Computational Analysis of Selection and Mutation
Probabilities in the Evolution of Chord Progressions
Eita Nakamura1?
Kyoto University
eita.nakamura@i.kyoto-u.ac.jp

Abstract. We build a model of cultural evolution and study the properties ofthe process in which new chord progressions are repeatedly generated by referencing and modifying past chord progressions. As an extension of the models
for biological molecular evolution, this model represents a stochastic process inwhich references are selected from an accumulating pool of chord segments andnew chord segments are created by mutation including insertion, deletion, andsubstitution of chord symbols. We used a dataset of Japanese popular music andanalyzed this evolutionary process by inferring the model parameters. A number of suggestive results regarding the evolution of the creative culture were obtained, including a strong recency bias, large mutation rates and large dynamicchanges in mutation probabilities, and correlations between fluctuations and mutation probabilities and between the diffusedness of mutant chord segments andtheir mutation probabilities. Model-based predictions of new chord progressions
were also made.
Keywords: cultural evolution; evolutionary model; symbolic music processing;chord progression; prediction of evolution; accumulating artifact pool1

Introduction

Cultural development is a key aspect of human’s intelligence, and musical culture provides a fruitful venue for studying its creative role. Quantitative studies on music evolution have revealed some interesting macroscopic phenomena. These include directionalchanges (trends) in average features continuing for decades [1–3] or centuries [4–6],
punctuational short time periods with rapid changes [1, 4, 7], concurrent and transient
cluster structure [1, 3, 4], and frequency-dependent selection bias [5, 8]. Since individual musical pieces are produced by creators who learn to create music from previous?

The author thanks Hitomi Kaneko and Daichi Kamakura for useful discussions. This work wasin part supported by JSPS KAKENHI Grant Numbers 21K12187, 21K02846, and 22H03661,
and JST FOREST Program Grant Number JPMJPR226X.

This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0).

2Nakamuracreators or musical pieces, revealing the microscopic processes of knowledge transmission and modification is essential for understanding the mechanisms underlying these
phenomena [9].
Transmission processes of musical knowledge can be classified into two types, direct and indirect. In direct transmission, a song or other musical data serves as a reference and is replicated for producing a new one. For example, folk songs are typicallytransmitted in this way [10]. Some models of direct transmission of musical scale [11]or music sampling [8] have been proposed for testing musicological hypotheses. Direct transmission of music is also studied in laboratory experiments [12, 13]. In indirecttransmission, on the other hand, knowledge for music creation is learned from a collection of past music or through teaching, and the acquired knowledge is used for creating(rather than replicating) new musical pieces. A dominant part of art music and popular music is considered to be created by indirectly transmitted knowledge, and there
is some evidence from studies on automatic music composition showing the relevanceof statistical learning [14]. Cultural evolution models incorporating indirect knowledgetransmission have been studied to explain empirical laws found in music data [5, 15].
Here we focus on the evolution of chord progressions in popular music. Chord progressions outline how accompaniments are played and are of prime importance in thecomposition process of tonal music. In popular music, they are usually notated together
with the melodies, forming a type of musical score called lead sheet. Unlike melodies,chord progressions are very often reused with possible modifications, and there arebooks [16] and websites [17] listing commonly used chord progressions, suggestingthat direct transmission is at work. It is commonly known and has been shown by a corpus analysis [18] that patterns of chord progressions have changed significantly over thelast decades while new chord progressions have continuously been invented. Therefore,
chord progressions are scientifically and practically interesting objects to study how acreative culture evolves by knowledge transmission and modification.
To reveal the basic characteristics of the transmission and modification processesof chord progressions, we construct a stochastic model of evolution and analyze adataset of chord progressions in popular music songs. We view a chord progression
as a sequence of chord symbols and consider chord segments (L-grams) as the unit ofknowledge transmission. The process of creating a new chord segment by (i) choosinga reference from the ‘artifact pool’ of previously created chord segments (i.e. selection) and (ii) possibly modifying it (i.e. mutation) is akin to that of biological molecular
evolution, where nucleotides or amino acids correspond to chord symbols.
We thus build a model similar to the models of molecular evolution [19, 20], withextensions to incorporate essential factors of the cultural evolution. First, we formulate amodel where created artifacts (chord segments) are accumulated in the artifact pool unlike individuals that are removed upon death from the population in biological models.
Second, we incorporate in the selection process the recency and frequency-dependent
biases, which are often relevant in cultural evolution [21, 22]. Third, we consider in themutation process insertions and deletions of chord symbols, which are often ignoredin molecular evolution models [20], as well as substitutions. These features also makeour model different from the one previously used for chord progression data [23], enabling us to harvest a number of suggestive results. Another study analyzed folk songs

Evolution of Chord Progressions

3and estimated note-wise mutation probabilities using a dataset without time information [10]. With the use of mathematical model and data with proper time information,
we here analyze more detailed properties of the evolutionary process such as dynamicchanges of mutation probabilities, correlations between evolutionary parameters, andthe characteristics of new chord segments that later become commonly used.

2

Method2.1

Data representation

We consider a dataset of chord progressions represented in a standard popular music notation, transposed to the natural key (C major or A minor), and labelled witha year of creation. The set of distinct chords in the data is denoted by Ω (e.g. Ω ={C, Am7, FM9, . . .}). From each progression, we extract L-grams (also called chord
segments), which are segments of L consecutive chords, where we remove repetitionsof chords. An L-gram so obtained is assigned a time stamp, which is the same as theyear of creation of its source progression. The collection of all L-grams obtained from(L)
progressions created in year t is denoted by St = {wi |ti = t} and its index set(L)
by It = {i|ti = t}, where i is used as an index for L-grams and wi = (wi` )L`=1
St−1 (L)(L)(wi` ∈ Ω) denotes the corresponding L-gram. We also define S<t = s=1 Ss ,where we take the starting time t = 1 as the earliest year of creation in the data. For
(L)
(L)
(L+1)
(L−1)
simplicity of notation, we define St = St , S<t = S<t , St+ = St
, St− = St,
etc. For the result in Sec. 3, we consider the case L = 4.
2.2

Evolutionary model

We consider that each L-gram w ∈ St is stochastically generated by selecting a reference segment w0 from past data and possibly mutating it. Three mutation modes are
considered: substitution, deletion, and insertion. In the substitution mode, a reference
w0 is taken from the set S<t of L-grams and mutated by changing one or more component chords, from w`0 to w` , according to the symbol-wise substitution probability
πsub (w` |w`0 ). The substitution probability from L-gram w0 to w is defined as
Prep/sub (w|w0 ) =LYπsub (w` |w`0 ),

(1)

`=1

where we also include the pure replication case (w` = w`0 for all `) in this probability.
+
In the deletion mode, a reference w0 is taken from the set S<tof (L + 1)-grams andmutated by removing one of its components. Since a removal of the first or last chord+in w0 ∈ S<tproduces an L-gram in S<t , we exclude such a case. Then, the deletion
probability can be defined as
L

Pdel (w|w0 ) =1 X
00δ(w, w1:(`−1)
w(`+1):(L+1)
),
L−1
`=2

(2)

4

Nakamura

where w`:`0 = w` w`+1 · · · w`0 , and δ(w1 , w2 ) = 1 if w1 = w2 and 0 otherwise. In the−insertion mode, a reference w0 is taken from the set S<tof (L − 1)-grams and mutated0by inserting a chord a after one of its chords w` according to the symbol-wise insertion
probability πins (a|w`0 ). The insertion probability is defined as
L−1

1 X
0
0
πins (w`+1 |w`0 )δ(w, w1:`
w`+1 w(`+1):(L−1)).
Pins (w|w ) =
L−1
0

(3)

`=1

Note that the mutation probabilities considered here are among the simplest choices
and we can generalize them to more elaborated models. For example, while we assumed that the symbol-wise substitution probabilities are context free, that is, the probability is independent of the preceding or succeeding chord symbols, it is possible toinclude context dependence by extending the probability πsub (w` |w`0 ) to such forms as000πsub (w` |w`−1, w`0 ) and πsub (w` |w`−1, w`0 , w`+1). Similarly, we can extend the insertion probability so that it also depends on the succeeding chord symbols. These refinements generally increase the complexity (the number of parameters) of the model andrequire a larger amount of data to reliably infer the parameters.
In the generative process, one of the mutation modes is first chosen according to
the mutation mode probability P (b) = λb where b ∈ {rep/sub, del, ins}. We again
note that the pure replication case is included in the mode b = rep/sub. Next, in thiscase, a reference segment w is chosen from S<t according to the selection probability
Psel (w|S<t ). We incorporate two biases in the selection probability to represent potential tendencies of creators. The first is the recency bias [21], which represents thecreators’ tendency to more likely choose a reference that appears in a recently createdsong. This bias can be represented by a weighting factor e−(t−ti )/τ for a segment i,
where the time constant τ represents the time scale for the bias. The second is thefrequency-dependent bias [22], which represents the creators’ tendency to more likely
choose a reference that is more (or less) frequently used in S<t . To formulate this bias,
let F (w; Is ) = #{j ∈ Is |wj = w}/#Is denote the relative frequency of w in Ss . The
frequency bias can be incorporated in a factor [F (w; Is )]α in the selection probability,
where α > 1 (α < 1) represents a positive (negative) frequency-dependency bias.
The selection probability incorporating the two biases is then given as
Psel (w; S<t ) ∝t−1
Xe−(t−s)/τ [F (w; Is )]α .

(4)

s=1

We note that this formulation removes a potential bias arising from the unbalanced num+bers of chord segments created in individual years. Similarly, we define Psel (w; S<t)−and Psel (w; S<t ) for choosing a reference in the deletion and insertion modes, respectively, where the same α and τ are used.
We can summarize the generative probability of w ∈ St as follows:X
P (w; St ) =λb Pb (w; St ),(5)b∈{rep/sub,del,ins}

Prep/sub (w; St ) =X
w0 ∈S<tPrep/sub (w|w0 )Psel (w0 ; S<t ),(6)

Evolution of Chord Progressions

Pdel (w; St ) =X

5+Pdel (w|w0 )Psel (w0 ; S<t
),

(7)

−
Pins (w|w0 )Psel (w0 ; S<t
).

(8)

+
w0 ∈S<tPins (w; St ) =X−
w0 ∈S<tWe can also separately define the (pure) replication and substitution probabilities as
Prep (w; St ) = Prep/sub (w|w)Psel (w; S<t ),X
Psub (w; St ) =Prep/sub (w|w0 )Psel (w0 ; S<t )(9)

w0 ∈S<t ,w0 6=w= Prep/sub (w; St ) − Prep (w; St ).
2.3

(10)

Inference method

The parameters of the evolutionary model, λb , πsub (a|a0 ), πins (a|a0 ), τ , and α, can be
estimated from the data by the maximum likelihood method. To estimate the first threesets of parameters, we apply the expectation-maximization (EM) algorithm by treating
the mutation mode b and reference w0 as latent variables for each observed w.
To estimate τ and α, we can apply a simple iterative grid search using the likelihood
as the objective function. Since the optimal values of these parameters depend on thevalues of the other parameters and vice versa, we iterate the EM step and grid search
step alternately until a convergence of the likelihood value. To evaluate their estimation
variances, we can apply a Bayesian method based on Monte Carlo Markov Chain sampling. Specifically, we used the Metropolis method with a flat prior distribution and alog-normal distribution as the proposal distribution.
2.4

Posterior analysis

Given the set of model parameters estimated as in Sec. 2.3, we can apply the method of
posterior analysis for analyzing possible dynamic changes in the evolutionary parameters. First, given an L-gram w at time t, its posterior probability of mutation modes
λ̃b (w, t) = P (b|w, t) (b ∈ {rep, sub, del, ins}) can be obtained as
λ̃rep (w, t) = λrep/sub Prep/sub (w|w)Psel (w; S<t )/P (w; St ),(11)

λ̃del (w, t) = λdel Pdel (w; St )/P (w; St ),(12)

λ̃ins (w, t) = λins Pins (w; St )/P (w; St ),(13)

λ̃sub (w, t) = 1 − λ̃rep (w, t) − λ̃del (w, t) − λ̃ins (w, t),(14)

where the right-hand sides of these equations can be calculated using Eqs. (5) to (8)and the replication and substitution probabilities are separately defined here. Then, themutation mode probabilities λ̃b (t) at time t can be estimated, for example, as
λ̃sub (t) =1 X
λ̃sub (w, t).
#St
w∈St

6

Nakamura

B

A 45
40

Num of 4-grams

Num of songs

35
30
25
20
15
10
5
0
1920

1940

1960 1980
Year

2000

2020

2000
1800
1600
all
1400
1200
1000
800
600
400
new
200
0
1950 1960 1970 1980 1990 2000 2010 2020
Year

Fig. 1. A: The distribution of composition years in the dataset used. B: The yearly numbers of
4-grams (including duplications) and that of newly appeared 4-grams.

We can also calculate the posterior probability P (w0 |w, t, b) of reference segments+−w ∈ S<t ∪ S<t∪ S<t
in a similar way. For example, in the insertion mode,0−
P (w0 |w, t, b = ins) ∝ Pins (w|w0 )Psel (w0 ; S<t).

The posterior probabilities obtained in this way can be used to estimate the mutation
probabilities π̃sub (a|a0 ; t) and π̃ins (a|a0 ; t) at time t.

3

Result3.1

DatasetWe used a dataset of Japanese popular music songs. The dataset was constructed by theauthor and comprised of 2419 songs. The songs were taken from top ranked songs in the
Oricon yearly charts and from a compiled collection of historical popular songs [24].
The composition years spanned the range [1927, 2019] and we applied the evolutionarymodel for analysis in a range of years t ≥ 1960 (Fig. 1A).
Before extracting L-grams of chords from a song, we transposed the song into thenatural key, converted a consecutive repetition of the same chord into a single chord,and converted a slash chord into a normal chord by removing the bass note. The target of the analysis was 4-grams (L = 4). The number of distinct chord symbols was
232, from which approximately 2.9 × 109 distinct 4-grams can be created in principle.
The numbers of distinct 3-grams, 4-grams, and 5-grams appearing in the dataset were
12 258, 27 237, and 44 204, respectively. Fig. 1B shows the yearly numbers of 4-gramsand those of newly appeared 4-grams; the average rate of new 4-grams was 39%.
3.2

Selection biases

The inferred value of the time constant was τ = 2.61 ± 0.53. This means that theprobability of a chord segment being chosen as a reference reduces by a factor of 10 inevery 6.1 years, when other conditions are equal. The inferred value of the frequencydependence parameter was α = 1.16 ± 0.16. The mean value indicates a slightly positive frequency-dependent bias, i.e., more common chord segments tend to be more
frequently chosen as a reference than its frequency expected for random selection.
However, the deviation of α from unity is small and the result is consistent with thefrequency-independent bias within the range of statistical error.

Evolution of Chord Progressions
λrep

λsub

λdel

λins

0.38

0.46

0.00

0.16

7

Table 1. Inferred values of mutation mode probabilities.

Mutation mode prob

0.6

substitution

0.5
0.4
0.3
0.2

rate of new 4-grams
insertion

0.1
deletion01960 1970 1980 1990 2000 2010 2020
Year

Fig. 2. Dynamic changes of mutation mode probabilities.

3.3

Mutation modes

The inferred values of mutation mode probabilities are listed in Table 1. We found thatthe probability of choosing the deletion mode tended to converge to zero. To understandthis result, we note that the probability of each mutation mode is of the same order,O(K −L ) for K = #Ω, if we suppose a uniform distribution over all chord symbols.
The result can then be explained by the fact that there are no tunable parameters forthe deletion operation whereas the substitution and insertion probabilities, πsub (a|a0 )and πins (a|a0 ), are trained so that the likelihoods of these modes will increase in thecourse of statistical inference. Consequently, in our model, the main mutation modes
are substitution and insertion.
We see that the sum of mutation probabilities, which is equal to 1 − λrep , is 62%
and larger than the average rate (39%) of new segments. This value is substantiallylarger than the mutation probabilities in typical biological evolution and leads to distinct
characteristics. For example, a significant proportion (38%) of reappeared segments are
estimated to be created through a mutation process according to the present model.
A posterior analysis over time showed that the substitution mode probability hadsome variations across years and a general trend of increase from the 1960s to the1990s (Fig. 2). Its temporal changes highly correlated with the the rate of new segments(ρ = 0.86, p < 10−10 ). On the other hand, the insertion mode probability had small
fluctuation and no notable trend was observed.
3.4

Symbol-wise substitution and insertion probabilities

The most frequent modes of symbol-wise substitution and insertion are shown in Fig. 3
with their yearly relative frequencies obtained by the posterior analysis. For substitution probabilities (Fig. 3A), we see that there are signifiant changes over years and thediversity of applied substitution modes considerably increased from the early period
to the late period. There is also a tendency that substitutions involving less frequent

8

A

Nakamura

B

Fig. 3. Dynamic changes of symbol-wise substitution (A) and insertion (B) probabilities. In eachpanel, the relative frequencies of the 25 most frequent modes are shown, and a smoothing with awindow of 5-year width is applied.

chords such as FM7 and Gsus4 became more frequent in later periods. From the listof substitution modes, we can find that most substitutions occur between chords withthe same harmonic function. These chords often share the root tone (e.g. E7 → Em,
Dm → Dm7, F → FM7, and G → Gsus4) or share constituent pitches (e.g. Dm → F,
C → Em, and Em7 → C). A similar relation between the substitutability of chords andthe harmonic function was also found in an analysis using hidden Markov model [25].
Significant changes over years were also found in the insertion probabilities (Fig. 3B).
The list of insertion modes mostly consisted of common chord transitions (see alsoSec. 3.5). We see a tendency that chord transitions used in the minor key (e.g. Dm →E7, E7 → Am, and Am → Dm) appear more frequently in the early period and thoseused in the major key (e.g. F → G, G → C, and FM7 → G) in the late period.
3.5

Correlation between fluctuation and mutation probabilities

We analyzed the correlations between evolutionary parameters to examine several expectations from the evolutionary theory. On the one hand, if new chord segments arestochastically generated by substitutions, we expect that the joint probability of different chord symbols Pvar (a0 , a) in variants of segments related by a single substitution
correlates with the joint probability of substitution πsub (a0 → a) = P (a0 )πsub (a|a0 ),where P (a0 ) is the prior probability of chord symbols. On the other hand, if we consider
the implicit effect of social selection in the data, among potential creators who generate new segments with different substitution probabilities, successful creators would
be those with substitution probabilities that are similar to the fluctuation probabilities
of chord symbols in past data. This also suggests that the probability πsub (a0 → a)correlates with the fluctuation represented by the probability Pvar (a0 , a) in past data.
To examine this expectation, we analyzed the correlation between the joint substitution probability πsub (a0 → a) in a time range [2010 : 2019] and the fluctuation
Pvar (a0 , a) observed in a time range [1927 : 2009]. The result in Fig. 4A supports the
expectation and shows a positive correlation (ρ = 0.18, p < 10−10 ). We also see asignificant amount of deviation: in particular, a high fluctuation probability does notalways indicate a high substitution probability.

Evolution of Chord Progressions

B 10-1

10-2

Insertion prob

Substitution prob

A

10-4
10-6
10-8
10-8

9

10-2
10-3
10-4
10-5
10-5

10-6 10-4 10-2
Fluctuation prob

10-4 10-3 10-2
Bigram prob

10-1

Fig. 4. Correlations between fluctuation and mutation probabilities. A: The fluctuation probability Pvar (a0 , a) and the joint substitution probability πsub (a0 → a). B: The bigram probability
Pbi (a0 , a) and the joint insertion probability πins (a0 → a).

Similarly, we expect that the insertion probabilities are related to the corresponding
fluctuation probabilities. More specifically, we expect that the joint probability of insertion πins (a0 → a) = P (a0 )πins (a|a0 ) correlates with the bigram probability Pbi (a0 , a)of chord symbols in past data. The result in Fig. 4B supports the expectation and showsa high correlation (ρ = 0.64, p < 10−10 ).
3.6

Diffusion of mutants

Characterizing the conditions of new mutant chord segments that will diffuse and become commonly used is important for understanding the macroscopic evolution ofchord progressions. For biological evolution, where mutations are rare, a similar problem of fixation has been studied, and the fitness of the mutant and the random sampling
in a finite population are studied as two major factors [26]. Our case of cultural evolution has two features that lead to an evolutionary process distinct from the typical caseof biological evolution. First, as we discussed in Sec. 3.3, the mutation rate is muchlarger than in biological evolution so that the chance that mutants of the same form are
independently generated is not negligible. Second, the evolutionary process is an accumulated process so that mutant segments will not be removed from the artifact pool.
Based on this consideration, we expect that the accumulation of independent mutants is relevant for the diffusion of a new chord segment. As a measure of diffusion ofa segment w, we can use its probability of replication. More specifically, we define thediffusedness D10yr (w, t?w ) of a mutant segment w first appeared in year t?w as?

t=tw +10
1 X
10yr
?
D
(w, tw ) =λrep/sub Prep (w, t),
10 t=t? +1

(15)

w

where the replication probability Prep (w, t) is given in Eq. (9). Our hypothesis is that amutant with a larger mutation probability P (w; St?w ) has a higher chance of repeatedly
introduced to the artifact pool and consequently has a higher diffusedness on average.
The relationship between the diffusedness and mutation probability analyzed for all
mutant segments that first appeared in years between 1960 and 2009 is shown in Fig. 5.
The observed high correlation (ρ = 0.57, p < 10−10 ) supports our hypothesis, andNakamura
10-4

Diffusedness

10

10-6
10-8

10-10
10-10 10-8 10-6 10-4
Mutation prob

Fig. 5. Diffusedness D10yr (w, t?w ) and mutation probability P (w; St?w ) for mutant segments w
with the year of first appearance t?w ∈ [1960 : 2009]. Segments that appeared four or more times
within ten years after the first appearance are indicated by red circles. A linear function y ∝ x(black line) is shown as a guide for the eyes.

A

1

B

recall0.1
precisionF-score0.01
0.001
0.0001
101

102

103
104
105
Size of predicted data

E7-G7-Abm7-FM7

Am7-Am-Gm7-FM7

F-Fm-E7-Dm7

Gsus4-Csus4-CM7-C7

G-F-D7-E7(#5)

Bb-Dm6-F-G

Asus4-D-C-F

Dm7-Em7-FM7-Asus4

FM7-FM7-Dm7-F

C-E7-Dm7-Em7

G7-A-C-F7

Fm6-G7-Fm7-Bb

Ebm7(b5)-Fm-Bsus4-G7

F-Bb-E-D

Fdim7-Em7-Em7-Eb

Dm7-Ebm7-Esus4-E

F-G7-G7-Am9

F-Gsus4-G-Fm7

E7-Am-A7-Abaug

F7-Am-Am7-Dm7

C-Esus4-D-F

F-Esus4-G-Dm7

Bb-Am7-G-G

FM9-Esus4-Am7-G7

G-Ab-Fm7-C(add9)

Am-FM9-A-F

F7-F-Dm7-E7

Fm-G-FM7-Em

Esus4-Em-FM7-Asus4

Gm7-Esus4-E7-Dm7

106

Fig. 6. Predictive ability of the evolutionary model. A: Prediction accuracies for varying sizes ofpredicted data. B: Most probable predictions by the evolutionary model.

the linear relation is particularly clear for mutant segments with highest diffusedness(marked in red circles in Fig. 5). The deviation from the linear relation in the small
mutation probability regime can be explained by the finite size effect. We can also seethat there is a variation of O(101 –102 ) in the diffusedness for those samples with amutation probability ∼ 10−6 , which indicates that the mutation probability cannot be
the only factor that determines the diffusedness.
3.7

Predictions

The present evolutionary model can be used for predicting new segments to appear in
the future. To examine its potential, we trained the model with a subset of the data ofsegments created in 1999 or before and evaluated its predictive ability using as test data
the remaining data of segments created in years 2000–2019. We randomly generated
106 4-grams by the model and obtained approximately 2.6 × 105 segments after removing duplications and the samples already appearing in the training data. The generated
4-grams were sorted by the mutation probability in the decreasing order. We compared
the predicted data with the test data by measuring the precision, recall, and F-score.
The recall achieved 48% on the whole predicted data (Fig. 6A). The precision wasapproximately 2%, which is much larger than the expectation value of 0.0004% byrandom sampling. Examples of predicted segments with highest mutation probabilities
that were not included in the analyzed dataset are shown in Fig. 6B.

Evolution of Chord Progressions

4

11

Conclusion and discussion

In this paper, we have studied the evolution of chord progressions based on a stochastic model of cultural evolution incorporating the selection and mutation processes. We
summarize the main findings and discuss implications. First, the inferred selection biases showed a strong recency effect with a time constant of 2.61 yrs. This indicates thatwhile chord segments are accumulated in the artifact pool, they will be effectively removed from the pool after some decades in the sense that their chance of being chosen
as a reference will decrease significantly in that time interval. On the other hand, no
significant sign of frequency dependence was observed.
Second, the analysis revealed large mutation rates and large dynamic changes in thesubstitution and insertion probabilities. The first feature reminds us of an interesting
phenomenon known as the survival-of-the-flattest effect [27], which suggests the possibility that a chord segment with a high probability of replication can be outcompetedby segments that have lower probabilities of selection but are robust in usability against
mutations. The second feature also suggests a selective advantage of chord segmentsthat are robust in usability against mutations toward multiple directions. While the significance of this effect depends on the mutation rate and other model configurations,this observation may provide a new perspective on understanding why certain chord
segments are more popular than others.
Third, the correlations found between fluctuation and mutation probabilities and between diffusedness and mutation probabilities support expectations from the evolutionary theory and may be useful for predicting the features of the evolutionary process.
It is also important to seek for possible explanations for the observed deviations ofO(101 –102 ) in the mutation probabilities for similar values of fluctuation probabilities.
We remark that although the present evolutionary model was build upon empirical knowledge on the process of music creation, the results of this study do not verify
that the assumed process is correct. We can think of other processes of creating chord
progressions, for example, a process involving data generation through statistical learning. To formulate a more realistic model, we should incorporate the multilevel structureof music, consisting of chord segments, musical piece, composer, etc.; reference andselection can occur at each of these levels. Japanese popular music is not a closed system and some chord progressions should have been imported from Western musicalcultures; such migrations were treated as mutations in this study. Future work should
experimentally test the model with other possibilities and address the aforementioned
theoretical issues. The present evolutionary model can also be applied to analyzing theorigins and relationships of chord segments in a similar way as the stochastic models ofmolecular evolution are applied to phylogenetic analysis.

References
1. Mauch, M., et al.: The evolution of popular music: USA 1960–2010. Royal Society OpenScience 2(150081), 1–10 (2015)
2. Interiano, M., et al.: Musical trends and predictability of success in contemporary songs inand out of the top charts. Royal Society Open Science 5(171274), 1–16 (2018)

12

Nakamura

3. Singh, R., Nakamura, E.: Dynamic cluster structure and predictive modelling of music creation style distributions. Royal Society Open Science 9(220516), 1–18 (2022)
4. Weiß, C., et al.: Investigating style evolution of Western classical music: A computational
approach. Musicae Scientiae 23(4), 486–507 (2019)
5. Nakamura, E., Kaneko, K.: Statistical evolutionary laws in music styles. Scientific Reports
9(15993), 1–11 (2019)
6. Moss, F.C., Neuwirth, M., Rohrmeier, M.: The line of fifths and the co-evolution of tonal
pitch-classes. Journal of Mathematics and Music (2022)
7. Zivic, P.H.R., Shifres, F., Cecchi, G.A.: Perceptual basis of evolving Western musical styles.
Proceedings of the National Academy of Sciences of the USA 110(24), 10034–10038 (2013)
8. Youngblood, M.: Conformity bias in the cultural transmission of music sampling traditions.
Royal Society Open Science 6(191149), 1–8 (2019)
9. Cavalli-Sforza, L.L., Feldman, M.W.: Cultural Transmission and Evolution. Princeton University Press, Princeton (1981)
10. Savage, P.E., et al.: Sequence alignment of folk song melodies reveals cross-cultural regularities of musical evolution. Current Biology 32(6), 1395–1402.e8 (2022)
11. McBride, J.M., Tlusty, T.: Cross-cultural data shows musical scales evolved to maximise
imperfect fifths. Preprint arXiv:1906.06171 (2019)12. Ravignani, A., Delgado, T., Kirby, S.: Musical evolution in the lab exhibits rhythmic universals. Nature Human Behaviour 1(0007), 1–7 (2016)
13. Anglada-Tort, M., et al.: Large-scale iterated singing experiments reveal oral transmission
mechanisms underlying music evolution. Current Biology 33(8), 1472–1486.e12 (2023)
14. Fernández, J.D., Vico, F.: AI methods in algorithmic composition: A comprehensive survey.
Journal of Artificial Intelligence Research 48, 513–582 (2013)
15. Nakamura, E.: Conjugate distribution laws in cultural evolution via statistical learning. Physical Review E 104(034309), 1–13 (2021)
16. Scott, R.: Chord Progressions for Songwriters. iUniverse, Bloomington (2003)
17. Hooktheory: Popular Chord Progressions, https://www.hooktheory.com/
theorytab/common-chord-progressions
18. de Clercq, T., Temperley, D.: A corpus analysis of rock harmony. Popular Music 30(1), 47–
70 (2011)
19. Felsenstein, J.: Evolutionary trees from DNA sequences: A maximum likelihood approach.
Journal of Molecular Evolution 17, 368–376 (1981)
20. Arenas, M.: Trends in substitution models of molecular evolution. Frontier in Genetics
6(319), 1–9 (2015)
21. Wang, D., Song, C., Barabási, A.L.: Quantifying long-term scientific impact. Science342(6154), 127–132 (2013)
22. Boyd, R., Richerson, P.J.: Culture and the Evolutionary Process. The University of Chicago
Press, Chicago (1985)
23. Warrell, J., Salichos, L., Gerstein, M.: Latent evolutionary signatures: A general framework
for analyzing music and cultural evolution. Preprint bioRxiv doi:10.1101/2020.10.23.352930(2020)
24. Shiiba, K., Kubo (ed.), S.: Japanese Songs Vols. 3–9 (in Japanese). Nobarasha, Tokyo(1998/1999/2001/2004/2014)
25. Uehara, Y., Nakamura, E., Tojo, S.: Chord function identification with modulation detection
based on HMM. In. Proc. CMMR, pp. 59–70. The Laboratory PRISM, Marseille (2019)
26. Kimura, M.: On the probability of fixation of mutant genes in a population. Genetics 47(6),713–719 (1962)
27. Wilke, C.O., et al.: Evolution of digital organisms at high mutation rates leads to survival of
the flattest. Nature 412, 331–333 (2001)