Overtone-based pitch selection in hermit thrush song: Unexpected convergence with scale construction in human music

Recordings and Acoustic Analysis.

High-quality recordings of hermit thrush songs from 14 birds were acquired from three sources: the Borror Laboratory of Bioacoustics, Kevin J. Colver (Kevin J. Colver Productions, Elk Ridge, UT), and Bernie Krause (Wild Sanctuary, Glen Ellen, CA). Recordings and passages in which a poor signal-to-noise ratio or overlapping vocalizations prevented accurate analysis of the songs, or in which the focal bird sang fewer than 12 songs, were discarded. The longest available uninterrupted passage of the hermit thrush’s vocalizations was selected from each recording. In the case of one bird there were two suitable uninterrupted passages and we used data from both. Acoustic analysis was performed using the software Praat (42) and custom Praat macros (written by W.T.F.). Recordings for each bird were segmented into individual songs, defined as bouts of vocalization separated by more than 2 s of silence. These extracted songs were saved as separate files, numbered consecutively, and labeled according to start time and duration. The accuracy of automated extraction of the fundamental pitch of each note was verified by ear and by visual inspection of spectrograms with overlaid pitch tracks.

Song Types and Ordering in Song Series.

Hermit thrush songs are made up of small number of song types (22, 23); to avoid pseudoreplication we analyzed only the first appearance of each song type of each bird. Song types were defined as identical sequences of elements, as determined by visual inspection of the sonograms, and by comparing the average frequencies of the introductory whistles and the frequencies of the postintroductory whistle notes of each song. Each of our birds sang between 6 and 10 song types, for a total of 114 different song types (Table 1). None of these song types was shared between birds. Each bird typically cycled through all his song types within about 14 songs. Song types were never immediately repeated. The largest observed repeat interval was 32 songs and the shortest 2 songs. Individual birds varied in the predictability with which they presented their song types.

Pitch Extraction.

Praat’s autocorrelation-based pitch extraction algorithm was used to estimate the average fundamental frequency (“F0” or “pitch” hereafter) of each note with a steady, determinable pitch and that lasted for at least 10 ms. The minimum F0 of a note measured from any bird was 1,545 Hz and the maximum was 7,925 Hz. The mean range for each song type was 1,190 cents (very close to an octave, which corresponds to 1,200 cents). The mean range covered by each bird (including all song types) was 2,537 cents, representing slightly more than two octaves, with a minimum range of 2,201 cents and a maximum of 3,002 cents.

Estimating the Harmonicity of the Songs.

To estimate the harmonicity of the hermit thrush songs we used two approaches, one based on an ordinary least-squares regression model and a second on a Bayesian estimator of harmonicity. Both models assume that f_i lies between 100 and 1,000 Hz (although identical results were obtained when considering a range from 20 to 5,000 Hz) and both consider only the first 16 overtones, given that higher overtones are less clearly separated on a logarithmic basis.

Ordinary Least-Squares Regression Model.

The ordinary least-squares regression model makes no particular assumptions about the distribution of f_i besides those outlined above. For each song, a best-fitting base frequency f_i is computed by finding the value of f_i that minimizes the mean square error (logarithm of the frequency deviation) of a linear regression that fits the pitches of each note in a song to integer multiples of f_i. To test the hypothesis H1 that a song is an exchangeable sequence of frequencies that are integer multiples of some implied f_i versus the null hypothesis H0 that songs are generated by drawing frequencies out of a log-normal distribution, we generated, for each song, 1,000 randomly generated songs with the same number of notes as the actual song, but using frequencies taken from a log-normal distribution (restricted between 0 and 10,000 Hz) modeling the frequency distribution of all hermit thrush songs in our sample. For each of these randomly generated songs, a best fit for f_i was computed using the least-squares approach described above. If fewer than 5% of the randomly generated songs showed a better fit (smaller mean square error) to an overtone series than the actual song, H0 was rejected at a significance threshold of 0.05.

Generative Bayesian Model.

The generative Bayesian model computes the posterior distribution of the implied base f_i in a manner analogous to performing a regression analysis over the measured frequencies of all individual song notes (dependent variable) relative to those predicted from integer multiples (predictor) of all possible f_i between 100 Hz and 1,000 Hz. Because the predictor values (represented by the integer multiples 1, 2, 3, 4, …, with f_i being the frequency associated with the multiple 1) are unknown, this is an instance of a latent variable problem. The model therefore generates distributions over predictor values, rather than point estimates.

A song is parameterized by a (strictly positive) base frequency b and an overtone distribution q = (q₁,…,q_m), where q_m is the probability of observing overtone m in a song, M is the maximal possible overtone and ∑m=1Mqm=1. To generate a song, we drew K notes n_k ∈ {1, …, M} out of q, where k = 1, …, K is the sequence index. These notes are latent variables, because they are not directly observable. In a perfectly harmonic song, the notes and the corresponding observable pitch frequencies f_k would be related byfk=b⋅nk.[1]

To allow for a certain amount of random frequency deviation (hereafter jitter), which might be attributed to inaccurate singing or measurement errors, we included this jitter ρ multiplicatively:fk=b⋅nk⋅ρ,[2]

where ρ is drawn from a log-normal distribution with mean 0 and SD σ. Factorizing the jitter from the “ideal” frequency b · n_k allows us to use a frequency-independent relative jitter model, because the frequency dependence is already captured by the first two factors on the right-and side of Eq. 2.

To complete the model, we specified priors on q and b, because they are a priori unknown and are thus treated as random variables. Because the notes n_k are multinomial variables (each note can take exactly one of M values), q is a multinomial distribution. The canonical prior on a multinomial distribution is the Dirichlet distribution (see ref. 43 for details). We used a symmetric Dirichlet distribution with concentration parameter α. To choose a prior on b, note that there is a scaling ambiguity in the model: Substituting b by b/z and n_k by z · n_k with z integer and positive would lead to the same observable frequencies f_k. To remove this ambiguity, we limited the range of b to b ∈ [b_min, b_max] with a densityp(b)∝b,[3]

that is, we prefer higher base frequencies linearly. The constant of proportionality in Eq. 3 was chosen to ensure normalization: ∫bminbmaxdb p(b)=1.

For base frequency inference we computed the posterior distribution of b given a song using standard Bayesian approaches (43). This computation involves an intractable integral over q, which we approximate by its value at the maximum (maximum-a-posteriori, or MAP, approximation). The one-dimensional integral over b is carried out numerically. Given the MAP estimate of b, inferring a note n_k is done by finding the n_k that maximizes the probability of generating the corresponding f_k. For the predictions, we used the MAP parameter estimates for b and q to compute predictive probability distributions for those frequencies that had not been used to make these estimates. We set b ∈ [100 Hz, 1,000 Hz] and α = 2. The value of α is not critical.

To determine a suitable value for the jitter SD σ, we recorded 10 human singers singing “Happy Birthday” two times each. Because this song is diatonic rather than purely harmonic, we used a song model assuming a chromatic scale, by replacing Eq. 2 withfk=b⋅rnk⋅ρ,[4]

where r=212 is the frequency ratio between two steps on a chromatic scale and nk≥0 an integer. The exact value of the SD of log ρ is not critical but should be small enough so that neighboring steps on the scale do not mix owing to jitter. We chose 0.01. For each recording, we inferred b and the n_k as described above and computed the reconstructed frequency f˜k=b⋅rnk. This allowed us to estimate the empirical SD of log⁡fk−logf˜k across all recordings and singers. We found a value of 0.014, which we subsequently used for σ.

The above generative model can be used not only to infer the base frequency and notes but also to determine how probable is an observed song under H0 versus H1. To avoid overfitting we used leave-one-out cross-validation (43), that is, we inferred b and q from all frequency observations in a song but one and then computed the predictive probability of the left-out observation.

Validity Testing of Both Models on a Ground Truth Dataset.

To verify that both statistical approaches used here produce sensible results we used a ground truth dataset (GT). A ground truth is a dataset that shares relevant features with the data of interest but has controllable statistical properties. Our GT is a random sequence generator (technically, an exchangeable sequence generator) that can be continuously adjusted between producing perfectly harmonic sequences of frequencies and sequences with a nearly log-normal frequency distribution.

We generated these sequences from our Bayesian model with q set to the average hermit thrush song overtone distribution and b = 393.21 Hz, which is the exponentiated average log-base frequency of the hermit thrush songs. The harmonicity of a generated sequence can be controlled by varying the jitter SD η. For very small η, the resulting frequency density is composed of distinct, evenly spaced peaks. This can be seen in Fig. 3 for η = 0.005 and η = 0.015. Drawing a sequence from such a density yields a harmonic sequence, because virtually all probability is contained in the peaks. For larger η, the peaks begin to overlap until the harmonic structure disappears and the density looks log-normal (η = 0.3 in Fig. 3). We drew sequences with K = 14 notes, which corresponds to the rounded average song length in our sample of hermit thrush songs. To validate that our hypothesis comparison is indeed sensitive enough to separate sequences whose frequency distribution corresponds to an overtone series from sequences whose frequencies are derived from a log-normal frequency distribution, we drew 100 sequences for several values of η between 0.005 and 0.3, for a total of 1,100 sequences.

We also used the GT to determine the minimum number of notes K that a sequence (or song) should contain so that we have enough statistical power to determine whether the detected harmonicity of the sequence corresponds to the true harmonicity. Specifically, we searched for a K such that both p(H1|harmonic song) >1–0.05 = 0.95 and p(H0|random song) >0.95. Given the value of 0.014 obtained for the empirical deviation σ with human singers, we treated sequences with a jitter of η ≤ 0.01 as clearly harmonic, and sequences with η ≥ 0.03 as clearly nonharmonic. From the GT, we computed the log-ratio that each note contributes on average in favor (or against) each hypothesis. We obtained a log-ratio of 0.290 for η = 0.01 in favor of H0 and. 1.015 for η = 0.01 in favor of H1. Assuming that we have no initial preference for either H0 or H1, these values imply K = 2.9 and K = 10.1, respectively. We therefore chose K = 10 as the minimal number of notes for which we have enough statistical power to determine the harmonicity of a song or sequence. Note that using a slightly larger or smaller value of K did not significantly affect our results.