1 Menzerath’s Law

1.1 Durations and Intervals

The distribution of element durations and inter-onset intervals from the whale vocal sequences included in this analysis. The times are z-scored within each study to enable direct comparison.

Figure 1: The distribution of element durations and inter-onset intervals from the whale vocal sequences included in this analysis. The times are z-scored within each study to enable direct comparison.

To assess whether the strength of Menzerath’s law in human language is similar when computed from element durations and inter-onset intervals, I conducted a supplementary analysis with a corpus of spoken language data that was collected to study efficiency in information encoding (1). DoReCo could not be used for this supplementary analysis because it includes the short gaps between phonemes in its measurements (2), making it impossible to analyze element durations and inter-onset intervals separately without a reanalysis of the raw data. Coupe et al.’s dataset is composed of 2,288 recordings of speech, ranging from three to five sentences in length, and representing 17 different languages (1). For each recording, they collected the number of syllables, the overall duration including vocalizations and silence, and the duration only including silence, making it possible to compute the average element duration and inter-onset interval of each recording.

Each sequence in Coupe et al.’s data is a set of sentences from a single speaker (1). Analyzing Menzerath’s law in utterances above the sentence-level is unusual but has precedent (3, 4). It would be ideal to conduct this analysis at the same level of the linguistic hierarchy as DoReCo (e.g., phonemes within words, words within sentences.), but Coupe et al.’s dataset is the only one I found that separates phonation from silence (1).

For each sequence, I analyzed (1) the length in syllables, (2) the average duration of syllables, and (3) the average inter-onset interval between syllables. These data were analyzed using the same linear model as in the main text, excluding the varying intercept because each point corresponds to the average duration/interval within a sequence:

\[\begin{equation} \ln(\textrm{duration}) \sim \ln(\textrm{length}) \tag{1} \end{equation}\]

The results indicate that the strength of Menzerath’s law for syllables in series of sentences is quite similar when computed from element durations (estimate = -0.216, 95% CI: [-0.256, -0.176]) and inter-onset intervals (estimate = -0.292, 95% CI: [-0.331, -0.253]).

1.2 Effects in Whale Data

Table 1: The effect of sequence length on element durations and inter-onset intervals for each whale species, computed from the base model that excludes position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
Group	Species	Type	Effect	2.5%	97.5%
Mysticete	Blue Whale	Elements	-0.255	-0.331	-0.178
	Bowhead Whale	Elements	-0.184	-0.318	-0.051
	Common Minke Whale	Elements	-0.278	-0.294	-0.261
	Humpback Whale	Intervals	-0.678	-0.692	-0.665
	North Pacific Right Whale	Elements	0.309	0.278	0.340
	Sei Whale	Elements	-0.194	-0.329	-0.059
Odontocete	Bottlenose Dolphin	Intervals	-0.242	-0.347	-0.138
	Commerson's Dolphin	Intervals	0.221	0.087	0.356
	Heaviside's Dolphin	Intervals	-0.119	-0.320	0.083
	Hector's Dolphin	Intervals	-0.008	-0.274	0.258
	Killer Whale	Elements	0.121	0.003	0.239
	Narrow-Ridged Finless Porpoise	Intervals	-0.304	-0.338	-0.270
	Peale's Dolphin	Intervals	-0.333	-0.489	-0.177
	Risso's Dolphin	Intervals	-0.420	-0.448	-0.392
	Sperm Whale	Intervals	-0.234	-0.241	-0.226

Table 2: The effect of sequence length and position on element durations and inter-onset intervals for each whale species, computed from the expanded model that includes both length and position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
			Length			Position
Group	Species	Type	Effect	2.5%	97.5%	Effect	2.5%	97.5%
Mysticete	Blue Whale	Elements	-0.255	-0.331	-0.178	-0.064	-0.087	-0.041
	Bowhead Whale	Elements	-0.179	-0.313	-0.046	-0.751	-0.789	-0.713
	Common Minke Whale	Elements	-0.278	-0.294	-0.261	-0.017	-0.023	-0.011
	Humpback Whale	Intervals	-0.678	-0.691	-0.665	-0.193	-0.201	-0.186
	North Pacific Right Whale	Elements	0.309	0.278	0.340	0.107	0.096	0.119
	Sei Whale	Elements	-0.194	-0.329	-0.058	-0.101	-0.132	-0.070
Odontocete	Bottlenose Dolphin	Intervals	-0.242	-0.346	-0.138	-0.084	-0.111	-0.056
	Commerson's Dolphin	Intervals	0.221	0.087	0.356	-0.106	-0.118	-0.095
	Heaviside's Dolphin	Intervals	-0.119	-0.320	0.083	0.019	0.010	0.027
	Hector's Dolphin	Intervals	-0.008	-0.274	0.258	-0.001	-0.010	0.008
	Killer Whale	Elements	0.121	0.021	0.221	0.528	0.428	0.628
	Narrow-Ridged Finless Porpoise	Intervals	-0.305	-0.339	-0.271	0.168	0.151	0.185
	Peale's Dolphin	Intervals	-0.333	-0.489	-0.177	-0.013	-0.017	-0.009
	Risso's Dolphin	Intervals	-0.420	-0.448	-0.392	-0.196	-0.200	-0.192
	Sperm Whale	Intervals	-0.234	-0.241	-0.226	0.028	0.026	0.031

1.3 Effects in Human Data

Table 3: The effect of sequence length on inter-onset intervals for each human language, computed from the base model that excludes position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
Language	Effect	2.5%	97.5%
Anal	-0.104	-0.113	-0.095
Arapaho	0.030	0.020	0.040
Asimjeeg Datooga	-0.063	-0.073	-0.053
Baïnounk Gubëeher	-0.102	-0.110	-0.093
Beja	-0.066	-0.073	-0.059
Bora	-0.127	-0.138	-0.116
Cabécar	-0.110	-0.120	-0.100
Cashinahua	-0.100	-0.108	-0.091
Daakie	-0.131	-0.141	-0.121
Dalabon	-0.079	-0.091	-0.066
Dolgan	-0.130	-0.139	-0.121
English (Southern England)	-0.053	-0.066	-0.040
Evenki	-0.101	-0.109	-0.092
Fanbyak	-0.091	-0.101	-0.080
French (Swiss)	-0.050	-0.063	-0.037
Goemai	-0.124	-0.138	-0.110
Gorwaa	-0.125	-0.136	-0.114
Hoocąk	-0.099	-0.109	-0.090
Jahai	-0.062	-0.073	-0.050
Jejuan	-0.093	-0.104	-0.083
Kakabe	-0.123	-0.135	-0.111
Kamas	-0.096	-0.111	-0.082
Komnzo	-0.081	-0.091	-0.071
Light Warlpiri	-0.144	-0.154	-0.133
Lower Sorbian	-0.078	-0.089	-0.067
Mojeño Trinitario	-0.147	-0.156	-0.137
Movima	-0.014	-0.023	-0.006
Nafsan (South Efate)	-0.072	-0.082	-0.062
Nisvai	-0.064	-0.074	-0.053
Nllng	-0.098	-0.111	-0.086
Northern Alta	-0.069	-0.079	-0.059
Northern Kurdish (Kurmanji)	-0.022	-0.033	-0.011
Pnar	-0.021	-0.035	-0.007
Resígaro	-0.079	-0.089	-0.069
Ruuli	-0.039	-0.048	-0.030
Sadu	-0.124	-0.136	-0.112
Sanzhi Dargwa	-0.135	-0.148	-0.122
Savosavo	-0.052	-0.062	-0.042
Sümi	-0.090	-0.101	-0.079
Svan	-0.066	-0.074	-0.057
Tabaq (Karko)	-0.213	-0.222	-0.203
Tabasaran	-0.138	-0.151	-0.125
Teop	-0.170	-0.181	-0.159
Texistepec Popoluca	-0.070	-0.081	-0.059
Urum	-0.126	-0.134	-0.118
Vera'a	-0.152	-0.163	-0.141
Warlpiri	-0.147	-0.157	-0.137
Yali (Apahapsili)	-0.191	-0.202	-0.179
Yongning Na	-0.128	-0.142	-0.114
Yucatec Maya	-0.067	-0.079	-0.056
Yurakaré	-0.198	-0.204	-0.191

Table 4: The effect of sequence length and position on inter-onset intervals for each human language, computed from the expanded model that includes both length and position. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
	Length			Position
Language	Effect	2.5%	97.5%	Effect	2.5%	97.5%
Anal	-0.105	-0.114	-0.096	0.100	0.091	0.109
Arapaho	0.030	0.020	0.039	0.099	0.090	0.109
Asimjeeg Datooga	-0.063	-0.074	-0.053	0.186	0.177	0.196
Baïnounk Gubëeher	-0.102	-0.110	-0.093	0.096	0.088	0.104
Beja	-0.066	-0.073	-0.059	0.065	0.058	0.072
Bora	-0.127	-0.138	-0.116	-0.131	-0.140	-0.123
Cabécar	-0.110	-0.120	-0.100	0.021	0.012	0.031
Cashinahua	-0.100	-0.108	-0.091	-0.019	-0.028	-0.010
Daakie	-0.131	-0.141	-0.122	0.164	0.155	0.173
Dalabon	-0.079	-0.091	-0.066	0.165	0.153	0.177
Dolgan	-0.130	-0.139	-0.121	0.043	0.034	0.052
English (Southern England)	-0.053	-0.066	-0.040	0.050	0.038	0.062
Evenki	-0.101	-0.109	-0.092	0.042	0.033	0.050
Fanbyak	-0.091	-0.101	-0.081	0.161	0.151	0.171
French (Swiss)	-0.050	-0.063	-0.037	0.160	0.151	0.169
Goemai	-0.124	-0.138	-0.110	0.063	0.052	0.074
Gorwaa	-0.125	-0.136	-0.114	0.009	0.000	0.018
Hoocąk	-0.099	-0.109	-0.090	0.141	0.132	0.151
Jahai	-0.062	-0.073	-0.050	0.142	0.131	0.153
Jejuan	-0.093	-0.104	-0.083	0.038	0.028	0.049
Kakabe	-0.123	-0.135	-0.111	0.103	0.091	0.115
Kamas	-0.096	-0.111	-0.082	0.003	-0.012	0.017
Komnzo	-0.081	-0.091	-0.071	0.026	0.018	0.035
Light Warlpiri	-0.144	-0.154	-0.133	0.078	0.067	0.088
Lower Sorbian	-0.078	-0.089	-0.067	0.046	0.037	0.056
Mojeño Trinitario	-0.147	-0.156	-0.137	-0.074	-0.083	-0.065
Movima	-0.014	-0.023	-0.006	-0.053	-0.062	-0.045
Nafsan (South Efate)	-0.072	-0.082	-0.062	0.094	0.085	0.103
Nisvai	-0.064	-0.074	-0.054	0.151	0.144	0.159
Nllng	-0.098	-0.111	-0.086	0.155	0.142	0.167
Northern Alta	-0.069	-0.079	-0.059	0.091	0.081	0.101
Northern Kurdish (Kurmanji)	-0.022	-0.033	-0.011	0.033	0.023	0.042
Pnar	-0.021	-0.035	-0.007	0.087	0.076	0.099
Resígaro	-0.079	-0.089	-0.069	0.012	0.003	0.022
Ruuli	-0.039	-0.048	-0.030	0.069	0.061	0.078
Sadu	-0.124	-0.136	-0.112	0.200	0.189	0.212
Sanzhi Dargwa	-0.135	-0.148	-0.122	0.012	0.000	0.023
Savosavo	-0.052	-0.062	-0.042	-0.125	-0.134	-0.117
Sümi	-0.090	-0.101	-0.080	0.109	0.098	0.120
Svan	-0.066	-0.074	-0.057	-0.026	-0.035	-0.017
Tabaq (Karko)	-0.213	-0.223	-0.203	-0.071	-0.080	-0.061
Tabasaran	-0.138	-0.151	-0.125	-0.025	-0.037	-0.012
Teop	-0.170	-0.181	-0.159	0.072	0.063	0.081
Texistepec Popoluca	-0.070	-0.081	-0.059	0.020	0.011	0.030
Urum	-0.126	-0.134	-0.118	-0.007	-0.015	0.001
Vera'a	-0.152	-0.163	-0.141	0.169	0.160	0.177
Warlpiri	-0.147	-0.157	-0.137	-0.026	-0.035	-0.017
Yali (Apahapsili)	-0.192	-0.203	-0.180	0.140	0.130	0.150
Yongning Na	-0.128	-0.142	-0.114	0.098	0.085	0.112
Yucatec Maya	-0.067	-0.079	-0.056	-0.075	-0.084	-0.066
Yurakaré	-0.198	-0.204	-0.191	-0.035	-0.040	-0.029

1.4 Words in Sentences

The 95% confidence intervals for the effect of sequence length (top) and position (bottom) on element durations and inter-onset intervals for the 16 whale species and 51 human languages. The human language data are comprised of words within sentences. The colors correspond to the taxonomic group and whether the data are element durations (ED) or inter-onset intervals (IOI).

Figure 2: The 95% confidence intervals for the effect of sequence length (top) and position (bottom) on element durations and inter-onset intervals for the 16 whale species and 51 human languages. The human language data are comprised of words within sentences. The colors correspond to the taxonomic group and whether the data are element durations (ED) or inter-onset intervals (IOI).

1.5 Production Constraint Model

James et al. (5) recently found that Menzerath’s law can be detected in pseudorandom sequences of birdsong syllables that are forced to match the durations of real songs. James et al. (5) interpret their model as approximating simple motor constraints, while stronger effects in the real data would indicate additional mechanisms (e.g., communicative efficiency through behavioral plasticity). I originally planned to compare the strength of Menzerath’s law in the real data with simulated data from the model of James et al. (5), as I recently did for house finch song (6), but analyses of language data suggest that it is far too conservative of a null model. 0 of the 51 of languages in the DoReCo dataset exhibit Menzerath’s law to a greater extent than simulated data. Even though many whale species exhibit Menzerath’s law to a greater extent than simulated data from the null model of James et al. (5) (75%; 12 out of 16 species), I do not want to over-interpret this result given the pattern in the human data. Upon further reflection I think that the fundamental assumption of James et al. (5), that sequence durations are governed by motor constraints alone, is unlikely to apply to many species with more complex communication systems. In humpback whales and sperm whales, for example, there appears to be significant inter-individual variation in song and coda length depending on social context (7, 8). More details about this analysis are below.

The production constraint model of James et al. (5) works as follows. For each iteration of the model, a pseudorandom sequence was produced for each real song in the dataset. Syllables were randomly sampled (with replacement) from the population until the duration of the random sequence exceeded the duration of the real song. If the difference between the duration of the random sequence and the real song was < 50% of the duration of the final syllable, then the final syllable was kept in the sequence. Otherwise, it was removed. Each iteration of the model produces a set of random sequences with approximately the same distribution of durations as the real data.

For each species, I generated 100 simulated datasets from the (1) random sequence model and the (2) production constraint model. Then, I fit Menzerath’s law separately to each of the 100 simulated datasets and pooled the model estimates for \(a\) and \(b\) using Rubin’s rule as implemented in the mice package in R. The results can be seen in Figure 3.

Most importantly, the estimated effects from the production constraint model tend to be more negative than those from the real human language data, suggesting that this null model is far too conservative to be informative about “language-like” efficiency.

The point estimates (points) from the real data alongside 95% confidence intervals (bars) from 10 simulated datasets from the production constraint model, for the effect of sequence length on element durations and inter-onset intervals for the 16 whale species and 51 human languages. The human language data are comprised of phonemes within words. The colors correspond to the taxonomic group and whether the data are element durations (ED) or inter-onset intervals (IOI).

Figure 3: The point estimates (points) from the real data alongside 95% confidence intervals (bars) from 10 simulated datasets from the production constraint model, for the effect of sequence length on element durations and inter-onset intervals for the 16 whale species and 51 human languages. The human language data are comprised of phonemes within words. The colors correspond to the taxonomic group and whether the data are element durations (ED) or inter-onset intervals (IOI).

1.6 Median Interpolation

The point estimates from the original datasets (orange) compared to median-interpolated datasets (blue). Interpolating sequences with the median inter-onset interval of each phoneme appears to systematically shift model estimates towards zero (in over 90% of cases).

Figure 4: The point estimates from the original datasets (orange) compared to median-interpolated datasets (blue). Interpolating sequences with the median inter-onset interval of each phoneme appears to systematically shift model estimates towards zero (in over 90% of cases).

1.7 Patterned Burst Pulses

Martin et al. (9) noticed that Heaviside’s dolphins sometimes produce temporally-patterned burst pulses with much more rhythmic variation, especially during social interactions. I analyzed 27 patterned burst pulses provided by Martin et al. (9) and found that they adhere to Menzerath’s law—there is a negative relationship between sequence length and inter-onset intervals (estimate = -0.186, 95% CI: [-0.308, -0.063]).

1.8 Plots with Transformed Axes

The baleen whale (Mysticete) species included in the study (left), alongside the distribution of element durations or inter-onset intervals and sequence lengths (middle) and the slope of Menzerath's law (right). The x- and y-axes have been log-transformed and z-scored to match the structure of the statistical model. Each point in the distribution plots (middle) marks the mean element duration or inter-onset interval, but the slopes on the right were computed from the full set of elements/intervals. The bars in the slope plots (right) mark the 95% confidence intervals around the point estimates.

Figure 5: The baleen whale (Mysticete) species included in the study (left), alongside the distribution of element durations or inter-onset intervals and sequence lengths (middle) and the slope of Menzerath’s law (right). The x- and y-axes have been log-transformed and z-scored to match the structure of the statistical model. Each point in the distribution plots (middle) marks the mean element duration or inter-onset interval, but the slopes on the right were computed from the full set of elements/intervals. The bars in the slope plots (right) mark the 95% confidence intervals around the point estimates.

The toothed whale (Odontocete) species included in the study (left), alongside the distribution of element durations or inter-onset intervals and sequence lengths (middle) and the slope of Menzerath's law (right). The x- and y-axes have been log-transformed and z-scored to match the structure of the statistical model. Each point in the distribution plots (middle) marks the mean element duration or inter-onset interval, but the slopes on the right were computed from the full set of elements/intervals. The bars in the slope plots (right) mark the 95% confidence intervals around the point estimates.

Figure 6: The toothed whale (Odontocete) species included in the study (left), alongside the distribution of element durations or inter-onset intervals and sequence lengths (middle) and the slope of Menzerath’s law (right). The x- and y-axes have been log-transformed and z-scored to match the structure of the statistical model. Each point in the distribution plots (middle) marks the mean element duration or inter-onset interval, but the slopes on the right were computed from the full set of elements/intervals. The bars in the slope plots (right) mark the 95% confidence intervals around the point estimates.

2 Zipf’s Law of Abbreviation

2.1 Effects in Whale Data

Table 5: The effect of count on element duration for each whale species. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
Group	Species	Effect	2.5%	97.5%
Mysticete	Blue Whale	-0.102	-0.139	-0.065
	Bowhead Whale	0.368	-0.386	1.121
	Humpback Whale	-0.696	-0.943	-0.450
	Sei Whale	0.249	-0.422	0.921
Odontocete	Killer Whale	-0.114	-0.314	0.086

2.2 Effects in Human Data

Table 6: The effect of count on element duration for each human language, at the level of phonemes. 2.5% and 97.5% denote the lower and upper bounds of the 95% confidence intervals.
Language	Effect	2.5%	97.5%
Anal	-2.104	-2.613	-1.594
Arapaho	-1.438	-1.704	-1.172
Asimjeeg Datooga	-2.295	-2.952	-1.638
Baïnounk Gubëeher	-1.936	-2.380	-1.492
Beja	-1.455	-1.813	-1.097
Bora	-1.983	-2.394	-1.572
Cabécar	-1.642	-1.935	-1.349
Cashinahua	-2.115	-2.568	-1.663
Daakie	-1.390	-1.788	-0.992
Dalabon	-1.957	-2.426	-1.487
Dolgan	-1.412	-1.731	-1.094
English (Southern England)	-1.065	-1.315	-0.815
Evenki	-1.578	-1.799	-1.357
Fanbyak	-1.230	-1.524	-0.936
French (Swiss)	-0.938	-1.141	-0.736
Goemai	-0.955	-1.238	-0.671
Gorwaa	-2.120	-2.681	-1.559
Hoocąk	-1.314	-1.539	-1.088
Jahai	-1.786	-2.138	-1.435
Jejuan	-1.618	-1.947	-1.289
Kakabe	-1.992	-2.539	-1.446
Kamas	-1.108	-1.376	-0.840
Komnzo	-1.656	-2.085	-1.227
Light Warlpiri	-1.624	-2.168	-1.079
Lower Sorbian	-1.343	-1.688	-0.999
Mojeño Trinitario	-1.436	-1.758	-1.113
Movima	-1.783	-2.134	-1.432
Nafsan (South Efate)	-1.565	-1.870	-1.259
Nisvai	-2.496	-3.049	-1.943
Nllng	-1.859	-2.408	-1.309
Northern Alta	-1.648	-1.981	-1.315
Northern Kurdish (Kurmanji)	-1.412	-1.904	-0.919
Pnar	-1.713	-2.278	-1.149
Resígaro	-1.271	-1.609	-0.934
Ruuli	-1.789	-2.048	-1.530
Sadu	-0.838	-1.178	-0.497
Sanzhi Dargwa	-2.009	-2.598	-1.420
Savosavo	-1.506	-1.990	-1.022
Sümi	-1.395	-1.680	-1.110
Svan	-1.695	-2.095	-1.294
Tabaq (Karko)	-1.727	-2.101	-1.352
Tabasaran	-1.506	-2.085	-0.928
Teop	-1.693	-2.145	-1.242
Texistepec Popoluca	-1.480	-1.755	-1.205
Urum	-1.822	-2.043	-1.600
Vera'a	-1.618	-1.933	-1.302
Warlpiri	-1.962	-2.662	-1.262
Yali (Apahapsili)	-1.414	-1.667	-1.162
Yongning Na	-0.930	-1.281	-0.578
Yucatec Maya	-1.056	-1.257	-0.855
Yurakaré	-2.145	-2.537	-1.753

2.3 Words

The 95% confidence intervals for the effect of count on element duration for the five whale species and 51 human languages. The human language data are comprised of words. The colors correspond to the taxonomic group and whether the data are element durations (ED) or inter-onset intervals (IOI).

Figure 7: The 95% confidence intervals for the effect of count on element duration for the five whale species and 51 human languages. The human language data are comprised of words. The colors correspond to the taxonomic group and whether the data are element durations (ED) or inter-onset intervals (IOI).

2.4 Plot with Transformed Axis

The whale species included in the study (left), alongside the distribution of element durations and counts (middle) and the slope of Zipf's law of abbreviation (right). The x-axis has been z-scored, and the y-axis has been z-scored and log-transformed, to match the structure of the statistical model. Each point in the distribution plots (middle) marks the mean duration of elements, but the slopes on the right were computed from the full set of elements. The bars in the slope plots (right) mark the 95% confidence intervals around the point estimates.

Figure 8: The whale species included in the study (left), alongside the distribution of element durations and counts (middle) and the slope of Zipf’s law of abbreviation (right). The x-axis has been z-scored, and the y-axis has been z-scored and log-transformed, to match the structure of the statistical model. Each point in the distribution plots (middle) marks the mean duration of elements, but the slopes on the right were computed from the full set of elements. The bars in the slope plots (right) mark the 95% confidence intervals around the point estimates.

References

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Supplementary information

Language-like efficiency in whale communication

Mason Youngblood

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