# Melodies as Maximally Disordered Systems under Macroscopic Constraints with Musical Meaning

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Microscopic Representation and Macroscopic Observables of Intervals

#### 2.1. Interval Size and Its Relation to the Fundamental Frequency of Pitches

- In the just scale, $b=12.0040\pm 6.8\times {10}^{-3}$ with a determination coefficient ${R}^{2}\approx 1$;
- In the Pythagorean scale, $b=11.9767\pm 4.9\times {10}^{-3}$ with ${R}^{2}\approx 1$;
- In the 12-TET, $b=12$.

- For the just scale, $c=2.632\times {10}^{-2}\pm 1.52\times {10}^{-4}$ with a determination coefficient ${R}^{2}=0.998$;
- For the Pythagorean scale, $c=2.642\times {10}^{-2}\pm 1.55\times {10}^{-4}$ with ${R}^{2}=0.998$;
- For the 12-TET scale, $c=2.635\times {10}^{-2}\pm 1.48\times {10}^{-4}$ with ${R}^{2}=0.998$.

#### 2.2. Expected Values with Musical Meaning

#### 2.3. Transposition Process

#### 2.4. Distinguishability of Pairs of Pitches

## 3. Connection with Tonal Consonance

#### 3.1. Measuring the Dissonance Levels of Intervals

#### 3.2. Expected Values of the Dissonance Levels Associated to Intervals

## 4. Melody and Expected Values of Melodic Intervals

#### 4.1. Concerning Melody

#### 4.2. Expected Values of Melodic Intervals

## 5. Materials and Methods: An Application to Melodic Lines

#### 5.1. Selection of Melodic Lines

- Brandenburg Concerto No. 3 in G Major BWV 1048. Johann Sebastian Bach: Polyphonic concerto for 11 musical instruments (three violins, three violas, three cellos, violone, and harpsichord).
- Missa Super Dixit Maria. Hans Leo Hassler: Polyphonic composition for four voices (soprano, contralto, tenor, and bass).
- First movement of the Partita in A Minor BWV 1013. Johann Sebastian Bach: This piece has just one melodic line for a flute.
- Piccolo Concerto RV444. Antonio Vivaldi (arrangement by Gustav Anderson): We selected the piccolo melodic line, owing to its rich melodic content.
- Sonata KV 545. Wolfgang Amadeus Mozart: We selected the melodic line for the right hand of this piano sonata, assuming that it drives the melodic content.
- Suite No. 1 in G Major BWV 1007 and Suite No. 2 in D Minor BWV 1008. Johann Sebastian Bach: The melodic lines of these pieces written for cello contain mainly successive pitches. In the cases of the few simultaneous pitches, the continuation of the melodic lines was assumed in the direction of the highest pitch.

#### 5.2. Procedure to Obtain the Probability and the Cumulative Distributions

- The MIDI files were generated from scores. Only successive pitches without rests between them were considered.
- The MIDI information was transformed into frequencies using the 12-TET scale with $A=440$ Hz. Supplementary Spreadsheet S1 contains the data ${f}_{t}$ and ${f}_{t+1}$ in Hz, corresponding to the melodic intervals of each melodic line.
- The PDs were obtained in three different cases:
- -
- Case 1: $|{f}_{t+1}-{f}_{t}|$ and $|{f}_{t+1}^{2}-{f}_{t}^{2}|$ not distinguishing between ascending and descending intervals. The complementary cumulative distribution (CCD) was also obtained.
- -
- Case 2: $|{f}_{t+1}-{f}_{t}|$ and $|{f}_{t+1}^{2}-{f}_{t}^{2}|$ for two different sets of intervals: Ascending and unisons, and descending and unisons. The CCD was also obtained for each set.
- -
- Case 3: ${f}_{t+1}^{2}-{f}_{t}^{2}$ for the set of ascending, descending, and unison intervals together. In this case, the sign of the descending intervals was considered as negative. The reason for only using the quantity ${f}_{t+1}^{2}-{f}_{t}^{2}$ is the quality of the experimental fits obtained in the two previous analyses for both quantities, and even more relevantly that the distinguishability analysis shows that ${f}_{t+1}^{2}-{f}_{t}^{2}$ has the best resolution properties for the case of 24 semitones in the 12-TET scale (see Table 1), which is the relevant range for melodic intervals in the analyzed melodic lines. The CCD was employed for the branch of the PD that contains the ascending intervals, and the cumulative distribution (CD) was utilized for the branch that contains the descending intervals.

- Because the number of melodic intervals in the studied melodic lines is at most one order of magnitude larger than the total number of possible pairs of successive pitches generated by the same ambitus (the range between the lowest and highest pitches) of the original melodic line, the PDs were constructed using histograms, in order to capture significant probabilities. Supplementary Table S3 shows the number of intervals of each melodic line, the number of ascending intervals, descending ones, and unisons, and the corresponding ambitus.
- As the number of possible melodic intervals for any melodic line is finite, independently of its length, the bin width in the histograms will be moderately dependent on the number of melodic intervals. This condition is satisfied by the Sturges criterion [45], and thus, this criterion was used to determine the bin width.
- In the third case, when ascending and descending PDs were combined in the same distribution for the quantity ${f}_{t+1}^{2}-{f}_{t}^{2}$, the bin width was taken as the average of those obtained separately using the Sturges criterion for ascending and descending distributions. The average bins were symmetrically located to the left and right, starting from the point ${f}_{t+1}^{2}-{f}_{t}^{2}=0$.
- In the experimental analysis, the contribution of unisons in the histograms is important for ascending intervals as well as descending ones, with different right-hand and left-hand limits at 0. In addition, if we attempt to split the unisons into the ascending and descending parts, this procedure reduces the determination coefficient ${R}^{2}$ of the fits for the histograms to an exponential function [46]. Hence, all unisons were included in the ascending part as well as the descending one, and then a correction of this double count was carried out in the procedure to obtain the expected values. In the histograms, the descending intervals are contained inside the bins labeled from 1 to $N/2$ (from left to right), and the ascending ones inside those labeled from $N/2+1$ to N (from left to right). Hence, all unisons have been taken into account inside the bin labeled $N/2$ as well as that labelled $N/2+1$. Notice that N is an even number.

## 6. Results and Discussion

#### 6.1. Experimental Results and Analysis

#### 6.2. Shannon Entropy of Melodic Intervals in Melodic Lines

#### 6.3. Statistical Model for Melodic Lines: Relative Entropy Minimization under Macroscopic Constraints

- Different registers of musical instruments and human voices can be distinguished using the Lagrange multiplier ${\lambda}_{1}$, allowing, for example, to discriminate between the same melodic line played in different parts of the register (a transposition). An example of a transposition is given in the Brandenburg Concerto No. 3 BWV 1048 by J. S. Bach, in which the harpsichord plays the same melodic line as the violone but transposed one octave higher (the fundamental frequency ratio of the transposition is equal to 2): While the entropy evolution in these melodic lines is the same, there is a change in the exponential decay parameters, characterized by the values of the Lagrange multipliers (see Table 2), and the numerical values of the expected values are related as:$$\begin{array}{c}{\langle \left|\epsilon \right|\rangle}_{Harpsichord}={2}^{2}{\langle \left|\epsilon \right|\rangle}_{Violone}\\ \langle |{f}_{t+1}^{2}-{f}_{t}^{2}{|\rangle}_{Harpsichord}={2}^{2}\langle |{f}_{t+1}^{2}-{f}_{t}^{2}{|\rangle}_{Violone}\\ {\left[\langle {\epsilon}_{>0}\phantom{\rule{4pt}{0ex}}\rangle +\langle {\epsilon}_{<0}\rangle \right]}_{Harpsichord}={2}^{2}{\left[\langle {\epsilon}_{>0}\phantom{\rule{4pt}{0ex}}\rangle +\langle {\epsilon}_{<0}\rangle \right]}_{Violone}\\ {\left[\langle {({f}_{t+1}^{2}-{f}_{t}^{2})}_{>0}\rangle +\langle {({f}_{{t}^{\prime}+1}^{2}-{f}_{{t}^{\prime}}^{2})}_{<0}\rangle \right]}_{Harpsichord}={2}^{2}{\left[\langle {({f}_{t+1}^{2}-{f}_{t}^{2})}_{>0}\rangle +\langle {({f}_{{t}^{\prime}+1}^{2}-{f}_{{t}^{\prime}}^{2})}_{<0}\rangle \right]}_{Violone},\end{array}$$
- With respect to the quantitative results of the model, the orders of magnitude of the fit parameters of the statistical model are in agreement with the corresponding results of the experimental fits. For each melodic line, Supplementary Table S8 contains the fit parameters to discontinuous asymmetric Laplace distributions, generated from the statistical model results. The average relative error in the histograms for the amplitude of the exponential distributions is $17.1\%$, and that for the decay coefficient is $20.6\%$. In the cases of the CD and CCD, the average errors of the amplitude and the decay coefficient are $7.2\%$ and $11.8\%$, respectively. Supplementary Table S9 contains the values of these errors for each melodic line.
- In most cases ($90\%$ of the melodic lines), Equation (35) takes positive values (corresponding to negative values of ${\lambda}_{2}$), and ${\tilde{p}}_{a}-{\tilde{p}}_{d}$ takes negative values (see Supplementary Table S3). This behavior is consistent with the asymmetry represented in Figure 4, in the sense that the magnitudes of ascending intervals are expected to be larger than those of descending ones, and the total number of descending intervals must be larger than that of ascending ones. Negative values of ${\tilde{p}}_{a}-{\tilde{p}}_{d}$ and ${\lambda}_{2}$ lead to different decay coefficients and different intercept points with the ordinate axis for the ascending and descending branches, which can be observed in the experimental fits of the CD and CCD through the comparison of the corresponding coefficients, ${F}_{+}^{C}<{F}_{-}^{C}$ and ${G}_{+}^{C}>{G}_{-}^{C}$ (see Supplementary Table S5). Figure 6 was created with the purpose of magnifying these particular asymmetries: ${P}_{1}>{P}_{2}$ and ${\alpha}_{1}>{\alpha}_{2}$ (implying that ${\lambda}_{2}<0$). The two exceptions are the Piccolo Concerto RV444 of Antonio Vivaldi, where ${\lambda}_{2}>0$ and ${\tilde{p}}_{a}-{\tilde{p}}_{d}>0$, and the melodic line of the tenor voice in Missa Super Dixit Maria, where ${\lambda}_{2}>0$ and ${\tilde{p}}_{a}-{\tilde{p}}_{d}<0$.
- Because the difference between ${\lambda}_{1}$ and ${\lambda}_{2}$ is between one and two orders of magnitude (i.e., the decay coefficients have the same order of magnitude), and the bin width selection affects the measure of the decay parameters, the asymmetry in the values of the decay coefficients is better observed in the cumulative distributions than in the histograms.
- Because in Figure 6, the limit ${P}_{1}$ of the CD (constructed for descending intervals) when ${f}_{t+1}^{2}-{f}_{t}^{2}\to {0}^{-}$ represents the probability of a value slightly smaller than 0, and in the CCD (constructed for ascending intervals), ${P}_{2}$ when ${f}_{t+1}^{2}-{f}_{t}^{2}\to {0}^{+}$ represents the probability of a value slightly larger than 0, the asymmetry ${\tilde{p}}_{a}-{\tilde{p}}_{d}\approx {P}_{2}-{P}_{1}$. This result can be observed in Figure 9 and represents the difference in the amplitudes of the exponential decay for the CD and CCD. In most cases, except for the Piccolo Concerto RV444, ${\tilde{p}}_{a}<{\tilde{p}}_{d}$, implying that ${P}_{1}>{P}_{2}$. In the case of the Piccolo Concerto RV444, it holds that ${\tilde{p}}_{a}>{\tilde{p}}_{d}$, implying that ${P}_{1}<{P}_{2}$.

#### 6.4. Transposition Processes and Mean Dissonance Level of Melodic Lines

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Relation between musical scale parameters and the interval size for the (

**a**) just, (

**b**) Pythagorean, and (

**c**) 12-TET scales, with an interval size from –87 to 87 semitones (representing a typical piano). The linear fit corresponds to interval sizes between –24 and 24 semitones.

**Figure 2.**Relation between the quantities ${f}_{j}-{f}_{i}$ and ${f}_{j}^{2}-{f}_{i}^{2}$ and the magnitude of the interval size $\left|L\right|$ in semitones for ${f}_{j}>{f}_{i}$, shown in panels (

**a**,

**b**), respectively. The register corresponds to a typical 88 key piano. The upper branch comes from $j=88$ (highest pitch), and i varies from 88 to 1. The tuning comes from the frequency relation for the 12-TET scale with $A=440$ Hz.

**Figure 3.**Relation between the dissonance level D and the locations of harmonic intervals in the register $X=({f}_{j}+{f}_{i})/2$ for the 12-TET scale. The spectrum of each complex tone contains six harmonics with amplitudes falling at a rate of 0.88. Each possible size L corresponds to a particular frequency ratio inside the octave in the 12-TET scale. The dissonance level has been normalized to 1 for the typical register of an 88 key piano.

**Figure 4.**Asymmetry in the use of ascending and descending intervals in melody. Fragment from the Fugue in D major BWV 850, of The Well-Tempered Clavier, Book 1 of J. S. Bach that begins and ends with the pitch D (red boxes), with an ascending jump (blue box) compensated using several small descending intervals.

**Figure 5.**Probability distributions of melodic intervals for the following melodic lines: Violin 1, viola 1, cello 1, and violone from the Brandenburg Concerto No. 3 in G Major BWV 1048, the first movement of the Partita in A Minor BWV 1013, and the Suite No. 1 in G Major BWV 1007. (

**a**) Quantity ${f}_{t+1}^{2}-{f}_{t}^{2}$ measure using bins ($\epsilon $). (

**b**) Traditional melodic interval size L in semitones.

**Figure 6.**General forms of the probability and cumulative distributions $P(\epsilon )$ and $P({f}_{t+1}^{2}-{f}_{t}^{2})$, respectively. In the symmetric case, ${P}_{1}={P}_{2}$ and ${\alpha}_{1}={\alpha}_{2}$.

**Figure 7.**(

**a**) Comparison between the Probability distributions (PDs) for the real melodic line of the first movement of the Partita in A minor BWV 1013 by J. S. Bach and for the corresponding bin degeneration for the same ambitus. (

**b**) Comparison between histogram for the melodic line of Suite No. 2 BWV 1008 by J. S. Bach and that produced by the statistical model.

**Figure 8.**(

**a**) Evolution of the Shannon entropy of melodic intervals for different melodic lines. (

**b**) Evolution of the Shannon entropy of melodic intervals for the melodic lines of the soprano, in the Missa Super Dixit Maria, and Suite 2 BWV 1008 with the corresponding random melodies constructed using the same ambitus. The maximum Shannon entropy of melodic intervals ${S}_{max}$ corresponds to the maximum possible value of the Shannon entropy of melodic intervals in a long random melodic line with the same ambitus as the original one.

**Figure 9.**Complementary cumulative distribution (CCD) (ascending branches) and cumulative distribution (CD) (descending branches) for the empirical distributions (

**a**,

**c**,

**e**) and the corresponding statistical model results (

**b**,

**d**,

**f**). (

**a**,

**b**) Brandenburg Concerto No. 3 in G Major BWV 1048 by J. S. Bach, (

**c**,

**d**) Missa Super Dixit Maria by Hans Leo Hassler, and (

**e**,

**f**) Piccolo Concerto RV444 by Antonio Vivaldi; First movement of the Partita in A Minor BWV 1013 by J. S. Bach; Sonata KV 545 by W. A. Mozart; Suite No. 1 in G Major BWV 1007 by J. S. Bach and Suite No. 2 in D Minor BWV 1008 by J. S. Bach.

**Figure 10.**(

**a**) Power law relation between the quantity $\langle \left|\epsilon \right|\rangle $ and the Lagrange multiplier ${\lambda}_{1}$. (

**b**) Relation between the mean dissonance $\langle D\rangle $ and the Lagrange multiplier ${\lambda}_{1}$. For 13 of the 20 melodic lines, a linear relation was observed.

Scale | Up to 24 Semitones | Up to 87 Semitones | ||
---|---|---|---|---|

${\mathit{f}}_{\mathit{j}}-{\mathit{f}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{j}}^{\mathbf{2}}-{\mathit{f}}_{\mathit{i}}^{\mathbf{2}}$ | ${\mathit{f}}_{\mathit{j}}-{\mathit{f}}_{\mathit{i}}$ | ${\mathit{f}}_{\mathit{j}}^{\mathbf{2}}-{\mathit{f}}_{\mathit{i}}^{\mathbf{2}}$ | |

Just | 52 for $d\ge 4$ | 2 for $d\ge 5$ | 208 for $d\ge 4$ | 5 for $d\ge 8$ |

Pythagorean | 8 for $d\ge 4$ | 0 for $d\ge 5$ | 47 for $d\ge 5$ | 2 for $d\ge 8$ |

12-TET | 0 for $d\ge 5$ | 0 for $d\ge 4$ | 0 for $d\ge 5$ | 0 for $d\ge 8$ |

**Table 2.**Final Shannon entropy of melodic intervals ${S}_{f}$, maximum Shannon entropy of melodic intervals reached by each melodic line ${S}_{max}^{*}$, maximum Shannon entropy of melodic intervals generated by the ambitus of the corresponding melodic line ${S}_{max}$, Lagrange multipliers ${\lambda}_{1}$ and ${\lambda}_{2}$, mean dissonance level $\langle D\rangle $, and mean dissonance level approximated using the Taylor expansion up to second order (Equation (25)) ${\langle D\rangle}^{*}$. Melodic lines marked with “⋆” do not satisfy a linear relation between ${\lambda}_{1}$ and $\langle D\rangle $.

Melodic Line | ${\mathit{S}}_{\mathit{f}}$ | ${\mathit{S}}_{\mathit{max}}^{*}$ | ${\mathit{S}}_{\mathit{max}}$ | ${\mathit{\lambda}}_{1}(\times {10}^{-5})$ $\left[{\mathit{Hz}}^{-2}\right]$ | ${\mathit{\lambda}}_{2}(\times {10}^{-7})$ $\left[{\mathit{Hz}}^{-2}\right]$ | $\langle \mathit{D}\rangle $ $(\times {10}^{-1})$ | ${\langle \mathit{D}\rangle}^{*}$ $(\times {10}^{-1})$ |
---|---|---|---|---|---|---|---|

Violin 1 | 7.358 | 7.378 | 10.089 | 0.550 | −1.870 | 1.282 | 1.278 |

Violin 2 | 7.213 | 7.234 | 10.000 | 0.570 | −0.189 | 1.215 | 1.211 |

Violin 3 | 7.253 | 7.285 | 10.000 | 0.660 | −0.895 | 1.242 | 1.240 |

Viola 1 | 6.941 | 6.953 | 9.615 | 1.330 | −1.860 | 1.339 | 1.333 |

Viola 2 | 6.935 | 6.944 | 9.510 | 1.500 | −1.280 | 1.381 | 1.375 |

Viola 3 | 7.022 | 7.053 | 9.716 | 1.540 | −2.200 | 1.364 | 1.357 |

⋆ Cello 1 | 6.888 | 6.904 | 9.716 | 6.300 | −18.700 | 2.795 | 2.788 |

⋆ Cello 2 | 6.884 | 6.899 | 9.716 | 6.400 | −17.200 | 2.797 | 2.790 |

⋆ Cello 3 | 6.862 | 6.879 | 9.716 | 6.500 | −15.100 | 2.816 | 2.812 |

Violone | 6.779 | 6.796 | 9.716 | 30.000 | −34.000 | 4.900 | 4.917 |

⋆ Harpsichord | 6.779 | 6.796 | 9.716 | 7.400 | −4.200 | 2.596 | 2.598 |

Soprano | 5.055 | 5.082 | 8.340 | 1.940 | −2.850 | 1.470 | 1.470 |

Contralto | 5.247 | 5.313 | 8.644 | 3.250 | −6.800 | 1.591 | 1.591 |

Tenor | 5.443 | 5.491 | 7.615 | 5.100 | −6.500 | 1.893 | 1.893 |

Bass | 5.723 | 5.787 | 8.644 | 7.300 | 6.450 | 2.219 | 2.218 |

⋆ Suite 1 | 7.069 | 7.073 | 10.000 | 3.500 | −5.100 | 2.528 | 2.509 |

⋆ Suite 2 | 7.235 | 7.248 | 10.000 | 3.700 | −5.800 | 2.653 | 2.631 |

Mozart sonata | 6.923 | 6.935 | 10.644 | 0.490 | −1.520 | 1.353 | 1.357 |

First mov. Partita | 7.145 | 7.145 | 10.000 | 0.295 | −1.760 | 1.293 | 1.294 |

⋆ Piccolo concerto | 7.087 | 7.182 | 9.288 | 0.056 | 0.175 | 0.749 | 0.747 |

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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Useche, J.; Hurtado, R.
Melodies as Maximally Disordered Systems under Macroscopic Constraints with Musical Meaning. *Entropy* **2019**, *21*, 532.
https://doi.org/10.3390/e21050532

**AMA Style**

Useche J, Hurtado R.
Melodies as Maximally Disordered Systems under Macroscopic Constraints with Musical Meaning. *Entropy*. 2019; 21(5):532.
https://doi.org/10.3390/e21050532

**Chicago/Turabian Style**

Useche, Jorge, and Rafael Hurtado.
2019. "Melodies as Maximally Disordered Systems under Macroscopic Constraints with Musical Meaning" *Entropy* 21, no. 5: 532.
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