It is well established that musical sounds comprising multiple partials with frequencies approximately in the ratio of small integers give rise to a strong sensation of pitch even if the lowest or fundamental partial is missing—the so-called virtual pitch effect. Experiments on thirty test subjects demonstrate that this virtual pitch is shifted significantly by changes in the spacing of the constituent partials. The experiments measured pitch by comparison of sounds of similar timbre and were automated so that they could be performed remotely across the Internet. Analysis of the test sounds used shows that the pitch shifts are not predicted by Terhardt’s classic model of virtual pitch. The test sounds used were modelled on the sounds of church bells, but a further experiment on seventeen test subjects showed that changes in partial amplitude only had a minor effect on the pitch shifts observed, and that a pitch shift was still observed when two of the lowest frequency partials were removed, so that the effects reported are of general interest.
Pitch is the characteristic that allows a set of sounds to be ordered in a musical scale. Sometimes, the pitch of a sound corresponds to a single frequency component present in the sound (so-called spectral pitch). However, a sound comprising a harmonic series with partial frequencies 2f, 3f, 4f etc. gives a strong perception of pitch at frequency f even if no energy at that frequency is present in the sound. This effect is variously known as the missing fundamental, residue pitch or virtual pitch. Terhardt [
The overview by Plack and Oxenham [
Experiments documented in [
Shifts in the frequencies of the partials of a sound from exact harmonic ratios give rise to a shift in the virtual pitch. Experiments reported by Patterson and Wightman in [
The term pitch shift is used for more than one effect in the literature. In particular in Terhardt’s work on virtual pitch [
The textbook definition of pitch given above mandates comparison by listeners as the only valid method of pitch estimation. Experiments show a variation in results between listeners―there is no absolute value for the pitch of a sound independent of a particular listener’s ability to judge it. Various methods for pitch estimation are documented in the literature:
・ comparison of complex tones against sine tones, either from tuning forks or electronically generated;
・ the method of post vocalisation described by Terhardt and Seewann [
・ comparison of complex tones against test tones with similar timbre.
This last method, used effectively by Järveläinen et al. [
Church bells make an interesting case study for investigations of virtual pitch because perception of their pitch is dominated by virtual pitch effects. Terhardt documents investigations into virtual pitch in bells in [
Due to the considerable importance of the nominal to the perceived pitch of bells, this partial is taken as the reference when considering the relationship
Ratios between partials | |||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Freq. (Hz) | Hum | Prime | Tierce | Quint | Nom. | p1 | p2 | p3 | Superquint | Oct. Nom. | I-7 | I-8 | |
Hum | 201.9 | 1.00 | |||||||||||
Prime | 323.7 | 1.60 | 1.00 | ||||||||||
Tierce | 424.2 | 2.10 | 1.31 | 1.00 | |||||||||
Quint | 562.8 | 2.79 | 1.74 | 1.33 | 1.00 | ||||||||
Nominal | 723.4 | 3.58 | 2.23 | 1.71 | 1.29 | 1.00 | |||||||
p1 | 862.1 | 4.27 | 2.66 | 2.03 | 1.53 | 1.19 | 1.00 | ||||||
p2 | 882.2 | 4.37 | 2.73 | 2.08 | 1.57 | 1.22 | 1.02 | 1.00 | |||||
p3 | 902.0 | 4.47 | 2.79 | 2.13 | 1.60 | 1.25 | 1.05 | 1.02 | 1.00 | ||||
Superquint | 1091.4 | 5.41 | 3.37 | 2.57 | 1.94 | 1.51 | 1.27 | 1.24 | 1.21 | 1.00 | |||
Oct. Nominal | 1507.3 | 7.47 | 4.66 | 3.55 | 2.68 | 2.08 | 1.75 | 1.71 | 1.67 | 1.38 | 1.00 | ||
I-7 | 1959.3 | 9.70 | 6.05 | 4.62 | 3.48 | 2.71 | 2.27 | 2.22 | 2.17 | 1.80 | 1.30 | 1.00 | |
I-8 | 2443.2 | 12.10 | 7.55 | 5.76 | 4.34 | 3.38 | 2.83 | 2.77 | 2.71 | 2.24 | 1.62 | 1.25 | 1.00 |
I-9 | 2948.3 | 14.60 | 9.11 | 6.95 | 5.24 | 4.08 | 3.42 | 3.34 | 3.27 | 2.70 | 1.96 | 1.50 | 1.21 |
I-10 | 3467.6 | 17.17 | 10.71 | 8.17 | 6.16 | 4.79 | 4.02 | 3.93 | 3.84 | 3.18 | 2.30 | 1.77 | 1.42 |
I-11 | 4000.8 | 19.82 | 12.36 | 9.43 | 7.11 | 5.53 | 4.64 | 4.54 | 4.44 | 3.67 | 2.65 | 2.04 | 1.64 |
I-12 | 4536.5 | 22.47 | 14.01 | 10.69 | 8.06 | 6.27 | 5.26 | 5.14 | 5.03 | 4.16 | 3.01 | 2.32 | 1.86 |
I-13 | 5077.1 | 25.15 | 15.68 | 11.97 | 9.02 | 7.02 | 5.89 | 5.76 | 5.63 | 4.65 | 3.37 | 2.59 | 2.08 |
between the frequencies of the various partials. The relationship between partial frequencies and the nominal frequency can be expressed either as a simple ratio or as a difference in cents (hundredths of a semitone) calculated using the following expression:
cents = 1200 ln ( 2 ) ln ( f p f n ) (1)
where fp is the partial frequency and fn is the nominal frequency.
In bell sounds, the frequencies of the upper partials are either compressed together or stretched apart relative to the nominal, according to the shape or thickness of the bell. Informal observations suggest that these changes alter the perceived pitch of the bell by one quarter or more of a semitone. A simple relationship which allows the intervals of all the upper partials to the nominal to be represented by the interval between the octave nominal and the nominal is documented in [
The intervals of the other named partials to the nominal do not display this simple relationship.
An experiment was designed to quantify the shifts in perceived pitch due to changes in various partials. The experiment involved comparison of the pitches of test tones against reference tones with similar timbre. Both test and reference tones had partial frequencies and amplitudes typical of bells, derived from the average of measurements on a number of bell recordings. Nine test sets in all were constructed, with nominal frequencies spaced at 1/3 octave intervals (i.e. increasing in frequency by a factor of 21/3 each time) from 315 Hz to 2000 Hz. To explore the effect of all the partial frequencies on the perceived pitch of a bell sound, in principle it is necessary to vary them all in relation to the nominal. However, because the intervals of all the upper partials to the nominal are related in bells, they must move together. This considerably simplifies the design of the experiment.
The independent variables chosen for the experiment, following some initial trials, were the intervals to the nominal of prime, tierce, quint, p1 to p3, and octave nominal (standing proxy for the intervals of all the upper partials). Intervals of each partial to the nominal occur over a range of values in bells seen in practice, and low, typical and high values of the interval were established for each partial. Values for the low and high values were chosen (based on examination of about 2000 bell recordings) to be extreme but realistic. To further simplify the experiment, all three of p1, p2 and p3 were assigned low, average or high values together even though, unlike the upper partials, no relationship exists between them. This gave five independent variables in total to be explored.
The amplitudes and intervals to the nominal (expressed as a frequency ratio) for the partials used in the tests are shown in
A simple amplitude envelope for the partials was used, based again on measurements on partials from a number of bell recordings. The envelope used was A/(1 + dt) where A is the initial amplitude (column labelled Amplitude in
Partial | Amplitude | Freq. ratio to nominal (low) | Freq. ratio to nominal (typical) | Freq. ratio to nominal (high) |
---|---|---|---|---|
Hum | 3.14 | 0.25 | ||
Prime | 4.02 | 0.45 | 0.50 | 0.50 |
Tierce | 6.78 | 0.59 | 0.60 | 0.63 |
Quint | 0.82 | 0.71 | 0.77 | 0.84 |
Nominal | 10 | 1.00 | ||
p1 | 1.25 | 1.12 | 1.19 | 1.26 |
p2 | 2.43 | 1.19 | 1.28 | 1.37 |
p3 | 1.83 | 1.28 | 1.36 | 1.43 |
Superquint | 6.88 | 1.44 | 1.48 | 1.51 |
Oct. Nominal | 5.15 | 1.93 | 2.01 | 2.09 |
I-7 | 3.71 | 2.45 | 2.59 | 2.74 |
I-8 | 2.39 | 3.00 | 3.21 | 3.43 |
I-9 | 1.75 | 3.56 | 3.85 | 4.16 |
I-10 | 1.13 | 4.13 | 4.51 | 4.92 |
I-11 | 0.76 | 4.71 | 5.19 | 5.71 |
Each of the nine test sets comprised 16 test sounds. All the test sounds in each set had the same nominal frequency, and the intervals of other partials were varied relative to this fixed nominal. The number of test sounds in each set was chosen to be 16 because this number of tests could be performed by each test subject in a reasonable period of time (no subject took longer than 45 minutes to complete a test set), while exploring the effect of four independent variables, each at either a high or low level; four variables each with two levels gives 16 combinations in total. To allow the effect of five independent variables to be explored, in different test sets either the quint or p1, p2 and p3 was varied, with the other held at the average value. These two variables were treated in this way because, although experience suggested that they did not affect virtual pitch, for completeness it was desired to explore their contribution.
The scheme used to explore the effect on virtual pitch of the five independent variables across the nine test sets is shown in
The sixteen test sounds, using the case of nominal = 1000 Hz as an example, involved the combinations shown in
To allow delivery of the tests across the internet, pre-created reference sounds were used in the tests. For each individual test, the subject was asked to select which of 16 reference sounds was closest in pitch to the test sound. The reference sounds for all 16 tests in a test set were the same. The reference sounds were built from the same partials as the test sounds, with the same amplitudes, but with their intervals to the nominal set to the typical values as shown in
Test nominal freq. (Hz) | Prime | Tierce | Quint | p1, p2, p3 | Upper partials |
---|---|---|---|---|---|
315 | low or high | low or high | typical | typical | low or high |
397 | low or high | low or high | low or high | typical | low or high |
500 | low or high | low or high | typical | low or high | low or high |
630 | low or high | low or high | low or high | typical | low or high |
794 | low or high | low or high | typical | low or high | low or high |
1000 | low or high | low or high | low or high | typical | low or high |
1260 | low or high | low or high | typical | low or high | low or high |
1587 | low or high | low or high | low or high | typical | low or high |
2000 | low or high | low or high | typical | low or high | low or high |
Prime | Tierce | Quint | p1, p2, p3 | Upper partials | |
---|---|---|---|---|---|
test 1 | low | low | low | typical | low |
test 2 | low | low | high | typical | low |
test 3 | high | low | low | typical | low |
test 4 | high | low | high | typical | low |
test 5 | low | high | low | typical | low |
test 6 | low | high | high | typical | low |
test 7 | high | high | low | typical | low |
test 8 | high | high | high | typical | low |
test 9 | low | low | low | typical | high |
test 10 | low | low | high | typical | high |
test 11 | high | low | low | typical | high |
test 12 | high | low | high | typical | high |
test 13 | low | high | low | typical | high |
test 14 | low | high | high | typical | high |
test 15 | high | high | low | typical | high |
test 16 | high | high | high | typical | high |
established during preliminary trials to ensure that the range of pitches experienced by subjects lay within the range of reference sounds for that test. In the trials it was found that the pitch shifts, and hence the spread required in the reference nominals, was rather greater at lower nominal frequencies. The details of the reference sounds for each test set appear in
The parameters for the test and reference sounds of all nine test sets were calculated in a spreadsheet and exported as csv files. These files of data were imported into bespoke software which generated each batch of test and reference sounds via additive synthesis of cosine waves.
The generated sounds were 16-bit with a sampling rate of 22,050 samples per second. The duration of all the sounds was set at 0.25 s. This short duration was intended to encourage pitch perception via virtual pitches rather than identification of individual partials (and is also typical of the spacing between bell sounds in English change ringing). During preliminary extensive trials of the tests, some tests with sounds 1 s long were conducted; the results were not significantly different from those at 0.25 s. In [
Conduct of the tests and reporting of the results was automated in software so
Test nominal freq. (Hz) | Lowest reference nominal freq. (Hz) | Highest reference nominal freq. (Hz) | Interval between lowest and highest reference nominal freq. (cents) |
---|---|---|---|
315 | 292.19 | 347.48 | 300 |
397 | 376.66 | 421.97 | 197 |
500 | 476.80 | 522.14 | 157 |
630 | 609.62 | 650.76 | 112.5 |
794 | 768.33 | 819.91 | 112.5 |
1000 | 968.03 | 1033.03 | 112.5 |
1260 | 1219.64 | 1301.53 | 112.5 |
1587 | 1553.37 | 1622.16 | 75 |
2000 | 1957.14 | 2043.79 | 75 |
that the tests could be carried out remotely across the internet. Test and reference sounds were played by the subject clicking on links on a webpage, using whatever sound reproduction equipment was attached to their PC. Each of the nine test sets was conducted separately (and not all subjects completed all nine sets). For each of the sixteen tests within a set, the subject was presented with links for the test sound and for the sixteen reference sounds, and asked to select which reference sound was closest in pitch (neither higher nor lower) to the test sound. Within each set of sixteen, the tests were displayed in a different random order on each test iteration. The subject was able to spend as long as required on each test but could not complete the set of sixteen tests without responding to them all. The result reported for each test was the nominal of the matched reference sound, which was taken to stand for the pitch the user had perceived. This was converted into a pitch shift in cents using Equation (1).
Extensive trials of the approach employed were carried out to validate the test procedure and documentation, analysis of the results, the effect on test results of quality of sound reproduction equipment and musical experience of the subject, and ranges of the various test parameters. The experiment design was proven to be robust against all these criteria, and in particular the quality of sound reproduction equipment was eliminated as a concern.
The test results were analysed using the standard analysis of variance (ANOVA) statistical technique [
30 test subjects (including one of the authors) completed 116 test sets, a total of 1856 individual tests. Subjects were recruited through bell-ringing newsgroups with a request for people with “a PC or Mac with sound, a broadband connection and a musical ear”. The statistical analysis of the results showed a small number of outliers (0.54% of the total) but all results were included in the analysis as the influence of the outliers on the test results was minimal.
The results were analysed using the standard ANOVA technique. Because the experiment design was orthogonal (i.e. all combinations of the independent variables were present; see
As an example of the results,
The statistical significance of each effect derived from the experiment results is expressed as a P-value. The P-value is the probability that a pitch shift of this size could have been observed (through chance variation) if in fact the independent variable did not give rise to a shift. A P-value of less than 0.05 indicates that the pitch shift is statistically significant at the 95% level.
・ the pitch shifts due to changes in the upper partials are both highly statistically significant, and musically significant. Pitch shifts range from 1/8 of a semitone at the highest frequency, through half a semitone in the middle range of frequencies, to well over a semitone at the lowest frequency;
Test nominal freq. (Hz) | Prime | Tierce | Quint | Middle partials | Upper partials | |||||
---|---|---|---|---|---|---|---|---|---|---|
Effect (cents) | P-value | Effect (cents) | P-value | Effect (cents) | P-value | Effect (cents) | P-value | Effect (cents) | P-value | |
315 | 12.5 | 0.047 | 7.1 | 0.256 | 134.0 | 6.2 × 10−53 | ||||
397 | 10.1 | 0.057 | 9.6 | 0.070 | 1.1 | 0.841 | 71.4 | 2.0 × 10−27 | ||
500 | 8.2 | 0.022 | 7.7 | 0.032 | 0.5 | 0.889 | 71.5 | 2.0 × 10−43 | ||
630 | 6.6 | 0.019 | 9.3 | 0.001 | 0.7 | 0.793 | 43.2 | 2.0 × 10−31 | ||
794 | 6.3 | 0.001 | 8.2 | 1.4 × 10−5 | 1.8 | 0.330 | 42.2 | 1.1 × 10−56 | ||
1000 | 6.2 | 3.9 × 10−6 | 8.3 | 1.0 × 10−9 | −0.9 | 0.499 | 36.0 | 2.4 × 10−92 | ||
1260 | 4.9 | 0.007 | 11.0 | 5.6 × 10−9 | 0.3 | 0.853 | 31.6 | 7.6 × 10−43 | ||
1587 | 6.3 | 0.001 | 5.9 | 0.001 | 0.3 | 0.849 | 18.8 | 1.2 × 10−18 | ||
2000 | 0.4 | 0.806 | 8.1 | 1.3 × 10−6 | 0.1 | 0.972 | 12.8 | 3.9 × 10−13 |
・ pitch shifts due to the quint and the middle partials are neither statistically nor musically significant at any frequency;
・ shifts due to prime and tierce are statistically significant at most frequencies, but musically quite small, generally a small fraction of the shifts due to the upper partials.
28 of the 30 test subjects gave an account of their musical experience, which ranged from near-professional experience to no experience at all. There was no correspondence between musical experience (or lack of it) and the size of the pitch shifts experienced by the test subjects.
As explained in Section 1.1, in bell sounds the spacing of the upper partials is uniquely parameterised by the interval of the octave nominal relative to the nominal. It is therefore convenient to show the upper partial pitch shift (in cents) for a one-cent shift in the octave nominal. The high and low frequencies of the octave nominal given in
The dotted line is a regression assuming a simple log relationship (specifically ln(pitch shift) = −1.10012 ln(fn) + 6.12396). No theoretical justification is proposed for this relationship; it just provides convenient shorthand allowing pitch shifts at an arbitrary frequency to be estimated. It is hypothesised that the very large pitch shifts at low nominal frequencies arise because, at this end of the spectrum, more subjects are hearing the pitch based on partial I-7 rather than the pitch based on the nominal.
A subsidiary experiment was conducted to explore the linearity or otherwise of the relationship between upper partial spacing and pitch shift. Test sounds were generated with the upper partial spacing in 16 steps from the low to the high values in
shows the results of eight test runs by a single subject, together with a linear regression on the results. The interval of the octave nominal to the nominal is, as ever, used as proxy for the spacing of all the upper partials. The average pitch shift between the case with upper partials at their low and high values is 23.0 cents. There is nothing in these results to suggest that the pitch shift effect due to upper partial spacing is not linear over the range of spacings used in the experiments.
Additional work also documented in [
A second experiment was constructed to investigate the effect of partial amplitude on the pitch shifts caused by changes in the upper partials. In this experiment, the amplitudes of superquint and octave nominal were varied from zero to 100% of the amplitudes in the previous experiment, while at the same time the frequencies of all the upper partials took on either the low or high values as defined in
To take advantage of the experimental procedure developed and proved in the previous experiment, this experiment followed a very similar approach and used the same software for delivery and execution.
The superquint and octave nominal amplitudes each took on four values; 0%, 33%, 67% and 100% of the amplitudes listed in
Test set 1 | Test set 2 | |||||||
---|---|---|---|---|---|---|---|---|
Superquint amplitude | Oct. Nom. amplitude | Upper partials | Superquint amplitude | Oct. Nom. amplitude | Upper partials | |||
test 1 | 0% | 0% | high | test 17 | 0% | 0% | low | |
test 2 | 0% | 33% | low | test 18 | 0% | 33% | high | |
test 3 | 0% | 67% | low | test 19 | 0% | 67% | high | |
test 4 | 0% | 100% | high | test 20 | 0% | 100% | low | |
test 5 | 33% | 0% | low | test 21 | 33% | 0% | high | |
test 6 | 33% | 33% | high | test 22 | 33% | 33% | low | |
test 7 | 33% | 67% | high | test 23 | 33% | 67% | low | |
test 8 | 33% | 100% | low | test 24 | 33% | 100% | high | |
test 9 | 67% | 0% | low | test 25 | 67% | 0% | high | |
test 10 | 67% | 33% | high | test 26 | 67% | 33% | low | |
test 11 | 67% | 67% | high | test 27 | 67% | 67% | low | |
test 12 | 67% | 100% | low | test 28 | 67% | 100% | high | |
test 13 | 100% | 0% | high | test 29 | 100% | 0% | low | |
test 14 | 100% | 33% | low | test 30 | 100% | 33% | high | |
test 15 | 100% | 67% | low | test 31 | 100% | 67% | high | |
test 16 | 100% | 100% | high | test 32 | 100% | 100% | low |
The arrangement of independent variables in each test set allowed statistically valid results to be derived if the subject only completed one of the two sets of tests. The reference sounds used were identical to those used in the previous experiment. All other aspects of the design and conduct of this experiment were identical to those previously described.
Subjects were again recruited through bell-ringing newsgroups. In total, 17 test subjects undertook the tests, one of whom was one of the authors of this paper. One subject only completed one of the two sets of tests, so the average results of the remaining subjects were used to provide the missing results. In total, 528 individual tests were completed.
As in the previous experiment, all subjects experienced a shift in pitch due to the change in the upper partials. An analysis of variance showed that the pitch shift due to change in upper partials averaged across all test subjects and partial amplitudes was 15.8 cents with a P-value of 2.7 × 10−74. As in the previous experiment, the shift was both highly statistically significant and musically significant.
The average values across all test subjects of the perceived pitch shift due to upper partial change were plotted against both superquint and octave nominal amplitude. The perceived pitch shifts at different superquint amplitudes are averaged across all octave nominal amplitudes, and vice versa. The plots appear as
These results suggest that pitch shifts due to changes in upper partials are relatively insensitive to the amplitude of the individual partials. The pitch shift averaged across all subjects with both superquint and octave nominal amplitudes at their maximum value is 21.1 cents. The pitch shift averaged across all subjects with both superquint and octave nominal absent is 8.8 cents. The shift with par-
tials at their maximum amplitude from this experiment is comparable with that shown in
The complexity of the partial structure in bells obscures the simplicity of the effect being experienced in this experiment. Using the terminology of the introduction, the set of upper partials from the nominal upwards form an approximate harmonic series with partial frequencies close to 2f, 3f, 4f, 5f, 6f etc. The shift in perceived pitch due to changes in the spacing of these partials survives even if those at 3f and 4f are removed from the sound. These results imply that all the partials in a harmonic series, not just the lowest in frequency, contribute to the virtual pitch effect.
Terhardt in [
The Terhardt algorithm has been implemented in software (C program ptp2svp) by Terhardt and Parncutt [
For the test sounds with nominal frequencies of 500 Hz and above, the algorithm predicted a single dominant pitch related to the nominal frequency. The effect on pitch of changes in individual partials was established by calculating contrasts and effects in the same manner as used in analysing the first experiment in this paper. The results obtained (quoted as pitch shifts in cents) are presented in
Comparison with the experimental results given in
The algorithmic results for test tones at nominal frequencies of 315 Hz and 397 Hz show multiple pitches with roughly equal perceived weight. This is in line with experience; due to the effect of the dominance region for virtual pitch, at lower nominal frequencies the higher harmonic series shown in
The pitch shifts seen in the experiment in Section 2 are an amalgamation of the various virtual pitches as predicted by the Terhardt algorithm, across the set
Nominal freq. (Hz) | Prime effect (cents) | Tierce effect (cents) | Quint effect (cents) | p1, p2, p3 effect (cents) | Upper partial effect(cents) |
---|---|---|---|---|---|
500 | 0.455 | 0.065 | 0.000 | 0.000 | |
630 | 0.450 | 0.040 | 0.000 | 0.000 | |
794 | 0.395 | 0.015 | 0.000 | 0.000 | |
1000 | 0.300 | 0.010 | 0.000 | 0.000 | |
1260 | 0.195 | 0.005 | 0.000 | 0.000 | |
1587 | 0.110 | 0.000 | 0.000 | 0.000 | |
2000 | 0.050 | 0.000 | 0.000 | 0.000 |
Predicted pitch (315 Hz nominal) | Prime effect (cents) | Tierce effect (cents) | Upper partial effect (cents) | Predicted pitch (397 Hz nominal) | Prime effect (cents) | Tierce effect (cents) | Upper partial effect (cents) | |
---|---|---|---|---|---|---|---|---|
Nominal/4 | 0.356 | 0.097 | 0.001 | Nominal/4 | 0.424 | 0.085 | 0.000 | |
Nominal/2 | 0.123 | 0.203 | 0.048 | Nominal/2 | 0.131 | 0.251 | 0.029 | |
Nominal | 0.245 | 0.410 | 0.095 | Nominal | 0.264 | 0.504 | 0.056 | |
I-7/4 | 0.330 | 5.225 | 23.685 |
of test subjects. However, for both the 315 Hz and 397 Hz test sounds, the large upper partial effect seen in the experiment results is not predicted by the algorithm; nor are the significant prime and tierce effects.
The Terhardt algorithm was also applied to the test sounds used in the subsidiary experiment, the results of which are shown in
On the basis of these trials one can confidently say that the pitch shifts seen in practice are not predicted by the Terhardt algorithm.
The two experiments reported in this paper have shown that the perceived pitch of a complex tone due to the virtual pitch effect is shifted from the expected value (of half the lowest frequency in the harmonic series) if the frequencies of the partials in the harmonic series are stretched or compressed. The second experiment also shows that the size of this shift is only moderately affected by the amplitude of the next two higher frequency partials in the series. A pitch shift is still observed if these two partials are removed completely from the sound. The pitch shifts are large enough (at considerable fractions of a semitone) to be musically significant.
These findings have a number of consequences:
・ although the complex tones used in the experiments were modelled on the sound of church bells, the relative insensitivity of pitch shifts to partial amplitude means that the principles, if not the exact numerical results, are of general applicability;
・ many partials in the harmonic series, not just the lowest in frequency, contribute to the formation of the virtual pitch in the ear;
・ the failure of Terhardt’s well-established model of virtual pitch to predict the pitch shifts observed suggests that further investigation into pitch formation mechanisms in the ear is desirable.
Hibbert, W.A., Taherzadeh, S. and Sharp, D.B. (2017) Virtual Pitch and Pitch Shifts in Church Bells. Open Journal of Acoustics, 7, 52-68. https://doi.org/10.4236/oja.2017.73006