Testing a spectral model of tonal affinity with microtonal melodies and inharmonic spectra

Abstract

Tonal affinity is the perceived goodness of fit of successive tones. It is important because a preference for certain intervals over others would likely influence preferences for, and prevalences of, "higher-order" musical structures such as scales and chord progressions. We hypothesize that two psychoacoustic (spectral) factors-harmonicity and spectral pitch similarity-have an impact on affinity. The harmonicity of a single tone is the extent to which its partials (frequency components) correspond to those of a harmonic complex tone (whose partials are a multiple of a single fundamental frequency). The spectral pitch similarity of two tones is the extent to which they have partials with corresponding, or close, frequencies. To ascertain the unique effect sizes of harmonicity and spectral pitch similarity, we constructed a computational model to numerically quantify them. The model was tested against data obtained from 44 participants who ranked the overall affinity of tones in melodies played in a variety of tunings (some microtonal) with a variety of spectra (some inharmonic). The data indicate the two factors have similar, but independent, effect sizes: in combination, they explain a sizeable portion of the variance in the data (the model-data squared correlation is r 2 = .64). Neither harmonicity nor spectral pitch similarity require prior knowledge of musical structure, so they provide a potentially universal bottom-up explanation for tonal affinity. We show how the model-as optimized to these data-can explain scale structures commonly found in music, both historical and contemporary, and we discuss its implications for experimental microtonal and spectral music.

Empirically testingTonnetz, voice-leading, and spectral models of perceived triadic distance

“…For the spectral pitch-class model, the optimized smoothing width falls within the expected range (3-13 cents), as does the roll-o↵, which corresponds approximately to the loudnesses of the partials in the string sounds used (similar optimized values were also obtained in the related models detailed in Sharp 2015 andSharp 2016). As shown in Figure 7, the optimized spectral pitch-class distance model can calculate values for any interval size, including microtonal.…”

“…The latter can easily be generalized to any possible scale tuning or spectral tuning (e.g. the non-harmonic partials produced by many percussion and non-Western instruments), as investigated in Milne, Laney, and Sharp (2016). It is not obvious how, or if, the Tonnetz could be extended to cover such non-harmonic timbres and non-standard tunings.…”

“…Other recent research has demonstrated that spectral models can successfully predict the perceived fit of chromatic probe tones to an established key (Krumhansl's (1982) tonal hierarchies) (Milne, Laney, and Sharp 2015) and the perceived a nity of microtonal melodies (Milne, Laney, and Sharp 2016). The spectral model tends to make chords with more common tones and more tones a perfect fifth/perfect fourth apart (between any voices) closer than those with fewer such intervals.…”

Perception of affect in unfamiliar musical chords

“…Harmonicity is generally described as the degree of similarity between the spectrum of a tone and a template harmonic complex tone, which consists of numerous frequencies all at integer multiples of a single fundamental frequency [40,24,41]. There are a variety of precise mathematical quantifications of harmonicity (see [42] for an overview of harmonicity models). Harmonicity is thought to contribute to pleasantness independently from roughness–consonance may therefore be not due only to a lack of unpleasant beating, but also to high harmonicity [43].…”

“…Spectral entropy, calculated like this, is a way of aggregating the spectral pitch (class) similarities [24,50,14,42,44] all pairs of sounds in a pitch (class) set, such as a chord or scale. This is because the greater the degree of overlap of the partials in two different sounds, the higher the spectral pitch similarity of the two sounds and the lower the overall entropy of the resulting spectrum.…”