Thresholds for hearing mistuned partials as separate tones in harmonic complexes

When a low harmonic in a harmonic complex tone is mistuned from its harmonic value by a sufficient amount it is heard as a separate tone, standing out from the complex as a whole. This experiment estimated the degree of mistuning required for this phenomenon to occur, for complex tones with 10 or 12 equal-amplitude components (60 dB SPL per component). On each trial the subject was presented with a complex tone which either had all its partials at harmonic frequencies or had one partial mistuned from its harmonic frequency. The subject had to indicate whether he heard a single complex tone with one pitch or a complex tone plus a pure tone which did not "belong" to the complex. An adaptive procedure was used to track the degree of mistuning required to achieve a d' value of 1. Threshold was determined for each ot the first six harmonics of each complex tone. In one set of conditions stimulus duration was held constant at 410 ms, and the fundamental frequency was either 100, 200, or 400 Hz. For most conditions the thresholds fell between 1% and 3% of the harmonic frequency, depending on the subject. However, thresholds tended to be greater for the first two harmonics of the 100-Hz fundamental and, for some subjects, thresholds increased for the fifth and sixth harmonics. In a second set of conditions fundamental frequency was held constant at 200 Hz, and the duration was either 50, 110, 410, or 1610 ms. Thresholds increased by a factor of 3-5 as duration was decreased from 1610 ms to 50 ms. The results are discussed in terms of a hypothetical harmonic sieve and mechanisms for the formation of perceptual streams.     

Grouping in pitch perception: effects of onset asynchrony and ear of presentation of a mistuned component.

Three experiments investigated how the onset asynchrony and ear of presentation of a single mistuned frequency component influence its contribution to the pitch of an otherwise harmonic complex tone. Subjects matched the pitch of the target complex by adjusting the pitch of a second similar but strictly periodic complex tone. When the mistuned component (the 4th harmonic of a 155 Hz fundamental) started 160 ms or more before the remaining harmonics but stopped simultaneously with them, it made a reduced contribution to the pitch of the complex. It made no contribution if it started more than 300 ms before. Pitch shifts and their reduction with onset time were larger for short (90 ms) sounds than for long (410 ms). Pitch shifts were slightly larger when the mistuned component was presented to the same ear as the remaining 11 in-tune harmonics than to the opposite ear. Adding a "captor" complex tone with a fundamental of 200 Hz and a missing 3rd harmonic to the contralateral ear did not augment the effect of onset time, even though the captor was synchronous with the mistuned harmonic, the mistuned component was equal in frequency to the missing 3rd harmonic of the captor complex tone and it was played to the same ear as the captor. The results show that a difference in onset time can prevent a resolved frequency component from contributing to the pitch of a complex tone even though it is present throughout that complex tone.     

Evidence against an effect of grouping by spectral regularity on the perception of virtual pitch

Two experiments investigated the role of the regularity of the frequency spacing of harmonics, as a separate factor from harmonicity, on the perception of the virtual pitch of a harmonic series. The first experiment compared the shifts produced by mistuning the 3rd, 4th, and 5th harmonics in the pitch of two harmonic series: the odd-H and the all-H tones. The odd-H tone contained odd harmonics 1 to 11, plus the 4th harmonic; the all-H tone contained harmonics 1 to 12. Both tones had a fundamental frequency of 155 Hz. Pitch shifts produced by mistuning the 3rd harmonic, but not the 4th and 5th harmonics, were found to be significantly larger for the odd-H tone than for the all-H tone. This finding was consistent with the idea that grouping by spectral regularity affects pitch perception since an odd harmonic made a larger contribution than an adjacent even harmonic to the pitch of the odd-H tone. However, an alternative explanation was that the 3rd mistuned harmonic produced larger pitch shifts within the odd-H tone than the 4th mistuned harmonic because of differences in the partial masking of these harmonics by adjacent harmonics. The second experiment tested these explanations by measuring pitch shifts for a modified all-H tone in which each mistuned odd harmonic was tested in the presence of the 4th harmonic, but in the absence of its other even-numbered neighbor. The results showed that, for all mistuned harmonics, pitch shifts for the modified all-H tone were not significantly different from those for the odd-H tone. These findings suggest that the harmonic relations among frequency components, rather than the regularity of their frequency spacing, is the primary factor for the perception of the virtual pitch of complex sounds.     

Processing simultaneous auditory objects: infants' ability to detect mistuning in harmonic complexes

The ability to separate simultaneous auditory objects is crucial to infant auditory development. Music in particular relies on the ability to separate musical notes, chords, and melodic lines. Little research addresses how infants process simultaneous sounds. The present study used a conditioned head-turn procedure to examine whether 6-month-old infants are able to discriminate a complex tone (240 Hz, 500 ms, six harmonics in random phase with a 6 dB roll-off per octave) from a version with the third harmonic mistuned. Adults perceive such stimuli as containing two auditory objects, one with the pitch of the mistuned harmonic and the other with pitch corresponding to the fundamental of the complex tone. Adult thresholds were between 1% and 2% mistuning. Infants performed above chance levels for 8%, 6%, and 4% mistunings, with no significant difference between conditions. However, performance was not significantly different from chance for 2% mistuning and significantly worse for 2% compared to all larger mistunings. These results indicate that 6-month-old infants are sensitive to violations of harmonic structure and suggest that they are able to separate two simultaneously sounding objects.     

Effect of duration on the frequency discrimination of individual partials in a complex tone and on the discrimination of fundamental frequency

Thresholds for the discrimination of fundamental frequency (FODLs) and frequency difference limens (FDLs) for individual partials within a complex tone (F0=250 Hz, harmonics 1-7) were measured for stimulus durations of 200, 50, and 16 ms. The FDLs increased with decreasing duration. Although the results differed across subjects, the effect of duration generally decreased as the harmonic number increased from 1 to 4, then increased as the harmonic number increased to 6, and finally decreased for the seventh harmonic. For each duration, FODLs were smaller than the smallest FDL for any individual harmonic, indicating that information is combined across harmonics in the discrimination of FO. FODLs predicted from the FDLs corresponded well with observed FODLs for the 200- and 16-ms durations but were significantly larger than observed FODLs for the 50-ms duration. A supplementary pitch-matching experiment using two subjects indicated that the contribution of the seventh harmonic to the pitch of the 16-ms complex tone was smaller than would be predicted from the FDL for that harmonic. The results are consistent with the idea that the dominant region shifts upward with decreasing duration, but that the weight assigned to individual harmonics is not always adjusted in an optimal way.     

The effect of inharmonic partials on pitch of piano tones

Piano tones have partials whose frequencies are sharp relative to harmonic values. A listening test was conducted to determine the effect of inharmonicity on pitch for piano tones in the lowest three octaves of a piano. Nine real tones from the lowest three octaves of a piano were analyzed to obtain frequencies, relative amplitudes, and decay rates of their partials. Synthetic inharmonic tones were produced from these results. Synthetic harmonic tones, each with a twelfth of a semitone increase in the fundamental, were also produced. A jury of 21 listeners matched the pitch of each synthetic inharmonic tone to one of the synthetic harmonic tones. The effect of the inharmonicity on pitch was determined from an average of the listeners' results. For the nine synthetic piano tones studied, pitch increase ranged from approximately two and a half semitones at low fundamental frequencies to an eighth of a semitone at higher fundamental frequencies.     

The effect of modulation rate on the detection of frequency modulation and mistuning of complex tones

Experiment 1 measured frequency modulation detection thresholds (FMTs) for harmonic complex tones as a function of modulation rate. Six complexes were used, with fundamental frequencies (F0s) of either 88 or 250 Hz, bandpass filtered into a LOW (125-625 Hz), MID (1375-1875 Hz) or HIGH (3900-5400 Hz) frequency region. The FMTs were about an order of magnitude greater for the three complexes whose harmonics were unresolved by the peripheral auditory system (F0 = 88 Hz in the MID region and both F0s in the HIGH region) than for the other three complexes, which contained some resolved harmonics. Thresholds increased with increases in FM rate above 2 Hz for all conditions. The increase was larger when the F0 was 88 Hz than when it was 250 Hz, and was also larger in the LOW than in the MID and HIGH regions. Experiment 2 measured thresholds for detecting mistuning produced by modulating the F0s of two simultaneously presented complexes out of phase by 180 degrees. The size of the resulting mistuning oscillates at a rate equal to the rate of FM applied to the two carriers. At low FM rates, thresholds were lowest when the harmonics were either resolved for both complexes or unresolved for both complexes, and highest when resolvability differed across complexes. For pairs of complexes with resolved harmonics, mistuning thresholds increased dramatically as the FM rate was increased above 2-5 Hz, in a way which could not be accounted for by the effect of modulation rate on the FMTs for the individual complexes. A third experiment, in which listeners detected constant ("static") mistuning between pairs of frequency-modulated complexes, provided evidence that this deterioration was due the harmonics in one of the two "resolved" complexes becoming unresolved at high FM rates, when analyzed over some finite time window. It is concluded that the detection of time-varying mistuning between groups of harmonics is limited by factors that are not apparent in FM detection data.     

Frequency discrimination of complex tones with overlapping and non-overlapping harmonics

These experiments address the following issues. (1) When two complex tones contain different harmonics, do the differences in timbre between them impair the ability to discriminate the pitches of the tones? (2) When two complex tones have only a single component in common, and that component is the most discriminable component in each tone, is the frequency discrimination of the component affected by differences in residue pitch between the two tones? (3) How good is the pitch discrimination of complex tones with no common components when each tone contains multiple harmonics, so as to avoid ambiguity of pitch? (4) Is the pitch discrimination of complex tones with common harmonics impaired by shifting the component frequencies to nonharmonic values? In all experiments, frequency difference limens (DLCs) were measured for multiple-component complex tones, using an adaptive two-interval, two-alternative, forced-choice task. Three highly trained subjects were used. The main conclusions are as follows. (1) When two tones have the first six harmonics in common, DLCs are larger when the upper harmonics are different than when the upper harmonics are in common or are absent. It appears that differences in timbre impair DLCs. (2) Discrimination of the frequency of a single common partial in two complex tones is worse when the two tones have different residue pitches than when they have the same residue pitch. (3) DLCs for complex tones with no common harmonics are generally larger than those for complex tones with common harmonics. For the former, large individual differences occur, probably because subjects are affected differently by differences in timbre. (4) DLCs for harmonic complex tones are smaller than DLCs for complex tones in which the components are mistuned from harmonic values. This can probably be attributed to the less distinct residue pitch of the inharmonic complexes, rather than to reduced discriminability of partials. Overall, the results support the idea that DLCs for complex tones with common harmonics depend on residue pitch comparisons, rather than on comparisons of the pitches of partials.     

Perception of the low pitch of frequency-shifted complexes

When all of the components in a harmonic complex tone are shifted in frequency by delta f, the pitch of the complex shifts roughly in proportion to delta f. For tones with a small number of components, the shift is usually somewhat larger than predicted from pitch theories, which has been attributed to the influence of combination tones [Smoorenburg, J. Acoust. Soc. Am. 48, 924-941 (1970)]. Experiment 1 assessed whether combination tones influence the pitch of complex tones with more than five harmonics, by using noise to mask the combination tones. The matching stimulus was a harmonic complex. Test complexes were bandpass filtered with passbands centered on harmonic numbers 5 (resolved), 11 (intermediate), or 16 (unresolved) and fundamental frequencies (FOs) were 100, 200, or 400 Hz. For the intermediate and unresolved conditions, the matching stimuli were filtered with the same passband to minimize differences in the excitation patterns of the test and matching stimuli. For the resolved condition, the matching stimulus had a passband centered above that of the test stimulus, to avoid common partials. For resolved and intermediate conditions, pitch shifts were observed that could generally be predicted from the frequencies of the partials. The shifts were unaffected by addition of noise to mask combination tones. For the unresolved condition, no pitch shift was observed, which suggests that pitch is not based on temporal fine structure for stimuli containing only high unresolved harmonics. Experiment 2 used three-component complexes resembling those of Schouten [J. Acoust. Soc. Am. 34, 1418-1424 (1962)]. Nominal harmonic numbers were 3, 4, 5 (resolved), 8, 9, 10 (intermediate), or 13, 14, 15 (unresolved) and F0s were 50, 100, 200, or 400 Hz. Clear shifts in the matches were found for all conditions, including unresolved. For the latter, subjects may have matched the "center of gravity" of the excitation patterns of the test and matching stimuli.     

Representation of concurrent acoustic objects in primary auditory cortex

Auditory scene analysis involves the simultaneous grouping and parsing of acoustic data into separate mental representations (i.e., objects). Over two experiments, we examined the sequence of neural processes underlying concurrent sound segregation by means of recording of human middle latency auditory evoked responses. Participants were presented with complex sounds comprising several harmonics, one of which could be mistuned such that it was not an integer multiple of the fundamental frequency. In both experiments, Na (approximately 22 ms) and Pa (approximately 32 ms) waves were reliably generated for all classes of stimuli. For stimuli with a fundamental frequency of 200 Hz, the mean Pa amplitude was significantly larger when the third harmonic was mistuned by 16% of its original value, relative to when it was tuned. The enhanced Pa amplitude was related to an increased likelihood in reporting the presence of concurrent auditory objects. Our results are consistent with a low-level stage of auditory scene analysis in which acoustic properties such as mistuning act as preattentive segregation cues that can subsequently lead to the perception of multiple auditory objects.     

Discriminating between coherent and incoherent frequency modulation of complex tones

A series of experiments measured the discrimination by human listeners of frequency-modulated complex tones which differed only in the coherence of frequency modulation (FM). For the coherently modulated tones all components were modulated by the same 5-Hz sinusoid, and by the same percentage of their starting frequencies, whereas for the incoherently modulated tones the modulation of one (target) component differed from that of the rest. When the 400-ms complex was composed of consecutive harmonics of a common fundamental, performance improved monotonically with increases in modulator delay, and was nearly perfect at the longest delays. When the complex was inharmonic, performance was near chance at all modular delays, both for component frequencies between 1500 and 2500 Hz, and for component frequencies between 400 and 800 Hz. It is argued that listeners detected incoherence in harmonic complexes by detecting the resulting mistuning of the target component. This conclusion was supported by the finding that listeners were usually at least as good at detecting a fixed mistuning of the center component of a harmonic complex as they were at detecting a modulator phase delay imposed on it. A final experiment, with a stimulus duration of 1 s and slower modulation rates, showed that listeners could detect incoherence for some inharmonic complexes. However, detection was worse than for harmonic complexes and was, it is argued, based on weak harmonicity cues. The results of all experiments point to the absence of an across-frequency mechanism specific to the detection of FM incoherence.     

Reduced contribution of a nonsimultaneous mistuned harmonic to residue pitch

Ciocca and Darwin [V. Ciocca and C. J. Darwin, J. Acoust. Soc. Am. 105, 2421-2430 (1999)] reported that the shift in residue pitch caused by mistuning a single harmonic (the fourth out of the first 12) was the same when the mistuned harmonic was presented after the remainder of the complex as when it was simultaneous, even though subjects were asked to ignore the pure-tone percept. The present study tried to replicate this result, and investigated the role of the presence of the nominally mistuned harmonic in the matching sound. Subjects adjusted a "matching" sound so that its pitch equaled that of a subsequent 90-ms complex tone (12 harmonics of a 155-Hz F0), whose mistuned (+/-3%) third harmonic was presented either simultaneously with or after the remaining harmonics. In experiment 1, the matching sound was a harmonic complex whose third harmonic was either present or absent. In experiments 2A and 2B, the target and matching sound had nonoverlapping spectra. Pitch shifts were reduced both when the mistuned component was nonsimultaneous, and when the third harmonic was absent in the matching sound. The results indicate a shorter than originally estimated time window for obligatory integration of nonsimultaneous components into a virtual pitch.     

The effect of hearing loss on the resolution of partials and fundamental frequency discrimination

The relationship between the ability to hear out partials in complex tones, discrimination of the fundamental frequency (F0) of complex tones, and frequency selectivity was examined for subjects with mild-to-moderate cochlear hearing loss. The ability to hear out partials was measured using a two-interval task. Each interval included a sinusoid followed by a complex tone; one complex contained a partial with the same frequency as the sinusoid, whereas in the other complex that partial was missing. Subjects had to indicate the interval in which the partial was present in the complex. The components in the complex were uniformly spaced on the ERB(N)-number scale. Performance was generally good for the two "edge" partials, but poorer for the inner partials. Performance for the latter improved with increasing spacing. F0 discrimination was measured for a bandpass-filtered complex tone containing low harmonics. The equivalent rectangular bandwidth (ERB) of the auditory filter was estimated using the notched-noise method for center frequencies of 0.5, 1, and 2 kHz. Significant correlations were found between the ability to hear out inner partials, F0 discrimination, and the ERB. The results support the idea that F0 discrimination of tones with low harmonics depends on the ability to resolve the harmonics.     

Effects of level and frequency on the audibility of partials in inharmonic complex tones

The effect of level and frequency on the audibility of partials was measured for complex tones with partials uniformly spaced on an equivalent rectangular bandwidth (ERB(N)) number scale. On each trial, subjects heard a sinusoidal "probe" followed by a complex tone. The probe was mistuned downwards or upwards (at random) by 4.5% from the frequency of one randomly selected partial in the complex. The subject indicated whether the probe was higher or lower in frequency than the nearest partial in the complex. The frequencies were roved from trial to trial, keeping frequency ratios fixed. In experiment 1, the level per partial, L, was 40 or 70 dB SPL and the mean frequency of the central partial, f(c), was 1201 Hz. Scores for the highest and lowest partials in the complexes were generally high for all spacings. Scores for the inner partials were close to chance at 0.75-ERB(N) spacing, and improved as the spacing was increased up to 2 ERB(N). For intermediate spacings, performance was better for the lower level used. In experiment 2, L was 70 dB SPL and f(c) was 3544 Hz. Performance worsened markedly for partial frequencies above 3544 Hz, consistent with a role of phase locking.     

Evidence for an analytic perception of multiharmonic sounds in the bat, Megaderma lyra, and its possible role for echo spectral analysis

For echolocation, the gleaning bat Megaderma lyra relies on short and broadband calls consisting of multiple harmonic components, each of which is downward frequency modulated. The harmonic components in M. lyra's calls have a relatively small frequency excursion and do not overlap spectrally. Broadband calls of other bat species, on the other hand, often consist of only a few harmonics which are modulated over broad and sometimes overlapping frequency ranges. A call consisting of narrow and nonoverlapping harmonic components may provide a less complete representation of target structure than a call which consists of broadly modulated components. However, a multiharmonic call may help the bats to perceive local spectral changes in the echo from shifts in the peak frequencies of single harmonics, and thereby to extract additional information about the target. To assess this hypothesis, the accuracy with which M. lyra can analyze frequency shifts of single partials in multiharmonic complex tones was investigated. A two-alternative, forced-choice behavioral task was used to measure M. lyra's frequency discrimination threshold for the third partial in complex tones whose spectral composition resembled that of the bat's sonar calls. The discrimination threshold for the third partial in a 21.5-kHz harmonic tone amounted to about 2% and was similar to the bat's pure-tone discrimination threshold at 64.5 kHz. Discrimination performance was essentially unaffected by random frequency changes of the other partials and by reducing stimulus duration from 50.5 to 1.5 ms. Both findings are in accordance with predictions made on the basis of the shape of M. Ivra's cochlear filters. The comparison between the observed frequency discrimination performance and a computational estimate of the expected frequency shift in the third harmonic of an echo reflected by a simple, two-front target showed that M. lyra's frequency resolution is sufficient for analyzing the target-specific information conveyed by shifts in the peak frequency of single echo components.     

Perception of the missing fundamental in nonhuman primates

In preparation for neurophysiological experiments aimed at mechanisms of pitch perception, four rhesus monkeys were trained to press a button when the fundamental frequencies (missing or present) of two complex tones in a tone pair matched. Both tones were based on a five-component harmonic series. Zero to three of the lowest components could be missing in the first tone, while the second (comparison) tone contained all five harmonics. The range of fundamentals tested varied from 200 to 600 Hz. Three monkeys learned to match tones missing their fundamentals to comparison harmonic complexes with the same pitch, whereas the fourth monkey required the physical presence of the fundamental. Consideration of several cues available to the monkeys suggests that the animals could perceive the missing fundamental.     

The integration of nonsimultaneous frequency components into a single virtual pitch

The integration of nonsimultaneous frequency components into a single virtual pitch was investigated by using a pitch matching task in which a mistuned 4th harmonic (mistuned component) produced pitch shifts in a harmonic series (12 equal-amplitude harmonics of a 155-Hz F0). In experiment 1, the mistuned component could either be simultaneous, stop as the target started (pre-target component), or start as the target stopped (post-target component). Pitch shifts produced by the pre-target components were significantly smaller than those obtained with simultaneous components; in the post-target condition, the size of pitch shifts did not decrease relative to the simultaneous condition. In experiment 2, a silent gap of 20, 40, 80, or 160 ms was introduced between the nonsimultaneous components and the target sound. In the pre-target condition, pitch shifts were reduced to zero for silent gaps of 80 ms or longer; by contrast, a gap of 160 ms was required to eliminate pitch shifts in the post-target condition. The third experiment tested the hypothesis that, when post-target components were presented, the processing of the pitch of the target tone started at the onset of the target, and ended at the gap duration at which pitch shifts decreased to zero. This hypothesis was confirmed by the finding that pitch shifts could not be observed when the target tone had a duration of 410 ms. Taken together, the results of these experiments show that nonsimultaneous components that occur after the onset of the target sound make a larger contribution to the virtual pitch of the target, and over a longer period, than components that precede the onset of the target sound.     

Asymmetry of masking between complex tones and noise: the role of temporal structure and peripheral compression

Thresholds for the detection of harmonic complex tones in noise were measured as a function of masker level. The rms level of the masker ranged from 40 to 70 dB SPL in 10-dB steps. The tones had a fundamental frequency (F0) of 62.5 or 250 Hz, and components were added in either cosine or random phase. The complex tones and the noise were bandpass filtered into the same frequency region, from the tenth harmonic up to 5 kHz. In a different condition, the roles of masker and signal were reversed, keeping all other parameters the same; subjects had to detect the noise in the presence of a harmonic tone masker. In both conditions, the masker was either gated synchronously with the 700-ms signal, or it started 400 ms before and stopped 200 ms after the signal. The results showed a large asymmetry in the effectiveness of masking between the tones and noise. Even though signal and masker had the same bandwidth, the noise was a more effective masker than the complex tone. The degree of asymmetry depended on F0, component phase, and the level of the masker. The maximum difference between masked thresholds for tone and noise was about 28 dB; this occurred when the F0 was 62.5 Hz, the components were in cosine phase, and the masker level was 70 dB SPL. In most conditions, the growth-of-masking functions had slopes close to 1 (on a dB versus dB scale). However, for the cosine-phase tone masker with an F0 of 62.5 Hz, a 10-dB increase in masker level led to an increase in masked threshold of the noise of only 3.7 dB, on average. We suggest that the results for this condition are strongly affected by the active mechanism in the cochlea.     

The perception of complex tones by a false killer whale (Pseudorca crassidens)

Complex tonal whistles are frequently produced by some odontocete species. However, no experimental evidence exists regarding the detection of complex tones or the discrimination of harmonic frequencies by a marine mammal. The objectives of this investigation were to examine the ability of a false killer whale to discriminate pure tones from complex tones and to determine the minimum intensity level of a harmonic tone required for the whale to make the discrimination. The study was conducted with a go/no-go modified staircase procedure. The different stimuli were complex tones with a fundamental frequency of 5 kHz with one to five harmonic frequencies. The results from this complex tone discrimination task demonstrated: (1) that the false killer whale was able to discriminate a 5 kHz pure tone from a complex tone with up to five harmonics, and (2) that discrimination thresholds or minimum intensity levels exist for each harmonic combination measured. These results indicate that both frequency level and harmonic content may have contributed to the false killer whale's discrimination of complex tones.     

Pitch shifts of mistuned partials: a time-domain model.

Mistuning one partial of a complex harmonic tone makes that partial easier to hear as a tone separate from the complex. At the same time, two pitch shifts may be observed. First, the low pitch of the complex is shifted in the direction of the mistuning, as if it were "pulled" by the partial. Second, the mistuning of the partial is perceptually exaggerated, as if the pitch of the partial were "pushed" away from the harmonic series defined by the complex. This paper shows how the latter effect can emerge within a hypothetical neural circuit. The circuit involves a gating neuron fed by three pathways, one direct and excitatory and the other two delayed and inhibitory. The neuron responds to any excitatory input spike unless it is accompanied by an inhibitory input spike on either delayed input, thus acting as a kind of "anticoincidence counter." The first delay is fixed and tuned to the period of the background harmonic complex. Its purpose is to weaken correlates of in-tune components and allow the mistuned partial to stand out. The second delay is variable and used to estimate the period of the mistuned partial, by searching for a minimum output as a function of delay. With an appropriate choice of parameters, the estimate is subject to shifts that are of the same sign as the mistuning and that peak at about 4% mistuning and decrease beyond, as observed experimentally.