The role of population size in folk tune complexity

The study examines how population size (effective cultural population size) relates to cultural complexity in music, focusing on Irish folk session tunes. Prior work shows population size often positively correlates with technological complexity due to more innovation opportunities and reduced stochastic loss, while in language larger populations show simpler grammars despite larger vocabularies, suggesting increased compressibility with frequent transmission. The arts may differ because aesthetic preferences and subjective enjoyment of novelty and complexity shape cultural selection; music often exhibits an inverse U-shaped relationship between complexity and aesthetic appeal. Folk session tunes are ideal because they are widely shared, memorized, and vary greatly in popularity (effective population size) and in the number and complexity of versions (settings). The paper asks: (1) Do larger effective cultural populations increase or decrease innovation/diversification (richness and variation across settings)? (2) Is the relationship between population size and musical complexity positive, negative, or inverse U-shaped? (3) Do larger populations promote convergence on preferred variants (reduced relative diversity and evenness)? (4) What social learning biases and population structures best explain observed patterns?

Across cultural domains, population size relates to complexity differently. Technological domains typically show a positive association with population size (Henrich 2004; Powell et al. 2009; Shennan 2001; Kline & Boyd 2010; Collard et al. 2013), with historical case studies (e.g., Tasmanian toolkit loss; Upper Palaeolithic innovation bursts) and experiments (Derex et al. 2013; Muthukrishna et al. 2014) supporting this. In language, larger speaker populations tend to simplify grammatical structures while expanding lexicons (Lupyan & Dale 2010; Bromham et al. 2015; Nettle 2012; Reali et al. 2018); experimental micro-societies produce simpler, more learnable systems in larger groups (Fay & Ellison 2013). Folktales show mixed patterns: positive with tale-type diversity but negative with motif diversity (Acerbi et al. 2017). The arts differ because aesthetic fitness landscapes feature trade-offs between novelty/complexity and predictability/structure, frequently yielding inverted U-shaped preference functions (North & Hargreaves 1995; Chmiel & Schubert 2017; Delplanque et al. 2019; Van Geert & Wagemans 2021). Folk scholars suggest enduring tunes balance interest and learnability (Vallely 2011; Hillhouse 2013). Session traditions emphasize memory-based transmission, enabling cumulative change but with conservative norms. This context motivates testing whether population size increases diversity yet drives convergence toward preferred, intermediate complexity in music.

Data source and scope: Weekly data dump from The Session (thesession.org; GitHub mirror), a community site for Irish traditional music. Initial dataset: 38,151 settings of 18,042 tunes (downloaded 01/02/2021). Popularity measures (proxies for effective cultural population size): tune-level 'tunebook adds' (available for tunes added to ≥10 tunebooks; n=9,782 tunes) and setting-level 'set adds' (settings added to at least one virtual set; n=13,615 settings of 7,089 tunes; downloaded 04/30/2021). After processing and exclusions, analyses used n=9,378 tunes and n=12,422 settings with complete measures.

Diversity measures (tune-level): (1) Richness: number of unique settings per tune. (2) Variation: inter-quartile range (IQR) in melodic complexity across settings of a tune.

Melodic complexity measures (setting-level, then aggregated by median for tune-level analyses): seven proxies reflecting learnability/playability: (a) number of bars; (b) number of notes; (c) expectancy-based model (EBM) of melodic complexity (Eerola & North 2000); (d) tone-transitions model (TTM; Simonton 1984, 1994); (e) first-order pitch-class entropy; (f) second-order (adjacent pitch-pair) entropy; (g) novelty via self-similarity (Foote 2000). Higher scores indicate greater complexity. Processing pipeline: ABC -> MusicXML (abc2xml) to extract bar counts; then XML -> MIDI (MuseScore Batch Convert) -> MIDI Toolbox 1.1 (MATLAB) to compute complexity metrics. Bars counted without expanding repeats; partial bars (anacruses) counted as separate bars. Exclusions: 7 settings <4 bars (fragments/format errors); 3,387 settings lacked measures due to corrupt/non-monophonic MIDI; 4 settings removed for implausible values.

Convergence measures: (1) Relative variation: model IQR in complexity predicted by popularity while controlling for richness (tests whether popularity reduces variation relative to richness); (2) Pielou’s evenness (Shannon diversity/log richness) computed on the distribution of setting-level popularity across settings of a tune (vegan package). Evenness analyses limited to tunes with >1 setting and complete popularity across settings (n=2,380 tunes).

Statistical analysis: Outcome distributions were right-skewed; thus nonlinear models with log10-transformed outcomes (exponential models). Evenness retained untransformed; log10-transform applied to popularity improved fit. Predictors included popularity and number of days since first upload (to control for time confound); VIF < 2 indicated low collinearity. Models run in a Bayesian framework (R, MCMCglmm): default diffuse normal priors for fixed effects; inverse-Wishart priors (V=1, v=0.002) for random effects and residuals. MCMC: 25,000 iterations, burn-in 5,000, thinning 10; convergence checked via trace plots and ESS ≥ 500. Reported posterior means, 95% credible intervals, and PMCMC. Model fit: marginal and conditional R2 (Nakagawa & Schielzeth 2013). Cross-validation: 100 repeats with 75% train/25% test; normalized RMSE (error as proportion of outcome range) reported.

Setting-level complexity models included tune identity as a random effect. Complexity variables were transformed to absolute deviation from the median prior to regression to test for inverse-U (intermediate complexity): 0 = median; positive values = deviation higher or lower than median.

Time-split robustness: Analyses repeated for subsets split at year 2005 to assess temporal consistency.

Simulations: Agent-based diffusion of tune settings across populations sizes 20–1000 (step 20). Each agent initialized with a setting complexity from N(3,1) (0 denotes intermediate complexity). At each timestep, with probability 0.1 an agent socially learns (copies another’s complexity with small error); else with probability 0.1 innovates (small self-alteration); otherwise retains current setting. Social learning conditions: (i) Unbiased copying: error ~ N(0, 0.01). (ii) Intermediate-biased copying: mean error ±0.005 toward 0 depending on sign of current setting. Alternative implementation of preference via selective copying: probability of being copied weighted by a negative quadratic in complexity (favoring values near 0). Population structure: (a) Homogeneous/global copying; (b) Structured into groups of 10 with local copying, with (b1) random group assignment or (b2) assortative grouping by initial complexity.

Diversity: Popularity positively predicts diversity. Tune richness (number of settings) and absolute variation (IQR) in complexity across settings increase with tune popularity. In the richness model (n=9,378 tunes), marginal R2 = 0.256; both popularity and days-since-upload had PMCMC < 0.001; normalized RMSE = 0.186 (CV mean 0.187 ± 0.003).

Complexity (inverse-U): Popular tunes tend to have intermediate melodic complexity. At the tune level (n=9,378), deviation from median complexity decreased with popularity for 6/7 measures (significant except note count): bar count (PMCMC < 0.001), melodic expectations (PMCMC < 0.001), tone transitions (PMCMC = 0.007), pitch entropy (PMCMC < 0.001), pitch-pair entropy (PMCMC < 0.001), novelty (PMCMC < 0.001); note count PMCMC = 0.070. Effects were small (marginal R2 ≤ 0.008) but robust with good out-of-sample RMSE (0.110–0.210). At the setting level (n=12,422; random intercept for tune), popularity predicted intermediate levels for bar count (PMCMC = 0.003), melodic expectations (PMCMC < 0.001), and pitch-pair entropy (PMCMC = 0.020); other measures nonsignificant. Marginal R2 ≤ 0.001; conditional R2 high (0.20–0.70) indicating strong tune-level effects.

Convergence: Controlling for richness, popularity negatively affected relative variation (IQR) in complexity, implying stronger convergence on preferred variants in larger populations. Pielou’s evenness decreased with popularity (log10 tunebook adds coefficient ≈ −0.282 [−0.301, −0.268], PMCMC < 0.001; marginal R2 = 0.386; normalized RMSE = 0.171), indicating more skewed preferences among settings of popular tunes.

Temporal robustness: Diversity effects held for tunes uploaded pre- and post-2005. Popularity–intermediate complexity effects were more consistent in older tunes at the tune level (novelty retained across periods), and broadly consistent at the setting level. Popularity reduced relative variation and evenness across periods for entropy/novelty measures; other measures showed effects mainly among older tunes.

Simulations: Across conditions, richness increased linearly with population size (more innovations). Mean complexity remained at the starting mean under unbiased copying and shifted toward 0 under intermediate-biased copying. Absolute variation (SD) increased with population size; relative variation (SD normalized by richness) declined with population size, especially in structured populations (random or assortative grouping). Unbiased copying can maintain intermediate complexity; a bias toward intermediate is required for initially simple/complex tunes to gravitate toward intermediate over time. Selective preference implementation (negative quadratic weighting) produced qualitatively similar outcomes, with faster convergence.

Findings indicate that larger effective cultural populations in folk music foster greater innovation and diversification (more settings and absolute complexity variation) but also drive convergence on preferred, intermediate complexity levels, producing an inverse-U relation between popularity and complexity. This pattern contrasts with technology (positive complexity–population size correlation) and language (simplification in larger populations), underscoring domain-dependent cultural selection pressures. In music, aesthetic fitness landscapes balance novelty/complexity against predictability, leading to optimal intermediate complexity being favored. Reduced relative variation and evenness for popular tunes indicate that larger populations facilitate stronger selection and coordination on favored variants. Simulations suggest these empirical patterns are most consistent with structured populations and biased social learning toward intermediate complexity; unbiased copying can sustain but not necessarily attract traditions toward intermediate levels from initially extreme states. The interplay of global vs local influence matters: local (structured) influence preserves diversity while still enabling convergence within groups. Overall, population size increases available variation, enabling selection to more effectively shape traditions toward domain-specific optima.

Popular Irish session tunes, as proxies for larger effective cultural populations, are more diverse yet converge on intermediate melodic complexity, with more uneven preferences across versions. This domain-specific pattern supports the view that the relationship between population size and cultural complexity is not universal but shaped by aesthetic selection pressures in the arts. Simulations reinforce that increased innovation with population size and biases toward intermediate complexity, especially in structured populations, best explain the empirical trends. The study contributes a large-scale, data-driven account of how demography shapes musical traditions and calls for cross-cultural corpora, controlled experiments on population size and learning biases in music, and deeper investigation into the roles of local versus global social influence in musical transmission.

Analyses are correlational; causality may be bidirectional—complexity may influence popularity as much as popularity influences complexity. Sampling and curation biases may inflate diversity measures for popular tunes (more effort to document well-known tunes). The proxy for effective cultural population size (tunebook/set adds) may not perfectly reflect actual learning/transmission behavior. Tune age and historical origins are often unknown, limiting control for age effects beyond time-since-upload. While simulations model population size directly, empirical data use popularity as a proxy; residual confounds may remain.