1 Introduction

According to available records, the relations between music pitches have been the subject of keen interest to philosophers and mathematicians for the past two and a half thousand years (Tenney, 1988). The Pythagoreans sought references to the harmony of spheres in those relations, manifesting in the perfect, inaudible music of the celestial bodies. The secret of perfect proportions has since then inspired researchers looking for a simple and consistent explanation of the world around us, a kind of theory of everything.

Numerous speculative theories have been proposed, combining musical harmony with mathematical proportions. The common theme of these theories is the simplicity of ratios corresponding to the perception of musical intervals. In this article, selected definitions of proportion simplicity will be confronted with music practice. The aim is to answer the question of whether the simplicity or complexity of mathematical ratios between pitches could be a determinant of their consonance degree.

2 Definitions

In the following discussion, pitch is referred to as the attribute of auditory sensation by which sounds are ordered on the scale used for melody in music (ASA, 2023). Hence, the relation of two pitches, represented by the frequency ratio of two or more tones played on musical instruments, denotes a mathematical proportion. The pitches can interact in a vertical or horizontal relation, corresponding to harmonic or melodic intervals, respectively.

Importantly, the proportions are derived from the so-called just tuning of musical intervals, an integer-ratio system described by Claudius Ptolemy in the 2nd century BC. Over the recent three centuries, the adjusted ‘equally tempered’ tuning has become widely used, however small deviations from just intervals do not change the perception of consonance, as already postulated by Aristoxenus in the 3rd century BC. Further, I will ignore phenomena related to the irregular (non-harmonic) spectrum of frequencies, as well as effects observed for very low or very high sounds, i.e., natural roughness or the decreasing accuracy of pitch perception above 2500 Hz (Langner, 2007), respectively. To exemplify the argument, I will only consider the proportions within one octave, i.e., between the ratios of 1:1 and 2:1.

Consonance will be herewith defined as a sound of two or more pitches that listeners consider to be in concordance with each other, evoking the sensations of fusion and stability. The opposite of consonance is dissonance, denoting an unconcordant, rough and unstable sound, the accumulation of which is regarded as less pleasant to listen to. Studies on human perception of musical intervals (e.g., Malmberg, 1918; Bowling et al., 2018) agree in this regard with the theory of the counterpoint, or the art of combining musical voices (Tinctoris, 1477), which assumes the following classification of musical intervals:

1. –consonances are considered to include the octave (12 semitones), perfect fifth (7), perfect fourth (5), major and minor third (4 and 3), major and minor sixth (9 and 8);
2. –dissonances are considered to include major and minor seconds (2 and 1), major and minor sevenths (11 and 10) and the tritone (6).

3 Historical Context

The Pythagoreans defined simplicity of proportions as the relation between the smallest natural numbers that formed the so-called tetraktys – the Arch-Four. The resulting proportions determined perfect consonances: 2:1 (octave), 3:2 (perfect fifth) and 4:3 (perfect fourth). All other proportions were considered dissonances.

Medieval and Renaissance theorists, starting with an anonymous thirteenth-century theorist and followed by Johannes de Garlandia, Walter Odington and Bartolomeo Ramis de Pareia, gradually acknowledged the role of thirds and even sixths as consonances. Thus, not only the proportions formed by the first four but also by further small natural numbers, came to be treated as consonances. The tetraktys ceased to be the determinant of proportion simplicity, replaced by the so-called senario (Zarlino, 1558), i.e. a series of natural numbers up to 6. Consequently, with the development of musical harmony and growing composer preference towards the major triad (4:5:6), consonances were extended to the major sixth (5:3) as well as the major and minor third (5:4 and 6:5).

Over time, the progress of the major-minor system meant that the minor sixth was added to the consonances, even though it did not fit into the senario with its 8:5 ratio. The inclusion of this interval, however, was natural as the minor sixth is an inversion (complement of the octave) of another interval already regarded as a consonance, i.e. the major third (Table 6.1).

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Geometric representation of simple ratios by Descartes

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Descartes attempted to represent the known musical intervals using geometrical proportions to explain proportion simplicity and consonance (Sepper, 1996) (Figure 6.1).

The points C, D, E and F bisect (divide in half) sections AB, BC, CD and CE, respectively. According to Descartes, only the proportions resulting from points A to E create consonances, while proportions containing point F form dissonances. This theory, however, suffers from inconsistencies, as it does not explain the major sixth’s 5:3 ratio, being one of the key consonances, while point F can also be included in the consonant proportion of AD:AF (4:3).

Benedetti (1585) suggested that the acoustic properties of harmonic sounds and the total period duration of musical intervals formed by them are related to simple natural number ratios and thereby constitute an explanation of consonance. According to Benedetti, the total period of an interval’s acoustic representation was proportional to the product of the numerator and the denominator of the corresponding ratio, expressed as an irreducible fraction. However, the minor sixth (8:5) remained a problem, because there exists a ‘simpler’ proportion – composed of smaller natural numbers – evoking a dissonance sensation. The problematic ratio is 7:5, equal approximately to the tritone, one of the strongest dissonances, dubbed ‘the devil in music” by counterpoint theorists. In conclusion, proportions formed by the smallest natural numbers cannot be directly equated with consonance.

where p stood for the factor, and k for its power.

For example, the factorisation of the minor sixth’s ratio (8:5) is 2³ * 5¹, which, based on (1), results in the gradus suavitatis: G = 1 + 3 (2 − 1) + 1 (5 − 1) = 8.

Notably, it was the first known theory that tackled the abovementioned problem of the tritone and the minor sixth (cf. Table 6.2). Unfortunately, the concept displays other shortcomings, such as a similar consonance degree of the minor third (6:5), the minor sixth (8:5) and the minor second (9:8). All of them are placed in the eighth gradus, which is at odds with the dominant perception of the first two intervals as consonances and the third one as a dissonance. Therefore, the simplicity of ratios proposed by Euler cannot be considered as an exhaustive explanation of consonance.

In the following centuries, studies in acoustics and signal processing by the human auditory system led to a better understanding of consonance and dissonance rooted in physical phenomena (von Helmholtz, 1863). Bohlen (1978) attempted to combine the simplicity of ratios with the phenomenon of combination tones, but the resulting consonance measures showed the same problem that had already manifested itself in Benedetti’s work.

An intriguing and consistent mathematical concept of consonance, based on the Farey series, was proposed by Erlich (1997). The Farey series of the F-th order includes all fractions with the numerator and denominator not larger than F. Using terms from the information theory, Erlich postulates that the dissonance sensation increases with the so-called harmonic entropy, defined as the cumulative probability of perceiving a musical interval as a mistuned version of another interval (ratio). The Farey series proves useful in this case, as one of its properties is a larger distance between its neighbouring terms for small-number ratios. This demonstrates lower ‘uncertainty’ of the cognitive system when dealing with simple, consonant intervals (Figure 6.2).

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Measure of harmonic entropy of order F = 64 for intervals between the unison and the octave + major third. Common intervals from ‘deeps’ in the graph and are marked accordingly. Notation of intervals: m – minor, M – major, P – perfect, Trt – tritone, Oct – octave. Cents (c) denote the proportion reflecting an interval as 2 ^c/1200: 1

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Closer analysis reveals that the harmonic entropy of the tritone (7:5) is quite low compared to the remaining intervals, including those commonly perceived as consonances. Yet, the hypothesis, that there is a certain cognitive complexity associated with the sensation of dissonance, is likely reasonable and reflects not only in the perceived degree of simplicity (ease) but also the stability of music intervals.

4 Hypothesis

Recent evidence suggests that the human cognitive system responds to the frequency of the incoming signal in a synchronised manner. This phenomenon has been observed at different stages of the so-called auditory pathway, i.e. the route of nerve impulses representing acoustic stimuli in the brain (Langner, 2007). Assuming that the cognitive system synchronises with the incoming signal, each period of the sound wave generates a (generalized) impulse, or discharge.¹ Importantly, there appears to be a separate cognitive axis responding to pitch (as an aggregate of incoming frequencies of a harmonic tone), on top of known responses to individual frequencies. Moreover, there exist ‘specialized’ units of the system, responding only to a particular pitch and its corresponding frequency. The mathematics of periodic signals provides an important observation here: a signal with the period (inverse of frequency) of p also ‘contains’ the periods of 2p, 3p, etc. Therefore, the signal with a particular period elicits responses of cognitive units (neuron groups) tuned to that period as well as to its integer multiples (Figure 6.3).

With regard to the conditions presented above, I have proposed the following mechanism:

1. –When two or more harmonic tones are heard simultaneously, the impulses (i.e., the model responses) form a specific pattern that can be analysed by the cognitive system.
2. –This pattern is repeated (or repeated approximately) in periods called pattern windows and the cognitive system is able to measure the duration of those periods by analysing the time between repeating patterns.
3. –Following Euler’s and Erlich’s intuitions, the perceived consonance degree is a result of pattern simplicity and stability. Both metrics represent the ease of processing a given pattern, and in turn, the corresponding ratio and musical interval, by the human cognitive system.

Pattern simplicity is naturally connected with the volume of model responses in a pattern. Its formal representation is the reciprocal of the average number of model responses in a pattern window. It can be conveniently expressed using so-called harmonic numbers, i.e., sums of reciprocals of natural numbers.

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Pattern window of model responses to two signals with periods of 4 ms and 5 ms (250 Hz and 200 Hz), forming the major third interval. The window length is 20 ms, and it approximately repeats every 20 ms. Note the responses of cognitive units (neuron groups) ‘tuned’ to multiples of the signal periods

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where p:q is the initial ratio and H_k is the sum of reciprocals of natural numbers up to k.

where LCM is the least common multiple.

Consequently, for a given mathematical proportion p:q, representing the respective musical interval, the product of pattern simplicity and stability forms the measure of consonance degree.

5 Discussion

It is worth noting that for intervals within one octave, the proposed metric is consistent with musical practice and theoretical intuition. It is the only concept among the described measures that makes a clear and appropriate distinction between consonances and dissonances (Table 6.2).

A further meta-analysis was conducted to quantify and confirm differences of presented measures in their discriminatory power between consonances and dissonances. Using ranking values as classifier inputs, we can calculate sensitivity, i.e., the share of correctly identified consonances, and precision, i.e. share of correct consonance indications, both computed with regard to a given ranking threshold. The F1 score, i.e., the harmonic mean of precision and sensitivity, was used as a unified measure of discriminatory power.

Results confirm that the proposed measure was the only one out of the examined metrics that – in alignment with experimental results – achieved an F1 score equal to 1 (Figure 6.4).

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The F1 score computed for four described measures and experimental results, based on ranking thresholds discriminating dissonances and consonances (see Table 6.2)

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This value denotes perfect discrimination between consonances and dissonances within one octave. We can therefore conclude that the suggested mechanism could potentially contribute to the formation of consonance and dissonance sensation.

As a further noteworthy fact, the measures of consonance/dissonance selected for this study represent only a small fraction of to-date mathematical, musical, psychoacoustic and neurophysiological concepts explaining consonance and dissonance (Foltyn, 2012). Curiously, however, multiple theories result in the same rank order of intervals as in the presented concepts. For example, interval orderings based on Kuile (1914), Bohlen (1978), Ebeling (2007), Gill & Purves (2009) and Trulla et al. (2018) coincide exactly with Benedetti’s ranking, thereby displaying the same advantages and shortcomings.

Reverting to the original idea of cognitive simplicity, there is only partial correspondence between small natural numbers constituting proportions of musical intervals and their perception as consonances. Further factors can impact perceived interval stability, for instance primeness of numbers forming a ratio. In line with (3), higher prime numbers result in larger LCM s, lower pattern stability, and so, lower consonance degree. This might explain why the 7:6 ratio (minor third reduced by approximately a quartertone, producing the blue note popularized by jazz music) is perceived as more consonant than the 7:5 tritone, a smaller-number ratio, but consisting of two highest primes among one-digit natural numbers.

It appears then that the simplicity of a musical interval’s proportion is only a prerequisite and not a sufficient condition for the consonance sensation to occur. Computational complexity associated with processing signals and their corresponding ratios depends on the pattern of neural responses it elicits, turning our attention to other mathematical concepts like harmonic or prime numbers. This idea sheds new light on the past and future quests of mathematicians and music theorists for an objective measure of consonance. Clearly, without understanding the mechanisms of processing pitch information by the human cognitive system it is difficult to speculate about generalised mathematical principles underlying these processes. However, tempting the vision of a single simple formula explaining the entire perception of consonance and dissonance might be, it is becoming increasingly evident that reality is at least partially more complex.