Biol. Cybernetics20, 27--36 (1975) 9 by Springer-Verlag 1975
An Intrinsic Structure of the Auditory Sensation Space with Special Reference to the Equal-Sensation Contours Te Sun Han University of Tokyo, Tokyo, Japan Received:January 5, 1975
Abstract An intrinsic structure of the auditory sensation space is studied with special referenceto the equal-sensation contours. An analysis of dependence relations among three kinds of equal-sensation contours for loudness, pitch and volume leads us to the concept of intrinsic structure as well as that of intrinsic coordinates. Mel, sone and 2-unit scales are interpreted from this standpoint.
Introduction The auditory sensation is one of the most important human sensations which are directly responsible for the voluntary and/or emotional communications. Therefore, extensive investigations have been developed from earlier times to clarify subjective and/or objective properties of auditory phenomena. Among them, physiological approaches to this problem have played a crucial role in studying the sensations due to pure tones. Fletcher and Munson (1933) were the first who experimentally defined the equal-loudness contours of pure tones from the point of view of subjective judgement. The contour so obtained has been named the F - M curve (Appendix II, Fig. 11) to their honour and therefrom the celebrated phon scale of loudness has been derived. Till the present time, the phon scale has widely been utilized in the design of diverse auditory instruments while its several improvements have been proposed [for instance, see Pollack (1948), Robinson et al. (1956)]. Equal-pitch contours were studied somewhat later by Stevens (1933), who has shown that the pitch sensation depends mainly upon the frequency of pure tone and also a little upon the intensity (Appendix II, Fig. 12). Thomas (1949) published the results of his experimental study of equal-volume contours (Appendix II, Fig. 13), thereby proving that the concept of volume sensation is quantitatively consistent. It seems
that the volume sensation of pure tone has, though not so much familiar as loudness and pitch, now become one of well established auditory attributes. In addition to these three attributes, the concepts such as density and brightness may be considered (Stevens, 1962) but the qualitative study as well as the establishment of their consistency remains for the future investigations. The present paper treats the problem how to clarify the dependence relations among the above three kinds of equal-sensation contours by introducing the 2-dimensional auditory sensation space as well as its intrinsic coordinate systems. We will show that the equal-loudness contours, the equal-pitch contours and the equal-volume contours form co-punctual straight lines passing through their own reference points, respectively, in reference to intrinsic coordinate systems. Furthermore, we determine the transformation group of intrinsic coordinate systems. It will also be pointed out that several subjective scales [such as mel scale (Stevens et al., 1940), sone scale (Howes, 1950) and X-unit scale (Garner, 1952) etc.] may be regarded as intrinsic coordinates. Finally, the result will be obtained that the auditory sensation space should be regarded as possessing a composite structure, i.e., consisting of two uniform parts. The psychological experiments to obtain the equal-sensation contours for loudness, pitch and volume seem to have been designed and handled in some or other experimental circumstances. Therefore, any theoretical treatment of dependence relations among them might be not to be undertaken unless the experimental results are established in a well controlled common setting. Nevertheless, our attempt to seek for an "intrinsic" structure of the auditory sensation space will be of methodological meaning from the standpoint of unifyingly investigating into various auditory attributes.
28
1. Preliminaries
1.1 Loudness, Pitch, and Volume Sensations to be adjoined to a physical stimulus need not necessarily be unique but several (Han, 1970). Owing to this uncertainty, the treatment of sensation will be of probability-theoretical character instead of deterministic one, as is the case the information theory in psychology. However, when the uncertainty of sensation may be considered to be so small that the responses to a stimulus can well be approximated by a representative sensation, we will be allowed to develop a deterministic theory. It is from this stanpoint that our study of the equal-sensation contours of pure tone I is made. According to Boring (1935), the number of distinctive attributes 2 is not necessarily in accord with the number of dimensions of the underlying physical space 3. In other words, he argues that the number of distinctive attributes may be more than the number of dimensions of the physical space. In the case of our auditory sensation space 3, the sensation of volume as well as those of loudness and pitch should be included among the distinctive attributes of pure tone: "a third attribute of tone (volume) is characterized by 'size of tone', 'extensity', 'spread-outness', 'massiveness' etc., and pure tones can be ordered on a scale from 'big', 'massive', and 'diffuse" to 'small', 'penetrating', and 'piercing' (Thomas, 1949)".
1.2. Equal-Sensation Contours The ensemble of pure tones forms a two-dimensional space with intensity and frequency as its coordinates. On the other hand, at least three attributes, i.e., loudness, pitch and volume are relevant in the auditory sensation space. The equal-sensation contour of an attribute A in the auditory sensation space is defined as a curve along which A has a constant magnitude of sensation forA" A = const.
(1)
According as A=loudness, pitch or volume, an equal-loudness contour, an equal-pitch contour or an equal-volume contour is obtained (see Appendix II, Figs. 11--13). 1 A pure t ~ is physicaily specified bY its intensity and frequency" 2 By attribute we, mean a constituent component of a psychological sensation. 3 Physical space is an aggregate of physical stimuli, whereas sensation space is that of psychological sensations caused by physical stimuli.
As is seen from the figures, these three kinds of contours consist of considerably curved lines on the intensity-frequency diagram, so that loudness, pitch and volume are expressed by particular functions of intensity and frequency: loudness = L(I, f ) , pitch = P(I, f )
and
(2)
volume = V(I, f ) ,
where I is the intensity and f is the frequency.
1.3. Formulation of the Problem The following three aspects should be taken into consideration in the study of sensations (Fig. 1). i) The physical aspect: Since the sensation space is an ensemble of sensations due to physical stimuli, what kinds of attributes of sensation come about is determined by the physical property of stimuli. In other words, the physical space is companied by its corresponding sensation space. ii) The transformational aspect: This aspect is concerned with how a physical stimulus is to be transformed to a sensation, i.e., the transformation from the physical space to the sensation space. This type of problem has been approached partly from the standpoint of "black box" by modern behaviourists and partly from the standpoint of the mechanism of nervous systems by physiologists. More concretely, it is the problem what particular form is given to the functions in (2). Since the forms of such functions are apt to vary with the contingent variations of characteristics of nervous systems and also with the coordinate transformations in the physical space, a particular form itself can not have any intrinsic psychological implication, iii) The intrinsic aspect of the sensation space: If a problem can be formulated in such a manner that neither the physical aspect nor the transformational aspect does intervene, by solving the problem we will obtain some intrinsic feature of the sensation space itself. Such a type of problem does not seem to have been considered with due concerns by modern psychologists, rather having been occupied in the investigations into the transformational aspects.
Physical space
~ [ Nervous system
--,
Sensation space
Fig, 1. Three aspects of sensation
29
Now let us formulate our main problem as follows. "Do there exist at all such a coordinate system of the sensation space that in reference to this system three kinds of equal-sensation contours above mentioned, i.e., the equal-loudness contours, the equal-pitch contours and the equal-volume contours all reduce to straight lines at the same time ?,.4 This has originated from the problem whether any dependence relation might be found among these three kinds of attributes and also from the primitive intuition that they are likely to be linearly dependent on one another if an appropriate coordinate system is chosen: Obviously, it is concerned with only the intrinsic aspect of the sensation space, hardly with any substantial relation to the physical or transformational aspect. Such a coordinate system, if exists, will describe the structure of the sensation space more essentially than the physical coordinate system, say, intensity and frequency of pure tone, and, therefore it will be called an intrinsic coordinate system of the sensation space.
2. Intrinsic Coordinates of the Auditor), Sensation Space
2.1. Scaling of Loudness and Pitch Sensation Since each of the constituent attributes of auditory sensation continously changes its magnitude with the intensity and frequency, the auditory sensation space is continuous. Hence, the auditory sensation space forms a two-dimensional manifold. Loudness and pitch are independent of each other in the sense that one can not be uniquely determined by another. Therefore, we shall tentatively adopt them as coordinate of the sensation space. For this purpose, we have to give concrete forms to real-valued functions (2). It is tantamount to the scaling problem of magnitude, of sensation, which may be solved from two different standpoints, i.e., one from the physical standpoint and the other from the psychological standpoint. Loudness is scaled, for instance, by phon (Fletcher et al., 1933) from the former standpoint and by sone (Howes, 1950) from the latter standpoint, whereas pitch is represented by musical scale from the former and by reel (Stevens et al., 1940) from the latter. From the time being, we shall adopt sone and reel scales for the scaling of loudness and pitch, respectively. Sone and reel are shown in Figs. 14 and 15 of Appendix It should be noted that in the two-dimensional space any two families of curves can always be transformed to two families of straight lines if an appropriate coordinate system is adopted.
II, respectively, where sone has been constructed by the equal-ratio method 5 and mel by the equal-distance method 5. 2.2 Representation of Equal-Volume Contours by Sone and Mel Scales In Fig. 2, the equal-volume contours by Thomas (1949) are represented by sone and mel, where the data (read from Fig. 13 of Appendix II) are shown by white or black circles. The values of sone used in Fig. 2 have been calculated by the following procedures. First, the frequency f and the intensity I of pure tone were read from Fig. 13. Next, the intensities 11 and 13 of the minimum audible threshold at f were read from Figs. 11 and 13, respectively, from which the value of I - 13 + 1i (= Io) was calculated (adjustment of minimum audible thresholds). Then, the value H of phon was determined by reading in Fig. 11 the value of parameter of the equal-loudness contour passing through the point with the frequency f and the intensity Io. Finally, the value of sone corresponding to H phon was read from Fig. 14 of Appendix II. On the other hand, the value of mel used in Fig. 2 was read from Fig. 15, where the equal-pitch contours were assumed to be approximated by straight lines parallel to the ordinate-axis, i.e., pitch is independent of the intensity 6. It should be noted that in Fig. 2 the met
4000.I 3ooo-
.
D C
2000~p~
Po -z,O -2'0
0
D,
o~
~,J~
2~)
4.;
6'0
B
8~3
sone 1;0
Fig. 2. Representation of equal-volume contours by sone and mel scales 5 Both methods are among the direct methods in the sense of Stevens. Let us denote the magnitude of sensation due to a physical stimulus x by re(x). For the construction of scales, the equal-ratio method demands an selection of stimuli x~ ..... x; ... such that m(xi+ O/m(x~)= const, from which m(xl)= ax ~ follows. On the other hand, if by the equal-distance method, we need to select stimuli such that m ( x ~ + t ) - m ( x ~ ) = c o n s t , then it follows that m(x~) = b + di. 6 It is reasonable, because the difference occurs in the domain to which the equal-volume contours in Fig. 13 do not stretch.
30
2500/reel
@
data to straight lines in Fig. 2 is justified, whereas the fact that the centers of the clusters are nearly co-linear in Fig. 3 means that the five straight lines (A, B, C, D and E), as has been anticipated above, are copunctual in Fig. 2. It is also seen from Fig. 3 that the equation of the straight line l0 passing through all the centers of the clusters is given by
y= (1.0/0.95)x+
/110
29.5/0.95
Po x ] 18 f mel:=80,
Y'= (1.0/0.95)X'+ 29.5/0.95,
(3)
from which we have the equation of the point Po through which the straight lines A, B, C, D, E all pass in Fig. 2: sone=-34 ,
I
I-i I - i 11
[0
0
2
12
4
6
lO0/sone
Fig. 3. Dual transform of Fig. 2
equal-loudness contours reduce to straight lines parallel t,o the ordinate-axis and the equal-pitch contours to straight lines parallel to the abscissa-axis. First, Fig. 2 shows that except for a neighbourhood of sone = 0 each of data (indicated by while circles) is considerably well fitted to one of the five straight lines A, B, C, D, E (equal-volume contours). Second, it is likely that these five straight lines all pass throug h a fixed point situated nearly at the abscissa-axis. This observation will further be ascertained from the following result obtained by transforming Fig. 2 by means of dual transformation theorem in projective geometry 7. Figure 3 is an alternative representation of Fig. 2 derived by the dual transformation where a point in Fig. 2 with sone = X and mel = Y as its coordinates corresponds to a straight line in Fig. 3 with x' = X/100 and y ' = Y/2500 as its line-coordinates, and vice versa. [In Fig. 3, the points in the neighbourhood of sone = 0 (indicated by black circles) are not drawn.] In Fig. 3, the intersection points of straight lines to belonging to the same family (corresponding to A, B, C, D or E in Fig. 2) are illustrated and the existence of five distinct clusters of intersection points corresponding to the five straight lines A, B, C, D and E, respectively, are clearly seen. The fairy good concentration of these clusters proves that the fitting of v This is the celebrated theorem in projective geometry. It says that the incidence relation between a (straight) line and a point remains invariant if we transform line coordinates to point coordinates and at the same time point coordinates to line coordinates (Veblen and Young 1938, 1946).
and
reel=80.
(4)
Similarly, the equal-loudness contours and the equalpitch contours pass through the respective fixed points at infinity s, which are given by reel = ~ and sone = ~ , respectively. Thus, as far as loudness, pitch and volume are concerned, all the equal-sensation contours pass through the fixed points of their own, which will be called the reference points. The problem presented in Section t.3 has now affirmatively solved at least in the domain excluding the above neighbourhood of sone = 0.
3. Intrinsic Structure of the Auditory Sensation Space
3.1, Reference Points of Attributes and their Co-Punctual Straight Lines So far it has been proved that sone and reel scales are intrinsic coordinates in the sense of Section 1.3. By using a projective transformation, such a structure of the sensation space can more compactly be represented as shown in Fig. 4. In Fig. 4, the equalsensation contours are all expressed by respective co-punctual straight lines. Thus, the structure of the sensation space in connection with the equal-sensation contours, if represented in reference to an intrinsic coordinate system, is completely characterized by designating three independent reference points R~, R 2, R 3. Such an simple structure has been brought about by the adoption of intrinsic coordinates, so that it will be called the intrinsic structure of the auditory sensation space.
3.2. Intrinsic Coordinate Transformations Let the set of all the intrinsic coordinate systems be denoted by Q. Then, the set of all the coordinate transformations among Q forms a group, which will 8 This statement is true in the sense of projective geometry.
31
R2
x and y can be uniquely expressed by a fixed Cartesian system Do with the coordinates X and Y:
al
X z
a
c2
y~
ai X" + a2 Y" + a3 Cl xn-~ r Y" + c3 '
(6)
bl X" + b2 Y" + b3 c I X" + c z Y" + c 3
R3
bl
b2
where a i, bi, ci (i = 1, 2, 3) and n are constants determined depending on the system K. To sum up, any intrinsic coordinate transformation is expressed as
R1
b3
Fig. 4. Intrinsic coordinate system (the points R1, Re, R 3 are the reference point of pitch, loudness, volume, respectively; the co-punctual lines al, bl, c~ are the equal-pitch, equal-loudness, equal-volume contours, respectively)
R2
X'= ai((atx +azy+a3)/(clx-l-c2yWc3))m cl ((al x + a2 y + a3)/(cl X + c 2 y + c3))" + #2 ((b t x + b2 Y + b3)/(ct x + c2 y + c3))" + a; + C~((b I x 4- b2 y + b3)/(ci x + c 2 y + c3))"~+ c;
R2
and
/
(7) bl ((an x + a z y + a3)/( q x + c2y + c3)) m
// R3 physicaI
K projective
R1
R3
/
/
C~I ((a 1 x
4-
a2 y + a3)/(cl x + c z y + c3)) m
+ b'~ ((bl x + b2 y + b3)/(cl x + c2 y + ca))" + b'3 + c'2 ((bl x + b2y + b3)/(c 1 x + cay + c3))m+ c; '
D(K) Cartesian
Fig. 5. Transformation from the physical coordinate system to the intrinsic coordinate systems
which completely determines the group G.
3.3 Quasi-Projective Group
be denoted by G and be called the group of intrinsic coordinate transformations. The purpose of this section is to study the property of G. Let the reference points of pitch, loudness and volume be denoted by R 1, R 2, R 3, respectively. Then, it is easily seen that given an intrinsic coordinate system K there always exists an appropriate projective transformation to derive from K a Cartesian coordinate system D(K) with the coordinates x and y such that in reference to D(K) R 1 is given by x = ov and y = 0, R 2 by x = 0 and y = ~ , and R 3 by x = y = 0. In reference to D(K), the equal-loudness contours, the equal-pitch contours and the equal-volume contours coincide with the straight lines parallel to y-axis, the straight lines parallel to x-axis and the straight lines passing through the origin ( x = y = 0 ) , respectively (Fig. 5). Such a Cartesian system D(K) may not be uniquely determined for a K b u t can be to within the transformations: x ' = ax",
y ' = by",
(5)
where a, b, n are arbitrary constants. (The proof is shown in Appendix I.) Therefore, it is concluded that any intrinsic coordinate system K with the coordinates
It will easily be seen that the inverse transformation of (7) is also of the same form as (7), m being replaced by its inverse 1/m. If it is demanded that both of the transformation (7) and its inverse are to be rational functions of their coordinates, only the case of m = 1 is allowed and consequently the group G coincides with the group of two-dimensional projective transformations (see Veblen and Young, 1946). Instead of such a rationality, if we assume that a certain straight line not passing through a n y of three reference points RI, Rz, R3 remains to be straight under every transformation in G, the group G reduces again to that of two-dimensional projective transformations. The group G, therefore, may be called the group of quasi-projective transformations and the space where the group of quasi-projective transformations is defined will be called a quasiprojective space. In this sense, our auditory sensation space is a quasi-projective one and suffers the quasiprojective transformations. On the other hand, the physical space is at most an affine one, of which coordinates, i.e., intensity I and frequency f suffer transformations of physical units such as I ' = b t I + b2, f ' = c f . Therefore, the scaling of the sensation space
32 is probably more flexible than that of the physical space. This suggests that the psychological or subjective distance between sensations might not a priori be regarded as an affine quantity. In order to be able to treat psychological distances as affine quantities, as is usually done without a reasonable justification in the field of psychophysics, detailed due discussions will be necessary for each concrete case.
pho~ scale is refered as parameter. It is seen from this figure that in the interval above about 95 phon the 2-unit scale is linearly related to the sone scale:
4. Non-Uniformity of the Auditory Sensation Space
y = x 3Is .
4.t. Sone Scale and 2-Unit Scale of Loudness
The remaining interval from 70 to 95 phon is of intermediate character. Thus, it is suggested that the quasi-projective transformation (7) above derived is realized in the coordinate transformations (8) and (9) from sone scale to 2-unit scale in that either (8) or (9) is a special form of (7).
Garner (1954) made a psychological experiment concerning loudness of pure tone, from which he derived another type of scale instead of sone scale and named it )~-unit scale (Fig. 14). In this experiment, he studied the relation of the scale constructed by the equal-ratio method to the scale by the equal-distance method. The sone scale is of the former type, whereas the 2-unit scale is of the latter. Figure 6 has been obtained by expressing the 2-unit scale as function of the sone scale, where the
x.-unit
117
10-
10,0,,71
8085~.~J
0
0
~0 . . . .
1
50
phon
. . . .
1
100
. . . .
I
150
. . . .
sone
F i g . 6. 2 - u n i t s c a l e
- 0.5[~ 7O 6
0.2
5~5 4 40-~ph~
0.0
one)S/8
1
y = 0.0405 x + 3.0,
where y is the magnitude of loudness measured in )~-unit and x is that in sone. On the other hand, in the interval from 20 phon to about 70 phon, as is shown in Fig. 7, the 2-unit is a power function of sone:
4.2.
9
.
9
I
-0.5
F i g . 7. R e l a t i o n
[
0.0 between
.
.
.
.
[
0.5
.
Ioglo(sone) 1.0 .
.
.
sone and 2-unit scales
[
(9)
Representation of Equal-Volume Contours by 2-Unit Scale and Mel Scale
In the preceding sections, we have confined ourselves to the discussions in the domain excluding the neighbourhood of sone = 0. In this section, we shall consider the problem in the remaining domain. For the sake of convenience, the neighbourhood of sone = 0 will be denoted by D t and the remaining domain by D z where the intrinsic structure in the latter has been established by sone and mel. Since in .Fig. 2 the equal-volume contours do not form straight lines in D1, sone and mel scales are not an intrinsic coordinate system in D 1. By representing the equal-volume contours in D1 by 2-unit and reel scales, we obtain Figs. 8 and 9 where the latter is the representation of the former by the dual transformation such that a point in Fig. 8 with A - u n i t = X and r e e l = Y as its coordinates corresponds to a straight line in Fig. 9 with x ' = X/5.0 and y ' = Y/1600 as its line-coordinates, and vice versa. Five clusters of points of intersections are seen in Fig. 9. The fairy good concentration of these clusters shows that the fitting to straight lines in Fig. 8 is justified, while, since the centers of clusters are nearly co-linear in Fig. 9, the five straight lines in Fig. 8 may be regarded as co-punctual. Quite similarly to in Section 2.2, we obtain the equation of the point Qo through which all the straight lines pass in Fig. 8: and
-0.2
(8)
),-unit = - 0.51 ( = - 0.17 in sone) reel = 112.
(10)
We conclude therefrom that the 2-unit and reel scales form a pair of intrinsic coordinates in the domain D1. It should be noted here that, though the
33
R 3000tme[
tangent
/E D2
2000- ~
/
r+dr r
/ D
D
1 B
IO00-
A
Oo~
it
I
-1
0
I
I
I
1
2
3
"~
I
I
I
4
5
6
Fig. 8. Representation of equal-volume contours by 2-unit and mel scales
- .
,
1600/melA~4
ze
E
e e+de
Fig. 10. Relation between equal-ratio and equal-distance scales
be remarked that D 1 nearly coincides with the domain which is frequently used through language sounds and/or music and so familiar to us, whereas the domain D2 is carried into effect mainly through noise sounds. 4.3. Adaption of the Zero Point
\XX\\
/I
/ (s~
In constructing the psychological distance by the equal-ratio method, it is an important matter how to identify the zero point (see Helson, 1948; Garner, 1954) where the magnitude of sensation is perceived to be null, because the judgement of ratio of magnitudes of two sensations would be impossible without identifying the zero point of sensation. Let an equal-ratio scale R be expressed as a function of an equal-distance scale E: R = f(E).
-10
-8
-6
-4
-2
0
2
Fig. 9. Dual transform of Fig. 8
If it is required that in a neighbourhood of E = e the equal-ratio scale R is to be in concordance with the equal-distance scale E, the following relation needs to hold: r+dr
e+de-ze+~e -
r
reference points of loudness and pitch for D1 nearly coincide with those for D2, respectively, the reference point of volume for Dj is considerably apart from that for D2. The position of Qo is considerably near to the origin (sone = 2-unit = mel = 0), compared to Po. Thus, though the auditory sensation space is not endowed with a unique intrinsic structure as a whole, the two parts of the space, i.e., D~ and D2 may be considered to possess their own intrinsic structures, respectively. In this sense, the auditory sensation space seems to be not uniform but to be divided into two uniform parts Dt and 0 2 . Incidentally, it should
(11)
, e--
(12)
Ze
where r = f (e), r + dr = f (e + de) and ze is the E-scaled value of the zero point to be assumed at the sensation point E = e. It follows therefrom that the zero point should be situated on the tangent line of the curve R = f ( E ) at the point E = e . Therefore, we have ze=e-
f(e) f'(e~-)'
(13)
where if(e) is the derivative of f(e). This shows that the zero point z e is not necessarily fixed at a certain point but rather moves with the sensation point E = e in consideration. Such a variability of the zero point
34 may be considered as a kind of adaptation widely recognized in psychological phenomena. Although it is conventionally assumed in studying of auditory sensations that zero point is fixed at a certain point, say, at the minimum audible threshold, we need to affirm the phenomenon of such an adaptation concerning the zero point if the consistency between the equal-ratio scale and the equal-distance scale is to be validated. By applying the Eq. (13) to the case where R = sone and E = 2-unit, we obtain the following results. i) In the interval from 20 to 70 phon where r = e 3/8, we have zx = (5/8)2.
(14)
Hence, the zero point does not coincide with the minimum audible threshold (2-unit= 0) but moves with the magnitude of loudness, which shows the continuous adaptation in this interval. ii) In the interval above 95 phon where r = 24.7e - 0.12, we have z~ = 3.0,
(15)
i.e., the zero poin t is fixed independently of the magnitude of loudness, the role of minimum audible threshold being played by the point 2 - u n i t = 3.0. It should be pointed out that the loudness specified by 2-unit = 3.0 nearly coincides with the most large one which can be attained in the domain D 1 . This means that the adaptation stops at the boundary olD1. Thus, the non-uniformity of the auditory sensation space mentioned above in Section 4.2 has been again confirmed from the standpoint of adaptation of zero point.
5. Concluding Remarks So far we have described an analysis of the auditory sensation space with special reference to the equalsensation contours. While the construction of various scales fo psychological sensation is usually based on the assumption that verbal responses for magnitude of sensation are to be consistent, that of equal-sensation contours can be made without such verbal responses. In this sense, the study of equal-sensation contours may be considered as more fundamental than that of psychological distances, so that any subjective scale of sensation may be given a more solid validity by clarifying its relation to the structure of equal-sensation contours. Therefore, the above result that the pair of 2,unit and mel scales as well as that of sone and mel scales forms intrinsic coordinate systems in D1 and D 2,
respectively, solidates the validity of these scales. Thus, the advantage to consider the attributes of pitch, loudness and volume not merely as fragmental separate ones but as constituent components of a psychological substance in the auditory sensation space seems to have been s h o w n to some extent. The intrinsic structure of the auditory sensation space studied in the foregoing sections consists of two kinds of elements, i.e., points and co-punctual straight lines therethrough. Since the equal-sensation contours are completely characterized by assigning its own reference point, the recognition of a psychological attribute is tantamount to that of its reference point if referred to an intrinsic coordinate system. By resorting to this intrinsic structure, the scaling problem of sensation for an attribute will be transformed in a natural manner to that of such co-punctual straight lines. In case the auditory sensation space reduces to a projective space (m = 1 in Section 3.2), an invariant scaling under the transformation group G for a bundle of co-punctual straight lines can be attained by means of line-coordinates, i.e., by the complex-ratio (see Veblen et al., 1954), as has well been established in projective geometry. Similarly for the general case. It should be noted that only three independent points are allowed in the two-dimensional quasiprojective space, so that the number of independent attributes is at most three. In this manner, the scaling of volume sensation, as has yet been untried from the standpoint of psychological experiments, may be undertaken from that of invariant scaling in the quasi-projective space. Here we emphasize the following point, which is so often overlooked: The scaling for an attribute need not to be restricted to a certain one but many altei-natives are possible depending on various psychological circumstances, where the problem of what forms of scale really take about does not matter so much as that of how one scale is related to others. The former aspect is concerned with the transformational one whereas the latter is with the intrinsic aspect of the sensation space. Since the data concerning the equal-sensation contours used in the above have been taken from the experiments by three psychologists under respective diverse conditions, the above results should not be considered as giving more than a tentative step to study the intrinsic structure of the auditory sensation space. In order to conclude definitely and to develop further the above results, it seems to be necessitated that detailed psychological experiments under a more unified condition are to be performed.
35 Appendix
Derivation
Appendix II
I
of the Transformation
(5)
Let us consider the coordinate transformation between two Cartesian systems x = f ( x ' , y'),
y = 9(x', y').
(A. 1)
In order for this transformation to satisfy the requirement mentioned in Section 3.2, it is necessary and sufficient that
i)
x' = const
whenever
x = const,
(A.2)
ii)
y' = const
whenever
y = const,
(A.3)
and iii)
i00 40 ~
y' = m' x '
whenever
y = rex,
o
(A.4)
where m is a constant and m' is a function of m. It follows from (A.2) that the function f ( x ' , y') is independent of y', so that we can put x = f(x').
4
20. 0 I
(1.5)
0.2
,
~
....
0.5
I
. . . . . . . .
1
2 3 4567 10 15
I
I
Similarly, from (A.3), we have Y = g(Y').
(A.6)
Fig. 11. EquaModness contours. [Fletcher, H., Munson, W.A.: J. acoust. Soc. Amer. 5 (1933) MAS = m i n i m u m audible threshold]
Then, the Eqs. (A.4) ~ (A.6) yield the relation f (x') = mg(y') ,
(A.7)
y' = G ( f (x')/m)= m' x' ,
(A.8)
from which it follows that
d b(pressure re.lbar)
where G(y') is the inverse function of g(y'). By putting x ' = 1 in (A.8), we have m'= G(f(1)/m).
(1.9)
40"
r
Hence, by substituting (A.9) into (A.8), we have
20G ( f ( x ' ) / m ) = x' G ( f ( 1 ) / m , from which it follows that G (uz/f(1))
0
=
Y(z) G(u),
(A. m)
o
/
.o
o
~II,~ ~ o
-20
Q o o
where we put z = f ( x ' ) , u = f ( l ) / m and F(z) is the inverse function of f ( z ) . By differentiating (A.10) with respect to z and putting z = f(1), we have uG'(u)/f(1)=F'(f(t))G(u).
(A.11)
o
-4Q
-60"
kHz |
0.2
Therefore, by integrating (A. 11), we have x' = ax",
y' = by n ,
where a, b, and n are arbitrary constants.
. . . . . . .
0.5
I
1
I
. . . . . . .
]
I
2 3 4 567 10 15
(A. 12) Fig. 12. Equal-pitch contours. [Stevens, S.S.: J. acoust. Soc. Amer. 7 (1935) MAS = m i n i m u m audible threshold]
36
00 800 db(intensityre.O.OOO2/abar)
20 ~ 0
k I
O2
z
. . . . . . .
I
0.5
1
. . . . . . . .
I
2 3 4567 10
Fig. 13. Equal-volume contours. [Thomas, G.J.: Amer. J. Psychol. 62 (1949) MAS = minimum audible threshold]
sone
[00
50
q---10 20
,
,
40
30
50
60
70
,
,
80
90
100
met
4000 mel(f)=4000f/(f §
"2000
"1000
0.0~
/
/
^ 3 ~ o.1
o2
o~
kHz 1
~ ~ ~,~1%
References Boring, E.G.: The relation of the attributes of sensation to the dimensions of the stimulus. Phil. Sci. 2, 236~245 (1935) Fletcher, H., Munson, W. A.: Loudness, its definition and calculation. J. acoust. Soc. Amer. 5, 8 ~ 1 0 8 (1933) Garner, W.R.: A technique and a scale for loudness measurement. J. acoust. Soc. Amer. 26, 73--88 (1954) Han, T. S.: A constructive approach to psychometric function from the information-theoretical point of view. RAAG Res. Notes, Third Series, No. 158 (1970) Han, T.S.: An information-theoretic and geometrical study of sensation metrices and its application to vowel perception. Doctor thesis, Univ. of Tokyo (1971) Howes, D.H.: The loudness of multicomponent tones. Amer. J. Psychol. 63, 1--30 (1950) Pollack, I.: Studies in the loudness of complex sounds. Ph. D. Diss. Harvard University (1948) Robinson, D~W., Davidson, M.A.: A re-determination of the equalloudness relation for pure tones. ISO/TC/43, 46 (1955) Stevens, S.S.: The surprising simplicity of sensory metrics. Amer. Psychol. 17, 29--39 (1962) Stevense, S.S., Volkmann, J.: The relation of pitch to frequency: a revised scale. Amer. L Psychol. 53, 329 353 (1940) Thomas, G.J.: Equal-volume judgements of tones. Amer. J. Psychol. 62, 182--201 (1949) Veblen, O., Young, J.W.: Projective geometry I, II. New York: Blaisdell Publishing Co., 1938, 1946 Dr. Te Sun Hart University of Tokyo Faculty of Engineering Tokyo 113, Japan
Fig. 14. Sone scale and 2-unit scale. [Sone from Howes, D.: Amer. J. Psychol. 63, (1950), 2-unit from Garner, W.R.: J. acoust. Soc. Amer. 26 (1954)]
-3000
Acknowledgements. The author wishes to thank Professor Masao Iri for helpful discussions. The idea of projective dual transformation used to obtain Figs. 3 and 9 is due to him.
1~
Fig. 15. Mel scale. [Data marked by white circles from Stevens, S.S., and Volkmann, J.: Amer. J. Psychol. 53 (1940); the theoretical curve from Han, T. S., Doctor Dissertation (1971)]
An Intrinsic Structure of the Auditory Sensation Space with Special Reference to the Equal-Sensation Contours Te Sun Han University of Tokyo, Tokyo, Japan Received:January 5, 1975
Abstract An intrinsic structure of the auditory sensation space is studied with special referenceto the equal-sensation contours. An analysis of dependence relations among three kinds of equal-sensation contours for loudness, pitch and volume leads us to the concept of intrinsic structure as well as that of intrinsic coordinates. Mel, sone and 2-unit scales are interpreted from this standpoint.
Introduction The auditory sensation is one of the most important human sensations which are directly responsible for the voluntary and/or emotional communications. Therefore, extensive investigations have been developed from earlier times to clarify subjective and/or objective properties of auditory phenomena. Among them, physiological approaches to this problem have played a crucial role in studying the sensations due to pure tones. Fletcher and Munson (1933) were the first who experimentally defined the equal-loudness contours of pure tones from the point of view of subjective judgement. The contour so obtained has been named the F - M curve (Appendix II, Fig. 11) to their honour and therefrom the celebrated phon scale of loudness has been derived. Till the present time, the phon scale has widely been utilized in the design of diverse auditory instruments while its several improvements have been proposed [for instance, see Pollack (1948), Robinson et al. (1956)]. Equal-pitch contours were studied somewhat later by Stevens (1933), who has shown that the pitch sensation depends mainly upon the frequency of pure tone and also a little upon the intensity (Appendix II, Fig. 12). Thomas (1949) published the results of his experimental study of equal-volume contours (Appendix II, Fig. 13), thereby proving that the concept of volume sensation is quantitatively consistent. It seems
that the volume sensation of pure tone has, though not so much familiar as loudness and pitch, now become one of well established auditory attributes. In addition to these three attributes, the concepts such as density and brightness may be considered (Stevens, 1962) but the qualitative study as well as the establishment of their consistency remains for the future investigations. The present paper treats the problem how to clarify the dependence relations among the above three kinds of equal-sensation contours by introducing the 2-dimensional auditory sensation space as well as its intrinsic coordinate systems. We will show that the equal-loudness contours, the equal-pitch contours and the equal-volume contours form co-punctual straight lines passing through their own reference points, respectively, in reference to intrinsic coordinate systems. Furthermore, we determine the transformation group of intrinsic coordinate systems. It will also be pointed out that several subjective scales [such as mel scale (Stevens et al., 1940), sone scale (Howes, 1950) and X-unit scale (Garner, 1952) etc.] may be regarded as intrinsic coordinates. Finally, the result will be obtained that the auditory sensation space should be regarded as possessing a composite structure, i.e., consisting of two uniform parts. The psychological experiments to obtain the equal-sensation contours for loudness, pitch and volume seem to have been designed and handled in some or other experimental circumstances. Therefore, any theoretical treatment of dependence relations among them might be not to be undertaken unless the experimental results are established in a well controlled common setting. Nevertheless, our attempt to seek for an "intrinsic" structure of the auditory sensation space will be of methodological meaning from the standpoint of unifyingly investigating into various auditory attributes.
28
1. Preliminaries
1.1 Loudness, Pitch, and Volume Sensations to be adjoined to a physical stimulus need not necessarily be unique but several (Han, 1970). Owing to this uncertainty, the treatment of sensation will be of probability-theoretical character instead of deterministic one, as is the case the information theory in psychology. However, when the uncertainty of sensation may be considered to be so small that the responses to a stimulus can well be approximated by a representative sensation, we will be allowed to develop a deterministic theory. It is from this stanpoint that our study of the equal-sensation contours of pure tone I is made. According to Boring (1935), the number of distinctive attributes 2 is not necessarily in accord with the number of dimensions of the underlying physical space 3. In other words, he argues that the number of distinctive attributes may be more than the number of dimensions of the physical space. In the case of our auditory sensation space 3, the sensation of volume as well as those of loudness and pitch should be included among the distinctive attributes of pure tone: "a third attribute of tone (volume) is characterized by 'size of tone', 'extensity', 'spread-outness', 'massiveness' etc., and pure tones can be ordered on a scale from 'big', 'massive', and 'diffuse" to 'small', 'penetrating', and 'piercing' (Thomas, 1949)".
1.2. Equal-Sensation Contours The ensemble of pure tones forms a two-dimensional space with intensity and frequency as its coordinates. On the other hand, at least three attributes, i.e., loudness, pitch and volume are relevant in the auditory sensation space. The equal-sensation contour of an attribute A in the auditory sensation space is defined as a curve along which A has a constant magnitude of sensation forA" A = const.
(1)
According as A=loudness, pitch or volume, an equal-loudness contour, an equal-pitch contour or an equal-volume contour is obtained (see Appendix II, Figs. 11--13). 1 A pure t ~ is physicaily specified bY its intensity and frequency" 2 By attribute we, mean a constituent component of a psychological sensation. 3 Physical space is an aggregate of physical stimuli, whereas sensation space is that of psychological sensations caused by physical stimuli.
As is seen from the figures, these three kinds of contours consist of considerably curved lines on the intensity-frequency diagram, so that loudness, pitch and volume are expressed by particular functions of intensity and frequency: loudness = L(I, f ) , pitch = P(I, f )
and
(2)
volume = V(I, f ) ,
where I is the intensity and f is the frequency.
1.3. Formulation of the Problem The following three aspects should be taken into consideration in the study of sensations (Fig. 1). i) The physical aspect: Since the sensation space is an ensemble of sensations due to physical stimuli, what kinds of attributes of sensation come about is determined by the physical property of stimuli. In other words, the physical space is companied by its corresponding sensation space. ii) The transformational aspect: This aspect is concerned with how a physical stimulus is to be transformed to a sensation, i.e., the transformation from the physical space to the sensation space. This type of problem has been approached partly from the standpoint of "black box" by modern behaviourists and partly from the standpoint of the mechanism of nervous systems by physiologists. More concretely, it is the problem what particular form is given to the functions in (2). Since the forms of such functions are apt to vary with the contingent variations of characteristics of nervous systems and also with the coordinate transformations in the physical space, a particular form itself can not have any intrinsic psychological implication, iii) The intrinsic aspect of the sensation space: If a problem can be formulated in such a manner that neither the physical aspect nor the transformational aspect does intervene, by solving the problem we will obtain some intrinsic feature of the sensation space itself. Such a type of problem does not seem to have been considered with due concerns by modern psychologists, rather having been occupied in the investigations into the transformational aspects.
Physical space
~ [ Nervous system
--,
Sensation space
Fig, 1. Three aspects of sensation
29
Now let us formulate our main problem as follows. "Do there exist at all such a coordinate system of the sensation space that in reference to this system three kinds of equal-sensation contours above mentioned, i.e., the equal-loudness contours, the equal-pitch contours and the equal-volume contours all reduce to straight lines at the same time ?,.4 This has originated from the problem whether any dependence relation might be found among these three kinds of attributes and also from the primitive intuition that they are likely to be linearly dependent on one another if an appropriate coordinate system is chosen: Obviously, it is concerned with only the intrinsic aspect of the sensation space, hardly with any substantial relation to the physical or transformational aspect. Such a coordinate system, if exists, will describe the structure of the sensation space more essentially than the physical coordinate system, say, intensity and frequency of pure tone, and, therefore it will be called an intrinsic coordinate system of the sensation space.
2. Intrinsic Coordinates of the Auditor), Sensation Space
2.1. Scaling of Loudness and Pitch Sensation Since each of the constituent attributes of auditory sensation continously changes its magnitude with the intensity and frequency, the auditory sensation space is continuous. Hence, the auditory sensation space forms a two-dimensional manifold. Loudness and pitch are independent of each other in the sense that one can not be uniquely determined by another. Therefore, we shall tentatively adopt them as coordinate of the sensation space. For this purpose, we have to give concrete forms to real-valued functions (2). It is tantamount to the scaling problem of magnitude, of sensation, which may be solved from two different standpoints, i.e., one from the physical standpoint and the other from the psychological standpoint. Loudness is scaled, for instance, by phon (Fletcher et al., 1933) from the former standpoint and by sone (Howes, 1950) from the latter standpoint, whereas pitch is represented by musical scale from the former and by reel (Stevens et al., 1940) from the latter. From the time being, we shall adopt sone and reel scales for the scaling of loudness and pitch, respectively. Sone and reel are shown in Figs. 14 and 15 of Appendix It should be noted that in the two-dimensional space any two families of curves can always be transformed to two families of straight lines if an appropriate coordinate system is adopted.
II, respectively, where sone has been constructed by the equal-ratio method 5 and mel by the equal-distance method 5. 2.2 Representation of Equal-Volume Contours by Sone and Mel Scales In Fig. 2, the equal-volume contours by Thomas (1949) are represented by sone and mel, where the data (read from Fig. 13 of Appendix II) are shown by white or black circles. The values of sone used in Fig. 2 have been calculated by the following procedures. First, the frequency f and the intensity I of pure tone were read from Fig. 13. Next, the intensities 11 and 13 of the minimum audible threshold at f were read from Figs. 11 and 13, respectively, from which the value of I - 13 + 1i (= Io) was calculated (adjustment of minimum audible thresholds). Then, the value H of phon was determined by reading in Fig. 11 the value of parameter of the equal-loudness contour passing through the point with the frequency f and the intensity Io. Finally, the value of sone corresponding to H phon was read from Fig. 14 of Appendix II. On the other hand, the value of mel used in Fig. 2 was read from Fig. 15, where the equal-pitch contours were assumed to be approximated by straight lines parallel to the ordinate-axis, i.e., pitch is independent of the intensity 6. It should be noted that in Fig. 2 the met
4000.I 3ooo-
.
D C
2000~p~
Po -z,O -2'0
0
D,
o~
~,J~
2~)
4.;
6'0
B
8~3
sone 1;0
Fig. 2. Representation of equal-volume contours by sone and mel scales 5 Both methods are among the direct methods in the sense of Stevens. Let us denote the magnitude of sensation due to a physical stimulus x by re(x). For the construction of scales, the equal-ratio method demands an selection of stimuli x~ ..... x; ... such that m(xi+ O/m(x~)= const, from which m(xl)= ax ~ follows. On the other hand, if by the equal-distance method, we need to select stimuli such that m ( x ~ + t ) - m ( x ~ ) = c o n s t , then it follows that m(x~) = b + di. 6 It is reasonable, because the difference occurs in the domain to which the equal-volume contours in Fig. 13 do not stretch.
30
2500/reel
@
data to straight lines in Fig. 2 is justified, whereas the fact that the centers of the clusters are nearly co-linear in Fig. 3 means that the five straight lines (A, B, C, D and E), as has been anticipated above, are copunctual in Fig. 2. It is also seen from Fig. 3 that the equation of the straight line l0 passing through all the centers of the clusters is given by
y= (1.0/0.95)x+
/110
29.5/0.95
Po x ] 18 f mel:=80,
Y'= (1.0/0.95)X'+ 29.5/0.95,
(3)
from which we have the equation of the point Po through which the straight lines A, B, C, D, E all pass in Fig. 2: sone=-34 ,
I
I-i I - i 11
[0
0
2
12
4
6
lO0/sone
Fig. 3. Dual transform of Fig. 2
equal-loudness contours reduce to straight lines parallel t,o the ordinate-axis and the equal-pitch contours to straight lines parallel to the abscissa-axis. First, Fig. 2 shows that except for a neighbourhood of sone = 0 each of data (indicated by while circles) is considerably well fitted to one of the five straight lines A, B, C, D, E (equal-volume contours). Second, it is likely that these five straight lines all pass throug h a fixed point situated nearly at the abscissa-axis. This observation will further be ascertained from the following result obtained by transforming Fig. 2 by means of dual transformation theorem in projective geometry 7. Figure 3 is an alternative representation of Fig. 2 derived by the dual transformation where a point in Fig. 2 with sone = X and mel = Y as its coordinates corresponds to a straight line in Fig. 3 with x' = X/100 and y ' = Y/2500 as its line-coordinates, and vice versa. [In Fig. 3, the points in the neighbourhood of sone = 0 (indicated by black circles) are not drawn.] In Fig. 3, the intersection points of straight lines to belonging to the same family (corresponding to A, B, C, D or E in Fig. 2) are illustrated and the existence of five distinct clusters of intersection points corresponding to the five straight lines A, B, C, D and E, respectively, are clearly seen. The fairy good concentration of these clusters proves that the fitting of v This is the celebrated theorem in projective geometry. It says that the incidence relation between a (straight) line and a point remains invariant if we transform line coordinates to point coordinates and at the same time point coordinates to line coordinates (Veblen and Young 1938, 1946).
and
reel=80.
(4)
Similarly, the equal-loudness contours and the equalpitch contours pass through the respective fixed points at infinity s, which are given by reel = ~ and sone = ~ , respectively. Thus, as far as loudness, pitch and volume are concerned, all the equal-sensation contours pass through the fixed points of their own, which will be called the reference points. The problem presented in Section t.3 has now affirmatively solved at least in the domain excluding the above neighbourhood of sone = 0.
3. Intrinsic Structure of the Auditory Sensation Space
3.1, Reference Points of Attributes and their Co-Punctual Straight Lines So far it has been proved that sone and reel scales are intrinsic coordinates in the sense of Section 1.3. By using a projective transformation, such a structure of the sensation space can more compactly be represented as shown in Fig. 4. In Fig. 4, the equalsensation contours are all expressed by respective co-punctual straight lines. Thus, the structure of the sensation space in connection with the equal-sensation contours, if represented in reference to an intrinsic coordinate system, is completely characterized by designating three independent reference points R~, R 2, R 3. Such an simple structure has been brought about by the adoption of intrinsic coordinates, so that it will be called the intrinsic structure of the auditory sensation space.
3.2. Intrinsic Coordinate Transformations Let the set of all the intrinsic coordinate systems be denoted by Q. Then, the set of all the coordinate transformations among Q forms a group, which will 8 This statement is true in the sense of projective geometry.
31
R2
x and y can be uniquely expressed by a fixed Cartesian system Do with the coordinates X and Y:
al
X z
a
c2
y~
ai X" + a2 Y" + a3 Cl xn-~ r Y" + c3 '
(6)
bl X" + b2 Y" + b3 c I X" + c z Y" + c 3
R3
bl
b2
where a i, bi, ci (i = 1, 2, 3) and n are constants determined depending on the system K. To sum up, any intrinsic coordinate transformation is expressed as
R1
b3
Fig. 4. Intrinsic coordinate system (the points R1, Re, R 3 are the reference point of pitch, loudness, volume, respectively; the co-punctual lines al, bl, c~ are the equal-pitch, equal-loudness, equal-volume contours, respectively)
R2
X'= ai((atx +azy+a3)/(clx-l-c2yWc3))m cl ((al x + a2 y + a3)/(cl X + c 2 y + c3))" + #2 ((b t x + b2 Y + b3)/(ct x + c2 y + c3))" + a; + C~((b I x 4- b2 y + b3)/(ci x + c 2 y + c3))"~+ c;
R2
and
/
(7) bl ((an x + a z y + a3)/( q x + c2y + c3)) m
// R3 physicaI
K projective
R1
R3
/
/
C~I ((a 1 x
4-
a2 y + a3)/(cl x + c z y + c3)) m
+ b'~ ((bl x + b2 y + b3)/(cl x + c2 y + ca))" + b'3 + c'2 ((bl x + b2y + b3)/(c 1 x + cay + c3))m+ c; '
D(K) Cartesian
Fig. 5. Transformation from the physical coordinate system to the intrinsic coordinate systems
which completely determines the group G.
3.3 Quasi-Projective Group
be denoted by G and be called the group of intrinsic coordinate transformations. The purpose of this section is to study the property of G. Let the reference points of pitch, loudness and volume be denoted by R 1, R 2, R 3, respectively. Then, it is easily seen that given an intrinsic coordinate system K there always exists an appropriate projective transformation to derive from K a Cartesian coordinate system D(K) with the coordinates x and y such that in reference to D(K) R 1 is given by x = ov and y = 0, R 2 by x = 0 and y = ~ , and R 3 by x = y = 0. In reference to D(K), the equal-loudness contours, the equal-pitch contours and the equal-volume contours coincide with the straight lines parallel to y-axis, the straight lines parallel to x-axis and the straight lines passing through the origin ( x = y = 0 ) , respectively (Fig. 5). Such a Cartesian system D(K) may not be uniquely determined for a K b u t can be to within the transformations: x ' = ax",
y ' = by",
(5)
where a, b, n are arbitrary constants. (The proof is shown in Appendix I.) Therefore, it is concluded that any intrinsic coordinate system K with the coordinates
It will easily be seen that the inverse transformation of (7) is also of the same form as (7), m being replaced by its inverse 1/m. If it is demanded that both of the transformation (7) and its inverse are to be rational functions of their coordinates, only the case of m = 1 is allowed and consequently the group G coincides with the group of two-dimensional projective transformations (see Veblen and Young, 1946). Instead of such a rationality, if we assume that a certain straight line not passing through a n y of three reference points RI, Rz, R3 remains to be straight under every transformation in G, the group G reduces again to that of two-dimensional projective transformations. The group G, therefore, may be called the group of quasi-projective transformations and the space where the group of quasi-projective transformations is defined will be called a quasiprojective space. In this sense, our auditory sensation space is a quasi-projective one and suffers the quasiprojective transformations. On the other hand, the physical space is at most an affine one, of which coordinates, i.e., intensity I and frequency f suffer transformations of physical units such as I ' = b t I + b2, f ' = c f . Therefore, the scaling of the sensation space
32 is probably more flexible than that of the physical space. This suggests that the psychological or subjective distance between sensations might not a priori be regarded as an affine quantity. In order to be able to treat psychological distances as affine quantities, as is usually done without a reasonable justification in the field of psychophysics, detailed due discussions will be necessary for each concrete case.
pho~ scale is refered as parameter. It is seen from this figure that in the interval above about 95 phon the 2-unit scale is linearly related to the sone scale:
4. Non-Uniformity of the Auditory Sensation Space
y = x 3Is .
4.t. Sone Scale and 2-Unit Scale of Loudness
The remaining interval from 70 to 95 phon is of intermediate character. Thus, it is suggested that the quasi-projective transformation (7) above derived is realized in the coordinate transformations (8) and (9) from sone scale to 2-unit scale in that either (8) or (9) is a special form of (7).
Garner (1954) made a psychological experiment concerning loudness of pure tone, from which he derived another type of scale instead of sone scale and named it )~-unit scale (Fig. 14). In this experiment, he studied the relation of the scale constructed by the equal-ratio method to the scale by the equal-distance method. The sone scale is of the former type, whereas the 2-unit scale is of the latter. Figure 6 has been obtained by expressing the 2-unit scale as function of the sone scale, where the
x.-unit
117
10-
10,0,,71
8085~.~J
0
0
~0 . . . .
1
50
phon
. . . .
1
100
. . . .
I
150
. . . .
sone
F i g . 6. 2 - u n i t s c a l e
- 0.5[~ 7O 6
0.2
5~5 4 40-~ph~
0.0
one)S/8
1
y = 0.0405 x + 3.0,
where y is the magnitude of loudness measured in )~-unit and x is that in sone. On the other hand, in the interval from 20 phon to about 70 phon, as is shown in Fig. 7, the 2-unit is a power function of sone:
4.2.
9
.
9
I
-0.5
F i g . 7. R e l a t i o n
[
0.0 between
.
.
.
.
[
0.5
.
Ioglo(sone) 1.0 .
.
.
sone and 2-unit scales
[
(9)
Representation of Equal-Volume Contours by 2-Unit Scale and Mel Scale
In the preceding sections, we have confined ourselves to the discussions in the domain excluding the neighbourhood of sone = 0. In this section, we shall consider the problem in the remaining domain. For the sake of convenience, the neighbourhood of sone = 0 will be denoted by D t and the remaining domain by D z where the intrinsic structure in the latter has been established by sone and mel. Since in .Fig. 2 the equal-volume contours do not form straight lines in D1, sone and mel scales are not an intrinsic coordinate system in D 1. By representing the equal-volume contours in D1 by 2-unit and reel scales, we obtain Figs. 8 and 9 where the latter is the representation of the former by the dual transformation such that a point in Fig. 8 with A - u n i t = X and r e e l = Y as its coordinates corresponds to a straight line in Fig. 9 with x ' = X/5.0 and y ' = Y/1600 as its line-coordinates, and vice versa. Five clusters of points of intersections are seen in Fig. 9. The fairy good concentration of these clusters shows that the fitting to straight lines in Fig. 8 is justified, while, since the centers of clusters are nearly co-linear in Fig. 9, the five straight lines in Fig. 8 may be regarded as co-punctual. Quite similarly to in Section 2.2, we obtain the equation of the point Qo through which all the straight lines pass in Fig. 8: and
-0.2
(8)
),-unit = - 0.51 ( = - 0.17 in sone) reel = 112.
(10)
We conclude therefrom that the 2-unit and reel scales form a pair of intrinsic coordinates in the domain D1. It should be noted here that, though the
33
R 3000tme[
tangent
/E D2
2000- ~
/
r+dr r
/ D
D
1 B
IO00-
A
Oo~
it
I
-1
0
I
I
I
1
2
3
"~
I
I
I
4
5
6
Fig. 8. Representation of equal-volume contours by 2-unit and mel scales
- .
,
1600/melA~4
ze
E
e e+de
Fig. 10. Relation between equal-ratio and equal-distance scales
be remarked that D 1 nearly coincides with the domain which is frequently used through language sounds and/or music and so familiar to us, whereas the domain D2 is carried into effect mainly through noise sounds. 4.3. Adaption of the Zero Point
\XX\\
/I
/ (s~
In constructing the psychological distance by the equal-ratio method, it is an important matter how to identify the zero point (see Helson, 1948; Garner, 1954) where the magnitude of sensation is perceived to be null, because the judgement of ratio of magnitudes of two sensations would be impossible without identifying the zero point of sensation. Let an equal-ratio scale R be expressed as a function of an equal-distance scale E: R = f(E).
-10
-8
-6
-4
-2
0
2
Fig. 9. Dual transform of Fig. 8
If it is required that in a neighbourhood of E = e the equal-ratio scale R is to be in concordance with the equal-distance scale E, the following relation needs to hold: r+dr
e+de-ze+~e -
r
reference points of loudness and pitch for D1 nearly coincide with those for D2, respectively, the reference point of volume for Dj is considerably apart from that for D2. The position of Qo is considerably near to the origin (sone = 2-unit = mel = 0), compared to Po. Thus, though the auditory sensation space is not endowed with a unique intrinsic structure as a whole, the two parts of the space, i.e., D~ and D2 may be considered to possess their own intrinsic structures, respectively. In this sense, the auditory sensation space seems to be not uniform but to be divided into two uniform parts Dt and 0 2 . Incidentally, it should
(11)
, e--
(12)
Ze
where r = f (e), r + dr = f (e + de) and ze is the E-scaled value of the zero point to be assumed at the sensation point E = e. It follows therefrom that the zero point should be situated on the tangent line of the curve R = f ( E ) at the point E = e . Therefore, we have ze=e-
f(e) f'(e~-)'
(13)
where if(e) is the derivative of f(e). This shows that the zero point z e is not necessarily fixed at a certain point but rather moves with the sensation point E = e in consideration. Such a variability of the zero point
34 may be considered as a kind of adaptation widely recognized in psychological phenomena. Although it is conventionally assumed in studying of auditory sensations that zero point is fixed at a certain point, say, at the minimum audible threshold, we need to affirm the phenomenon of such an adaptation concerning the zero point if the consistency between the equal-ratio scale and the equal-distance scale is to be validated. By applying the Eq. (13) to the case where R = sone and E = 2-unit, we obtain the following results. i) In the interval from 20 to 70 phon where r = e 3/8, we have zx = (5/8)2.
(14)
Hence, the zero point does not coincide with the minimum audible threshold (2-unit= 0) but moves with the magnitude of loudness, which shows the continuous adaptation in this interval. ii) In the interval above 95 phon where r = 24.7e - 0.12, we have z~ = 3.0,
(15)
i.e., the zero poin t is fixed independently of the magnitude of loudness, the role of minimum audible threshold being played by the point 2 - u n i t = 3.0. It should be pointed out that the loudness specified by 2-unit = 3.0 nearly coincides with the most large one which can be attained in the domain D 1 . This means that the adaptation stops at the boundary olD1. Thus, the non-uniformity of the auditory sensation space mentioned above in Section 4.2 has been again confirmed from the standpoint of adaptation of zero point.
5. Concluding Remarks So far we have described an analysis of the auditory sensation space with special reference to the equalsensation contours. While the construction of various scales fo psychological sensation is usually based on the assumption that verbal responses for magnitude of sensation are to be consistent, that of equal-sensation contours can be made without such verbal responses. In this sense, the study of equal-sensation contours may be considered as more fundamental than that of psychological distances, so that any subjective scale of sensation may be given a more solid validity by clarifying its relation to the structure of equal-sensation contours. Therefore, the above result that the pair of 2,unit and mel scales as well as that of sone and mel scales forms intrinsic coordinate systems in D1 and D 2,
respectively, solidates the validity of these scales. Thus, the advantage to consider the attributes of pitch, loudness and volume not merely as fragmental separate ones but as constituent components of a psychological substance in the auditory sensation space seems to have been s h o w n to some extent. The intrinsic structure of the auditory sensation space studied in the foregoing sections consists of two kinds of elements, i.e., points and co-punctual straight lines therethrough. Since the equal-sensation contours are completely characterized by assigning its own reference point, the recognition of a psychological attribute is tantamount to that of its reference point if referred to an intrinsic coordinate system. By resorting to this intrinsic structure, the scaling problem of sensation for an attribute will be transformed in a natural manner to that of such co-punctual straight lines. In case the auditory sensation space reduces to a projective space (m = 1 in Section 3.2), an invariant scaling under the transformation group G for a bundle of co-punctual straight lines can be attained by means of line-coordinates, i.e., by the complex-ratio (see Veblen et al., 1954), as has well been established in projective geometry. Similarly for the general case. It should be noted that only three independent points are allowed in the two-dimensional quasiprojective space, so that the number of independent attributes is at most three. In this manner, the scaling of volume sensation, as has yet been untried from the standpoint of psychological experiments, may be undertaken from that of invariant scaling in the quasi-projective space. Here we emphasize the following point, which is so often overlooked: The scaling for an attribute need not to be restricted to a certain one but many altei-natives are possible depending on various psychological circumstances, where the problem of what forms of scale really take about does not matter so much as that of how one scale is related to others. The former aspect is concerned with the transformational one whereas the latter is with the intrinsic aspect of the sensation space. Since the data concerning the equal-sensation contours used in the above have been taken from the experiments by three psychologists under respective diverse conditions, the above results should not be considered as giving more than a tentative step to study the intrinsic structure of the auditory sensation space. In order to conclude definitely and to develop further the above results, it seems to be necessitated that detailed psychological experiments under a more unified condition are to be performed.
35 Appendix
Derivation
Appendix II
I
of the Transformation
(5)
Let us consider the coordinate transformation between two Cartesian systems x = f ( x ' , y'),
y = 9(x', y').
(A. 1)
In order for this transformation to satisfy the requirement mentioned in Section 3.2, it is necessary and sufficient that
i)
x' = const
whenever
x = const,
(A.2)
ii)
y' = const
whenever
y = const,
(A.3)
and iii)
i00 40 ~
y' = m' x '
whenever
y = rex,
o
(A.4)
where m is a constant and m' is a function of m. It follows from (A.2) that the function f ( x ' , y') is independent of y', so that we can put x = f(x').
4
20. 0 I
(1.5)
0.2
,
~
....
0.5
I
. . . . . . . .
1
2 3 4567 10 15
I
I
Similarly, from (A.3), we have Y = g(Y').
(A.6)
Fig. 11. EquaModness contours. [Fletcher, H., Munson, W.A.: J. acoust. Soc. Amer. 5 (1933) MAS = m i n i m u m audible threshold]
Then, the Eqs. (A.4) ~ (A.6) yield the relation f (x') = mg(y') ,
(A.7)
y' = G ( f (x')/m)= m' x' ,
(A.8)
from which it follows that
d b(pressure re.lbar)
where G(y') is the inverse function of g(y'). By putting x ' = 1 in (A.8), we have m'= G(f(1)/m).
(1.9)
40"
r
Hence, by substituting (A.9) into (A.8), we have
20G ( f ( x ' ) / m ) = x' G ( f ( 1 ) / m , from which it follows that G (uz/f(1))
0
=
Y(z) G(u),
(A. m)
o
/
.o
o
~II,~ ~ o
-20
Q o o
where we put z = f ( x ' ) , u = f ( l ) / m and F(z) is the inverse function of f ( z ) . By differentiating (A.10) with respect to z and putting z = f(1), we have uG'(u)/f(1)=F'(f(t))G(u).
(A.11)
o
-4Q
-60"
kHz |
0.2
Therefore, by integrating (A. 11), we have x' = ax",
y' = by n ,
where a, b, and n are arbitrary constants.
. . . . . . .
0.5
I
1
I
. . . . . . .
]
I
2 3 4 567 10 15
(A. 12) Fig. 12. Equal-pitch contours. [Stevens, S.S.: J. acoust. Soc. Amer. 7 (1935) MAS = m i n i m u m audible threshold]
36
00 800 db(intensityre.O.OOO2/abar)
20 ~ 0
k I
O2
z
. . . . . . .
I
0.5
1
. . . . . . . .
I
2 3 4567 10
Fig. 13. Equal-volume contours. [Thomas, G.J.: Amer. J. Psychol. 62 (1949) MAS = minimum audible threshold]
sone
[00
50
q---10 20
,
,
40
30
50
60
70
,
,
80
90
100
met
4000 mel(f)=4000f/(f §
"2000
"1000
0.0~
/
/
^ 3 ~ o.1
o2
o~
kHz 1
~ ~ ~,~1%
References Boring, E.G.: The relation of the attributes of sensation to the dimensions of the stimulus. Phil. Sci. 2, 236~245 (1935) Fletcher, H., Munson, W. A.: Loudness, its definition and calculation. J. acoust. Soc. Amer. 5, 8 ~ 1 0 8 (1933) Garner, W.R.: A technique and a scale for loudness measurement. J. acoust. Soc. Amer. 26, 73--88 (1954) Han, T. S.: A constructive approach to psychometric function from the information-theoretical point of view. RAAG Res. Notes, Third Series, No. 158 (1970) Han, T.S.: An information-theoretic and geometrical study of sensation metrices and its application to vowel perception. Doctor thesis, Univ. of Tokyo (1971) Howes, D.H.: The loudness of multicomponent tones. Amer. J. Psychol. 63, 1--30 (1950) Pollack, I.: Studies in the loudness of complex sounds. Ph. D. Diss. Harvard University (1948) Robinson, D~W., Davidson, M.A.: A re-determination of the equalloudness relation for pure tones. ISO/TC/43, 46 (1955) Stevens, S.S.: The surprising simplicity of sensory metrics. Amer. Psychol. 17, 29--39 (1962) Stevense, S.S., Volkmann, J.: The relation of pitch to frequency: a revised scale. Amer. L Psychol. 53, 329 353 (1940) Thomas, G.J.: Equal-volume judgements of tones. Amer. J. Psychol. 62, 182--201 (1949) Veblen, O., Young, J.W.: Projective geometry I, II. New York: Blaisdell Publishing Co., 1938, 1946 Dr. Te Sun Hart University of Tokyo Faculty of Engineering Tokyo 113, Japan
Fig. 14. Sone scale and 2-unit scale. [Sone from Howes, D.: Amer. J. Psychol. 63, (1950), 2-unit from Garner, W.R.: J. acoust. Soc. Amer. 26 (1954)]
-3000
Acknowledgements. The author wishes to thank Professor Masao Iri for helpful discussions. The idea of projective dual transformation used to obtain Figs. 3 and 9 is due to him.
1~
Fig. 15. Mel scale. [Data marked by white circles from Stevens, S.S., and Volkmann, J.: Amer. J. Psychol. 53 (1940); the theoretical curve from Han, T. S., Doctor Dissertation (1971)]
