The mixtures we tested with ∼30 components were highly similar, but are they an olfactory white? To address this question, we conducted a discrimination experiment: Twenty participants performed a three-alternative forced-choice discrimination task between a grand mixture made of 35 components and nonoverlapping component mixtures of various sizes. Even when selecting the mixtures spread in perceptual space, the mixtures remained discernible. Although the accuracy of discrimination [Kendall correlation (KC), τ = −0.51, P < 0.04; Fig. 2B ] and self-rated confidence in the discrimination (KC, τ = −0.76, P < 0.001; Fig. 2C ) decreased as the number of components increased, the accuracy of discrimination for even 30-component mixtures remained well above chance. Does this retained discriminability argue against an analogy between such large odorant mixtures and the color white or the sound white noise?

Increasing the number of nonoverlapping spanned components in two mixtures renders them more similar and less discernible. (A) The average rated similarity for mixture pairs differing in number of components, where mixtures were selected from perceptual space (red) or physicochemical space (black). Component number is expressed as the square root of the product of the two mixture sizes. (B and C) Discrimination accuracy (B) and confidence level (C) ratings for a 35-component mixture and nonoverlapping mixtures of various sizes. Error bars indicate SE.

Consistent with our hypothesis, there was a significant relationship between the number of components in each of two mixtures and their perceived similarity, in both perceptual [F(1,17) = 124.8, P < 0.0001] and physicochemical [F(1,28) = 34.1, P < 0.0001] space, reflecting increased similarity with an increased number of components (r = 0.94 in perceptual space and r = 0.75 in physicochemical space, both P < 0.0001) ( Fig. 2A ). Looking at each target mixture size independently revealed that this relationship was consistently true for target mixtures of 20 or more components (correlation between number of components and similarity score, all r > 0.58, all P < 0.03; SI Appendix, Figs. S1 A–D and S2 A–C ) but not for target mixtures of fewer than 10 components (all r < 0.18, all P > 0.26; SI Appendix, Figs. S1 F–H and S2 E and F ). In other words, the more components there were in each of two mixtures, the more similar the smell of those two mixtures became, even though the mixtures had no components in common ( Fig. 2A ). This trend implies that if more and more nonoverlapping components are added to each of two mixtures, these two mixtures eventually should smell the same, despite having no components in common. Indeed, given a sufficient number of equal-intensity spanned components, this trend implies that eventually all mixtures should smell the same. We call this predicted ultimate point of perceptual convergence “olfactory white.”

We conducted pairwise similarity tests (using a nine-point visual analog scale, VAS) of 191 mixture pairs, in 56 participants (average of 14 participants per comparison). Each target mixture (1, 4, 10, 15, 20, 30, 40, or 43 components) was compared with all other mixtures (1, 4, 10, 15, 20, 30, 40, or 43 components) and, as a control, with itself. Other than comparisons of a mixture with itself, all comparisons were nonoverlapping; in other words, each pair of mixtures being compared had no components in common.

Odorants plotted in stimulus space. (A) Perceptual space: 144 odorants commonly used in olfaction research projected onto a 2D space made of principal component 1 (PC1; 30.8% of the variance) and principal component 2 (PC2; 12% of the variance) of perception. (B) Physicochemical space: 1,492 odorants commonly modeled in olfaction research projected onto a 2D space made of principal component 1 (33.4% of the variance) and principal component 2 (10% of the variance) of structure. The 86 odorants we used are plotted in red. Considerations of human safety prevented us from including odorants that were at the extremes of physicochemical space, because these odorants often are toxic.

We obtained 86 monomolecular odorants that were well distributed in both perceptual ( Fig. 1A ) ( 1 , 11 ⇓ – 13 ) and physicochemical ( Fig. 1B ) ( 1 , 14 , 15 ) stimulus space. We then diluted each of these odorants separately to a point of about equal perceived intensity as estimated by an independent group of 24 participants ( SI Appendix, Table S1A ) and prepared various odorant mixtures containing various numbers of such equal-intensity odorant components. Importantly, to prevent the formation of novel compounds, odorant mixtures were not mixed in the liquid phase; instead, each component was dripped onto a common absorbing pad in a sniff-jar, so that their vapors alone mixed in the jar headspace. To select the components of each mixture, we used an algorithm that automatically identified combinations of molecules spread out in olfactory stimulus space ( SI Appendix ). We prepared several different versions for each mixture size containing 1, 4, 10, 15, 20, 30, 40, or 43 components, so that half of the versions were spread optimally in perceptual space, and half of the versions were spread optimally in physicochemical space. Note that although Fig. 1 is limited to a 2D representation, our algorithm selected mixtures based on their multidimensional features ( SI Appendix ).

Mixtures with Many Equal-Intensity Spanned Components Are Identified as Olfactory White.

Visually, humans can discriminate easily between many different “whites,” but all these whites retain the color-gestalt identity of white. To determine whether odorant mixtures of ∼30 spanned components similarly obtain a gestalt identity, we conducted an odor-identification experiment. Selecting from physicochemical space, we generated four versions of 40-component mixtures. To prevent any cognitive influences of the label “white,” we labeled these mixtures with the meaningless name “Laurax.” Each of the four versions of Laurax was assigned to three different participants from a group of 12. To acquaint themselves with the odor, each participant came to laboratory on three consecutive days, and every day repeatedly smelled and rated the applicability of 146 verbal descriptors (16) to only their version of Laurax. On the fourth day, test day, participants performed a four-alternative forced-choice identification task for 23 different, novel, but partially overlapping target odorant mixtures of 1, 4, 10, 20, 30, or 40 components, all selected to span physicochemical space. Each target mixture was provided with four alternative labels: Three labels were assigned by an expert perfumer (coauthor D.G., who was blinded to experimental aims and conditions) as optimal identifiers for each mixture (SI Appendix, Table S2), and the fourth label was “Laurax.” Consistent with our prediction, the probability of assigning the name “Laurax” to a novel mixture increased as the number of components increased (KC, τ = 0.73, P < 0.05) (Fig. 3A). Moreover, novel target mixtures with fewer than 20 components were significantly less likely to be identified as Laurax than novel targets with 20 or more components [t (10) = 5.54, P < 0.001; Fig. 3A, Inset]. Finally, although chance application of a label in the four-alternative task is 25%, the label “Laurax” was applied to novel 40-component mixtures 57.6% of the time [t (10) = 3, P < 0.02].

Fig. 3. Mixtures made of many equal-intensity spanned components are identified as olfactory white. A and B show the average probability of identifying a novel odorant mixture as white (Laurax) as a function of the number of components in the mixture in a four-alternative (A) or five-alternative (B) forced-choice identification test (B). Each dot represents the average rating for four versions of a given number of components. Error bars indicate SE. (Insets) The average probability of identifying a novel odorant mixture as white (Laurax) for mixtures of fewer than 20 components, compared with mixtures of 20 or more components in four-alternative (A) or five-alternative (B) forced-choice identification tests.

The descriptors we used for Laurax, although provided by a professional perfumer, nevertheless may lack universal applicability. To address the possibility of “dumping” (17), namely the assignment of inappropriate labels in the face of limited alternatives, we repeated the experiment with a different group of 13 participants and with the additional response option of “other.” Again, the probability of assigning the name “Laurax” increased as the number of components increased (KC, τ = 0.93, P < 0.01) (Fig. 3B), and novel target mixtures with fewer than 20 components were significantly less likely to be identified as Laurax than novel targets with 20 or more components (t (11) = 5.68, P < 0.001; Fig. 3B, Inset). Finally, although the chance application of a label in the five-alternative task is 20%, the label “Laurax” was applied to novel 40-component mixtures 50% of the time (t (11) = 3.35, P < 0.007).

Because we were limited by the available components for which we had equated intensity, but we wanted to have meaningful differences across the various target mixtures, in the two above experiments there was inevitable, minimal overlap of the components in the learned Laurax mixture and in target mixtures. Moreover, despite the addition of ”other” as a viable response, dumping remained possible. With these considerations in mind, we applied even stricter conditions in the following experiments: After 2-d acquaintance with 30-component Laurax, 12 participants smelled 21 target mixtures of various numbers of components but with no component overlap with the mixture they had learned to identify as Laurax, and judged whether these mixtures were or were not Laurax; i.e., no alternative labels were provided in this delayed match-to-sample task.

Consistent with the previous experiments, the probability of discriminating a mixture from the Laurax participants had learned decreased as the number of components increased (KC, τ = −0.68, P < 0.05; Fig. 4A), and novel target mixtures with fewer than 20 components were significantly more likely to be discriminated from Laurax than novel targets of 20 or more components [t(11) = 3.52, P < 0.005; Fig. 4A, Inset]. Moreover, chance at this task is 50%. Although participants were significantly above chance (all P < 0.05) in discriminating smaller novel mixtures (with 1, 5, 10, or 15 components) from the percept of Laurax, they did not differ from chance in discriminating novel 20-, 25-, and 30-component mixtures from the percept of Laurax (all P > 0.05). It is important to acknowledge the perceptual memory component in the delayed match-to-sample task. Had participants been provided simultaneously with the Laurax they had learned previously and the novel 30-component mixtures, they likely would have discriminated them (as in Fig. 2B). However, when participants were given only the gestalt percept of the Laurax they had learned (which indeed was sufficient for 97% accuracy in identifying the learned Laurax), novel mixtures with ∼30 components were deemed not significantly different from Laurax.

Fig. 4. Mixtures made of many equal-intensity spanned components match the perceptual memory of olfactory white. (A) Delayed match-to-sample between a learned Laurax and novel nonoverlapping mixtures of various sizes. Each dot represents the average rating for three versions of a given number of components. Error bars indicate SE. (Inset) The average probability of discrimination for mixtures of less than 20 components or mixtures of 20 or more components. (B and C). Delayed match-to-sample between a learned Laurax and a novel nonoverlapping 25-component mixture that was or was not spanned in space, was not equated for intensity, or was the very same mixture learned as Laurax (100% accuracy). (B) Continuous similarity score as reflected by location on the VAS (median ± median absolute deviations across participants). (C) Probability of discriminating a mixture from Laurax (scores below 25%). *P < 0.05. Error bars indicate SE among participants.

One may raise the possibility that Laurax became a percept associated with “large mixtures,” despite the mixtures’ olfactory identity. Ideally, to test this possibility, one would compare large mixtures that are spanned in olfactory space (Laurax) with equally large mixtures clustered in olfactory space. However, because we initially selected 86 molecules that span space, one could not extract from them a cluster of ∼30 components. Any cluster of ∼30 of these 86 components would, by definition, be relatively well spanned. To mitigate this limitation partially, we equated the perceived intensity of an additional 58 molecules, so that we had a pool of 144 molecules to choose from (SI Appendix, Table S1B). We then repeated the above strict delayed match-to-sample task with an additional 16 participants, and used test-target mixtures with only 25 components. Thus we had nine versions that spanned physicochemical space, five versions that, although not perfectly clustered, were not spanned in physicochemical space, and two versions with components identical to the Laurax the participants had learned but that were not equated for perceived intensity (SI Appendix). Moreover, rather than a strict yes/no selection, we asked participants whether the odor was Laurax and provided participants with a VAS ranging between “yes” and “no.” This approach allowed us to conduct two complementary analyses: In the first we simply extracted the median VAS scores as a continuous measure of similarity (Fig. 4B), and in the second we arbitrarily parsed the VAS scale into dichotomous “yes” (above 75%) or “no” (under 25%) answers (Fig. 4C).

The similarity analysis revealed that tested Laurax was rated as identical to the remembered percept of Laurax (median VAS = 100% ± 0); nonequated Laurax was less similar (when comparing Laurax with itself we used a one-tailed test because similarity cannot exceed 100%; Wilcoxon one-tailed test, median VAS Laurax = 100% ± 0, median VAS nonequated = 89.64% ± 6, P < 0.03); spanned mixtures were even less similar (median VAS spanned = 40.54% ± 12.75, median VAS nonequated = 89.64% ± 6, t (15) = 7.56, P < 0.001); and nonspanned mixtures were even further less similar (Wilcoxon test, median VAS spanned = 40.54% ± 12.75; median VAS nonspanned = 1.5% ± 1.5, P < 0.03) (Fig. 4B).

In the alternative analysis of the same experiment, nonspanned mixtures were more discernible from the remembered percept of Laurax than were novel spanned mixtures (Wilcoxon test, nonspanned = 70% ± 27.3, spanned = 47.2% ± 25.9, P < 0.01) (Fig. 4C), and the Laurax not equated for intensity was more discernible from the remembered percept of Laurax than was Laurax itself (Wilcoxon one-tailed test, not equated = 12.5% ± 22.36, Laurax = 0% ± 0, P < 0.04) (Fig. 4C). Repeating this analysis but parsing at the 50% mark (rating above 50% as “yes” and below 50% as “no”) revealed similar results.

Taken together, these analyses imply that the percept of Laurax was dependent more on spanning olfactory space than on equating intensity (Fig. 4B) and that only nonspanned mixtures were decidedly “not Laurax” (Fig. 4C). The full influence of spanning olfactory space is likely much greater than revealed here, because the possibility of selecting independent clusters comprising 25 out the 144 molecules for which we have equated intensity remains limited. In other words, our “clusters” still were quite spanned. Thus, this result, although significant in itself (P < 0.03 in similarity or P < 0.01 in yes/no selection), likely underestimated the differences between spans and clusters.

Finally, to verify again the limits of this phenomenon, we repeated the delayed match-to-sample task in 18 participants testing only mixtures of up to 15 components (i.e., mixtures that on average should not converge) and in 14 participants testing mixtures of up to 60 components (i.e., mixtures well beyond our estimated point of convergence). Although participants were significantly better than chance in discriminating the novel 15-component mixtures from the percept of Laurax (Wilcoxon test, 10.1% ± 22.33, P < 0.002), they were not different from chance in discriminating novel 60-component mixtures from the percept of Laurax (median VAS = 52.8% ± 16.3, t (11) = 0.012, P > 0.99). Taken together, these experiments are consistent with the notion of a gestalt percept following combinations of ∼30 equal-intensity components or more that are well distributed in physicochemical space. We call this percept “olfactory white.”