.375684

11 r 4t 0.37688 0.14094 0·91117 1 0.26253 .-------.----------~----.--- --_. -

<1 = s49 = 12 Sum. 0 0 0 4

0.169584 -0.1l8363 .-"-~------".--.-------..-. ---.-.. ------------------~---.

13 r 1tX5 0.556754 0.048499 0.236737 0.209829 -0.155941

14 r 2t Y5 0.026269 0.301564 0.049001 0.042502 -0.070008

15 r3t Z5 -0~565980 -0.216284 -1.331061 -1.212823 -0·544883

16 r1tT5 0.263048 0.098371 0.635961 0.697961 0."83236

17 r9t -0.28009 -0.23215 0.40936 0.26253 1

18 Sum 0 0 0 0 <15 ""

0.412404

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"Escalator" Computations for Linear _~B?:'ession C2L

(b)

i c::: 1 2 3 4 9

1 -0.08711 -0.414199 0.012754 0.556754 0 1 -0.126409 0.007074 0.301564 0 0 1 -0.917734 -1.331061 0 0 0 1 0.697961

-' ----,' -.--..

(0 ) ith column of (b), divided by <li

i "" 1 2 3 4 -~

1 -0.087776 -0·515597 0.075090 Xi!<li 0 1.007646 -0.157355 Q.041714 Yi/<li 0 0 1.244806 -5·411677 Zi/<li 0 0 0 5.896782 T)<l° ,1 ).

(d) r 1j (j>1) and Sij(J;>i)

i = 1 2 3 4 j

----,-----.---.--,~-----------,-.-.-

0.08711 0.42521 0037688

:"'0.28009

0.125450 0.108110

-0.207751 0.737251 0.554719 -0.118363

2 3 4 9

--------------------~----~-------------~-----

Xi Yi Zi Ti

- 9 -

We first choose X2 = -0.08711 so that X2 + r12 = O. This choice enables us to complete the first three rows of Table 3(a) completely? but only the first two columns of Tables 3(b), 3(c) and 3(d). To obtain X3? Y3 , we multiply the elements of the third row of Table 3(d) by the corresponding elements of the first and second rows of Table 3(c) respec­tively, and add. With these values of X39 Y3 we can complete rows 4 - 7 of Table 3(a) and hence the third column of Tables 3(b), 3(c) and 3(d). As a check? the sum of the first two columns of Table 3(a)9 row ~must be zero, and

To obtain X4 , Y4 , Z4' we now multiply the el@rnorrrs of the fourth row ?f Table 3(d) by the corresponding elements of the first, second and third rows of Table 3(c), and add. This enables us to complete rows 8 - 12 of Table 3(a) and hence the fourth column of Tables 3(b), 3(c) and 3(d). The sum of the first three columns in row 12 must now be zero, and

X5 , Y59 Z5 and T5 are obtained similarly.

We now find that, ~ith the .yalues of X9 Y, Z etc. given in Table 3(b), the var~ables ~ l' g Z 9 g 3 and ~ 4 given by (8) are uncorrelated, and the regress~on formula (5) required becomes

(ll)

The expected correlation coefficient R between ~ and (X5xl + Y5x2 + Z5x3 + T5x4) is (1 - q5)~ = 0.76655, whereas the correlation coefficient r actually found by direct comparison of

. (-X5x1 - Y5x2 - Z5x3 - T5x4) and ~ is 0.7664. In terms of the original variables the linear regression formula is

g = -1.33<{ D - 0.134 R + 3.097 .Tm - 0.0382 Vl /3 + 5.994

C;' If we had tried to obtain (11) by working with5i ' or any other linearly independent set of linear combinations of the x's, we should have obtained an apparently different formula (11), but when we substituted back in terms of the original variables, we should obtain the same formula (12). If however we try to obtain a quadratic regression formula instead of a linear one, as in Section 3, the result will depend on the variables used to obtain it, unless all possible product terms are included, in which Case we have to solve equations like (7)9 but involving fourteen unknovms.

(12)

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3. SMadratic Regression Formulae

The object of obtaining a quadratic regression formula is to see whether any of the variables involved have special or "preferred" values.

As the solution of simultaneous equations involving fourteen unknowns is laborious 9 we must first consider whether almost as satisfactory a result could be obtained by omitting some of the terms. Let us suppose that the best-fitting quadratic regression formula with all possible product terms in the variables (1) present is

(13)

and that the correlation coefficient between the values g* of g derived from (13) and the observed values is P. Then (13) expresses a purely geometrical relation. For suppose we have a 5-dimensional space in which Xl? X29 x3 9 x4 and Ji are rectangular coordinates. From any observed value of g associated with y~own values of Xl x2 x3 x49 we derive a point P(x1 9 x29 x39 x49 g)9 while a corresponding point q(x19 X29 x39 ,x4 9 g*) can be derived from (13). The distance between.£. and ..9. is g - g*9 and (13) was chosen so that the sum of the squares of these distances should be a mJ.nJ.mum. The fact that the sum of the squares of these distances is a'minimum is a property of a particular quadric Q in this fivG­dimensional space 9 not of the system of coordinates in which the equation of that quadric is. expressed in (13). If therefore we now replace Xl x2 x3 and xL). by linearly independent combinations Yl Y2 Y3 Y4 of these variables 9 and recalculate our quadratic regression formula in terms of the yts? we shall obtain the y-equation of the same quadric Q. As Q is a quadric 9 the result must necessarily be of the form

g 4

a l + L.. i=l

4 b'i Yi + ~

i=l

4 T... C'ij Yi Yj j=l

In the above argument it was assumed that no Change was made in the coordinate by means of which g and g* were measured. If any of the bi or ci' in (13) are missing, it does not follow that the corresponding b¥~ or C'ij is missing. J3ut we could in various ways express (13) OT (14) as a sum of squares, thus

g (15)

- 11 -

so that if by some means we knew or guessed the variables Zi? we would be able to obtain a value of ~ having correlation coefficient P with ~ by assuming only terms in 2. i 2 without product terms.

If we try to obtain a quadric (13) with the xi and omit any terms, we shall obtain a value of ~ whooe correlation coefficient with g is less than or equal to P? it would be equal to P if the Xi were the same as the 2 i and the term omitted was a product term. As there is no means of deter­mining the 2 i in advance, we have made an interim assumption that the Xi are not significantly different from the 2: i9 and as from the practical point of view we do not expect there to be a· "preferred volume", we have also omitted any term in x 2. On this assumption we now obtain a simpli­fied quadratic regression iormula, and for the present we postpone any attempt to obtain (13) in full, though we do later consider how to reduce the labour of finding (13). To obtain ths simplified quadratic regression, we introduce new variables

(16)

where

and then proceed to find a linear regression formula for x9 in terms of Xl'.' x7 as though these seven variables were completely independent. The correlation-coefficient matrix is now as in Table 2, and the quadratic regression equation is then found to be

X9 + 0.508xl + 0.16lx2 - 1.3535x3 + 0.752x4

0.1125x5 + 0.364x6 + 0.122x7 :; 0 (18) .

In terms of the original variables, this becomes

g :; 0.366 (D - 4.77 )2 - 0.0341 (R - 2.91)2

- 0.374 (Tm - 5.37 )2 - 0.0412 Vl/ 3

+ 11 (19)

The correlation coefficient between g as observed and as found by (19) is expected to be 0.83895, (i.e. 0.83895 is the value. of the quantity corres­ponding to R for the linear regression equation (5)) whereas the value I'

obtained by direct comparison for the twenty-five studios is 0.8388. (19) suggests that there may be speCial values for D9 Rand Tm? namely

R :; 2.91 (20 )

- 12 -

2.91 is within the range of values of R for the present set of observations9 and (19) does therefore suggest that 2.91 is a genuine "preferred value" for R? since g is reduced by any deviation of R from 2.91. On the other hand, 5.37 is more than double the highest observed value of Tm' The term in (19) involving Tm tnerefore always makes a large contri-bution to the reduction of g. In this case (19) merely indicates that if an increase of Tm can be achieved without altering the other variables 9 it should increase g5 Vfuen we consider the contribution to (19) from the term involving D? vie find that not only is 4.77 greater than all the observed values of D, but also that deviations of D from 4.77 tend to increase g instead of decreasing it. The actual contribution of this term to g is usually small, and its ~

positive sign we regard as having little practical importance. ~

The special values (20) cannot be regarded as very reliable, since they might be modified by inclusion of the remaining terms of (13). A positive coeffiCient, like that associated with the D-term in (19), is surprising, and might well disappear when all terms of (13) are included. Raising the correlation coefficient from 0.767 for linear regression to 0.839 for (19), however, is significant. It suggests that a very high correlation coefficient would be obtained if all the terms of (13) were included. On the other hand, with fifteen adjustable constants in (13) and only twenty-five readings at present available, the correlation coefficient may have to be very high to be statistically significant~. This suggests that (19) is adeQuate at present, though perhaps capable of improvement by the addition of a term in X429 while the full-scale determination of

. (13) should not be undertaken until more studios have been measured.

When the full-scale determination of (13) is undertaken, it will be better to use (14) with Yi replaced by ~ i in (8) than to determine (13) direct with the original Xi. For from the fifteen eQuations determining the coefficients aI, b l

i ? C'ij of (14), it will be easy to eliminate aI, b l

i leaving ten eQuatlons in the ten unknowns C'ij.

Table 4 gives the values of G2 (based on (12)) and G3 (based on (19)) tygether itith X, the observed g, and certain functions of X discussed in Appendix 2.

~ The correlation coefficient for twenty-five observations and fifteen disposable constants must reach about 0.90 to be significant at the "5 per cent level"? and about 0.95 to be significant at the 1% level.

- 13 -

~a;,ervi~iterion numb~r X2 and various calculated counter~arts of the obseryed subjective qualitx

Studio No. X g G1 HI G2 ~-""------"----~-

1 0.83 5 1.03~36_ 3.8420 4·7193 2 1.02 5 4.9843 4·7070 4.2088 3 1. 07 4 4.8223 4·5809 3·5921 4 1.06 5 4.8671 4.6158 2.9192 5 0.93 4 4·7966 4.5608 3.4610

6 1.01 3 4.9960 4.1.161- 3.7897 7 0·78 4 3.2532 3.3588 4.0007 8 0·94 5 4.8506 4.6029 4.6105 9 1.12 4 4 .. 5267 403506 4.0558

10 2.28 2 003696 1.1129 2·5444 ---~--,

11 1.46 2 2.0992 2.4600 3.2529 12 1.53 2 1. 7661 2.2005 2.5367 13 2.30 1 0.3569 1.1030 0.4871 14 2.08 1 0.5327 1.2399 1. 8262 15 0·51 "I 0.6734 103495 2-1248

16 1.40 3 2·4378 2.7237 203712 17 1.12 4 4·5267 403506 3.1453 18 1.28 5 3. 2J)4 303122- 3.1313 19 00334 3 0.1244 0·9219 3·9317 20 1.12 4 4·5267 4.3506 4.2430

21 1. 51 2 1.8541 2.2691 2.4737 22 1. 07 5 4.8223 4.5809 5.4694 23 _ 0·96 4 4·9341 4.6679 3.6541 24 0·72 2 2·5064 2·7711 2.2565 25 0.87 3 403133 4.1844 3.5287

-----------,----~----~---

Theoretical correlation (R) with g 0·7666

Actual correlation (r) with g 0.8120 0.8120 0.7664 ------

G3

4.8989 4·5157 3.9188 3.1608 3.6074

4.0306 3.9273 4.8157 4.4093 1.1888

3.2660 208283 1.0011 1·7478 1. 5471

2.6026 303154 3.439:2 3.2063 4.2518

2.2973 5·0234 3.9377 2.0618 3.6992

0.8390

0.8388

- 14 -

4. ~N.?mo_gram ~jvale!!!".jio the ·Quad~atic RE?_~~ssion ~rm~laJ].21

Equation (19) is already in the form

and 9 as shown in reference (5)9 this means that (19) can be expressed in the form of a. It triple alignmenttl nomogram. We write (20) in the form

fl(R) + f 2 (Tm) - RI 0

f 3 (D) + f 4 (V) - R2 0

RI + R2 + f5(g) 0

(21)

(22)

(23)

By suitable choice of scale-factors fli and scale-separations 0i9 (21) oan be made equivalent to the condition for the points r -019 fll fl (R)1 (q9 fl2 R~] and [°3 9 fl3f2 (Tm)) to be collinear 9 and similarly for (22) and ~23) • The scale associated with each. parameter .depends upon the nature of the corresponding f. The fli and 0i are not com­pletely arbitrarY9 but are subject to the restriotions

111 --=-+-fl2 fll fl3

In this way Fig. 3 has been constructed to have overall size 20 cm. by 15 Cm. For any studio the following sets of points are collinear 9 if the origin is taken at the lowest point of the R-scale on the extreme left of the diagram and the axes ax'e along and perpendicular to the scalesg

(a)

.. (b)

(c)

[0 9 0.182 [82063 -(R - 2.9l)2)] 9 [5.44 9 - 1.60R~ and [7.77 9 0.856 {25.20 - (Tm - 5.37)2B

[11.119 2.0~2{ (4.77 - D)2 - 0-325}) 9 (150919 2.57R2]

and [2qO.l96 (90•86 _ vl / 3 )}

[5 0 449 - 1.60Rl ] 9 [90449 0.986(g + 4 0878 9

and (15•91 9 2. 57RJ .

There is no need for the scales of RI and R2 to be calibrated. The remaining vertical scales are calibrated with the value of the variable concerned at a given scale height. In the case of the

- 15 -

R-sca1e, this is not unique for values of R between -2 and 7.82, since in (a) above, Rand (5.82 - R) will give the same scale height, and both Rand (5.82 - R) are within the relevant range 12 ~ R ~ -2. The scales for R9 Tm an~/~ are nonli~ear because (19) contains quadratic terms; the scales for V and g are linear, but the V-scale can also be calibrated directly in terms of V~ it ceases to be linear if this is done,

To find the value of ~ for any particular hall, (e.g. St. Andrews, Glasgow), join the appropriate R-scale point (R = 6) to the appropriate Tm-scale point (Tm = 2.6) and determine the intersection (X) with the uncalibrated RI-scale. Also jOin. t~e appropriate D-scale pOint (D= 3.4) to the appropriate V-scale point (Vl/3 = 84034) and determine the intersection (Y) with the uncalibrated R2-scale. Join these uncalibrated­scale intersections (X and Y) and the value of g (5) predicted by (19) is read off the G3-scale. The correspondin€:, line; for Manchester 1 are also drawn on Fig. 3.

From Fig. 3 it is clear that (if perfect prediction of subjective quality by (19) could be assumed, and if the range of the variables provided on the nomogram is adequate) for high quality R should be near 2.91, Tm as large as possible, D as small as possible and V as small as possible. In practice, high Tm will inevitably be associated with high V9 so that the line corrosponding to XY will then slope steeply downwards. We can also see clearly the effect of varying the parameters one at a time. Thus a reduction of R for Manchester 1 from 12 to 9 would raise g to about 2.7, whereas a reduction of R for st. Andrews from 6 to 3 would only slightly raise g. Again, the nomogram can be used "backwards", that is to say, if Tm alone is varied, we can find from the nomogram the value of Tm which would reduce g to 4. Joining Y to the point marked 4 on the G3-scale, it meets the RI-scale in a point ~ below X which when joined to R = 6 gives Tm ~ 2.13, whereas if R were varied and Tm kept at 2.6 9 R would have to rise to about 9.6 to produce the same effect.

Again, to improve the quality of Manchester 1 the parameters must be altered in such a way that either XI or Y' is raised. If XI is raised a given distance, it' has more effect on g than raising yl by an equal distance since the G:),-scale is nearer the Rl-line than the R2-line. Alteration of volume-would be difficult, so that if Y' is to be raised, D must be reduced. The R-scale is very open at the lower end, so that a reduction of R can, as we have seen, raise g considerably. XI could also be raised by an increase in Tm, and the Tm-scale is close to the Rl-scale, but Tm must be raised to about 1.67, a large increase, in order to produce the same effect as red~cing R to 9.

These examples, chosen at random 9 show that, if (19) is assumed to predict subjective quality perfectly? Fig. 3 will indicate clearly and quickly the general way in which the parameters would have to be varied

- 16 -

to improve Cluality. Although as we have seen the correlation between G3 and the obserVed g is high9 it is not high enough to guarantee that improvements suggested by Fig. 3 may not occasionally fail to fulfil expectations. But Fig. 3 should be regarded as a means of determi­ning Cluickly what possible improvement is likely to be worth trying first.

Conclusions

The values of the following quantities are tabulated in Table 4 for the twenty-·five studios and concert halls? together with the relevant correlation coefficientsj when the correlation coefficients can be derived theoretically as woll as directly from the tabulated values? both values are included? and are in excel­lent agreement~-

X, the Somerville criterion number 1£9 the observed studio quality (as in Table 1) G19 H~ calculated values of ~ based on the assumption that

g is a function of X alone. (The formula for X and the reasons for the ohoioe of G19 HI are disoussed in Appendix 2.)

G2' obtained from the linear regression equation (12) G3, obtained from the simplified quadratio regression

equation (19).

Calculated values of Gl , H19 G29 G3 which differ from the observed value by more than 1 are underlined, and we notice that there are four or five such values in each case? and always for Studio No, 18 (Glasgow 8).

In view of the difficulty of making the subjective assess­ments, the general agreement is remarkably satisfactory. Our primary objective of finding a formula based on measurable quantities which enables quality to be predicted has to a first approximation been achiev~d. Statistical determination of a linear regression yields E- less good correlation than the best obtainable on the assumption that quality is only a function of Somervi11e's X. A quadratic regression can yield a higher oorrelation than that obtained by means of X, but the adjustment· of fifteen constants is required to obtain the best possible correlation. This involves a laborious calculation? and the statistical significance of the result may be doubtful unless a oonsiderable addition is made to the number of studios for which data are available. Every possible opportunity should be

- 17 -

taken of comparing the values of HI and G3 vvith subjective oplnlons, whether of the two observers who determined g or of other people, both for studios and 00ncert halls listed in Table 1 and for others. The calculations described will yield a definite result of the form (13) whatever initial values of g are assumed. The results given ~n Appendix 1 suggest? however, that individual observers will differ considerably in their opinions, and it may therefore happen that in practice the Cluality value obtained by a formula like that for HI or for G3 1s more useful than that obtained subjectively by any observer or grou:p of observers. Hitherto it has only been possible to compare observers with each other. Now it is also possible to compare observers with a calculated Cluality­value, and as a result of th1s compar1son we may be able to obtain a better formula of the type HI or G3•

6. Possible Burther Work

(a) A better Cluantity like X can probably now be obtained by tren.t1ng the thirteen stUdios for vvhich g is 4 or 5 in Table 4 (Nos. 1 - 5,7 .- 9 9 17,18 9 20 9 22,23) in the same way as studios Nos. I, 29 3 9 49 22 and 23 were treated for the determination of X described in Appendix 2.

(b) The full Cluadratic regression formula ((13) or (14)) should be determined at least once 9 partly in order to see what value of the multiple correlation coefficient P is thus obtainable, and partly in order to obtain genuine "preferred values" if these exist. When (15) was derived from (14)9 it was not necessarily assumed that zi + ~ i = 0 (i = I, 29 3 9 4) were orthogonal, but the mcst signi­ficant "preferred values" rimy well be those associated with the principal axes of (14). But for this determination of (15) it is probably advisable to await further data to ensure that the results are statistically significant.

(c) The labour of a full~scale determination of (14) is certainly reduced by usingmcorrelated variables like S i in (8). It is possible that an advantage Can be gained by using the uniClue set of uncorrelated variables 11 i which are also orthogonal and normal, 1. e. such that if ""1 i = 0( i Xl + ~i x2 + Y i x3 + bi x4 the.n

d.~i 2 + Pi 2 -:- Y i 2 + 0i 2 = 1 and 0( i oZ j + Pi f3 j + Y i Y j + 6i 6 j = 0 for i t j9 in that a good approximation to (14) can be obtained with fewer terms than when arbitrary uncorrelated § i are used. The Quanti ties 'Yt i are associated with the latent roots of the matrix If rij 11 where rii = 1 and i9 j = I, 29 3, 4·

- 18 -

(d) As the correlation between g and HI is so high, perhaps a regression should be sought for g - HI as a lineir or quadratic function of other variables 9 possibly only of VI 3 and any new variable not hitherto considered. We should then assume that D9 Rand Tm contributed to the final result only through X.

(e) After equation (4), it was suggested that the variables D, R9 frm 9 VI/3 and g could possibly be replaced by et> 1 (D)? cP2(R) etc., but that such replacement? in the few cases tried 9 deemed to have li tUe effect. This matter could be more fully investigated. The fitting of a regression of the form (12) or (19) may imply certain tacit assumptions about the general nature of the distribution of D? B9 Tm 9 Vl/3 and g9 such as that the associated sets of values of these variables can be regarded as a random sample of a popUlation normally doistri buted!

(f) Instead of taking the "observed" values of .£ given by a pair of observers? we might have started with observations obtained in some sunh ,vay as the 11 average tr column on the extreme right of Table 5. This would overcome the diffi­culty that few observers are likely to know all the studios and concert halls for which the necessary measurements are likely to be availaoble shortly.

References

(1) T. Somerville "An Empirical Acoustic Criterion tr

(To be published in "Acustica lf)

(2) T. Somerville trSubjective Comparison of Concert Halls " ?

(3) ,

B.B.C. Quarterly? Vo1. VIII, No. 29 Summer 1953

J. Morris 9 "An :Cscala tor Process for the Solution of Linear Simultaneous Equationstr, Philosophical Magazine 9 Vol, 37, pp, 106 - 120 9 February 1946

(4) H..G. Kendall, "Advanced Theory of Statisticstr (Griffin and Co'? London), Vol. I, pp. 368 ff.

(5) H.J. Allcock and J.B. Jones 9 liThe Nomogram tr , Pitman and Sons, London, 4th edition (1950) revised by JoG.L. Michel, pp. 167 ff.

- 19 -

APPENDIX 1

Table 5 gives the subjective assessments of various observers for Studios 1 - 19; the last column but one gives the corresponding values of G3 for comparison 9 rounded off to one place of decimals. The last column gives the average value of g for all the observers who expressed an opinion on a particular studio. A and B were the same observers as gave the values used in Tables 1 and 4. Observer E asked the opinion of observer F on the stUdios 12 and 15. The correlation coefficients included at the foot of the table were based on values for Studios 1 - 17 onlY9 since what is here required is a general idea of the degree of agreement between competent observers. Observers E and li' could not be satisfactorily included in this calculation 9 except through their contributions to the last "average" column.

The high correlation between G3 and A and B is due to the fact that only the observations of A and B were used to determine G30 (The high correlation between observers C and D is probably due to the fact that they work in the same office.) We thus expect in general a correlation of the order of 0.7 between the estimates of individual observers 9 and one of the order of 0.56 between G3 and observer's other than those whose results were used to obtain G3" The correlation coefficient for Studios 1 _. 17 between G3 and the i1average!! opinion in the last column of Table 5 is 0.75.

- 20 -

studio No. A and B C D E and F Gi Average

-.-.- ..... --................. ~ • _. ___ ... __ .". >. __ •• _ •••••• _ •••• ___ J?a.l_c~_a._~_dJ._ .... ___ "'_' _______

1 5 4 4 5 4·9 4·5 2 5 4 5 4 4·5 4·5 3 4 4 4 3·9 4 4 5 4 (?) 4 4 3.2 4·25 5 4 4 3 3 3.6 3·5 6 .3 3 2 3 4.0 2·75 7 4 4 4 3 3·9 3·75 8 5 3 3 3 4.8 3·5 9 4 4 3 3 4·4 3·5

10 2 .3 2 1 1.2 2.0 11 2 3 3 3 3.6 2·75 12 2 3 3 1 (],) 2.8 2.25 13 1 1 1 1.0 1.0 14 1 1 1 1.7 1.0 15 1 3(?) 3 2(F) 1·5 2.25 16 3 5 5 2.6 4·333 17 4 4 4 3.3 4.0

___ ___ - __________ --..-. ~ ____ ............ _______ "'4_ ,, ____ ............... _____ ~ _ ......... ~ ____ ~ ________ ......... __

18 19

A and B C D

5 4 3 3 3

Correlation Coefficients

A and B C D E and F

0.70 0.69 \ - 0.89

3.4 4·0 3.2 3.0

- 21 -

APPENDIX 2

'J.1he Somerville Cri terion_~nd--.the_ Derived Fount~r..Ea;rJ..l G1 9 · HI of the Observed Quality g

The Somerville oriterion number

was originally devised with the limited objective of separating good studios from less good 9 as indicated in Fig. la The observed ~·~values used here had not then been determined 9 but Studios Nos. 19 29 3 and 49 and concert halls Nos. 22 and 23 were regarded as IIgood". For these 9 it was found that there was a high correlation between D + o(.R and Tm for values of cL.... in the. neighbourhood of 0.5 9 but that if cJ... was taken as 0.7 the correlation was still very nearly as high. X ~ 1 was the regression equation of D +oZR on Tm when 0(= 0.7. Values of X for good studios (whether among those

enumerated above or not) almost all fell in the range 0.8 to 1~29 while values of X for less good studios fell outside these limits. No significanoe was attached to variations of X between 0.8 and 1.2 9

or to the values of X - 1.2 or 0.8 - X when X lay outside the "good ll

range. No distinction was made between studios for which X ;> 1.2 and those for which X < 0.8.

If we try to compare the effectiveness of X as a criterion with that of values of ~ calculated from equations (12) or (19) l we find that the correlation between g and X is ~0.4459. This is misleadingly small 9 because extremely low or extremely high values of X associated with low values of g make large contributions to the value of 2. (g-g)(X-X) which are of opposite signs. '1he first possible source of improvement is to compare Y it X + X with ~9 since extreme values of X in either direction will make Y large 9

while values of X near 1 will make Y near its minimum value 2. The correlation is thus (numerically) increased to -0.562. A further possible improvement is to consider

1 '7. = xn + ~ '-11 Xn

as the effective parameter instead of X. The objection to Zn is that Zn changes little if X is near 19 while a few extreme values of X have disproportionate influence if ~ is large. The latter objection is much reduced by considering

S n = 10/~

- 22 -

as the effective parameter. If X = l~ Sn is 5 for all ~9 and we can choose n so as to make 5 n have a specified value for any suitable value of X? 5 n will tend to zero when X tends to zero or infinity for all~. If n = 41 g 4 is about 4 when X is 1.2 or 0.83 9 so that all studios for WhlCh 0.83 ~ X ~ 1.2 will be associated. with values of ~ 4 between 4 and. 5. In Fig. 2 the (D + 0.7R1 Tm) plane is divided up into regions for each of which 54 lies between two consecutive integers.

In Table 4 we have tabulated

and the correlation between Gl and g is 0.8120. This correla­tion value would be unaffected if we replace Gl by

(A9 B constants)

and we can choose A9 B so that

and

The appropriate values of A9 B are then

B = 0.779

HI can be regarded as probably the best avail~ble simple function of X for a calculated counterpart to ~.

A possible improvement of the function HI would be to choose a function

B A + Xn + X-n

such that

(i) 11 = 5 when X = 1 therefore

B A + ~. -~--.-. ---- = 4

(1.2)n + (1.2)-n (ii) I = 4 when X = 1.2 therefore

(iii) I 1 when X 2.1 therefore A + B

1

(X = 2.1 is the mean of the X's for the three studios for which g = 1.)

- 23 -

In this Case

4 1.5

and the root n = 0 is clearly inadmissible 9 when!l is 4 the equation is very nearly satisfied j so that any improvement in correlation in this way is likely to be slight 9 the correlation coefficient between 11 and .£ depends only on n and not on A9 B.

DIW

1 MAIDA VALE 1 2 CONCERT HALL, B.H. :5 SWANSEA 1 4 EDINBURGH I 5 MAIDA VALE 2 6 MAIDA VALE:5 7 CHARLES STREET, CARDIFF 8 CRITERION RESTAURANT 1 9 PARIS CINEMA

10 MANCHESTER 1 II BIRMINGHAM 4 12 BRISTOL 1 13 PORTlAND PLACE 3 14 PORTLAND PLACE 5 15 BRISTOL 5 16 PORTLAND PLACE 3 (MODIFIED) 17 PORTLAND PLACE 5 (MODIFIED) 18 GLASGOW 8 19 BELFAST 2 20 FREE TRADE HALL,MANCHESTER 21 ROYAL FESTIVAL HALL, LONDON ZZ ST. ANDREWS HAll, GLASGOW Z3 USHE R HALL, EDINBURG H 24 CIVIC HALL, WOL VERHAMPTON 25 BELFAST 2 (MODIFIED CALCULATION)

X GOOD ACOUSTICS [J PROBABLY GOOD ACOUSTICS o OTHER ACOUSTICS A CONCERT HAllS "" /'

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15

10

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+ 0

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"" ,;

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12 0

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FIG.l Tm IN SECONDS

SOMERVlllES CRIT ERION: THE "GOOD CORRIDOR"

X" 1·2

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the British Broadcasting Corporation and mt C)

not be reproduced or disclosed to • t t--~-----------~:..!--, ~ _ party in any form without the written k

mission of the Corporation. ::::

I MAIDA VALE 1 2 CONCERT HAll, B.H. :5 SWANSEA 1 4 E DINRURG H 1 5 MAIDA VALE 2 6 MA I DA V ALE :5 7 CHARLES STREET. CARDIFF 8 CRITERION RESTAURANT 1 9 PARIS CIHE MA

10 MANCHESTER 1 II BIRMINGHAM 4 12 BRISTOL 1 13 PORTLAHD PLACE:5 14 PORTlAND PLACE 5 15 BRISTOL 5 16 PORTlAND PLACE 3 (MODIFIED) 17 PORTLAND PLACE 5 (MODIFI ED) 18 GLASGOW 8 19 BELFAST2 20 FREE TRADE HALL. MANCHESTER 2r ROYAL FESTIVAL HALL, LONDON 27 ST. ANDREWS HAll, GLASGOW 23 USHER HALL, EDINBURGH 24 CIVIC HALL) WOLVERHAMPTON 25 BELFAST 2 (MODIFIED CALCULATION)

X GOOD ACOUSTICS o PROBABLY GOOD ACOUSTICS o OTHER ACOUSTICS ~ CONCERT HALLS

9:5 9=4 9,=3 9=2 9=1

NOTHING SYMBOL SYMBOL HUMBER NUMBER

FILLED BLACK WITH BLACK DOT RINGED OHCE RINGED TWICE

FIG.2

0'--------,,01"'1

®----/

o @~ __ -..I"'I ~

10 r-

O~==~'I====T===~I~==~'I====~==~ o 2

T m IN SECONDS

QUALITY PREDICTION ,

IN GREATER DETAIL BASED UPON SOMERVILLE S CRITER ION

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R

2-91 -r:2·91 4-, Z

5 I

6

-I 7

7-5

-2 8

8-5

9

9·5

10

10 -5

11

11· 5

I Z R

'·8 I' 7

1-6

10'35 10

9

8

7

1-5 4

1·4

I- 3

I- 2 2

o 0-9

0-8 -I

0-7 -2

0-6 -3

0 -5

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G 3 2·0

2· I

Z· 2

3'9 4-0 4-1 4-2

o

o

FIG.3

APP 0

I V3 V

14·26 2923 15 =t- 3375

20 8000

2515625

30 27000

42875

40 64000

45 91 125

50 125000

55 166375

60 216000

65 274625

70 343000

75 421875

80

90

vi

512 000

614125

729000

V

NOMOGRAM FOR QUALITY PREDICTION BY MEANS OF THE QUADRATIC REGRESSION FORMULA (19)