Stimulus Range Effects in Temporal Bisection by Humans J.H. Wearden A. Ferrara

Stimulus Range Effects in Temporal Bisection by Humans J.H. Wearden A. Ferrara
Stimulus Range Effects
in Temporal Bisection by Humans
J.H. Wearden
Ma nchester University, Ma nchester, U.K.
A. Ferrara
University of LieÁge, LieÁge, Belgium
Two experiments with human subjects, using short-duration tones as stimuli to be judged,
investigated the effect of the range of the stimulus set on temporal bisection performance. In
Experiment 1, six groups of subjects were tested on a temporal bisection task, where each
stimulus had to be classi® ed as ``short’’ or ``long’’ . For three groups, the difference between
the longest (L) and shortest (S) durations in the to-be-bisected stimulus set was kept constant
at 400 msec, and the L/S ratio was varied over values of 5:1 and 2:1. For three other groups,
the L/S ratio was kept constant at 4:1 but the L 2 S difference varied from 300 to 600 msec.
The bisection point (the stimulus value resulting in 50% ``long’ ’ responses) was located
closer to the arithmetic mean of L and S than the geometric mean for all groups except
that for which the L/S ratio was 2:1, in which case geometric mean bisection was found. In
Experiment 2, stimuli were spaced between L and S either linearly or logarithmically, and the
L/S ratio took values of either 2:1 or 19:1. Geometric mean bisection was found in both cases
when the L/S ratio was 2:1, but effects of stimulus spacing were found only when the L/S
ratio was 19:1. Overall, the results supported a previous conjecture that the L/S ratio used in
a bisection task played a critical role in determining the behaviour obtained. A theoretical
model of bisection advanced by Wearden (1991) dealt appropriately with bisection point
shifts discussed above but encountered dif® culties with stimulus spacing effects.
The technique of tempora l bisection has been one of the most commonly used methods for
the study of animal timing (e.g. Church & DeLuty, 1977; Maricq, Roberts, & Church,
1981; Meck, 1983; Platt & Davis, 1983; Raslear, 1983, 1985). Although some studies (e.g.
Platt & Davis, 1983) use different methods, most have followed the technique originally
developed by Church and Deluty (1977) which, in schematic form, is as follows: An
animal subject is reinforced for making one response (e.g. a press on the left lever of a
standard operant chamber) after a stimulus of a certain length (e.g. 2 sec), de® ned as the
Requests for reprints should be sent to J.H. Wearden, Department of Psychology, The University,
Manchester, M13 9PL, UK.
We are grateful to members of groups participating in ``Travaux Pratiques’’ both at Manchester and LieÁge for
running subjects for us in Experiments 1 and 2. The collaboration from which this article resulted was supported
by grants from the CGRI of Belgium and the British Council.
1996 The Experimental Psychology Society
standard short stimulus (S), and another response (e.g. on the right lever) after a longer
stimulus (e.g. 8 sec long, a standard long stimulus, L). If S and L are suf® ciently different,
as they are with the example values given above, this discrimination is not dif® cult for rats
and pigeons, who come to emit more than 90% correct responses after a few sessions of
training. What ``suf® ciently different’’ means in the context of temporal bisection is
discussed more fully in the paper.
After training, subjects receive stimuli that differ in length between (and usually
including) S and L, and the response to each stimulus (which is unreinforced) is
noted. The near-universal ® nding is a monotonic increase in the number of ``long’’
responses (i.e. the response that would have been reinforced after L) with increases in
stimulus length. The psychophysical function relating proportion or number of ``long’’
responses to stimulus length is of ogival shape, ranging from zero or just a few percent of
``long’’ responses when S is actually presented to 90% or more when L is presented. One
focus of experimental interest has been the location of the bisection point, the stimulus
value giving rise to 50% ``long’’ responses, and most studies using animal subjects agree
that this is located at the geometric mea n of S and LÐ that is, the square root of the
product of S and L, 4 sec with the example values used above.
Many of the studies of bisection in animals have been inspired by, or related to, scalar
timing theory (Gibbon, 1977; Gibbon, Church, & Meck, 1984), currently the leading
theory of animal timing, and there has been some recent interest in the application of
this theory to timing in humans (for a non-technical review see Wearden & Lejeune,
1993). The method of Church and Deluty (1977) can easily be adapted to produce
techniques for studying temporal bisection in humans (as in Allan & Gibbon, 1991;
Wearden, 1991, 1993; Wearden & Ferrara, 1995), and these quickly produce very orderly
data. Unfortunately, there are some disagreements about the characteristics of temporal
bisection performance in humans. These can be summarized in terms of two issues, and
the ® rst of these is the location of the bisection point. As noted above, studies with
animals with only a few exceptions (which are discussed later in this article) ® nd this
bisection point at the geometric mean of S and L, and Allan and Gibbon (1991) duplicated this result with humans. On the other hand, work by Wearden (1991) and Wearden
and Ferrara (1995) found that the bisection point was better described as being closer to
(usually just less than) the arithmetic mean of S and L, rather than nearer to the
geometric mean.
A second issue is whether the spacing of non-standard stimuli between a constant S
and L makes a difference to bisection performance. Most animal studies (as reviewed in
Wearden and Ferrara, 1995) ® nd that stimulus spacing makes no difference: for a constant
S and L, the number of ``long’’ responses made to some intermediate stimulus, t, is not
affected by the other non-standard stimuli around t. This result was also duplicated by
Allan and Gibbon (1991), at least in the sense that arithmetic or geometric stimulus
spacing between a constant S and L made little obvious difference to bisection performance. On the other hand, Wearden and Ferrara (1995) found large and clear effects of
stimulus spacing in some of their conditions. For example, when S was 100 msec and L
900 msec, the psychophysical function was shifted markedly to the left when intermediate
stimuli were logarithmically spaced compared with arithmetic spacing. The effect was
much smaller, or non-existent, when the L/S ratio was smaller (4:1).
There were numerous procedural differences between the studies of Allan and Gibbon
(1991) and those of Wearden (1991) and Wearden and Ferrara (1995), which prompt the
pessimistic conclusion that bisection performance in humans is highly susceptible to small
procedural changes, and that it will be dif® cult to draw any coherent conclusions about it.
However, Wearden and Ferrara (1995) found that bisection performance was robust in
their studies in the face of a number of procedural variants, suggesting that some
resolution of the con¯ icting results might be possible, even when methods used for
temporal bisection vary between studies. Wearden and Ferrara (1995) examined the
temporal bisection literature, including experiments with both humans and animals,
and suggested that a critical variable operating in the different studies was the relative
value of S and L, expressed either as the L 2 S difference or as the L/S ratio. When L
and S were close together, the bisection point was located at the geometric mean, and
stimulus spacing effects were small or absent. On the other hand, when L and S were far
apart, the bisection point was located at or near the arithmetic mean, and effects of
stimulus spacing were found. If the close or far relation is expressed as the L/S ratio,
values of 2:1 or less (as used by Allan & Gibbon, 1991) appear to produce geometric mean
bisection and absence of stimulus spacing effects in humans; values greater than 2:1 (as
used in Wearden, 1991, and Wearden & Ferrara, 1995) produce the other pattern of
results. In studies with animals, on the other hand, values of 4:1 can still produce
geometric mean bisection.
The aim of the present article is to report data relevant to this stimulus range
conjecture, which, more speci® cally, states that small L/S ratios (or L 2 S differences)
should produce evidence for geometric mean bisection and absence of stimulus spacing
effects, whereas large L/S ratios should produce arithmetic mean bisection points (or
bisection points close to the arithmetic mean) and evidence for spacing effects. It should
be noted that this proposal is advanced on the basis of observation of apparent regularities
in previous data (as discussed by Wearden & Ferrara, 1995). It is not a prediction that
derives from any formal theory of bisection and, indeed, as will be discussed later, may
provide a challenge for current theories if found to be supported.
As L and S become more different, both the L/S ratio and L 2 S difference increase,
and it is unclear in which terms the psychological difference of L and S is better
expressed, although data from some animal experiments indicate very similar bisection
performance when the L/S ratio is constant and the L 2 S difference varies (see Church
& Deluty, 1977, and Maricq et al., 1981). Experiment 1 below thus investigated the
bisection performance of humans faced with six different stimulus ranges. In three of
these, the L/S ratio remained constant at 4:1 but the L 2 S difference varied; for the
other three, the L 2 S difference remained constant but the L/S ratio varied, from 5:1 to
2:1. The spacing of non-standard stimuli between L and S was arithmetic in all cases, so
the focus of interest in Experiment 1 was on possible variation in the location of the
bisection point with respect to the two putative candidates (the geometric and arithmetic
means) as the stimulus range varied.
In all procedural details the present study closely followed the partition method of
bisection used by Wearden and Ferrara (1995). In this method, which closely resembles
the classical method of constant stimuli, no speci® c stimulus in the to-be-bisected set is
identi® ed as a standard, either ``short’’ or ``long’’. Subjects simply receive repeated
presentations of the whole stimulus set, with the members arranged in different random
orders, and have to classify each stimulus as ``short’’ or ``long’’, using whatever criteria they
wish to employ. The whole stimulus set is repeated 20 times, and data are taken from the last 10
repetitions. Wearden and Ferrara (1995) found that this method produced data very similar to
those obtained with the Church and DeLuty (1977) variant used by Wearden (1991).
M ethod
Subjec ts
Seventy-two Manchester University undergraduates, participating for course credit, were arbitrarily allocated to six equal-sized groups.
App aratus
A Hyundai TSS8C personal computer (IBM compatible) controlled all experimental events. The
computer keyboard served as the response manipulandum. All stimuli in the sets to be bisected were
500-Hz tones produced by the computer speaker. The experiment was controlled by a Turbo Pascal
program, but specially written routines derived from assembly language were used to time the
stimuli, which were timed to an accuracy of at least 1 msec.
The procedure for all groups was identical except for the stimuli in the to-be-bisected set. Each
group received a different stimulus set, with values shown in Table 1.
For three groups (D1, D2, and D3), the difference between the shortest (S) and longest (L)
stimulus in the set used varied, but the L/S ratio remained constant at 4:1. For the other three
groups (R1, R2, and R3), the difference between L and S remained constant at 400 msec, but the
L/S ratio varied, over values of 5:1 ± 2:1.
All subjects received a single experimental session, lasting from 15 to 20 min, conducted as
follows: Subjects started the experiment by pressing the spacebar after an appropriate prompt
appearing on the screen, then presented each trial by again pressing the spacebar after a Press
spacebar for next trial prompt. A stimulus presentation then followed after a delay that was a value
randomly chosen from a uniform distribution ranging from 1 to 3 sec. Each stimulus set (i.e. D1 to
Stim u lus V alu es U sed fo r the Six D iffere nt G ro ups in Ex pe rim en t 1
Stimulus Values
Note: Also shown are L
given in msec.
S difference (in msec), and L/S ratio. Stimulus values are
D3, or R1 to R3Ð see Table 1 for the values) was presented 20 times, with each of the 20 series
involving presenting the stimuli in the set in a different random order. After each stimulus presentation, subjects were required to classify each presented stimulus either as ``short’’ or ``long’’ by
pressing appropriate keys on the computer keyboard. As there is no ``right answer’ ’ on the task,
no feedback was given after the response. Subjects had previously been told that all stimuli had
durations of less than 1 sec, but that some tones would be clearly shorter than others. Data were
collected from the last 10 presentations of each stimulus set.
Results and D iscussion
Figure 1 shows the mean proportion of ``long’ ’ classi® cations of each stimulus, plotted
against stimulus duration, for each of the six groups used in Experiment 1. Data points
are shown as unconnected ® lled circles, and the line ® tted to them comes from a
theoretical model to be discussed later.
Inspection of data in the different panels of Figure 1 immediately reveals that the
method used produced orderly data in all the groups, with the proportion of ``long’’
responses rising, usually monotonically, from near-zero ``long’’ responses at S to near
100% at L, as stimulus length increased. This pattern of results very closely resembles
that obtained in previous bisection experiments with human subjects (e.g. Wearden, 1991,
1993; Wearden & Ferrara, 1995).
The bisection pointÐ the stimulus value that would yield 50% ``long’’ responsesÐ was
calculated by three different methods, as by Wearden and Ferrara (1995), and results are
shown in Table 2. Each method used the psychophysical function of the proportion of
``long’’ responses versus stimulus duration (i.e. plots in the form shown in Figure 1) as its
basis. Mean interpolation (MINT) used the averaged psychometric function to interpolate the bisection point by eye; the individual interpolation method (IINT) performed
an identical operation on the psychometric functions of individual subjects, then averaged
out the resulting bisection points. The regression method (REG), used originally by
Maricq et al. (1981) and followed by Wearden (1991), used linear regression of the
data points yielding the line of steepest slope. The resulting regression line was then
used to calculate the bisection point. Table 2 shows the bisection points, as well as the
arithmetic and geometric means of S and L, for comparison.
Allan and Gibbon (1991) suggested that the bisection point might in some cases not be
located exactly at the geometric mean, but at some consistent fraction of this mean. One
way of testing this suggestion is to divide the bisection points by the geometric mean
value, thus expressing the bisection points as some fraction of the geometric mean, with
values greater than 1.0 indicating bisection points above the geometric mean and values
lower than 1.0 below it. The results of this calculation are shown in Table 2, as K 1 values.
It is also possible that the bisection points are some consistent fraction of the arithmetic
mean, and the K 2 values shown in Table 2 test this suggestion in a similar manner.
Several points are obvious on inspection of the values given in Table 2. (1) The three
different methods used to calculate the bisection point yielded values within a few msec of
each other (a result also obtained by Wearden & Ferrara, 1995), and so could be aggregated to produce a mean, suggesting that the method used to calculate the bisection point
makes little difference to the outcome of the calculation. (2) For most groups, the
FIG. 1.
Proportion of ``long’ ’ responses plotted against stimulus duration, shown as unconnected ® lled circles,
for the six groups of Experiment 1. The text within each panel shows the group identi® er and the shortest and
longest stimulus (values in msec) in the to-be-bisected set. The continuous line in each panel indicates the best® tting theoretical values generated by the modi® ed difference model discussed in the text, with parameter values
shown in Table 3.
Bisectio n Po ints Calculated A ccordin g to the Th ree D iffe rent M etho ds and th e M ean o f A ll Thre e
fo r the Six D iffer ent Gr oups o f Ex pe rim ent 1 and the Fo ur D iffere nt G ro ups o f Ex pe rim ent 2
Experiment 1
Experiment 2
Note: MINT: mean interpolation; IINT: Individual interpolation; REG: Regression
Also shown are the geometric (GM) and arithmetic (AM) means of S and L. K 1 is the mean bisection
point divided by GM, K 2 the bisection point divided by AM. Values greater than 1.0 indicate that the
bisection point was greater than GM or AM; values less than 1.0 indicate that the bisection point was
bisection point, however calculated, was closer to the arithmetic mean of S and L than to
the geometric mean; for example, almost all K2 values were closer to 1.0 than were K 1 values,
(where K1 and K2 are as described above). In general, this ® nding replicates those obtained
in previous bisection experiments we have conducted with humans (Wearden & Ferrara,
1995; see also Wearden, 1991, 1993). The exception to this generalization came from group
R3, which had the smallest L/S ratio, 2:1. In this case, the bisection point was not only much
closer to the geometric mean than to the arithmetic mean but, uniquely in the stimulus
ranges we used in Experiment 1, was actually slightly below the geometric mean.
A common result in studies of human timing inspired by scalar timing theory is
conformity of data to Weber’ s law, the requirement that the sensitivity of timing remains
constant as the absolute durations timed vary. Some data manifest such Weberian properties directly (e.g. Wearden & McShane, 1988), but in other studies Weber’ s law is tested by
examining the property of superimposition (also called superposition), the ® nding that data
from different conditions superimpose when response measures are plotted against time
expressed on some relative scale (e.g. see Church & Gibbon, 1982). Allan and Gibbon
(1991) discuss the appropriate method of testing superimposition in bisection experiments
(pp. 45± 46) and advocate plotting the proportion of ``long’’ responses against stimulus
duration, where stimulus duration is expressed as a fraction of the bisection point appropriate for the condition plotted. This type of manipulation allows data sets with different
L/S ratios to be directly compared, because if Weber’ s law holds, the proportion of ``long’’
responses from all conditions should superimpose when plotted in this way.
We treated the psychophysical functions from the six groups of Experiment 1 in this
way, using the mean bisection point (MEAN in Table 2) for each condition as a divisor for
the stimulus durations, and results are plotted in Figure 2. Inspection of Figure 2 shows
that superimposition was clearly manifested in our data, indicating that our bisection
procedure produced constant-sensitivity Weberian timing. Comparison of our Figure 2
with superimposition data from Allan and Gibbon (1991, their Figures 5 and 9) suggests
that the quality of superimposition in our Experiment 1 was about equal to theirs.
Data from Experiment 1 clearly supported Wearden and Ferrara’ s (1995) conjecture
that the range of the stimulus set used plays some role in determining the location of the
bisection point in studies of temporal bisection with humans. Inspection of data in Table 2
also suggests that range effects are better expressed in terms of the L/S ratio than the
L 2 S difference. Although the bisection point did decline relative to the arithmetic mean
of S and L, suggesting some effect of L 2 S difference (e.g. comparisons of groups D1,
D2, and D3), it only reached the geometric mean of S and L when the L/S ratio was at its
lowest (in group R3). Although our data are not conclusive as to how the difference
between ``short’’ and ``long’’ standards should be represented in terms of difference or
ratio, ratios do have clearer effects, and using ratios to represent the dif® culty of the time
discrimination task has been proposed in previous studies with humans (e.g. Allan &
Gibbon, 1991). Furthermore, studies with animals have found no clear effects of the L 2 S
difference on bisection performance when the L/S ratio was kept constant (e.g. Church &
Deluty, 1977; Maricq et al., 1981). Ideally, this issue would be examined experimentally
by using a wider range of differences and ratios than we have used, but certain problems
arise. For example, having a large L 2 S difference while maintaining a small L/S ratio
necessitates using longer-duration stimuli than we have used (such as stimuli several
seconds long), and human subjects would almost certainly employ chronometric counting in this situation, thus eroding any possible psychological continuity between humans
and animals. Likewise, large L/S ratios and small L 2 S differences could be arranged
only by using very short stimuli. Some evidence suggests that Weber’ s law may not be
obeyed in such circumstances (e.g. Fetterman & Killeen, 1992), and so the data generated
FIG. 2.
Proportion of ``long’ ’ responses from the different groups (D1, D2, D3, R1, R2, R3) plotted against
stimulus duration expressed as a fraction of the bisection point for each group (MEAN in Table 2). Data points
from the different groups are indicated by different characters in the ® gure, as shown in the key.
may not resemble those obtained from other studies of bisection where Weberian timing
operates (e.g. Allan & Gibbon, 1991, and Figure 2 of this article).
In their studies of temporal bisection in humans, which invariably obtained geometric
mean bisection, Allan and Gibbon (1991) also used small L/S ratios, with their largest
value being 2:1. As mentioned above, there are many procedural differences between the
studies of Allan and Gibbon (1991) and those conducted in our laboratory (reported in
Wearden, 1991, and Wearden, & Ferrara, 1995). We used a larger subject population, and
group mean data, whereas Allan and Gibbon used a much smaller subject population, but
each subject received far more intensive training and testing than did ours, and individual
subjects’ data were presented. Furthermore, in Allan and Gibbon’s study, subjects were
presented with examples of the standard S and L and asked whether each presented
stimulus was S or L, with occasional feedback being given. Our method in Experiment
1 above was very different, without explicit identi® cation of S and L and without feedback. Nevertheless, in spite of all these differences, when we used an L/S ratio in the
same range as that of Allan and Gibbon (1991) rather than the larger values we ordinarily
use in our laboratories (up to 9:1, see Wearden, 1991, for example), we obtained the same
result as they did, geometric mean bisection, whereas we replicated our previous results of
near-arithmetic mean bisection at larger L/S values. This suggests that there is no real
contradiction in the literature on temporal bisection in humans but, rather, an effect of
stimulus range. Why such an effect might occur is considered further in the General
Experiment 1 provided support for the ® rst part of Wearden and Ferrara’ s conjectureÐ
namely, that geometric mean bisection might be obtained when L/S ratio was small but
not when it was larger. The second part of their conjecture, that stimulus spacing effects
(e.g. different effects of linear or logarithmic spacing of stimuli between S and L) can be
obtained when L/S ratios are large but not when they are small was the focus of interest
of Experiment 2. In this experiment, four groups of subjects were used. For two of these
groups (LIN450 and LOG450), S was 450 msec and L was 900 msec, producing an L/S
ratio of 2:1, as for group R3 of Experiment 1. Following on from the results of Experiment 1, we would anticipate near-geometric mean bisection in this case. In the LIN450
group, non-standard stimuli were spaced linearly between S and L; in the LOG450 group,
the spacing of the stimuli between S and L was logarithmic. According to Wearden and
Ferrara’s conjecture, stimulus spacing effects should be small or absent in this case,
because of the low L/S ratio.
For the other two groups (LIN50 and LOG50), S was 50 msec and L was 950 msec, a
19:1 L/S ratioÐ by far the largest used in any of our previous studies (Wearden 1991,
1993; Wearden & Ferrara, 1995). In this case we would expect marked stimulus spacing
effects (the non-standard stimuli being linearly spaced in LIN50 and logarithmically
spaced in LOG50) and a bisection point well above the geometric mean of S and L.
As mentioned above, Wearden and Ferrara (1995) used two different bisection methods,
which they termed similarity and partition, although they found that the two methods
produced very similar results when the same stimulus set was used. The partition method
was used in the present Experiment 1; for Experiment 2, we reverted to the similarity
method, which involves explicit identi® cation of S and L (as in Wearden, 1991). The
main aim of making this procedural change was to see whether geometric mean bisection at small L/S ratios would be obtained with the similarity method, which is
procedurally much closer both to our original bisection study (Wearden, 1991), and
also to the method used by Allan and Gibbon (1991) who used explicitly identi® ed
stimuli as S and L.
M ethod
Subjec ts
Eighty undergraduate students from the University of LieÁge, Belgium, participated for course
credit. All were native French speakers or had near-native competence in French.
App aratus
A PC8088 (IBM-compatible) computer controlled all experimental events. Apart from the
different model of machine, all procedural details were as in Experiment 1.
In the description given here, all relevant events are described as if they had been presented for
English-speaking subjects, although in fact the use of the French language necessitated some
differences (e.g. the keys used for the responses were L (``long’ ’ ) and C (``court’’ ), not L and S, as
described here). The general procedure was identical in all respects to that used in Wearden (1991),
except that continuous tones served as stimuli to be timed. The subject pressed the spacebar of the
computer keyboard to start the experiment, which commenced with 5 presentations of each of the
standard short and long durations (S and L) identi® ed by an appropriate display (e.g. This is the
standa rd short dura tion). Stimulus presentations were separated by a 5-sec interpresentation interval.
The subjects then received the display Press spacebar for next trial, and a press on the spacebar was
followed by a delay that was randomly chosen from a uniform distribution running from 1 to 3 sec. A
stimulus was then presented, and stimulus offset was followed by the display, Was tha t more similar to
the standa rd short stimulus ( Press S) , or the standa rd long ( Press L) ?. The response was followed by a 3sec delay, followed by the reappearance of the Press spaceba r for next trial prompt. This procedure was
repeated until all stimuli in a stimulus set (S and L, plus all the intermediate durations) had been
presented once in a random order. Subjects then received two ``refresher’’ presentations of S and L,
conducted as at the start of the experiment, and this was followed by another series of stimulus
durations to be classi® ed as similar to S and L. The experiment continued until the stimulus had been
presented 10 timesÐ that is, the subject made 10 judgements about each stimulus in the stimulus set
in the course of the experiment.
The stimulus sets used for the different groups of Experiment 2, with all values in msec, were:
50, 150, 250, 350, 450, 550, 650, 750, 850, 950
50, 69, 96, 133, 185, 256, 356, 493, 684, 950
450, 500, 550, 600, 650, 700, 750, 800, 850, 900
450, 486, 523, 567, 612, 658, 714, 771, 829, 900
Here, the ® rst stimulus in the above lists served as S, the last as L.
Results and D iscussion
Figure 3 shows the psychometric functions (mean proportion of ``long’’ responses plotted
against stimulus duration) for the LIN50 and LOG50 groups (left panel) and LIN450 and
LOG450 groups (right panel). Data are shown by unconnected points (open circles for
linear spacing conditions, ® lled circles for logarithmic spacing); the solid and dotted lines
in each panel come from a theoretical model to be discussed later.
Inspection of the data in Figure 3 makes it immediately obvious that the effect of
stimulus spacing (linear or logarithmic in the present case) was marked for stimuli
intermediate between S and L when the L/S ratio was large (as in the LIN50 and
LOG50 groups: left panel of Figure 3), but small or non-existent when the L/S ratio
was small (LIN450 and LOG450 groups: right panel of Figure 3). The fact that the
spacing of stimuli between S and L was different in the different conditions meant that
the same intermediate stimuli did not occur in relevant comparison groups (e.g. LIN50
and LOG50), but the leftward shift of the psychophysical function can be tested by
comparing the number of ``long’’ responses at an intermediate stimulus from the
LOG50 group with the number made to the stimulus from the LIN50 group with the
next-longest duration. Thus we compared the following pairs of stimuli (® rst stimulus
from the LOG50 group, second stimulus from LIN50 group): 185/250 msec, 256/
350 msec, and 356/450 msec. Signi® cantly more ``long’’ responses occurred to the former stimulus than to the latter in the ® rst and third cases, t (38) 5 2.122 and 2.29, p <
0.05, respectively, and approached signi® cance, t 5 2.002, in the second. It should be
noted that this comparison is an extremely conservative one, as it compares a stimulus
duration from the LOG50 group with one from the LIN50 group, which can be nearly
100 msec longer. Similar comparisons between intermediate stimuli from the LIN450 and
LOG450 groups never yielded statistical signi® cance. Overall, therefore, this analysis
FIG. 3.
Left panel: proportion of ``long’ ’ responses plotted against stimulus duration for the LIN50 and
LOG50 groups of Experiment 2 (lin50dat and log50dat); the solid and dashed lines connect points derived from
the modi® ed difference model (lin50sim and log50sim), with parameter values shown in Table 3. Right panel:
data and simulated values from the LIN450 and LOG450 groups, with all details as in the left panel.
supports the impression gained from inspection of Figure 3 that stimulus spacing effects
occur when the L/S ratio is large, but not when it is small.
As for the data in Experiment 1, bisection points for the four different stimulus sets of
Experiment 2 were calculated according to three different methods (MINT, IINT, and
REGÐ see description of Experiment 1, above), and results are shown in the lower part of
Table 2. As in Experiment 1 and in Wearden and Ferrara (1995), all three methods for
calculating the bisection point produced values within a few percentage points of each
other, so these values are averaged to produce a MEAN in Table 2.
According to the results of Experiment 1, the bisection point should be above the
geometric mean (and nearer to the arithmetic mean) of S and L when the L/S ratio is
large (groups LIN50 and LOG50) but close to the geometric mean when the L/S is small
(as in groups LIN450 and LOG450). The second prediction was certainly supported, as,
in the LIN450 and LOG450 groups, the mean bisection point was almost identical to the
geometric mean of S and L. In the LIN50 group, the bisection point greatly exceeded the
geometric mean and was much closer to the arithmetic mean of S and L, although it was
still some distance from it. The LOG50 group produced results that at ® rst appear
ambiguous, as the bisection point was very different from both the arithmetic and
geometric means of S and L and lay between the two. The result may make more sense
if we consider the mean of all the stimuli in the to-be-bisected set. When stimuli are
spaced linearly between S and L, the arithmetic mean of S and L is also the arithmetic
mean of the set, but this is not true when stimuli are logarithmically spaced between S and
L. In the LIN50 group, the arithmetic mean of the entire set is 327 msecÐ a value much
closer to the mean bisection point (295 msec) than either the arithmetic or geometric
means of S and L. In their discussion of stimulus spacing effects in temporal bisection,
Wearden and Ferrara (1995) explicitly assumed that subjects bisected stimulus sets by
using the arithmetic mean of all the stimuli in the set, not just S and L. Their model,
which reacted appropriately to changes in stimulus spacing even though it did not ® t
data perfectly in all respects, basically assumed that subjects responded ``long’’ if they
decided that some just-presented stimulus was discriminably longer than the mean and
``short’ ’ if the stimulus was discriminably shorter, and they responded at random if the
stimulus could not be discriminated from the mean. Stimulus spacing effects become
less marked as the S to L stimulus range decreases, as the arithmetic mean of S and L
becomes a better descriptor of the set mean in these cases (e.g. the difference between
the mean of an arithmetically and logarithmically spaced set decreases as the ratio of L
and S decreases).
Thus, overall, the data obtained in Experiment 2 veri® ed the ® nding of near-geometric
mean bisection when the L/S ratio is small and also illustrated that stimulus spacing
effects are only clearly found when the L/S ratio is large (e.g. group LIN50 compared
with LOG50) and are small or absent when the L/S ratio is small (e.g. group LIN450
compared with LOG450).
The results of the two experiments above can be summarized with respect to two features
of the bisection task: bisection point location and stimulus spa cing effects.
Bisection Point Loca tion. In data from both experiments, the bisection point was
located close to the geometric mean of S and L (or even slightly below the geometric
mean in some cases) when the L/S ratio was small, but closer to the arithmetic mean of S
and L or the arithmetic mean of the stimulus set (e.g. the LOG50 group of Experiment 2)
when the L/S ratio was larger. Although the line of division between ``large’’ and ``small’’
remains to be determined by more exhaustive parametric analyses, ``small’’ seems to be
2:1 (and probably also less, see Allan & Gibbon, 1991), and ``large’ ’ values greater than
2:1. Data from Experiment 1 also suggested that there was a small effect of L 2 S
difference on the bisection point when L/S ratio was held constant at 4:1. As noted above,
this result suggests that there is no important empirical contradiction between the data of
the Wearden (1991) and Wearden and Ferrara (1995) studies, which found arithmetic
mean bisection in humans (for another example, see Wearden, 1993), and the ® nding of
geometric mean bisection by Allan and Gibbon (1991). In spite of the myriad procedural
differences between the studies, when stimulus sets with ``small’’ L/S ratios were used,
geometric mean bisection was obtained with human subjects in all cases.
Stimulus Spacing Effects. The results of Experiment 2, above, taken together with
those from Wearden and Ferrara (1995), strongly suggest that stimulus spacing effects on
temporal bisection in humans are obtained only when the L/S ratio is ``large’’ (i.e. greater
than 2:1).
The ® nding of arithmetic mean bisection from humans when ``large’’ L/S ratios are
employed is supported by ® ndings from a study that pre-dates recent interest in bisection
and is rarely quoted in the English-speaking literatureÐ that of Bovet (1968). Bovet
employed tones with durations of S 5 0.3 sec, L 5 1.65 sec, and S 5 0.3 sec and
L 5 3.00 sec. After hearing examples of these tones, subjects were required to produce a
time interval (by button pressing:production method), or to adjust the length of a third
tone (adjustment method), to result in a duration exactly half-way between the two
standards. For the ® rst case cited above (L/S ratio 5 5.5), the average value obtained
from subjects (0.91 sec from production, 0.90 sec from adjustment) was closer to the
arithmetic mean of S and L (0.975 sec) than to the geometric mean (0.70 sec); for the
second case (L/S ratio 5 10), the results were even clearer (bisection point by production
1.58 sec, by adjustment 5 1.69 sec), with both values being much closer to the
arithmetic mean (1.65 sec) than to the geometric mean (0.95 sec).
In the light of the ® nding that the location of the bisection point appears to depend on
the L/S ratio in studies with humans, and to be close to the arithmetic mean when the
L/S ratio is larger than about 2, we were prompted to attempt to characterize the location
of the bisection point in animal studies when ``large’’ L/S ratios were used.
The vast majority of studies of temporal bisection in animals have followed Church
and Deluty’s original article (1977) in using an L/S ratio of 4.0 or less, most commonly
with S 5 2 sec and L 5 8 sec (e.g. see Maricq et al., 1981; Meck, 1983; Meck & Church,
1982; Roberts, 1982, for just some examples). In these cases, the bisection point was
invariably located very close to the geometric mean of S and L, regardless of the absolute
values of S and L.
A few experiments have used L/S ratios larger than 4.0, and some conditions of this
type come from Platt and Davis (1983). These workers used a bisection technique that is
somewhat different from that of Church and Deluty (1977) in that pigeons were reinforced, on each trial, for a response occurring on one of two keys in a pigeon chamber. On
a random 50% of trials, the ® rst response more than S sec after the start of the trial was
reinforced on one key; on the other 50% the ® rst response more than L sec after the start
of the trial was reinforced on the other key. L was, by de® nition, greater than S, and the
absolute values of S and L, as well as their ratio, were varied across conditions. The
bisection point was calculated either from the crossover of the averaged response rate
versus time functions on the two keys, or by a measure of switching between the keys.
When the L/S ratio was less than 4.0, the geometric mean of S and L described the
bisection point well, but at larger ratios (e.g. with S 5 10 and L 5 160 sec) the bisection
point deviated from the geometric mean. However, the different methods of measuring
the bisection point (from response rate or switching) disagreed as to the direction of the
deviation (i.e. whether the bisection point was above or below the geometric mean). A
major problem with studies of this type is that subjects may emit few or no responses for a
period between S and L when the L/S ratio is large, rendering measurement of the
bisection point impossible on most trials (Platt & Davis, 1983, p. 168).
An experiment with rats by Siegel (1986) used a method close to that of Church and
Deluty (1977), with white noise signals of values S 5 1 sec, L 5 16 sec. The bisection
point was located well below the geometric mean of S and L (e.g. at the geometric mean,
4 sec, 81.7% of responses were ``long’’). Unfortunately, the testing situation used ``outside’ ’ signals (i.e. signals shorter than S or longer than L)Ð a manipulation that according
to a second experiment reported by Siegel, made a difference to the location of the
bisection point even when a ``standard’’ 2/8 sec bisection condition was run. This,
obviously, makes it dif® cult to interpret the results from the 16:1 L/S ratio used in the
® rst experiment.
A number of studies by Raslear and colleagues have used rats in bisection studies with
an L/S ratio greater than 4. For example, Shurtleff, Raslear, and Simmons (1990) used
tones with S 5 0.5 and L 5 5.0 sec. For both the rats used in their Experiment 1, the
bisection point was considerably above the geometric mean of S and L (1.58 sec), and
nearer to the arithmetic mean (2.75 sec). In a similar study (Shurtleff, Raslear, Genovese,
& Simmons, 1992) visual or auditory stimuli were used (S 5 0.5, L 5 5.0 sec), and the
drugs physostigmine and scopolamine were administered in some conditions. When
saline was administered as a control, the bisection point was invariably closer to the
arithmetic mean than to the geometric mean, with results being particularly clear when
tones were used as stimuli. Increasing doses of physostigmine had little effect on the
bisection point, but increasing doses of scopolamine decreased the bisection point, which
thus became closer to the geometric than arithmetic mean at larger doses. Overall,
therefore, data from ``normal’’ (i.e. undrugged) conditions of both studies resembled
those obtained from humans in suggesting bisection points much closer to the arithmetic
than geometric mean when a ``large’’ L/S ratio, like 10:1, was used.
Finally, Raslear (1983) provided data on visual stimulus duration bisection in rats with
a range of L/S ratios much greater than used in other studiesÐ up to 100:1 (e.g. S 5
0.1 sec, L 5 10 sec). Unfortunately, the 8 individual rats used in the study did not all
receive the same S and L values, nor the same intermediate stimulus lengths, rendering a
simple summary of results from the multitude of different conditions very problematical.
For example, one problem is that some L/S ratios are unique to individual rats, so it is
unclear how reliable results from these conditions are. In general, however, the bisection
point was not always well described by the geometric mean of S and L, and deviations
were usually in the direction of the bisection point being above the geometric mean (i.e.
towards the arithmetic mean), although bisection points were sometimes well below the
arithmetic mean.
Overall, therefore, this survey of animal experiments using L/S ratios greater than 4.0
suggests the rather negative conclusion that although geometric mean bisection appears
to break down in these conditions, it is impossible, because of methodological problems
with the various studies, to decide exactly where the bisection point does lie. Some studies
suggest near-arithmetic mean bisection in these cases (e.g. Shurtleff et al., 1990, 1992),
whereas others ® nd bisection points below the geometric mean (e.g. Siegel, 1986), and
others have results that are complex and dif® cult to interpret clearly (e.g. Platt & Davis,
1983; Raslear, 1983).
M odelling S tim ulus Range Effects
Although results from experiments with animals regarding possible L/S range effects on
bisection point location are ambiguous, our Experiments 1 and 2 above with humans
found clear effects, so the obvious theoretical problem that arises is how to account for
them. Some algebraic approaches to bisection (e.g. as discussed in Gibbon, 1981) simply
use relations between values of S, L, and some just-presented stimulus, t, to derive
bisection points, and thus cannot account either for stimulus spacing or stimulus range
effects. One model that may have some promise, however, is the modi® ed difference model
used to ® t bisection data in Wearden (1991).
The starting point of this model is the basic idea that subjects form memory representations of the ``short’’ and ``long’’ standards of the bisection task (S and L) and calculate
the difference between each of these memory representations and the just-presented
stimulus, t. The response in any particular case is (usually) based on which of these
two differences is the smallerÐ that is, the model uses a similarity rule based on absolute
difference. If S and L are represented on average accurately, and the just-presented
stimulus is also timed accurately, then the differences (t 2 S) and (L 2 t) will be equal
at the arithmetic mean of S and L. To account for deviations from arithmetic mean
bisection, the modi® ed difference model assumed further that when the two differences
de® ned above were suf® ciently similar (i.e. the difference between them was less than
some threshold), the subject was biased to respond ``long’’. Trial-to-trial variability in the
model arose from the memory representations of S and L, which were represented as
distributions with scalar propertiesÐ that is, distributions with accurate means (S and L)
and with standard deviations that were proportional to these means.
Speci® cally, the model generated memory representations of S and L in the form of
Gaussian distributions with means S and L, and some constant coef® cient of variation
(standard deviation/mean), c, thus yielding a standard deviation of cS for S and cL for L.
On any particular trial, the model sampled from the distribution of S and L, to produce
two values s* and l*, which varied from trial to trial by random sampling. The currently
presented stimulus, t, was assumed to be timed without variance. The basis of behavioural
output was the absolute difference between t and s* and t and l* [symbolized as D(s*, t)
and D(l*, t)]. If these differences were ``similar’’ (i.e. expressed casually, the subject
cannot tell whether t is closer to S or to L), then the model responded ``long’’; specifically, ``long’’ responses occurred if the absolute difference between D(s*, t) and D(l*, t) was
less than some threshold, x. If the differences were suf® ciently different (i.e. greater than
x), then the model used the smallest difference as the basis of output, by responding
``short’ ’ when D(s*, t) < D(l*, t) and ``long’’ when D(l*, t) < D(s*, t). The model thus has
two parameters, c, the coef® cient of variation of memory representations of S and L, and
x, the difference threshold.
It is immediately obvious how this model might deal with stimulus range effects, as
when the S to L range is small, a substantial proportion of intermediate stimuli might
tend to be ambiguous (i.e. differences from S and L would be similar), and the model is
biased towards responding ``long’’ in these cases. When the stimulus range is larger, on
the other hand, fewer intermediate stimuli are ambiguous, and on most trials the model
uses the smallest difference to generate responding on most trials. Operating by itself, the
difference rule will yield arithmetic mean bisection, as discussed above, but the ``respond
long’ ’ bias occurring under ambiguous conditions will tend to shift the psychophysical
function to the leftÐ that is, towards geometric mean bisection.
We applied this model to data from Experiments 1 and 2. Two questions arise: (1) Can
parameter values be found that enable the model to ® t data adequately in terms of goodness
of ® t (with a subsidiary question being whether the parameter values change coherently
with changing conditions)? (2) With parameter values held constant, does the model
behave appropriately as conditions are changed (e.g. with changes in the L/S ratio)?
The lines in the six panels of Figure 1 and the two of Figure 3 show the ® ts of the
modi® ed difference model to data, and Table 3 gives parameter values and mean absolute
deviation (MAD) between data and the model’ s output (de® ned as the sum of the absolute
deviation between data and ® tted points, divided by the number of points). The ® tted
values were derived from simulation of 1000 trials at each stimulus duration modelled.
Inspection of Figures 1 and 3 and the MAD values shown in Table 3 indicates that the
model ® tted data well in most cases. The largest MAD value was 0.04, but in some cases
most of the deviation arose from a single discrepant point. Overall, MAD values compare
well with those obtained on other tasks, such as temporal generalization (Wearden, 1992;
Wearden & Towse, 1994) and categorical timing (Wearden, 1995). Inspection of parameter
values (c and x) reveals a less reassuring picture. In some cases, for example, threshold
values were exceptionally low (0.01 for group R2 and even 0 for group D1), but in these
instances the model ® tted data where the bisection point exceeded the arithmetic mean,
and thus the ``long’’ response bias, which would have shifted the psychophysical function
to the left and the bisection point below the arithmetic mean, was not needed. Another
problematical case comes from the LOG50 group, where the very high difference threshold value (0.35) seems extremely implausible, as subjects can make much ® ner discriminations than this. However, data from this group also present a bisection point anomaly, as
the location of this point was, uniquely in the data in Experiments 1 and 2, not well
described by either the arithmetic or geometric mean of S and L, so the problems
experienced by the modi® ed difference model, which ® ts normal bisection data well,
are perhaps unsurprising.
Pa ram ete r V alues fo r Fits
o f the M od i® ed D iffe ren ce M o de l
to D ata fro m E xp erim ents 1 an d 2
Note: c is the coef® cient of variation of representations of S and L; x is the difference threshold (see text
for details); MAD is the mean absolute deviation between the data points and the predictions of the model:
the total absolute deviation divided by the number of
points ® tted.
We next tested whether the modi® ed difference model behaved appropriately when the
stimulus range used in a bisection task was varied, with constant c and x values (0.2 and
0.1). We made no attempt to ® t any particular data set, but instead simulated four
different stimulus ranges. For all four, L was 900 msec and S varied over values of
100, 300, 450, and 600 msec. Stimuli were approximately linearly spaced between S
and L, except that the geometric mean value was always included in the set so that the
proportion of ``long’’ responses occurring at this value could be measured. The stimulus
ranges used (ranging from L/S ratios of 9:1 to 1.5:1) spanned most of the range used with
humans, in experiments from Wearden (1991, 1993), Wearden and Ferrara (1995), the
present study, and Allan and Gibbon (1991). One thousand trials at each stimulus value
used were simulated, and the results are shown in Figure 4.
The left panel of Figure 4 shows the psychophysical functions produced in the four
cases simulated. It is obvious that all exhibited the monotonically increasing proportion of
``long’’ responses found in data when stimulus duration was increased, and that all
functions had the approximately ogival shape found in data. The right panel abstracts
from the overall results just the proportion of ``long’’ responses occurring in the simulations at the arithmetic and geometric mean values, as the L/S ratio changed. It is clear
from Figure 4 that the main effect of L/S ratio in the simulation is on the proportion of
``long’’ responses occurring at the geometric mean. When the L/S ratio was 9:1, virtually
no ``long’’ responses were predicted at the geometric mean of S and L, and, in fact, very
few are found in the data (Wearden, 1991). As the L/S ratio declined, however, an
increasing proportion of ``long’’ responses occurred at the geometric mean of S and L,
with the 50% barrier being crossed somewhere between L/S ratios of 3:1 and 2:1. Of
course, the L/S ratio that yields a bisection point at the geometric mean in the simula-
FIG. 4.
Left panel: psychophysical functions produced by the simulations described in the text; in all cases L
was 900 msec, and S varied from 100 to 600 msec, generating the 4 different L/S ratios indicated in the panel.
Right panel: the proportion of ``long’ ’ responses predicted by the simulation at the arithmetic mean of S and L
(am: ® lled circles) and the geometric mean (gm: open circles) as the L/S ratio was changed.
tions depends upon the choice of parameters used, and the crossover from less than to
greater than 50% ``long’ ’ responses at the geometric mean will occur at different L/S
values in other cases. In addition, the overall shape of the function relating the proportion
of ``long’’ responses at the geometric mean to the L/S ratio also depends on simulation
parameter values, although the function will be continuous in all cases. In spite of these
quali® cations, the model accurately captured the increasing proportion of ``long’’
responses occurring at the geometric mean as stimulus range decreases, with constant
parameter values.
The principal fault of the modi® ed difference model when applied to data from the
present study is that it cannot simulate stimulus spacing effects when parameter values are
kept constant. For example, for some constant c and x values, simulated data points from
conditions used in the LIN50 and LOG50 groups of Experiment 2 superimpose, rather
than exhibiting the leftward shift of the psychophysical function in the LOG50 group
found in data (e.g. left panel of Figure 3). Although we do not present any results here, to
save space, it is intuitively obvious that a model like the modi® ed difference model has no
mechanism for taking into account the context in which some stimulus duration, t,
occurs, so for ® xed S and L, a given value of t will always produce the same proportion
of ``long’’ responses. Wearden and Ferrara’s model (from Wearden & Ferrara, 1995) can
simulate stimulus spacing effects with reasonable accuracy but cannot deal with the
stimulus range effects noted in data, above. Overall, therefore, it seems that neither
model deals with all aspects of the data in a satisfactory way, although some parameter
values can be found that will ® t the modi® ed difference model reasonably well to all the
psychophysical functions obtained, as shown in Figures 1 and 3.
Whilst a coherent theoretical treatment of all the effects obtainable in temporal bisection with humans remains at present elusive (and we have explored some more complex
models than that outlined above without obtaining an accord between data and theory that
is better than the modi® ed difference model provides), data in the present article strongly
suggest that the literature on temporal bisection in humans is empirically consistent, with
geometric mean bisection and absence of stimulus spacing effects occurring when the
L/S ratio is low (see also Allan & Gibbon, 1991; Wearden & Ferrara, 1995) but with
arithmetic mean bisection and marked spacing effects when the L/S ratio is larger. As
these results seem reliable across studies conducted with different procedures and in
different laboratories, they seem a reasonable basis for the development of some more
coherent and comprehensive theory of bisection than exists at present.
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Manuscript received 18 August 1994
Accepted revision received 13 J uly 1995
Effets de la m arg e de dure e des stim uli dans la bisection
tem porelle chez le sujet hum ain
Deux expe riences investiguent l’effet de la marge de stimuli sonores de courte dure e sur la
performance de bisection temporelle chez le sujet humain. Dans l’Expe rience 1, six groupes
de sujets son testeÂs avec une taÃche de bisection temporelle ouÁ chaque stimulus doit eÃtre classeÂ
comme ``court’’ ou ``long’’ . Pour trois groupes, la diffeÂrence entre les dure es la plus longue
(L) et la plus courte (S, Short) de l’ensemble des stimuli qui doit eÃtre bisecte reste constante aÁ
400 ms., et le rapport L/S varie de 5:l aÁ 2:1. Pour trois autres groupes, le rapport L/S est
maintenu constant aÁ 4:1 alors que la diffeÂrence L± S varie de 300 aÁ 600 ms. Le point de
bisection (la valeur de stimulus produisant 50% de reÂponses ``long’ ’ ) est plus proche de la
moyenne arithmeÂtique des stimuli L et S que de la moyenne ge omeÂtrique chez tous les
groupes, excepte celui dont le rapport L/S est 2:1 et pour lequel on obtient la bisection aÁ
la moyenne ge omeÂtrique. Dans l’Expe rience 2, 1’ espacement des stimuli entre L et S est
lineÂaire ou logarithmique et le rapport L/S est 2:1 ou 19:1. Une bisection aÁ la moyenne
geÂomeÂtrique est obtenue dans les deux cas ouÁ le rapport L/S est 2:1, mais les effets de
l’espacement des stimuli sont observeÂs seulement lorsque le rapport L/S est 19:1.
Globalement, les donne es vont dans le sens d’ une supposition ante rieure, selon laquelle le
rapport L/S utilise dans la taà che de bisection est un de terminant majeur de la performance.
Un modeÁ le theÂorique de la bisection propose par Wearden (1991) rend ade quatement compte
du de placement des points de bisection discute cËidessus, mais est mis en dif® culteÂs par l’effet
de l’espacement des stimuli.
Efectos del ra ngo de estõ m ulos sobre la biseccio n tem poral
en sujetos hum anos
En dos experimentos con sujetos humanos, utilizando tonos de corta duracio n como
estõ mulos, se estudio el efecto del rango del conjunto de estõ mulos sobre la actuacio n en
una tarea de biseccioÂn temporal, en la que cada estõÂ mulo habõÂ a de ser clasi® cado como ``corto’’
o ``largo’ ’ . En tres grupos de sujetos, la diferencia entre la duracio n maÂs larga (L) y maÂs corta
(C) del conjunto de estõÂ mulos a biseccionar se mantuvo constante en 400 msegs., mientras
que la razo n C/L variaba entre 5:1 y 2:1. En otros tres grupos, la razo n C/L se mantuvo
constante, mientras que la diferencia L± C vario entre 300 y 600 msegs. El punto de biseccioÂn
(valor de estõ mulo al que se daba un 50% de respuestas ``largo’’ ) se situo maÂs cerca de la
media artimeÂtica de C y L que de la media geome trica en todos los grupos, excepto an aque l
para el que la razo n C/L fue 2:1, en el que se observo biseccioÂn en funcio n de la media
geomeÂtrica. En el Experimento 2, los extõ mulos se espaciaron entre C y L segu n una escala
lineal o segu n una escala logarõ tmica y el valor de la razo n C/L fue de 2:1 o de 19:1. Se
observo biseccioÂn geome trica en ambos casos cuando la razo n C/L fue de 2:1, aunque soÂlo
hubo efectos del espaciamiento de estõ mulos cuando la razo n C/L fue de 19:1. En general, los
resultados apoyan conjeturas anteriores sobre el papel determinante que la razo n C/L
desempenÄ a en las tareas de biseccioÂn. Un modelo teoÂrico propuesto por Wearden (1991)
explica adecuadamente las desviaciones del punto de biseccioÂn observadas, aunque encuentra
di® cultades para explicar los efectos de espaciamiento.
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