Consonance Preferences Within an Unconventional Tuning System

Participants were 110 undergraduate students at the University at Albany (55 female; Age: M = 18.81, SD = 1.39) who completed the study for course credit in an introductory psychology course. Thirty (27.27%) reported having at least one year of formal training in music theory and sixty-one (55.45%) reported at least one year of formal training on a musical instrument.

Scale degrees and frequencies of the tones in each dyad and triad are reported in Tables 1 and 2. Following Bowling et al. (2018a), all chords were equated for average pitch height at 261.63 Hz (C4). Test stimuli were created from single-note piano and bass clarinet samples from the University of Iowa's Electronic Music Studios digital archive (Fritts, 2019). (Power spectra of examples of the piano and clarinet tones are presented in Figure 2.) These were pitch shifted and combined into chords using Audacity (v. 2.3.0) software. Specifically, the archived sample with the closest frequency to each BPCJ target pitch was shifted to the frequency of that pitch using Audacity’s “Change Pitch” function. This function enables shifting the frequency of an audio sample by manually inputting the desired frequency value into a dialog box. Pitch-shifted samples were then combined into chords using Audacity and then RMS-equated in PRAAT (v. 6.0.43; Boersma & Weenink, 2018). Due to the naturalistic quality of the samples, the length of the finished chords varied for the clarinet condition (from 1.9–2.4 s, depending on how long the original note was played) but remained constant for the piano condition (2.5 s). All audio files, including samples from the digital archive, were mono. Synthesized chords were saved as uncompressed WAV files (sampling rate = 44,100 Hz, bit depth = 32) and were presented via headphones (Koss UR-20) at a fixed volume deemed comfortable in informal testing with undergraduate and graduate research assistants.

Following Bowling et al. (2018a), we computed harmonicity scores for each chord by first finding the greatest common divisor (GCD) of the F0s of the tones of a chord made up of intervals with the exact same frequency ratios (e.g., 5/3 and 3/1), yet with tone frequencies set at integer values (e.g., 100 Hz for an exemplar with tones of 300, 500, and 900 Hz) so as to simplify calculations. We then determined the complete harmonic series beginning on this GCD, assigning the highest frequency in this series to the least common multiple (LCM) of the F0s of the chord tones (e.g., 4500 Hz). (As discussed by Bowling et al., 2018a, the LCM is used as a cut-off for purposes of computing harmonicity because it represents the point after which the pattern of harmonics repeats for any given chord. Using a different value risks truncating or extending the harmonic pattern arbitrarily). Finally, we calculated the percentage of harmonics in this complete series that were actually present in the complex spectrum produced by the chord (see Bowling et al., 2018a, and Gill & Purves, 2009, for additional details and see Appendix for a rudimentary worked example). The resulting harmonicity values are presented in the Supplementary Materials (see https://mp.ucpress.edu).

As a secondary means of assessing the VSH, Bowling et al. (2018a) also tested whether chords were perceived as more consonant when the absolute frequency intervals between their tones were more akin to the absolute frequency intervals that tend to occur between harmonics within human vocalizations. According to Bowling et al. (2018a), the minimum absolute frequency interval between adjacent harmonics in human vocalizations is approximately 50 Hz. Consequently, when the F0 s of the tones making up a particular chord form intervals smaller than 50 Hz, this reduces the vocal similarity of this chord, leading it to be perceived as less consonant.

Based on this assumption, Bowling et al. (2018a) made a series of comparisons of consonance ratings among pairs of Western chords in which the minimum interval between the F0 s of at least one of the chords was less than 50 Hz. For to-be-compared chords that met this criterion, they predicted that: (1) when one chord featured a minimum interval between F0 s of less than 50 Hz whereas the second chord featured a minimum interval between F0 s of greater than 50 Hz, the second chord would be perceived as higher in consonance; and, (2) when both chords featured a minimum interval between F0 s of less than 50 Hz, the chord featuring a greater minimum interval would be perceived as higher in consonance. In the present study, we conceptually replicated this frequency intervals analysis using BPCJ dyads and triads.

Following Bowling et al. (2018a), consonant chords were defined for participants as “relatively pleasant or attractive” and dissonant chords as “unpleasant or unattractive.” Participants then heard a set of conventionally consonant and dissonant chords, after which they rated two 12-TET “practice” chords (a perfect 4th and a minor 7th) on a 7-point Likert scale anchored at 1 (very dissonant) and 7 (very consonant). Afterward, they were randomly presented with and asked to rate each BPCJ dyad using the same scale. Upon completing 2 blocks of 13 dyad ratings, participants rated all BPCJ triads in 2 consecutive blocks of 78. Participants were randomly assigned to hear all chords in either a piano or a clarinet timbre.

Intraclass correlations (ICC; two-way random effects, consistency, average measures model) were first computed to assess interrater reliability in average consonance ratings. These were uniformly high for both dyads (ICC_piano = .96; ICC_clarinet = .92) and triads (ICC_piano = .92; ICC_clarinet = .85), suggesting that dataset provides fairly reliable estimates of the consonance values for those chords.

Average consonance ratings for dyads appear in Table 1. A scatterplot of the relationship between harmonic similarity scores and consonance ratings of BPCJ dyads appears in Figure 3. Shapiro-Wilk tests revealed that harmonicity scores for dyads were non-normally distributed. Therefore, Spearman rank order correlations were used for our analyses. Inconsistent with the original findings and subsequent predictions of Bowling et al. (2018a, 2018b), initial analyses revealed no statistically reliable association between harmonicity and consonance, irrespective of whether a piano or clarinet timbre was used, ρ_overall = .19, p = .534.

In trying to account for this failure to replicate, we conducted a number of post hoc exploratory analyses. First, we found that two of the BPCJ dyads that we had presented were particularly similar in their tone ratios to twelve-tone equal tempered (12-TET) dyads. Specifically, the ratio for dyad 2 is 25:21 (1.19) is extremely close to that of the 12-TET minor third (1.189) and the 63:25 (2.52) tone ratio of dyad 11 is within rounding error of that of the 12-TET major tenth (2.5198). Indeed, given their close resemblance to existing 12-TET intervals, these two specific dyads have been previously classified as “quasi-tempered” BPCJ intervals (Haluška, 2004) and both were rated by participants as very high in consonance despite their relatively low harmonicity.

To confirm that the two dyads at issue were indeed more similar to 12-TET intervals than the remaining dyads in the present set, we calculated the absolute difference between each BPCJ dyad’s frequency ratio and that of the closest chromatic 12-TET dyad. Scores on this ad hoc “chromatic distance” measure appear in Table 1. They reveal that the ratios for the two so-called quasi-tempered dyads (2 and 11) were indeed the most similar to 12-TET dyads among the intervals tested. A subsequent correlational analysis showed that chromatic distance was reliably predictive of consonance ratings, ρ = -.58, p < .05; however, when dyads 2 and 11 were dropped from the analysis, this correlation was no longer reliable, p = .318. This is consistent with the possibility that consonance ratings for the two quasi-tempered dyads may have been unduly influenced by their familiarity, undermining our ability to detect an effect of harmonicity on consonance.

Based on these findings, we recomputed the raw correlations between harmonicity and consonance, excluding ratings for dyads 2 and 11 from the analysis. Consistent with the correlational analysis reported by Bowling et al. (2018a), the results showed that increased harmonicity was now robustly associated with higher consonance ratings for BPCJ dyads, both overall and within each timbre condition separately, ρ_overall = .65, p < .05; ρ_piano = .61, p < .05; ρ_clarinet = .68, p < .05 (see Figure 3). Following Bowling et al.’s (2018a) analytical strategy, we also computed a series of Bonferroni-adjusted post hoc comparisons between consonance ratings of all dyads included in the aforementioned correlational analysis. In line with results reported for Western dyads by Bowling et al. (2018a), for significant comparisons, the harmonic similarity analysis correctly predicted the chord perceived as more consonant in 89% (24 out of 27) of cases.

Average consonance ratings for dyads appear in Table 1. A scatterplot of the relationship between harmonic similarity scores and consonance ratings of BPCJ triads appears in Figure 4. Shapiro-Wilk tests revealed that consonance ratings as well as harmonicity scores for the triads were non-normally distributed, therefore Spearman rank correlations were used for the analyses reported below. As in the case of BPCJ dyads, these analyses revealed no statistically significant associations between harmonicity and consonance, irrespective of whether a piano or clarinet timbre was used, ρ_overall = .14, p = .217. These findings fail to replicate Bowling et al.’s (2018a) results, which showed a highly reliable rank order correlation between harmonicity and consonance ratings of conventional triads played in a piano timbre (ρ = .66, p < .001, according to our calculations from their published data).

In line with our analysis of BPCJ dyads, we recomputed the harmonicity analyses excluding the 23 of 78 chords that contained dyads 2 and dyad 11 (i.e., the intervals corresponding to the frequency ratios 25/21 and 63/25, respectively; see Table 1). Following these exclusions, correlations between harmonicity and average consonance ratings were now reliable both overall, ρ = .32, p < .05, and within the piano timbre condition, ρ = .30, p < .05, but did not attain significance within the clarinet timbre condition, ρ = .26, p = .060 (see Figure 4). These correlations remained substantially weaker than those reported by Bowling et al. (2018a) for conventional triads.

Another possible contribution to the relative weakness of the foregoing results may be tied to differences between the harmonicity distributions for conventional and BPCJ triads (see Figure 5). Specifically, the mean harmonicity of conventional triads, based on Bowling et al.’s (2018a) data is 0.20 (SD = 0.16), whereas that for BPCJ triads is only 0.09 (SD = 0.11). Moreover, harmonicity values for BPCJ triads are more positively skewed (skewness = 2.24), that is, more concentrated in the lower end of the distribution, than are conventional triads (skewness = 1.06). Indeed, the first quartile of harmonicity scores for conventional triads (0.103) exceeds the third quartile of harmonicity scores for the BPCJ triads (0.099). This suggests that the association between harmonicity and consonance may emerge more robustly at higher levels of harmonicity. As a post hoc method of evaluating this possibility, we recomputed the regression analysis separately for BPCJ triads with harmonicity values that were at or above the first quartile of harmonicity scores for conventional triads. This value is virtually identical to the mean harmonicity for conventional triads for which common labels (e.g., major, minor, diminished, augmented) are available, which we computed as 0.102 from the data of Bowling et al. (2018a). Within this more conventional range of harmonicity there was a markedly stronger positive correlation between harmonic similarity scores and average consonance ratings, ρ_overall = .49, p < .05; ρ_piano = .51, p < .05; ρ_clarinet = .41, p = .103. The emergence of this more pronounced linear relationship at higher levels of harmonicity is visually apparent in Figure 4. Mirroring the results reported by Bowling et al. (2018a) for Western triads within a similar range of harmonicity, for pairs of BPCJ triads that were significantly different in consonance according to Bonferroni-adjusted post-hoc comparisons, the harmonic similarity analysis correctly predicted the chord perceived as more consonant in 81% (17 out of 21) of cases.

Altogether, the results of our harmonicity analyses do not straightforwardly replicate the findings of Bowling et al. (2018a). However, a similar pattern of results can be detected with unconventional triads, given the important provision that quasi-tempered intervals are excluded or that harmonicity is within a range relatively akin to that of conventional triads. Notably, throughout the series of harmonicity analyses reported above, there were nominal differences between the correlations obtained in the piano and clarinet timbre conditions. However, across analyses, these differences were inconsistent in their direction and magnitude and not a single difference between correlations approached statistical significance (using independent samples tests based on Fisher’s r to z transformation). This fails to support our hypothesis that the use of a clarinet timbre should strengthen the correlation between consonance and harmonicity using BPCJ-based chords. However, it does support Bowling et al.’s (2018b) contention that these associations should be relatively unaffected by timbre.

Using the full set of BPCJ stimuli, the frequency interval-based vocal similarity analysis devised by Bowling et al. (2018a) was applicable to 29% of BPCJ dyads and 75% of BPCJ triads. By way of comparison, 44% of Western dyads and 93% of Western triads were eligible to be analyzed using this measure in Bowling et al.’s (2018a) original dataset. (BPCJ-based intervals are generally larger than Western intervals as they are based on the span of a tritave as opposed to an octave. As such, BPCJ chords are less likely to contain intervals smaller than 50 Hz between the F0 s of constituent tones.) The results of the analysis were consistent with the predictions and original findings of Bowling et al. (2018a): For both BPCJ dyads and triads, the chord with higher vocal similarity according the frequency interval measure was rated as more consonant in the majority of cases (61% or 14 out of 23 dyads and 77% or 1735 out of 2262 triads).

Although the primary objective of our study was to conceptually replicate the procedure of Bowling et al. (2018a) using unconventionally tuned chords, we also broadened our analyses to incorporate additional measures that were not originally taken into account in their study. First, although Bowling et al.’s (2018a) pattern matching-based harmonicity measure is elegant in its simplicity, as noted by Harrison and Pearce (2020), its validity may be undercut by a failure to take into account how the auditory system accommodates small deviations in the frequencies of partials when fitting a complex sound wave to a harmonic template during the pitch estimation process. This means that very small changes in the tuning of chord tones may translate into inordinately large changes in harmonicity estimates for that chord. Harrison and Pearce (2020) address this shortfall using a more comprehensive measure that models the information-theoretic uncertainty involved in estimating a fundamental frequency from a set of simultaneously presented complex tones. Essentially, the model builds an idealized representation of the combined spectrum of a chord—one that allows for some modicum of variance in the frequencies of constituent partials—and estimates the extent to which this representation converges upon a candidate for a matching harmonic spectrum among competing harmonic templates (see Harrison & Pearce, 2018, for computational details). In addition, unlike Bowling et al.’s (2018a) harmonicity measure, which provides distinct harmonicity estimates for different voicings of the same chord (e.g., different inversions of a C major triad), Harrison and Pearce’s (2020) measure operates on representations of pitch class, rather than on representations of specific pitches, leading it to compute voicing-invariant estimates of harmonicity. This may better reflect the possibility that the auditory system computes similarity between spectra (including that between the complex spectrum of a chord and that of a harmonic template) using pitch class-based spectral representations. In their recent study comparing various harmonicity-based models, Harrison and Pearce (2020) found that their own harmonicity measure was best able to predict consonance ratings. In light of these findings, we reanalyzed consonance ratings for BPCJ chords in our dataset using this alternative measure, assuming that it might offer a more powerful test of the VSH by more realistically capturing how harmonic, and thereby vocal, similarity may be represented psychologically.

Second, theoretically speaking, harmonicity should be negatively correlated with spectral interference, as the frequency distance between harmonic partials renders them less likely to activate overlapping critical bands along the basilar membrane. As discussed earlier, whereas the notion that consonance may result from an absence of auditory roughness due to spectral interference was proposed over a century ago, Bowling et al. (2018a; Bowling & Purves, 2015) have argued against the viability of this hypothesis on both conceptual and empirical grounds. Accordingly, they have excluded measures of spectral interference from their analyses.

However, the results of several recent studies suggest that it may be premature to dismiss the role of spectral interference in judgments of consonance. For instance, in contrast to earlier findings by Cousineau et al. (2012), Marin, Thompson, Gingras, and Stewart (2015) demonstrated that congenital amusics do in fact show systematic preferences between conventionally consonant and dissonant chords and suggested that this effect may be fully attributed to spectral interference. In addition, Parncutt, Reisinger, Fuchs, and Kaiser (2019) proposed that prior studies (e.g., McDermott et al., 2010) may have found relatively weak and inconsistent relationships between spectral interference and consonance because they restricted their investigations to relatively “smooth” chords such as major and minor triads. In line with this possibility, they put forth evidence from a sweeping corpus study of seven centuries of Western polyphonic music, showing that chord prevalence—an indirect measure of consonance—is reliably and uniquely associated with both roughness as well as harmonicity. Similar conclusions have been drawn by Harrison and Pearce (2020), who returned to Bowling et al.’s (2018a) dataset and found that spectral interference was in fact a nominally stronger predictor of consonance than harmonicity, leading them to advocate for a composite model of consonance judgments for isolated chords (see also, Goffinet, 2018).

Most pertinent to the present study, Smit, Milne, Dean, and Weidemann (2019), recently reported the results of a study in which they asked participants to make consonance judgments of both BPCJ as well as Bohlen-Pierce equal tempered (BPET) triads. In contrast with our procedure, Smit et al. (2019) did not attempt to replicate the procedure of Bowling et al. (2018a) and, as such, did not take into account their measure of harmonicity, did not test all possible triads within a tritave, and did not equate the average pitch height of component chord tones (although they did statistically control for absolute pitch height). In addition, they did not manipulate chord timbre and played a series of 13 randomized individual BP chromatic scale tones prior to presentation of each target triad, whereas chords in our study were rated without any added musical context. Despite these differences, Smit et al. (2019) also found evidence that consonance ratings of BP triads were positively correlated with harmonicity, as gauged using a relatively comprehensive measure akin to that of Harrison and Pearce (2020). Moreover, consistent with the work of both Harrison and Pearce (2020) and Parncutt et al. (2019), Smit et al.’s (2019) analyses revealed that that spectral interference was independently (negatively) associated with judgments of consonance, once again supporting a composite model.

Given these recent advances in the field, we conducted a series of supplementary regression analyses including both measures of harmonicity and spectral interference as predictors of consonance ratings. The spectral interference produced by each BPCJ chord in our stimulus set was estimated using Hutchinson and Knopoff’s (1978) measure, as implemented by Harrison and Pearce (2020). This models the overall roughness engendered by the interaction of the partials comprising chord tones within critical bands on the basilar membrane (Roederer, 2008). Harrison and Pearce (2020) found that this classic interference measure was in fact the strongest predictor of consonance ratings in Bowling et al.’s (2018a) dataset, contributing to unique variance in these ratings above and beyond harmonicity.

Zero-order correlations between all predictor and outcome variables in our supplementary analyses appear in Table 3. Harmonicity and spectral interference scores for all BPCJ chords appear in the Supplementary Materials (see https://mp.ucpress.edu). Scatterplots of the relationships between these scores and consonance ratings appear in Figures 6 and 7. In the case of BPCJ dyads, when substituting the harmonicity measure of Harrison and Pearce (2020) for that of Bowling et al. (2018a), the positive, zero-order correlation between harmonicity and consonance was marginally reliable for chords played in a piano timbre, ρ = .52, p = .071, and significant for those played in a clarinet timbre; ρ = .64, p < .05, conceptually replicating Bowling et al.’s (2018a) results. We next performed multiple regression analyses on consonance ratings for dyads, entering scores on both Harrison and Pearce’s (2020) harmonicity measure as well as Hutchinson and Knopoff’s (1978) interference measure as predictors. As shown in Table 4, these analyses revealed negative effects of interference on consonance ratings of chords appearing in both a piano, β = -.60, t = -2.76, p < .05, as well as a clarinet timbre, β = -.51, t = -1.97, p = .076, suggesting that chords featuring more overall beating between constituent partials were judged as sounding less consonant. However, the effects of harmonicity were no longer statistically significant when shared variance with interference was taken into account.

With respect to BPCJ triads, when substituting the harmonicity measure of Harrison and Pearce (2020), reliable positive correlations between harmonicity and consonance once again emerged both for chords played in a piano timbre, ρ = .31, p < .01, as well as in a clarinet timbre, ρ = .26, p < .05. We next conducted multiple regression analyses on consonance ratings for triads, again entering the supplementary measures of harmonicity and interference as predictors. As shown in Table 4, these analyses revealed significant effects of interference on consonance ratings of chords appearing in both a piano, β = -.68, t = -8.43, p < .001, as well as a clarinet timbre, β = -.70, t = -8.21, p < .001. In addition, there was a significant positive effect of harmonicity on ratings of consonance in triads played in a piano timbre, β = .17, t = 2.12, p < .05. The effect of harmonicity on ratings of triads in a clarinet timbre did not approach statistical significance when shared variance with interference was taken into account, p = .554.^{1, 2}

In this study, we tested whether the positive correlation between harmonicity and consonance, as recently reported by Bowling et al. (2018a), generalizes to stimuli produced using an unconventional tuning system. The results failed to straightforwardly replicate this association using the measure of harmonicity originally employed by Bowling et al. (2018a). However, the hypothesized effect did reliably emerge following the post hoc exclusion from the analysis of a set of chords containing intervals that were particularly similar to existing 12-TET dyads. This is consistent with the possibility that consonance ratings for these stimuli were inflated due to familiarity effects, thereby distorting the overall predictive model. Post hoc analyses also indicated that the correlation between harmonicity and consonance for BPCJ triads only began to approach the magnitude of that reported by Bowling et al. (2018a) within a range of harmonicity typical of conventional triads. This may suggest that listeners have difficulty discriminating between chords on the basis of harmonicity when it falls within the comparatively low range characteristic of BPCJ triads. Alternatively, these relatively weak correlations may reveal limitations of the harmonicity measure employed by Bowling et al. (2018a). For instance, the measure does not tolerate near-misses from perfect harmonicity, failing to take into account that the auditory system responds similarly to intervals that deviate slightly from small integer ratios (e.g., Tramo et al., 2001). It also assumes that the constituent tones of a chord contain energy at all partials and in equal measure. These aspects of Bowling et al.’s (2018a) harmonicity measure may diminish its predictive utility when modeling how the auditory system processes naturalistic chord stimuli with a full range of timbral variation.

In line with this proposition, the findings of Bowling et al. (2018a) were conceptually replicated with both BPCJ dyads and triads and newly shown to generalize across both piano and clarinet timbres when harmonicity was measured using the more comprehensive measure recently introduced by Harrison and Pearce (2020). Multiple regression analyses confirmed an independent effect of harmonicity on consonance ratings of BPCJ chords, but only for triads presented in a piano timbre. The majority of variance in consonance ratings across all conditions was not uniquely accounted for by harmonicity, but by spectral interference. Alongside evidence that listeners may be induced to prefer inharmonic spectra when these are generated to minimize beating between partials (Sethares, 2005), these results argue against the contention that “roughness models do not explain consonance” (Bowling et al., 2018b, p. E4958). Rather, in line with the proposals of Harrison and Pearce (2020) and Parncutt et al. (2019), as well as the recent empirical findings of Smit and colleagues (2019), they support a composite model of consonance in which spectral interference plays a prominent role, as do harmonicity and cultural familiarity (McLachlan et al., 2013).³ From this perspective, the principal question currently facing consonance researchers is not which of these three factors explains consonance, but when and how do each of them contribute to affective responses to isolated chords.

Secondary analyses using Bowling et al.’s (2018a) frequency interval measure also conceptually replicated their findings using BPCJ chords. However, the implications of these results are relatively unclear due to limitations of this secondary vocal similarity measure. First, Harrison and Pearce (2020) noted that the measure is difficult to distinguish from overall spectral interference, which is also associated with the presence of F0 intervals smaller than 50 Hz. Second, as it currently stands, the frequency interval measure categorizes chord stimuli built on either pure or complex tones as equally similar to human vocalizations, although the latter are invariably composed of complex waveforms.

Finally, the present results failed to support our own hypothesis that the use of a clarinet timbre would lead to a stronger correlation between consonance and harmonicity for BPCJ-based chords. Again, we had proposed that such an effect might arise due to the shared emphasis of BPCJ tones and clarinet-like timbres on odd integer frequency ratios/odd-numbered partials. In retrospect, upon examination of the power spectra for our clarinet-based stimuli (see e.g., Figure 2), it is apparent that these naturalistic sounds, which were generated from recorded samples of individual notes on a clarinet, featured considerable energy in even-numbered harmonics. As such, future research examining how the “fit” between timbre and tuning influences consonance judgments may have to rely on computer-generated stimuli (e.g., synthesized square waves), the spectra of which can be more precisely manipulated (see Sethares, 2005, for a discussion of this approach). As noted above, the failure to detect an effect of timbre is consistent with Bowling et al.’s (2018b) proposal that associations between harmonicity and consonance should be relatively consistent across timbres, provided that the latter are harmonic.

Taken together, the present results converge with those of Bowling et al. (2018a) to offer additional support for the VSH, yet suggest that further research is needed to assess whether preferences for isolated chords are indeed at least partly driven by an evolved preference for sounds that resemble human vocalizations. In conjunction with recent studies reporting independent effects of harmonicity on consonance preference based on both behavioral data (Harrison & Pearce, 2020; Smit et al., 2019) and corpus analyses (Parncutt et al., 2019), the present findings using BPCJ chords do lend credence to the notion that the preference for harmonic, and thereby voice-like, musical sounds is rooted in human auditory neurobiology and does not require prior exposure to these sounds (Bowling & Purves, 2015; Purves, 2017).

Nonetheless, proponents of the VSH will also need to contend with ostensibly inconsistent evidence, including McDermott et al.’s (2016) provocative recent findings of weaker or non-existent preferences for harmonic chords in individuals with limited exposure to Western harmony (cf. Bowling, Hoeschele, Gill, & Fitch, 2017) as well as the well-documented prevalence of inharmonic timbres within certain musical systems (e.g., Indonesian gamelan, Thai classical music; Sethares, 2005). These observations would suggest that the preference for harmonicity does require enculturation within a musical system that reinforces attention to and valuation of harmonic sounds (see also, Plack, 2010).

In conclusion, the results of the present study suggest that Bowling et al.’s (2018a) correlational findings in support of the VSH were not merely an artifact of the familiarity or timbre of the chords that they used in their procedure. Our findings also converge with those of Harrison and Pearce (2020), Parncutt et al. (2019), and Smit et al. (2019), by demonstrating that consonance ratings of unconventionally tuned chords are independently associated not only with harmonicity, but with spectral interference as well. We hope that the behavioral data available from this large-scale study, which uniquely includes responses to both dyads and triads in two distinct timbres, will help inspire additional research aimed at assessing the VSH and more generally assist scholars working to develop more comprehensive models of musical consonance.

We are grateful to Carly Feldstein, Faith Gaillard, Anthony Leva, and Ryley Scott for their assistance in running these experiments and to Daniel Bowling for his invaluable comments and suggestions.

Assuming a triad made up of tones of 300, 500, and 900 Hz, the greatest common divisor (GCD) of the tones is 100 Hz and the least common multiple (LCM) is 4500 Hz.

A full harmonic series based on a F0 of 100 Hz with a cutoff of 4500 Hz would contain partials with the following frequencies:

For each of the constituent chord tones, harmonic series with the same LCM-based cutoff would be as follows:

300 Hz

500 Hz

900 Hz

Combining the frequencies present within the latter 3 harmonic series results in a chord spectrum including the following 21 partials:

This set of 21 partials overlaps with 46% of the full set of 45 partials present in the full harmonic series based on the GCD of 100 Hz, leading to a harmonic similarity score of 0.46 (21/45). The overlapping partials are highlighted in bold in the full harmonic series above.