Modeling Consonance and its Relationships with Temperament, Harmony, and Electronic Amplification

TL;DR

This paper presents a computational model of musical consonance based on Helmholtz's theory, applied to various scales, chords, and electronic amplification, revealing insights into scale classification and effects of nonlinearities.

Contribution

It introduces a simple yet effective mathematical model of consonance, representing temperaments as graphs, and analyzes the impact of electronic nonlinearities on musical perception.

Findings

Model aligns well with observed properties of scales and chords

Graph representation reveals two main groups of temperaments

Nonlinear electronic amplification significantly alters consonance patterns

Abstract

After briefly revising the concepts of consonance/dissonance, a respective mathematic-computational model is described, based on Helmholtz's consonance theory and also considering the partials intensity. It is then applied to characterize five scale temperaments, as well as some minor and major triads and electronic amplification. In spite of the simplicity of the described model, a surprising agreement is often observed between the obtained consonances/dissonances and the typically observed properties of scales and chords. The representation of temperaments as graphs where links correspond to consonance (or dissonance) is presented and used to compare distinct temperaments, allowing the identification of two main groups of scales. The interesting issue of nonlinearities in electronic music amplification is also addressed while considering quadratic distortions, and it is shown that such nonlinearities can have drastic effect in changing the original patterns of consonance and dissonance.

Tables

Table 1. Table 1 : The intervals characteristic of tonal music, with respective ratios and types of consonance/dissonance.

interval	just ratio	just cents	type of cons.
unison	1/1	0	absolute cons.
perfect octave	1/2	111.73	absolute cons.
fifth	2/3	203.91	perfect cons.
forth	3/4	315.64	medium cons.
major sixth	3/5	386.31	medium cons.
major third	4/5	498.04	medium cons.
minor third	5/6	582.51	imperfect cons.
minor sixth	5/8	701.96	imperfect cons.
major second	8/9	813.69	diss.
major seventh	8/15	884.36	diss.
minor seventh	9/16	996.09	diss.
minor second	15/16	1088.27	diss.
tritone	32/45	1200	diss.

Equations

$$x=a_{1}cos({\theta }_1)+a_{2}cos({\theta }_2)$$

$$y=a_{1}sin({\theta }_1)+a_{2}sin({\theta }_2)$$

$$a=\sqrt{x^2+y^2}$$

$$\theta =atan\left(\frac{y}{x}\right).$$

$$a_{1}cos(\omega t+{\theta }_1)+a_{2}cos(\omega t+{\theta }_2)=acos\left(\omega t\theta \right).$$

$$cent({\omega }_2,{\omega }_1)=cent(f_2,f_1)=1200lo{g}_2\left(\frac{w_2}{w_1}\right).$$

$$s(t)=a_{1}cos({\omega }_{1}t+{\theta }_1)+a_{2}cos({\omega }_{2}t+{\theta }_2).$$

$$consonance(X,Y)=\sum _{cons.(i,j)}M(i)M(j)\delta f_{i,j}$$

$$dissonance(X,Y)=\sum _{diss.(i,j)}M(i)M(j)\delta f_{i,j}$$

$$s(t)=a_{1}cos({\omega }_{1}t)+a_{2}cos({\omega }_{2}t)+a_{3}cos({\omega }_{3}t)+a$$

$$s^2(t)=a_3^{2}cos(2{\omega }_{3}t)+a_2^{2}cos(2{\omega }_{2}t)+2a_1^{2}cos(2{\omega }_{1}t)+$$

$$+2a{a}_{3}cos({\omega }_{3}t)+2a{a}_{2}cos({\omega }_{2}t)+2a{a}_{1}cos({\omega }_{1}t)+$$

$$+2a_2{a}_{3}cos(({\omega }_2+{\omega }_3)t)+2a_2{a}_{3}cos(({\omega }_2-{\omega }_3)t)+$$

$$+2a_1{a}_{3}cos(({\omega }_1+{\omega }_3)t)+2a_1{a}_{3}cos({\omega }_1-{\omega }_3)t)+$$

$$+2a_1{a}_{2}cos(({\omega }_1+{\omega }_2))+2a_1{a}_{2}cos(({\omega }_1-{\omega }_2)t)+$$

$$+5a^2+a_1^2+a_2^2$$

Full text

‘Geometry is in the humming of strings.’

Pythagoras.

1 Introduction

Sound and music are intrinsically human concepts and activities. In other words, there are no definitions to be found in nature regarding what sound and music are, or how they should be. These definitions and properties are relative to us, humans. As such, these concepts are inherently subjective, and they may vary in time and space. Indeed, the history of music has continuously witnessed big changes in instruments, styles, and theories, contributing to ever increasing diversity of results.

Nevertheless, there is an aspect somewhat stable in sound and music, and this has to do with consonance and dissonance. These concepts have permeated the history of tonal music, while also playing important roles in many other musical traditions. Basically, consonance has to do with our sensation of ‘harmony’ between two simultaneous sounds, or notes. As such, consonance/dissonance is at the very basis of sound combinations along time and space, which underlies music .

One of the first mathematical approaches to consonance was developed by Pythagoras (e.g. [1]), by considering ratios between the length of strings. Afterwards, the historic records mostly go back to monophony, especially Gregorian chant. With time, drones were added, initiating a very simple kind of harmony. Subsequent developments would lead to two parallel voices, contrary/oblique voices, and then to ever increasing types of polyphony. With the incorporation of more and more voices, the problem of combining sounds in time and space became ever more critical.

Consonance has played an important role in many musical traditions. In Indian classical music (e.g. Bharata), for instance, discussions on consonance seem to go back at least to the $VIII$ century $bC$, involving the concepts of vadi (‘sonant’), samvadi (‘consonant’), vivadi (‘dissonant’) and anuvadi (‘assonant’) (e.g. [2])

To make a long (and interesting) history short, the progression to polyphony followed, or even implied, the development of harmony and counterpoint, which can be informally understood as the activity of combining simultaneous or subsequent sounds, respectively (e.g. [3, 4]). One of the first more formal approaches to consonance was developed by J. P. Rameau, who argued strongly against the current view overlooking differences between consonance and dissonance. To him, consonance was a kind of repose, so that dissonance would imply a depart from that state (e.g. [5]).

Because of its central role in sound and music, consonance motivated several theories, such as based on interval ratios or interaction between harmonics/partials. In particular, the latter was developed mainly by H. L. F. Helmholtz (1821–1894). Backed by careful experiments, Helmholtz advanced the idea that consonance derives from interrelationships between partials, producing constructive combination of frequencies or, alternatively, potentially unpleasant beats (e.g. [6]).

Recall that, given a fundamental tone, the accompanying higher frequency tones are called partials. In case these partials follow the harmonic series (i.e. string wavelengths of $1,1/2,1/3,1/4,\ldots$), they are called harmonics.

In this text (a previous version of which appeared recently [7]), we approach the interesting issues of consonance/dissonance by considering Helmholtz’s respective theory. More specifically, we derive a simple mathematic-computational model (e.g. [8]) capable of, given two sounds, providing degrees of consonance and dissonance. In addition to considering the frequencies of the involved partials, the respective amplitudes are also taken into account their magnitudes. Then, we use this model to revisit some of the main temperaments (informally speaking, ways of assigning frequencies to notes) as well as some aspects of basic harmony, especially triad consonance.

The interesting and important issue of nonlinearity in electronic amplification is also addressed with respect to quadratic effects, and the results show that nonlinear amplification can substantially change the original patterns of consonance and dissonance found in the original sounds.

2 On the Mixture of Sinusoidals with a Same Frequency

A first interesting issue to be addressed regards what happens when two sinusoidal sounds with a same frequency $\omega=2\pi f$, but distinct amplitudes ($a_{1}$, $a_{2}$) and phases ($\theta_{1}$, $\theta_{2}$), are combined, i.e. $a_{1}cos(\omega t+\theta_{1})+a_{2}cos(\omega t+\theta_{2})$.

This interesting question can be simply addressed by considering the phasor representation (e.g. [9]) of these two sounds, respectively $a_{1}e^{i\theta_{1}}$ and $a_{2}e^{i\theta_{2}}$, as illustrated in Figure 1.

Phasor representation allows us to add, in complex or vector fashion, the involved phasors, resulting the phasor shown in red in Figure 1, which has real and imaginary parts, as well as its magnitude $a$ and phase $\theta$, easily calculated as

[TABLE]

.

So, we have that

[TABLE]

Mixtures of sines and cosines can also be handled by using Equation 1 while recalling that $sin(\alpha)=cos(\alpha-\pi/2)$

Thus, the linear combination of any number of sines and cosines with the same frequency, no mattering how different are their magnitudes and phases are, will always yield a simple cosine with effective magnitude $a$ and phase $\theta$ as a result. In the next section we show that more interesting results arise when linearly combining two or more sinusoidals with different frequencies.

3 Mixture of Sinusoidal Signals with Distinct Frequencies

Given two sinusoidal tones with respective distinct frequencies $\omega_{1}=2\pi f_{1}$ and $\omega_{2}=2\pi f_{2}$, $\omega_{1}<\omega_{2}$, it immediately becomes interesting to quantify the relationship between these frequencies. One possibility is in terms of their ratio $f_{2}/f_{1}$, and another is in terms of the measurement called cent, which is defined as

[TABLE]

Observe that 100 cents equate a equally-tempered half-tone, corresponding to each subsequent note along a scale.

Let’s sum these two signals as

[TABLE]

Again, the phasor representation turns out to be particularly useful in analysing this signal addition. Figure 2 illustrates this situation.

Because the two frequencies are now different, for simplicity’s sake we can keep the slower-rotating phasor (shown in blue) parallel to the real axis. Observe that the original phases $\theta_{1}$ and $\theta_{2}$ are therefore lost, but this does not matter for our following analysis.

Observe that the second phasor (shown in red) now rotates with angular velocity $\delta\omega=\omega_{2}-\omega_{1}$. So, as it moves, the magnitude of the resulting phasor (projection onto the real axis) varies from a maximum value of $a_{1}$, for $\delta\omega=0$, to its smallest value, observed at delta $\delta\omega=\pi$ . This oscillating effect on the amplitude takes place with angular frequency $\omega_{2}-\omega_{1}$. Figure 3 illustrates the amplitude oscillation produced by the addition of two signals with respective frequencies $f_{1}=440Hz$ and $f_{2}=450Hz$ and identical unit magnitudes, so that a beat frequency of $10Hz$ is obtained.

It is interesting to observe that, though there is a difference of angular velocities, the sum of the two signals only incorporates the two original frequencies $\omega_{1}$ and $\omega_{2}$, so that $\omega_{2}-\omega_{1}$ does not belong to the respective frequency spectrum. This can be readily appreciated by taking into account this addition in the Fourier domain, where only these two frequencies can be present.

Now, we come to an interesting issue: the way in which humans perceive this frequency difference also depends on its magnitude. For $\delta\omega$ very small, (e.g. $<2Hz$, approximately), we tend to perceive the two sounds as a single, tuned note. For slightly larger difference values (typically $10Hz>\delta\omega\geq 2Hz$), a single frequency with a relatively slow varying amplitude tend to be heard. Indeed, difference in this range are smaller than our minimal frequency limit of $20Hz$.

For larger values of $\delta\omega$ (e.g. $60Hz>\delta\omega>10$), the added sinusoidal signals tend to cause an unpleasant sensation, as the resulting sound cannot be distinguished between a single or double frequency stimulus. The two latter situations are often described as the beat effect, and are typically avoided in music and instrument construction and tuning. Recall that this phenomenon, jointly with tuned sounds, provided some of the main motivation for Helmholtz’s consonance theory.

The situation where $\delta_{\omega}>60Hz$ typically leads to two separate sounds, which are typically called intervals. We will discuss this concept further in the next section.

4 Intervals

Table 1 presents the main intervals often considered in tonal music, as well as the respective just ratio (in cents) and equal-temperament counterpart. The type of consonance typically perceived for each case is also provided in the last column of this table.

It should be observed that this classification of consonances is not definitive, and some of these intervals have been perceived differently along time and space. This may have to do with the fact that consonance, as we will see further on this text, depends on the partial features of each type of sound, as well as on the type of assumed temperament.

The just intonation, commonly used to tune some types of instruments, represents the basis for some temperament systems, such as the pythagorean.

Going back to the previous table, we have that only three or four intervals – namely those corresponding to unison, octave, and fifth – are understood as presenting more definite consonance. It is thus hardly surprising that the fifth, and to a lesser extent the forth, represent such an important reference in occidental music composition.

Four intervals are understood as having intermediate consonance, and the remainder five are typically understood to be dissonant, with emphasis on the tritone (e.g. $C-F\#$).

Classification of consonance/dissonance has traditionally relied on relatively subjective human appreciation. In the following we will describe Helmholtz’s experimental/mathematical approach to consonance and then consider it for developing a simple, and yet relatively effective, model capable of automatically estimating degrees of consonance and dissonance.

5 Helmholtz’s Theory of Consonance, and a Simple Model

Helmholtz’s theory of consonance was presented mainly in the second part of his Sensations of Tone (e.g. [6]). The two involved sounds are understood in terms of their respective harmonic series, and these are pairwise compared in order to identify respective tuning and beats. The resulting sensation of consonance/dissonance would be related to the intensity in which these constructive interactions and beats occur given each the two original sounds. Interestingly, we have that two pure sounds (i.e. without partials) could hardly be compared regarding their consonance.

Here, we consider Helmholtz’s main concepts to develop a simple mathematic-computational model, capable of estimating in a quantitative way the consonance/dissonance between two sounds $s_{1}(t)$ and $s_{2}(t)$ with respective fundamental frequencies $f_{0,1}$ and $f_{0,2}$.

A first important issue regards the frequency distribution of partials. For simplicity’s sake, and without loss of generality, we assume that these follow the harmonic series, leading to partials of $f_{0},2f_{0},3f_{0},4f_{0},\ldots$. This is not to say that real sounds necessarily follow this structure, but other partial models can be immediately considered in the described model.

As there are virtually infinite harmonics, we also have to delimit the frequency extent of these sounds. For this purpose, we consider the fact that the partials of most sounds decay steadily with frequency (and also along time), so that too high harmonic partials result inaudible. For simplicity’s sake, we consider that the partials decay in terms of a negative exponential. Figure 4 illustrates the adopted partial decay with respect to the frequency index $i$ corresponding to $f_{0}=1Hz$.

Though other configurations can be adopted, we henceforth assume a partial decay scheme as in Figure 4, incorporating a maximum of $N=50$ harmonic partials (observe that the limits of human frequency perception, varying roughly as $20Hz\leq f\leq 20kHz$, should also be considered).

Now, given two signals represented in terms of their partial content, we need to devise some means to compare these partials in pairwise fashion and then estimate, for each case, the respective degrees of consonance and dissonance.

Figure 5 illustrate the spectrum of two sounds to be compared $X(f)$ and $Y(f)$ (red and blue, respectively). Observe that $X(f)$ has higher fundamental frequency, implying in more widely spaced partial intervals as well as effective frequency extent.

Let’s assume the lower fundamental signal $Y(f)$ to be our reference. For each of its partials, with respective frequency $f_{i}$, the closest partial $f_{j}$ belonging to the other signa, $X(f)$, is identified. The frequency difference $\delta f_{i,j}=|f_{j}-f_{i}|$ is then taken, being then classified as: (i) consonance if $\delta f_{i,j}<f_{c}$; (ii) dissonance if $f_{c}\leq\delta f_{i,j}<f_{d}$; and (iii) neutral for $\delta f_{i,j}\geq f_{d}$.

So, many consonance or dissonance matches, as well as several neutral cases, can be identified for a same pair of sounds. In order to derive an overall degree of consonance and dissonance between the two original tones $X$ and $Y$, we can respectively apply

[TABLE]

Observe that the sum takes place over the pairs $(i,j)$ of consonant harmonic partials in the case of Equation 3, and dissonant pairs of partials in Equation 4. Other ways of combining the partial relationships are possible and can be immediately adapted into the developed model.

Number theory, as well as the convolution mathematical operation, can also be applied to identify the potential consonances and dissonances arising from distinct partial distribution and specific sound intervals.

So, we end up with a model that, as illustrated in Figure 6, is capable of automatically assigning degrees of consonance or dissonance to pairs of presented sounds represented in terms of partials and magnitudes (spectral).

Observe that, according to the adopted assumptions, the same pair of sounds (interval) has intrinsic levels of both consonance and dissonance (this happens rarely, though). It is also possible to adopt a threshold for classifying an interval as being consonant/dissonant. In the following we adopt 5 and 4 as thresholds for consonance and dissonance. An exponential decay of $0.08$, as well as $f_{c}=10Hz$ and $F_{d}=60Hz$ have been also considered, as these tended to yield more appropriate results.

Some interesting implications can be observed about the above outlined approach to consonance. First, we have that consonance tends to decrease for larger difference of fundamental frequencies because of resulting smaller overlap between the partials. Then, we also have that even linear filtering can influence the resulting consonance/dissonance, as a consequence of changes in the partial amplitudes. So, for instance, low-pass filtering can potentially contribute to reduction of dissonance as a consequence of the respectively implied attenunation of higher partials.

All in all, we also have that consonance/dissonance ultimately depends on a series of factors, including temperament type, distribution of partials (other than harmonic series), and different decay profiles of partial magnitudes.

Nevertheless, the developed model allows us, to some limited degree of precision that also depends on the configuration of the involved parameters and roles (which can be adapted), to study several issues in sound and music. In the remainder of this text, we apply the simple developed model to investigate temperaments and basic harmony, especially triads.

6 Temperaments

Because scales are so important in music, several such systems have been proposed along centuries. Here, we will be limited to five representative cases: (i) equal temperament; (ii) pythagorian; (iii) just major; (iv) mean-tone; and (v) Weckmeister.

Figure 7 illustrates the bipartite graph of consonance (blue) and dissonance (red) relationships between the 12 tones. This scale is characterized by having fixed relationship between each subsequent half-tone, in the sense that $f_{i+1}/f_{i}=2^{1/12}$. Therefore, any minor or major modes will be characterized by identical intervals, providing uniform subsidy for modulations and transpositions.

The effect of the uniform interval characteristic of this type of scale becomes evident in this figure. Except for the fact that the consonance tends to change as one of the two frequencies becomes larger than the other, the obtained consonance/dissonance patterns are mostly similar for every note, confirming the properties often expected from this scale. However, observe that the interval relationships are mostly dissonant, with the main exceptions of the major forth and major fifth. So, interval uniformity is achieved at the expense of consonance.

The consonance/dissonance results obtained for the pythagorean scale are shown in Figure 8(a). This scale is founded on the $2/3$ ratio interval, corresponding to the pure perfect fifth, possibly the third most consonant interval after the unison and octave.

Interestingly, the consonance/dissonance patterns turn out to be very similar to those obtained for the equal temperament, but the consonances related to the major forth and fifth now alternates for almost every node, sometimes vanishing or being exchanged for other intervals.

Figure 8(b) presents the consonance/dissonance patterns obtained for the mean-tone temperament. The presence of a substantially increased number of consonances can be readily observed. For instance, most of the intervals become consonant in the case of $C$. The intervals involving $G$, an important tone, also tend to be consonant. However, the aforementioned effect is achieved at the expense of consonance uniformity. Indeed, the intervals based on $C\#$ are mostly dissonant, but they become slightly more consonant for $D$. It is also interesting to observe the variation of the intensities of the dissonances, represented by the edges width. Particularly strong dissonances are observed for the intervals $D-D\#$ and $D\#-E$. These result are in agreement with the fact that the mean-tone will provide best results for the first modes, e.g. $C$-major and $G$-major, becoming less suitable for further mode transpositions.

An even more consonant overall pattern is observed for the just major scale, depicted in Figure 8(d). Indeed, this results is similar to that observed for the mean-tone case.

The consonance/dissonance patterns obtained for the intervals of the Werckmeister temperament, shown in Figure 8(e) seem to be similar to those obtained for the equal temperament and pythagorean.

7 Scales as Graphs

Visualizing the consonances as graphs (e.g. [10]) where each node corresponds to a single node can provide further insights about the properties and relationships between diverse temperaments. Figures 9 and 10 present, respectively, the consonance and dissonance graphs regarding the results discussed in the previous section.

Remarkably, this type of visualization allows a nice complementation of the information provided by the previously considered bipartite interrelationships. Now, it becomes evident that we have two main groups of scales: one similar to the equal temperament (also including the pythagorean and Werckmeister), and the other corresponding to the mean-tone and just major.

The fact that the mean-tone and just major scales tend to promote overall consonance is directly reflected in the greater interconnectivity exhibited by the respective graphs. The more uniform interval distributions underlying the other three types of scales define a chain of consonances between several notes, with a cluster of connections being observed between $C$, $G$, $C2$, and $F$.

In the Werckmeister temperament, this cluster also includes $A$ and $E$. The consonances between non-adjacent notes in these three temperaments tend to be little consonant, with the pythagorean scale providing possibly improved consonance in this case.

8 Triads and Harmony

Harmony, the art of combining simultaneous sounds, is largely related to the consonance/dissonance properties characteristic of the involved intervals. The minor (major) ds, corresponding to the basis tone plus the minor (major) triad and fifth, provides one of the most important foundations of traditional harmony.

Let’s also apply the developed simple model of consonance/dissonance to characterize the first minor and major triads obtained for the considered five temperament types. Figure 11 present the respective consonance (left part of the table) and dissonance (right portion) levels.

The intrinsic uniformity of the equal temperament approach is evident from these results, characterized by constant horizontal intensities. The other cases are characterized by the respective triads having varying consonance/dissonance patterns. This would be reflected in a different effects while transposing or modulating along a music piece played according to these temperaments.

9 Electronic Amplification

Let’s us now illustrate the application of the suggested consonance/dissonance model to the characterization of electronic amplification (e.g. [11, 12, 13, 14]). Basically, an electronic amplified applies a mapping $y(t)=f\left[s(t)\right]$ of the input signal $s(t)$. Ideally, in order to avoid distortion, the mapping $f\left[s(t)\right]$ should correspond to be linear (e.g. $f\left[s(t)\right]=As(t)$, for some gain $A$)

However, it is virtually impossible to obtain a completely linear amplification, so that we have to cope with some level of nonlinearity. Here, we consider quadratic nonlinearity, i.e. $f\left[s(t)\right]=\left[s(t)\right]^{2}$. Unlike what happens in linear amplification, new frequencies corresponding to sums and subtractions of the original frequencies, are obtained as part of the resulting amplified signal $y(t)$. In case the input signal contains three frequencies $\omega_{1}$, $\omega_{2}$, and $\omega_{3}$, as shown in Equation 5, the output will be as given in Equation 9.

[TABLE]

[TABLE]

The new frequencies induced by the nonlinearity provide new opportunities for obtaining consonance and dissonance, so it is interesting to compare the effect of the nonlinearity on the resulting sound. Figure 12 depicts the quadratic amplification curve adopted in this section. The level of nonlinearity can be controlled by selecting different biasing intensities, so that the linearity increases with this parameter, i.e. $L=a$.

Figure 13 shows the output signals obtained for the input signal $s(t)=cos(\omega_{1}t)+1/3\,cos(\omega_{2}t)+1/5\,cos(\omega_{3}t)+a$ considering the three involved frequencies as corresponding to the just major fundamental, major third and major fifth.

The effects of nonlinearity are evident in the first two or three cases (i.e. $L=0,1and2.5$), and are much less perceptible in the other cases. However, the consonance graphs shown in Figure 14 reveal that the effects of the nonlinearity proceed until the last case (i.e. $L=50$), where the resulting simulated sound is nearly identical to what would be otherwise obtained by using a fully linear amplification.

If follows from these simulations that amplification nonlinearity can substantially chance the consonance (similar results were observed for the dissonances, not shown here) relationships between pairs of notes. For instance, the interval $D-G$, which exhibit a considerable consonance originally, results disconnected for the largest dissonance case $a=0$. Even more pronounced effects are obtained as a consequence of the introduction of several harmonics not present in the original signal, which give rise to new consonances and dissonances.

10 Concluding Remarks

We have briefly addressed the interesting issues of consonance and dissonance in sound and music, with special focus on Helmholtz’s consonance approach. A simple model was developed that allowed us to quantify the levels of consonance and dissonance between two tones according to comparisons of their respective partials.

The potential of this approach was then illustrated with respect to an analysis of temperaments and triads, and interesting results were obtained that are often in agreement with what is commonly believed. In particular, the visualization of consonance relationships estimated for five types of temperaments resulted to be particularly useful for identifying specific features and comparing different scales. The potential of the suggested consonance/dissonance model was also illustrated with respect to the characterization of the strong effects that nonlinear electronic amplification can have on changing the patterns of consonance and dissonance of the amplified sounds.

It should be kept in mind that the obtained results are preliminary and dependent on the model parameter configurations. Substantial more efforts need to be developed in further assessing the simple proposed approach, and new results should be reported in the future.

We believe that, to any extent, this work motivated the potential of consonance/dissonance models with respect to sound and music aspects.

In addition to the study of temperaments and triads, it would be also possible to consider a large number of possible applications including: provide subsidy to design of scales and composition, musical instrument construction and characterization, acoustics, design and characterization of electronic amplifiers considering other types of nonlinearities, and filters, to name but a few possibilities.

It would be also promising to try to quantify the intrinsic sound consonance of a single note by considering interval relationships between its harmonics through an adaptation of the described modeling approach.

Acknowledgments.

Luciano da F. Costa thanks CNPq (grant no. 307085/2018-0) for sponsorship. This work has benefited from FAPESP grant 15/22308-2.