Pierre Pestieau

Gregory Ponthiere
email: gregory.ponthiere@ens.fr

Optimal Lifecycle Fertility in a Barro-Becker Economy

Keywords: fertility, birth timing, population, dynastic altruism, OLG model. JEL classi…cation codes: D10, J13, O40

Parenthood postponement is a key demographic trend of the last three decades. In order to rationalize that stylized fact, we extend the canonical model by Barro and Becker (1989) to include two -instead of onereproduction periods. We examine how the cost structure of early and late children in terms of time and goods a¤ects the optimal fertility timing. Then, focusing a stationary equilibrium with stationary population, we provide two alternative explanations for the observed postponement of births: (1) a fall of the direct cost of late children (thanks to medical advances); (2) a rise in hourly productivity, which increases the (relative) opportunity costs of early children in comparison to late children.

Introduction

As this is well-known among demographers, human fertility has exhibited a globally decreasing pattern during the last century. At the world's level, the total fertility rate (TFR) has decreased from about 5.2 children per women in 1900 to about 2.7 children per women in 2000. 1 Another well documented stylized fact concerns not the number, but the timing of births: there has been, over the last three decades, a global tendency towards the postponement of parenthood. That stylized fact is illustrated by Figure 1, which shows the rise in the mean age at motherhood in several advanced economies, over 1985-2005. 2 Another way to illustrate the postponement of births consists of decomposing the total fertility rate into the average number of children per women in the early part of the reproduction period (i.e. between ages 15 and 34 years), and the average number of children per women in the late part of the reproduction period (i.e. between ages 35 and 49 years). 3 As shown on Figure 2, whereas the total fertility rate has remained relatively constant over the period 1985-2005, late fertility has signi…cantly increased over time. For instance, whereas children born from a mother older than 35 years amounted to only 7.5 percents of US births in 1985, that proportion jumped to about 14 percents in 2005.

Can economic models of fertility, such as the Barro Becker (1989) framework, provide some plausible explanation or rationalization of that stylized fact? 4 At …rst glance, the answer seems to be negative, since that model includes, as most dynamic models with endogenous fertility, a unique reproduction period. However, that model can be extended, so as to introduce several reproduction periods, instead of a single one. The goal of this paper is precisely to provide such an extension of the Barro Becker (1989) model. For that purpose, we keep the fundamental structure of the Barro Becker model: overlapping generations, physical capital accumulation, dynastic altruism, but we add one reproduction period, in such a way as to be able to examine whether the observed postponement of births can or cannot be rationalized by an economy à la Barro and Becker. Given that we are here only interested in the timing of fertility, we will, when discussing how to rationalize the postponement of births, assume that the total fertility rate equals the replacement level, and study the lifecycle distribution of births. 5 Anticipating on our results, we …rst show that, at the temporary equilibrium, the cost structure of early and late children, in terms of goods and time, is a key determinant of the optimal fertility pro…le. If late children are more costly than early children during their childhood, and if early children are no longer timeconsuming once adult, the optimal fertility pro…le only involves early children. If, however, early children are, once adult, still time-consuming for their parents, that additional opportunity cost provides some incentive for postponing fertility. Then, focusing on a stationary equilibrium with stationary population size, we show that improvements in total factor productivity tend, under general conditions, to shift steady-state optimal fertility pro…les towards more late births. Hence the extended Barro Becker model proposes two explanations for the postponement of births: either a change in the direct cost of early and late children, or, alternatively, a rise in total factor productivity.

In sum, the extended Barro Becker model is shown to be able to rationalize the observed postponement of births. As such, that model complements all static frameworks studying the optimal timing of births (Happel et al 1984, Cigno and Ermisch 1989, Gusta¤son 2001), as well as dynamic frameworks also focusing on lifecycle fertility patterns, but using alternative assumptions, such as D'Albis et al (2010) and Pestieau and Ponthiere (2011). 6 The rest of the paper is organized as follows. Section 2 presents the model. Section 3 studies lifecycle fertility at the temporary equilibrium. Section 4 examines the timing of births at a stationary equilibrium with stationary population. Empirical material on total factor productivity and fertility timing in the U.S. and the OECD is studied in Section 5. Section 6 concludes.

The model

Let us consider a three-period OLG model. The three periods are: childhood, young adulthood and old adulthood. Agents work the entire time-period in young adulthood and old adulthood. Reproduction is monosexual. Agents can have children either at the beginning of young adulthood (i.e. second period), or at the beginning of old adulthood (i.e. third period). The number of early children for a young adult at time t is denoted by n t , while the number of late children for a young adult at time t is denoted by m t+1 .

Following Barro and Becker (1989), we assume that parents are altruistic towards young family members. Parents'utility depends on own consumption at young adulthood (denoted by c t ) and at old adulthood (denoted by d t+1 ), and on the number and the utility of younger family members. The utility of an agent who is a young adult at time t is represented by the utility function:

U t = v(c t ) + v(d t+1 ) + a (N t+1 ; N t ) N t+1 U t+1 (1)
where v( ) is the utility from consumption, is a time-preference factor, and the weighting function a( ) represents the weight assigned by a young adult belonging to the adult cohort at period t, of size N t , to the welfare of each family member belonging to the adult cohort t + 1, of size N t+1 . The number of young adults at time t + 1, N t+1 , includes his own children, whose number is N t n t , as well as the late children of his parents, i.e. his young brothers and sisters, whose number is N t 1 m t . Thus the number of family members who are young adults at time t > 0 is denoted by N t+1 = N t n t +N t 1 m t . [START_REF] Ayiar | Total factor productivity revisited: a dual approach to development accounting[END_REF] That number of descendants can also be rewritten as:

N t+1 = N t n t + N t 1 m t N t = N t n t + m t g t = N t g t+1 (2)
where g t+1

Nt+1

Nt is the (adult) cohort growth factor, equal to n t + mt gt . The family weighting function a( ) has the following form:

a(N t+1 ; N t ) = N 1 " t N " t+1
(3) 6 D'Albis et al (2010) focus on optimal lifecycle fertility in a continuous time OLG model, under the assumption of a decreasing relationship between the total number of children and the timing of births. Pestieau and Ponthiere (2011) focus on the optimal fertility timing in an OLG model à la Samuelson (1976), and examine whether the Serendipity Theorem still holds in a broader demographic environment with several reproduction periods instead of one. [START_REF] Ayiar | Total factor productivity revisited: a dual approach to development accounting[END_REF] We assume, to avoid any period without births, that N 1 = N 0 = 1, and that m 1 = 0.

where 0 < < 1 and 0 < " < 1. That functional form is a generalization of the one in Barro and Becker (1989), where g t+1 = n t (as m t = 0). 8 Note also that substituting for a(N t+1 ; N t ) in expression (1) yields:

U t = v(c t ) + v(d t+1 ) + (g t+1 ) 1 " U t+1 (4)
That expression generalizes the utility function used in Barro and Becker (1989), where all children are born from young parents (i.e. g t+1 = n t ), yielding:

U t = v(c t ) + v(d t+1 ) + (n t ) 1 " U t+1 (5)
In the Barro Becker setting, the unique fertility rate n t serves as the investment rate in the future. Here, in our framework with two reproduction periods, that investment rate in the future equals g t+1 n t + mt gt . Substituting repeatedly for a cohort's utility in the utility function of the dynastic head, we obtain the following dynastic utility function:

U 0 = 1 X t=0 t (N t ) 1 " [v(c t ) + v(d t+1 )] (6)
That expression collapses to Barro and Becker's standard dynastic utility function when old adulthood does not matter ( = 0) and when there are no births in third period, i.e. m t = 0 for all t, in which case

N t = N t 1 n t 1 = t 1 Y s=0 n s . 9
The dynastic head treats equally the welfare of two descendants born at the same point in time, even though one is related to him through a lower number of family ties than the other. The underlying intuition is that, even though there may be di¤erences, in the short-term, between grand children and grandgrand-grand children, those di¤erences appear minor once the dynasty includes a su¢ ciently large number of generations.

The elasticity of v ( ) with respect to consumption is assumed to be constant:

v(c t ) = c t (7)
where < 1. That assumption is also made in Barro Becker (1989).

8 Indeed, in Barro-Becker, we have

N t+1 = ntNt, so that a( ) = = h N 1 " t n " t N " t i = =Ntn " t , implying a ( ) N t+1 = g 1 " t+1 = n 1 " t .
Here that formula is generalized, to obtain a ( ) N t+1 = g 1 " t+1 even when g t+1 6 = nt as a consequence of mt > 0. 9 Note that the above dynastic utility function keeps the standard properties of the one derived by Barro and Becker (1989). If one had used, instead of expression (1), another utility function, such as, for instance:

Ut = v(ct) + v(d t+1 ) + a(nt)ntU t+1 + a(m t+1 )m t+1 U t+2
with a( ) concave in its argument, one would not have been able to derive a dynastic utility function with the same simple form as expression (6). Indeed, the altruistic weighting function is non linear, so that a (n 0 n 1 ) + a(m 1 ) 6 = a(n 0 n 1 + m 1 ), implying that the dynastic utility function can hardly be reduced to an in…nite sum of lifetime welfare terms weighted by the (concavi…ed) size of the di¤erent successive cohorts, as in Barro Becker (1989) and in (6).

Each adult earns a wage w t on time supplied to the market, in young adulthood and old adulthood. Since we de…ne an adult's total time available per period to be 1 unit, w t and w t+1 denote the total labour income in, respectively, young adulthood and old adulthood, in the case of no children.

Following Barro and Becker (1989), we assume that a child costs, during his childhood, a fraction b of the time unit of parents, so that parents having n t children can only work during a period 1 bn t at young adulthood. Similarly, parents who have m t+1 late children can only work a fraction 1 Bm t+1 of their second working period. It is widely documented that a late child is more costly than an early child, so that we have: B b. 10 However, the duration of coexistence of the parent with a child is longer with an early child than with a late child, and this may induce some additional time costs at older adulthood. We represent those extra time costs by the fraction h 0. The intuition is that early children are not only "enjoyed" longer than late children, but they also require some time once they are young adults. The source of asymmetry in comparison with late children is that late children become adults when their parents are retired, unlike early children, who become adults when their parents are still working. This di¤erence creates an asymmetry in the lifetime opportunity costs of early and late children.

Within the Barro-Becker (1989) setting, there is only one adult life-period. As a consequence, there is no savings: the only way to transfer resources over time is through bequests to children, each adult parent giving, in the last period of his life, a bequest to his children. Here, on the contrary, adulthood includes two periods, during which the agents works and can have children. The introduction of an additional life-period tends to complicate the analysis. Hence, we will make here three assumptions, which insure analytical solvability:

Young individuals can save resources for the future. 11 All children, whatever these are born from young or old parents, receive their bequests at the same point in their lifetime, that is, at the beginning of young adulthood (i.e. the majority age). 12 All children who live in the same time period receive exactly the same bequest from their parents, whatever those parents are young or old. 13 Under those assumptions, a young adult at time t, whatever he is born from young or old parents, faces the budget constraint:

w t (1 bn t ) + (1 + r t )k t = c t + s t + n t e t + n t k t+1 (8)
where (1 + r t )k t is the bequest received by the young adult from his parent, s t is savings, e t is the cost of an early child in terms of resources, b is the fraction of time dedicated to the rearing of each early child, and n t k t+1 are the resources he saves to provide a bequest to his early children. The amount n t k t+1 is invested in the production process, and then given, with interests, to the children born at t once these are adults. Thus each of those children receives:

(1 + r t+1 )k t+1 .
At the next period, the adult agent faces the budget constraint:

w t+1 (1 Bm t+1 hn t ) + s t (1 + r t+1 ) = d t+1 + m t+1 E t+1 + m t+1 k t+2 (9)
where E t+1 is the cost of a late child in terms of resources, B is the fraction of time dedicated to each late child, and h is the fraction of time dedicated to early children once adults. Given that the cost of a late child is generally larger than the cost of an early child, the assumption E t e t is reasonable. 14 The amount m t+1 k t+2 constitutes the resources that the parent saves to provide a bequest to his late children. The amount m t+1 k t+2 is invested in the production process, and then given, with interests, to the children born at t + 1 once these are adults. Each of those late children receives:

(1 + r t+2 )k t+2 .
Substituting for savings in the young age budget constraint yields the individual's intertemporal budget constraint:

w t (1 bn t ) + w t+1 (1 Bm t+1 hn t ) 1 + r t+1 + (1 + r t )k t = c t + n t e t + n t k t+1 + d t+1 + m t+1 E t+1 + m t+1 k t+2 1 + r t+1 (10)
That intertemporal budget constraint is the same for all agents, whatever these are born from young or old parents. That result would not prevail if children received their bequests at the death of their parents. 15 Substituting repeatedly for k t in the intertemporal budget constraint, we obtain the following dynastic budget constraint:

k 0 + 1 X t=0 z t N t (1 bn t )w t + 1 X t=1 z t N t 1 (1 Bm t hn t 1 )w t = 1 X t=0 z t N t c t + N t d t+1 1 + r t+1 + 1 X t=0 z t N t n t e t + 1 X t=0 z t+1 N t m t+1 E t+1 (11)
1 4 On the cost di¤erential between early and late children, see Gusta¤son (2001). 1 5 If children received their bequest at the death of their parent, the overall budget constraints of agents born at the same point in time would di¤er, depending on whether they are born from young or from old parents. Individuals born from old parents receiving the bequests earlier in their lifecycle, they bene…t, thanks to a larger opportunity set for savings, from more resources on the lifetime, ceteris paribus, than individuals born from young adults, who need to wait more to get the bequest. See the Appendix on this.

where

z t = t Y s=0 1 (1+rs)
. That constraint states that the present value of all resources (LHS) must be equal to the present value of all expenditures. In comparison with the standard Barro-Becker setting, the LHS and the RHS di¤er by the presence of third-period consumption on the RHS, and, also, by the de…nition of the number of descendants, N t , equal here to N t 1 n t 1 + mt 1Nt 2 Nt 1 .

Temporary equilibrium

The optimization problem faced by the dynastic head consists of maximizing utility U 0 subject to the dynastic budget constraint, subject to the initial assets k 0 , and subject to the initial young adult cohort size N 0 = 1. Through that maximization problem, each agent takes as given the path of wage rates w t and of interest rates r t , and selects a path for consumption at the young age and old age for all descendants, c 0 , c 1 , c 2 , ::: and d 1 , d 2 , d 3 , :::, bequests per young adult, k 1 , k 2 , ::: and age-speci…c fertility rates, n 0 ; n 1 ; n 2 ; ::: and m 1 ; m 2 ; m 3 ; :::. 16 The dynastic head's problem can be written as follows:

max ct;dt;nt;mt;kt 1 X t=0 t (N t ) 1 " [v(c t ) + v(d t+1 )] s.t. k 0 + 1 X t=0 z t N t (1 bn t )w t + 1 X t=1 z t N t 1 (1 Bm t hn t 1 )w t = 1 X t=0 z t N t c t + N t d t+1 1 + r t+1 + 1 X t=0 z t N t n t e t + 1 X t=0 z t+1 N t m t+1 E t+1
First-order condition for optimal consumption of a young adult at time t is:

t (N t ) 1 " c 1 t = z t N t
where is the Lagrange multiplier associated with the dynastic budget constraint. The FOC for optimal old-age consumption at period t + 1 is:

t (N t ) 1 " d 1 t+1 = z t N t 1 + r t+1
Combining those two FOCs yields:

d t+1 c t 1 = (1 + r t+1 ) (12)
Combining the FOC for optimal consumption of a young adult at t + 1 with the FOC for optimal consumption of a young adult at t, we have:

c t c t+1 1 = 1 n t + m t g t " 1 1 + r t+1 (13)
Thus the intergenerational consumption pro…le is increasing in the parameter , and decreasing in the cohort growth factor g t = Nt

Nt 1 and in the interest rate. Similarly, for old-age consumption, we have:

d t d t+1 1 = 1 n t 1 + m t 1 g t 1 " 1 1 + r t+1 (14)
Regarding the optimal demography, we know from the FOC for consumption that the optimal cohort growth factor g t+1 satis…es the condition:

g t+1 = " (1 + r t+1 ) c t c t+1 1 # 1 " (15)
The optimal cohort growth factor is increasing in parental altruism, and in the interest rate, but decreasing in the parameter ". If the interest rate is constant, and if the intergenerational consumption path is also invariant over time, the optimal cohort growth factor is also constant over time. But beyond those properties, that condition does not, on its own, su¢ ce to characterize, for any period, a unique optimal fertility pro…le (n t ; m t ), since various fertility pro…les (n t ; m t ) can bring the optimal cohort growth factor g t+1 , as long as (n t ; m t ) is such that n t + mt gt = g t+1 . Hence, to characterize the optimal fertility path, it is necessary to focus on the FOCs for optimal age-speci…c fertility.

Keeping in mind that the number of persons born at period t can be written as

N t = N t 1 g t = N t 1 n t 1 + mt 1Nt 2 Nt 1
, the FOCs for the optimal n t and m t are, respectively:

1 X j=0 t+1+j (1 ")(N t+1+j ) " @N t+1+j @n t [v(c t+1+j ) + v(d t+2+j )] = 1 X j=0 z t+1+j @N t+1+j @n t c t+1+j + d t+2+j 1 + r t+2+j + z t N t bw t + z t+1 N t hw t+1 + z t N t e t + 1 X j=0 z t+1+j @N t+1+j @n t n t+1+j e t+1+j + 1 X j=0 z t+1+j @N t+1+j @n t m t+1+j E t+1+j 1 X j=0 z t+1+j @N t+1+j @n t (1 bn t+1+j )w t+1+j 1 X j=0 z t+2+j @N t+2+j @n t (1 Bm t+2+j hn t+1+j )w t+2+j (16) 1 X j=0 t+1+j (1 ")(N t+1+j ) " @N t+1+j @m t [v(c t+1+j ) + v(d t+2+j )] = 1 X j=0 z t+1+j @N t+1+j @m t c t+1+j + d t+2+j 1 + r t+2+j + z t N t 1 Bw t + z t N t 1 E t + 1 X j=0 z t+1+j @N t+1+j @m t n t+1+j e t+1+j + 1 X j=0 z t+1+j @N t+1+j @m t m t+1+j E t+1+j 1 X j=0 z t+1+j @N t+1+j @m t (1 bn t+1+j )w t+1+j 1 X j=0 z t+2+j @N t+2+j @m t (1 Bm t+2+j hn t+1+j )w t+2+j (17)
Those two FOCs are almost identical, except that these depend on how respectively n t and m t in ‡uence the number of individuals born at all subsequent periods t + 1; t + 2; :::. Although that e¤ect is complex -since any change in fertility is followed by changes in the number of descendants at all next generations -, the relationship

N t = N t 1 g t = N t 1 n t 1 + mt 1Nt 2
Nt 1 can help us to quantify the relative levels of the derivatives @Nt+1+j @ni and @Nt+1+j @mi .

Lemma 1 The partial derivatives of the number of descendants with respect to early and late fertility satisfy the following equation, for all t, j > 0:

@N t+1+j @n t = N t N t 1 @N t+1+j @m t = g t @N t+1+j @m t
Proof. See the Appendix. Thus whether the marginal e¤ect of a change in early fertility on the total number of descendants is superior or inferior to the marginal e¤ect of a change in late fertility depends on whether the population is growing or not, that is, on whether g t exceeds 1 or not. Substituting for this in the FOC for optimal m t , and simplifying yields:

1 X j=0 t+1+j (1 ")(N t+1+j ) " @N t+1+j @n t [v(c t+1+j ) + v(d t+2+j )] = 1 X j=0 z t+1+j @N t+1+j @n t c t+1+j + d t+2+j 1 + r t+2+j + z t N t Bw t + z t N t E t + 1 X j=0 z t+1+j @N t+1+j @n t n t+1+j e t+1+j + 1 X j=0 z t+1+j @N t+1+j @n t m t+1+j E t+1+j 1 X j=0 z t+1+j @N t+1+j @n t (1 bn t+1+j )w t+1+j 1 X j=0 z t+2+j @N t+2+j @n t (1 Bm t+2+j hn t+1+j )w t+2+j (18)
That FOC can then be compared with the FOC for optimal n t . Several cases can arise, depending on the relative levels of early and late children in terms of time and goods.

Let us start with the simplest case. If there is perfect equality of all costs, i.e. b = B, e t = E t and h = 0, then the two FOCs are exactly identical. Thus, in that special case, the optimal early fertility rate n t and late fertility rate are determined jointly, by the same condition. The dynastic head's planning problem is thus underdetermined.

If, on the contrary, there is perfect equality of all costs, except that the early children still recommend some time at the adult age, i.e. h > 0. Then, it follows that the marginal welfare loss from increasing early fertility rate is always larger, ceteris paribus, than the marginal welfare loss from increasing the late fertility rate, since the RHS of the FOC for optimal n t always exceeds the one for optimal m t . We thus have the corner solution:

n t = 0, m t > 0.
Alternatively, if there is no time costs of early children at the adult age (h = 0), but if late children have a larger cost, either in time or in goods, i.e. b

B and e t < E t or b < B and e t E t , the marginal welfare loss from increasing early fertility is always smaller, ceteris paribus, than the marginal welfare loss from increasing late fertility, since the RHS of the FOC for optimal m t always exceeds the one for optimal n t . Hence, we have: n t > 0 and m t = 0.

Finally, when none of those conditions are satis…ed, the two FOCs are satis-…ed simultaneously, and we have the interior optimum for the fertility pro…le as a whole: n t > 0 and m t > 0. The following proposition summarizes our results.

Proposition 1

If b B and e t < E t or b < B and e t E t , as well as h = 0, the optimal fertility pro…le involves n t > 0 and m t = 0.

If b = B, e t = E t and h > 0, the optimal fertility pro…le involves n t = 0 and m t > 0.

If b = B, e t = E t and h = 0, the optimal fertility pro…le is indeterminate.

Otherwise, the optimal fertility pro…le involves n t > 0 and m t > 0.

Proof. See the above FOCs.

In sum, various fertility pro…les (n t ; m t ) can constitute the solution to the dynastic head's optimization problem, depending on the structure of early and late children costs in terms of goods and in terms of time. Therefore, the evolution of children cost structure is likely to have had a signi…cant impact on the timing of births. One can think, for instance, that recent medical advances, such as the development of assisted reproductive technologies, have strongly reduced the cost of late children, leading to the observed postponement of births. However, as we shall now see, that kind of "cost-based explanation" is not the unique possible one. Other explanations can account for the postponement of births, without relying on external changes in children costs structure.

Births postponement: theory

In order to identify other potential determinants of the change in the timing of births, let us …rst make some additional assumptions, concerning the production process, from which wages w t and interest rates r t are determined. In this paper, we assume, like Barro and Becker (1989), a closed economy, where the wages and interest rates depend on the level of the stock of productive capital.

Production

Production takes place via a standard one-sector production function that exhibits constant returns to scale:

Y t = F (K t ; L t ) (19)
where Y t is total output, K t is total capital, and L t is total labour. As Barro Becker (1989), we assume no depreciation of physical capital.

The total labour at period t can be written as:

L t = (1 bn t )N t + (1 Bm t hn t 1 )N t 1 = N t 1 bn t + 1 Bm t hn t 1 g t ( 20
)
The total capital stock at period t can be written as:

K t = (N t 1 n t 1 + N t 2 m t 1 ) k t + N t 1 s t 1 = N t k t + N t g t s t 1 (21)
where the …rst term of the RHS coincides with the bequest received by individuals who are young adults at time t, whereas the second term is the savings from young adults at the previous period.

In intensive terms, we can denote the capital per worker t as:

t K t (1 bn t )N t + (1 Bm t hn t 1 )N t 1 = k t + st 1 gt 1 bn t + 1 Bmt hnt 1 gt (22)
Given constant returns to scale, we can rewrite the production process as:

y t Y t (1 bn t )N t + (1 Bm t hn t 1 )N t 1 = F ( t ; 1) = f ( t ) (23)
with, as usual, f 0 ( t ) > 0, f 00 ( t ) < 0.

Factors are assumed to be paid at their marginal productivities:

r t = f 0 ( t ) (24) w t = f ( t ) t f 0 ( t ) (25)
The interest rate is decreasing in t , while the wage rate is increasing in t .

Long-run fertility timing

The dynamics of our economy is quite complex, because there are not one, but three demographic variables: early fertility rate n t (like in Barro-Becker), but, also, late fertility rate m t , and the cohort growth factor g t = n t 1 + mt 1 gt 1 . Hence, to facilitate our analysis of the change in fertility timing, we will here assume that there exists a unique stable stationary equilibrium with a stationary population. That assumption can be justi…ed on the ground of analytical tractability, but, also, on the ground that an ever increasing or decreasing population is hardly plausible on a …nite Earth. Moreover, in the light of Pestieau and Ponthiere (2011), it appears that, when the cohort growth factor g di¤ers from unity, stationary equilibria can, under some fertility pro…les, turn out to be unstable, making steady-state analysis irrelevant. Thus imposing a stationary population protects us against the possibility of unstable stationary equilibria, but without preventing the study of the optimal fertility timing. 17 When there exists a stationary equilibrium, we know, since:

g t+1 = " (1 + r t+1 ) c t c t+1 1 # 1 " 1 7
On the existence and stability of a stationary equilibrium with exogenous age-speci…c fertility rates n and m, see Pestieau and Ponthiere (2011). It is shown there that, although there exists, under mild conditions, a stationary equilibrium in an OLG model with two reproduction periods, the stability of that equilibrium requires n > 0. When n = 0, the economy exhibits cyclical dynamics around a stationary equilibrium, except when g equals 1, in which case there is a stable stationary equilibrium. that the cohort growth rate g t must be constant over time, and equal to:

g = [ (1 + r( ))] 1 " (26)
Note that, for a given, constant, cohort growth factor g = 1, there exist various possible fertility pro…les (n; m). Indeed, any pair (n ; m ) such that:

1 = n + 2 p n 2 + 4m 2
is compatible with the stationary equilibrium with stationary cohort size.

To study the optimal fertility pro…le, let us go back to the conditions describing the optimal levels of n and m , and substitute for steady-state wages, interest rate, and child costs. Making those substitutions in the two FOCs and substituting for the Lagrange multiplier in the …rst FOC by means of the other FOC, we obtain the optimality condition:

e + bw + hw 1 + r = E + Bw ( 27
)
That condition is necessary for the two FOCs to be satis…ed, that is, for an interior optimal fertility pro…le with n > 0, m > 0. It states that the marginal cost from an early child (LHS), in terms of goods (1st term) or in time (2nd and 3rd terms), must be exactly equal to the marginal cost from a late child (RHS), in terms of goods (1st term) and time (2nd term). On its basis, we can characterize the steady-state fertility pro…le.

Proposition 2 Consider an economy with a unique stable stationary equilibrium and a stationary population.

(i) if Bw + E > e + bw + hw 1+r , we have n = 1, m = 0;

(ii) if Bw + E = e + bw + hw 1+r , we have n > 0, m > 0;

(iii) if Bw + E < e + bw + hw 1+r , we have n = 0, m = 1:

Proof. See the Appendix. Proposition 2 allows us to propose a …rst explanation for the observed postponement of births. In the early stages of history, late children were extremely costly. Hence, the large levels of B and E push the economy towards case (i). However, as medical science has developed, the costs of late children have decreased substantially, pushing the economy towards case (ii), with interior optimal early and late fertility rates. But that stage may not be the end of the story: as the costs of late parenthood become closer and closer to those of early parenthood (i.e. B ! b and E ! e), the existence of a positive time cost for early children who become adults (i.e. h > 0) may push the economy towards case (iii), with a corner solution n = 0 and m = 1.

In sum, Proposition 2 proposes a cost-based explanation to the observed postponement of births. Over time, the structure of child costs would have evolved, with, as a consequence, a shift from early fertility to late fertility. Note that, whereas the shift from stage (i) (no late births) to stage (ii) (mixed fertility) as a result of medical advances reducing late children costs E could have been anticipated a priori, the shift from stage (ii) to stage (iii) is less trivial. That shift relies on the fact that children are durable consumption goods, who also involve a durable opportunity cost. Thus the model has the virtue to highlight the key role played by that durable time costs of children for the optimal fertility timing issue.

Although attractive, that explanation is not the unique possible one. Actually, we can also propose another explanation, which relies not on the structure of child costs, but on the evolution of the production process. To explore that explanation further, let us impose a particular functional form on the production process, i.e. the standard Cobb-Douglas production function:

Y t = AK t L 1 t (28)
where A > 0 is a productivity parameter, while is the elasticity of output with respect to capital. In intensive terms, the production process can be written as:

y t = A t (29)
Hence, at the stationary equilibrium with stationary population, the steadystate capital level is:

= A 1 1 1 (30)
from which we can deduce that the wages and interest rates are:

w = (1 ) 1 A 1 1 1 1 (31) r = 1 (32)
Substituting for wages and interest rates in the conditions of Proposition 2 allows us to study the impact of productivity shocks (through A) or changes in capital intensity (through ) on the optimal timing of births. The impact of factors a¤ecting the wage and the interest levels on the timing of fertility is particularly worth being studied in the case where B b h < 0, that is, when the total (discounted) proportion of labour time dedicated to early children along the whole career exceeds the total time dedicated to late children.

Corollary 1 Consider an economy with Cobb-Douglas production technology, with a unique stable stationary equilibrium and a stationary population. Assume e E < 0 and B b h < 0.

(i) if (1 ) 1 A 1 1 h 1 i 1 < e E (B b h) , then n = 1, m = 0; (ii) if (1 ) 1 A 1 1 h 1 i 1 = e E (B b h) , then n > 0, m > 0; (iii) if (1 ) 1 A 1 1 h 1 i 1 > e E (B b h) , then n = 0, m = 1;
Proof. See the Appendix.

In the light of Corollary 1, we are able to propose a second explanation for the observed postponement of fertility. That alternative explanation relies on the changes in the characteristics of the production process. Indeed, we have, ceteris paribus: @w @A > 0; @w @ < 0

Thus, when the economy bene…ts from a positive total factor productivity (TFP) shock, the wage, which is the LHS of the condition in Corollary 1, goes up. As a consequence, case (i) becomes less likely, and cases (ii) or (iii) more likely. If there is also a rise in the elasticity of output with respect to capital , the LHS of the condition goes down, which counteracts the change induced by the rise in TFP. But in periods of strong technological progress with quasi constant elasticity , there has to be a shift from early births towards late births.

In sum, beyond a change in the structure of children costs in terms of goods (i.e. a fall of E), one can also rationalize the observed tendency towards later fertility on the mere basis of a rise in TFP. A rise in TFP increases the opportunity cost of early children more than the one of late children, since the former, once adult, still consume some working time of their parents, contrary to late children, who only become adults when their parents are retired. Hence, provided other things remain equal (including children costs in terms of goods, e and E), the timing of births is modi…ed in the direction of later fertility. This constitutes an alternative explanation of the postponement of births. 18 5 Births postponement: back to the data Let us now turn back to empirical facts, to investigate whether the productivitybased explanation of the observed postponement of births is compatible with the data. For that purpose, we will proceed in two stages, and focus …rst on cross section data for OECD countries, and, then, on time series for the U.S.

TFP and fertility timing in the OECD

A …rst, simple way to check the plausibility of the TFP-based explanation of the postponement of births consists of investigating whether economies with the largest TFP are also characterized by later births in the lifecycle. Naturally, such a correlation is only a clue supporting the above explanation provided the economies under study are close to each others on all relevant dimensions (i.e. same structural parameters in our model except TFP parameter A).

For that purpose, we use the study by Aiyar and Dalgaard (2005), which provides estimates of TFP for 20 advanced economies over 1980-1990. The speci…city of Aiyar and Dalgaard is to use two distinct methodologies for TFP estimations: besides the standard estimation method based on data on stocks of inputs (which they call the "primal" approach), they also develop an alternative method, based on factor price data (called the "dual" approach). 19 On the fertility side, we rely here on age-speci…c fertility from the United Nations Population Division World Fertility Data 2008, which provides estimates of age-speci…c (period) fertility rates around the world, on the period 1970-2006. 20 Figure 3 plots 19 economies in the space (TFP rank, m/TFR rank) for 1990, using the primal TFP estimates approach. 21 Countries in the bottom-left corner are characterized by higher TFP and later births, relative to other countries. The trend line suggests that there exists a positive relationship between the ranks in terms of TFP and in terms of late motherhood. Highly productive economies, such as France, Italy and the Netherlands, are also characterized by a larger share of late births. On the contrary, less productive economies, such as Japan and South Korea, are also characterized by less late motherhood. But while Figure 3 supports a positive relationship between TFP and late fertility, Figure 4, which relies on the dual TFP estimates, does not exhibit an 1 9 That dual approach relies on the well-known equivalence, under constant returns to scale, between total output and total factor payments, implying that the growth rate of TFP can be deduced either from the comparison of output growth with input growth (primal approach), or from the growth of factor prices (dual approach), provided one knows the share of each input in the production process.

2 0 Note that the 19 countries under study have total fertility rates that are sometimes close to the replacement rate (for instance for the US and New Zealand), but sometimes signi…cantly lower (for instance in Germany and Italy). That di¤erence is somewhat problematic, as the hypothesis we test concern economies with replacement fertility (see below).

2 1 Late fertility rate m includes here fertility between ages 35 and 49 years included.

increasing pattern. The trend line is here (weakly) decreasing, which does not support the TFP-based explanation of birth timing. Hence, the observed relationship between TFP and late motherhood does not seem to be robust to the methodology used for productivity measurement. When we take the average TFP ranking on the two approaches to productivity measurement (primal and dual), we obtain a weakly increasing relationship (Figure 5). But this can hardly be regarded as a strong support for a TFP-based explanation of the observed fertility timing. The reason is that many countries, such as Ireland, the Netherlands, Norway and Austria are very close in terms of productivity, but very distant in terms of fertility timing. Hence TFP di¤erentials cannot be the unique factor driving the timing of births. If one restricts the sample to countries whose TFR is su¢ ciently close to the reproduction rate in 1990 (Figure 6), the positive relationship between TFP and late motherhood becomes slightly steeper, suggesting that intercountry heterogeneity in terms of total fertility may have caused some noise in our comparisons. More generally, those simple cross country comparisons could only inform us on the validity of the TFP-based explanation of birth timing provided all countries are identical in all other aspects, which is hardly the case. In order to avoid the di¢ culties raised by unobserved heterogeneity across countries, the next subsection will focus on a single country, the United States, and try to compare late fertility patterns and TFP patterns.

TFP and fertility timing in the U.S.

In order to assess the TFP-based hypothesis for the U.S., we will rely on TFP estimates by Fernald (2011), and on age-speci…c fertility rate from the Human Fertility Database. Regarding the period under study, we will focus on the period after 1988, in such a way as to be as close as possible to a period with replacement fertility (TFR close to 2). As shown on Figure 7, the proportion of births beyond age 35 has been growing strongly during that period. If we plot, in a simple diagram, the annual growth rate of the share of late fertility in total fertility, against the annual growth rate of TFP, one obtains an unambiguously increasing relationship, whatever the concept of TFP that is used. That robust increasing relationship appears on Figure 8, which relies on TFP estimates for the whole business sector, and on Figures 9 and 10, which are based on (respectively unadjusted and utilization-adjusted) TFP estimates for consumption goods (i.e. non-equipment goods). Those simple graphs suggest that, over the period 1988-2007, the higher the annual TFP growth was, the larger the postponement of births was. This is exactly in conformity with the TFP-based rationalization outlined from the extended Barro-Becker framework.

In sum, this section does not have the pretension to show that the TFP-based explanation is valid, nor that it is better than a cost-based explanation. 22 More modestly, that section suggests that the extended Barro-Becker model is, at …rst glance, not incompatible with the data, since one of its major predications …ts the data. When one reminds that such a qualitative test sometimes su¢ ces to question the relevancy of a model -such as Doepke's (2005) rejection of the Barro-Becker model for the mortality/fertility relationship -the material provided in this section is at least encouraging for further empirical tests of the extended Barro-Becker model in the context of changing fertility timing.

Conclusions

The postponement of births constitutes a major demographic stylized fact of our times. The present paper proposed to extend the canonical fertility model of Barro and Becker (1989), to examine whether standard economic fertility models can help us to rationalize observed fertility timing tendencies.

For that purpose, we introduced an additional reproduction period in the Barro-Becker OLG model. That extension allowed us to examine how individuals can, under dynastic altruism, solve the dilemma between early and late children. Whereas the former are consumed earlier, they are also, in general, associated with a longer period of coexistence, which may lead to additional opportunity costs in comparison to later children.

In a …rst stage, we showed that, at the temporary equilibrium, the optimal timing of births is strongly dependent on the cost structure of early and late children, in terms of goods and in terms of time. If late children are more costly than early children during their childhood, and if early children are no longer time-consuming once adult, the optimal fertility pro…le only involves early children. But if early children who become adults are, to some extent, still time-consuming for their parents, then there is some room for late children, especially if medical advances, such as assisted reproductive technologies, have reduced the direct cost of late children.

In a second stage, focusing on a stationary equilibrium with stationary population size, we showed that an improvement in TFP tends, in general, to shift steady-state optimal fertility pro…le towards more late births. Hence the extended Barro Becker model proposes not one, but two distinct plausible explanations for the postponement of births: either a change in the direct costs of early and late children, or, alternatively, a rise in TFP.

Cross country comparisons in the OECD suggest that the empirical testing of the TFP hypothesis depends signi…cantly on how productivity is measured. Focusing on the U.S., we showed that there exists a positive correlation between TFP annual growth and the annual growth of late fertility, in conformity with what the extended Barro-Becker model suggests. But further empirical investigations will have to be done in the future, to be able to assess the plausibility of the cost hypothesis and the TFP hypothesis for changes in fertility timing. within a given cohort, depending on the age of parents, let us derive the intertemporal budget constraint in the two cases. When the bequest is received at the death of the parent, the budget constraint of a young adult born from a young parent is:

(1 bn t )w t = c t + s t + n t e t

If he receives a bequest k t+1 from his parent at his death, his budget constraint at t + 1 is: Thus the intertemporal budget constraint di¤ers for agents born at the same point in time, from young or old parents. The overall lifetime spending (RHS) is the same in the two cases, but the lifetime resources (LHS) di¤ers: those born from older parents bene…t from the bequests earlier in their own lifecycle, which makes them richer if one considers an economy at the steady-state (under r > 0). As a consequence, the mere passage of time would lead, under those alternative budget constraints, to an explosion of intracohort heterogeneity, between children from parents, grand-parents etc. of di¤erent ages.

w t+1 (
Proof of Lemma 1 In order to compare the derivatives @Nt+1+j @nt and @Nt+1+j @mt , remember that N t+1 = N t g t+1 = N t n t + mtNt 1

Nt

. Hence we have:

@N t+1 @n t = N t @N t+1 @m t = N t 1 @N t+2 @n t = N t g t+2 + N t+1 m t+1 N t N t N 2 t+1 = N t g t+2 m t+1 g t+1 = N t n t+1 @N t+2 @m t = N t 1 g t+2 + N t+1 m t+1 N t N t 1 N 2 t+1 = N t 1 g t+2 m t+1 g t+1 = N t 1 n t+1 @N t+3 @n t = N t n t+1 g t+3 + N t+2 m t+2 N t N t+2 m t+2 N t+1 N t n t+1 N 2 t+2 = N t [n t+1 n t+2 + m t+2 ] @N t+3 @m t = N t 1 n t+1 g t+3 + m t+2 N t 1 N t+2 N t+1 n t+1 N t+2 = N t 1 [n t+1 n t+2 + m t+2 ] :::
or, in general terms: @N t+1+j @n t = N t N t 1 @N t+1+j @m t Thus, whether the marginal impact of raising early fertility is larger than the marginal impact of raising late fertility depends on whether the cohort size is growing over time or not. If N t > N t 1 or g t > 1, we necessarily have, everything else being unchanged, that:

@Nt+1+j @nt > @Nt+1+j
@mt .

Proof of Proposition 2 Note that, in general, we have:

@Nt+1+j @Nt+1 = j+1 Y k=1
g t+k , and @Nt+1+j @nt = @Nt+1+j @Nt+1 @Nt+1 @nt = N t j+1 Y k=1 g t+k . Hence, imposing g t = g = 1 for all g t leads to @Nt+1+j @Nt+1 = 1, and to N t+j = 1 for all t. Hence, the FOC for optimal early fertility is, at the stationary equilibrium with stationary population:

Figure 1 :

 1 Figure 1: Mean age at motherhood (years), 1985-2005.

Figure 2 :

 2 Figure 2: Decomposition of the total fertility rate (before and after 35 years), 1985-2005.

Figure 3 :Figure 4 :

 34 Figure 3: TFP rank (primal) and m/TFR rank, 1990.

Figure 6 :

 6 Figure 5: TFP rank (average) and m/TFR rank, 1990.

Figure 7 :

 7 Figure 7: Fertility rate before age 35 and after age 35, and total fertility rate, US, 1933-2007.

  Fig. 8: TFP growth (business) and m/TFR growth.

  1 Bm t+1 hn t )+s t (1+r t+1 )+(1+r t+1 )k t+1 = d t+1 +m t+1 E t+1 +(n t +m t+1 )k t+2Hence his intertemporal budget constraint is:w t (1 bn t )+ w t+1 (1 Bm t+1 hn t ) 1 + r t+1 +k t+1 = c t +n t e t + d t+1 + m t+1 E t+1 + (n t + m t+1 )k t+2 (1 + r t+1 )For an agent born at the same time, but from an old parent, the two budget constraints are:w t (1 bn t ) + (1 + r t )k t = c t + s t + n t e t w t+1 (1 Bm t+1 hn t ) + (1 + r t+1 )s t = d t+1 + m t+1 E t+1 + (n t + m t+1 )k t+2From which the intertemporal budget constraint is w t (1 bn t )+ w t+1 (1 Bm t+1 hn t ) 1 + r t+1 +(1+r t )k t = c t +n t e t + d t+1 + m t+1 E t+1 + (n t + m t+1 )k t+2 1 + r t+1

That stationary population assumption, which allows us to focus only on the birth timing problem, is actually quite plausible given the …niteness of the Earth. One could hardly imagine an ever growing population living on a …xed space.

0 On the cost di¤erential between early and late children, see[START_REF] Gustafsson | Optimal age at motherhood. Theoretical and empirical considerations on postponement of maternity in Europe[END_REF]. 1 1 The introduction of savings allows us to characterize the agent's intertemporal budget constraint even if he has no children early in life.1

One can interpret this as parental will to give an equal chance to all children, early and late. On the analytical side, that assumption helps a lot. If children received their bequests at the time of the death of their parent, then their lifetime budget constraint would di¤er, depending on whether they are born from young or old parents (see below).1

According to that assumption, only the timing of life determines the bequest one receives, and not the age of parents. The underlying idea is that, from the point of view of the dynastic head, a late child and an early grand child are similar, since they are alive at the same point in time, and, thus, equally "distant" from the dynastic head.

6 Note that, since the objective function is time-consistent, the paths chosen by the dynastic head will also be chosen by any other following descendants facing the same problem.

8 That result is a comparative statics result, where A is the unique changing parameter. If, alternatively, one assumes that children costs e and E vary with TFP, then additional e¤ects have to be taken into account before drawing conclusions on the net e¤ect of TFP growth on fertility timing. But given that the e¤ect of changes in children costs in terms of goods was discussed above, we focus here on the pure e¤ect of a rise in TFP, under …xed e and E.

2 Further empirical assessments will have to be made, while taking into account identi…cation di¢ culties due to unobserved heterogeneity, endogeneity and other potential biases.

Substituting for on the basis of the FOC for optimal m , we obtain

which can be simpli…ed to:

Proof of Corollary 1 Substituting for wages and interest rate:

in the condition:

Bw + E = e + bw + hw 1 + r yields:

From which the three cases of Proposition 2 can be expressed as:

Assuming that (B b h) < 0 and e E < 0, those conditions can be rewritten as the conditions in Corollary 1.


Frédéric Bimbot

Emmanuel Deruty

Gabriel Sargent

Emmanuel Vincent

Methodology and conventions for the latent semiotic annotation of music structure

This document proposes and describes guidelines for the annotation of music structure into elementary units called semiotic blocks, reflecting high-level and high-scale similarities across the music piece.

The annotation process requires both the localization of block boundaries (segmentation) and the determination of block classes (labeling).

The proposed methodology is based on concepts and principles inspired and adapted from semiology. It combines elements resulting from the analysis of internal properties forming systems within blocks (morphological analysis), from the characterization of distinctive properties across blocks (paradigmatic analysis) and from the location and context of blocks within the music piece (syntagmatic analysis).

The implementation of these principles has resulted in a set of notations which are also developed in the document.

Méthodologie et conventions pour l'annotation sémiotique latente de structure musicale

Résumé : Ce document propose et décrit un cadre méthodologique pour l'annotation de structure musicale sous forme d'unités élémentaires dénommées blocs sémiotiques, reflétant les ressemblances de haut-niveau et à grande échelle dans un morceau de musique.

Le processus d'annotation consiste à la fois à localiser les frontières de blocs (segmentation) et à déterminer leurs classes d'équivalence (étiquetage).

La méthodologie proposée repose sur des concepts et des principes inspirés et adaptés de la sémiologie. Elle combine des éléments d'analyse résultant de l'étude des systèmes de propriétés internes au sein des blocs (analyse morphologique), de la caractérisation des propriétés distinctives entre blocs (analyse paradigmatique) et de la position et du contexte des blocs dans le morceau (analyse syntagmatique) La mise en oeuvre de ces principes a débouché sur un système de notation, lui aussi développé dans ce document.

Presentation

This document presents principles [START_REF] Bimbot | Decomposition into Autonomous and Comparable Blocks : A Structural Description of Music Pieces[END_REF] , notations and methodological guidelines designed at IRISA for the description and annotation of the latent semiotic structure of musical pieces.

Music structure, as understood in this work, does not correspond to the acoustic structure (who/what plays/sings when) nor to the functional structure (verse / chorus / bridge).

The latent semiotic structure reflects high-level and high-scale systematic relationships and distinctive properties across segments of comparable size (and of same-scale morpho-syntagmatic organization) within the music piece [START_REF] Bimbot | Methodology and Resources for The Structural Segmentation of Music Pieces into Autonomous and Comparable Blocks[END_REF] .

General concepts

The semiotic description and annotation process consists in representing the music piece as a sequence of labels (transcription) from a limited set of symbols called (semiotic) alphabet. Each occurrence of the labels corresponds to a segment of the musical piece and each symbol represents metaphorically a class of musical segments.

Structural blocks constitute the elementary units to describe the semiotic structure. They are primarily identified by considering morpho-syntagmatic patterns resulting from local relationships on one or several musical information layers Block boundaries are located at instants when morpho-syntagmatic patterns are re-actualized, under regularity assumptions.

Each structural block is assigned a semiotic label which denotes : (i) Properties which are common to segments of a same class within the piece (one label per class) (ii) Properties which are distinctive across classes (two different labels for two distinct classes) (iii) Variations of the ending of the morpho-syntagmatic pattern of the block (subscripts) (iv) Variations of surface properties which are considered as expressive and which develop over the semiotic structure (superscripts) but which do not affect the morpho-syntagmatic pattern of the block Some specific semiotic labels are used to render situations when structural blocks show properties which are clearly ambiguous or undecided between two classes.

The annotation process consists in :

-Identifying which musical layers are relevant to the structural description of the piece -Determining which musical layers are relevant to locate structural block boundaries (structuring information layers)

3

-Determining which properties of these layers are relevant to identify the alphabet of semiotic classes (distinctive semiotic properties) -Transcribing the piece with this alphabet (labels, subscripts and optionally, superscripts) -Identifying a structural pattern to which the structure of the piece is assumed to correspond.

Trying to achieve all these tasks on the only basis of the musical piece is an ill-posed (under-determined) problem.

Assumptions are introduced to reduce the indetermination of the solution, which boils down to considering that :

-Components of the high-level musical process which are periodic, cyclic, regular, recurrent, repeated, synchronized, correlated, quantizable, etc… are prone to be linked to the structural level of the process.

-Components of the high-level musical process which are aperiodic, spurious, occasional, unique, asynchronous, erratic, continuous, etc… are less likely to pertain to the structure.

This general dichotomy must not be used to decide blindly on the nature of an observation, but rather as a guideline to orientate investigations and arbitrate situations in combination with a careful analysis of the actual musical content.

Note that, considering that a particular element or layer of musical information is not relevant to the structural level does not mean that it is not important to the musical piece. It simply means that it should be considered as relevant to another level of the musical discourse, which we call the expressive level.

As a result of the aforementioned constraints, the annotation process is based on some optimal compromise between the following objectives :

1. The largest possible number of musical layers should be encompassed in the structural description 2. The choice of labels should render as accurately as possible the distinctive properties of semiotic information layers 3. The semiotic alphabet should be reasonably compact (in terms of number and distribution of symbols) 4. The transcription should be as regular as possible and relate as much as possible to a simple structural pattern Semiotic blocks are defined at a high time-scale (typically 16 seconds). We therefore consider the downbeat (or snap) as the unit of time at which the musical content is sampled at the "low" level (typically 1 s).

Musical information layers

The musical discourse is based on a variety of processes and properties which operate at different levels of description.

The musical content generally results from the activity of a number of musical sources, such as : Musical information layers refer to the organized evolution followed by abstract musical properties, which can generally be coded as discrete observations from a quantized set of configurations.

o
Note that a source signal does not constitute per se an information layer, but that the behavior of the source (i.e. the organized pattern created by its activity) does. Conversely, several sources can contribute (simultaneously, successively or alternatively) to a same information layer.

To complete the overall picture, we may also consider macroscopic properties of the piece (or a section of the piece) when dealing with :

o tonality (key, reference tonic) o modality (scale) o tempo (execution pace) o language (reference phonetic system of the lyrics) o volume (execution strength, number of active instruments) o timbre (nature and properties of the voice and/or the instruments)

Macroscopic properties are usually evolving slowly over the whole piece and can usually be inferred from the long-term behavior of musical sources and/or the time-integration of musical information layers.

Definition of structural blocks and semiotic labels

Semiotic blocks are defined by time boundaries and a label denoting an equivalence class. Their localization and their characterization relies on the internal (morpho-syntagmatic) organization of one or several of their musical information layers.

Definition of blocks boundaries relies on what we call structuring properties. Structuring properties participate to the cohesion of the musical content within the blocks.

In a second step, semiotic properties are considered for characterizing and comparing blocks in the song3 .

Not all structuring properties act as semiotic properties (and vice-versa) : some properties may strongly contribute to identify block boundaries but not be relevant to define classes of block, and conversely.

4.a -Morpho-syntagmatic properties of blocks

We assume that a block is composed of a stem (usually 16 snaps) and a number of additional (suppressible) elements called affixes [1,2]. In most cases, a stem decomposes into 4 morphological units (called sub-blocks) of 4 snaps each4 . We further define the first three sub-blocks of the stem as the carrier system and the last one as the contrast (sometimes, a "flat" contrast) 5 .

The (usually simple) network of relationships between the sub-blocks form a graph which represents syntagmatic relations within the block and are the basis of its internal musical consistency. These relationships vary from pure identity to translation, transposition, inversion, conjugation, … A structural block is therefore characterized by the properties of its morphological units and by a graph which internally relates these units. These systems exist on several musical layers in different forms and at different scales simultaneously, which indeed further contributes to the musical consistency of the segment. The layers taking part to the morpho-syntagmatic cohesion are called structuring layers.

We admit here that a musical information layer is considered as a structuring layer if the morphosyntagmatic relationships observed in the block stem for this particular layer is affiliated to one of the following morphological patterns :

• aa'a"c (c can be equal to a)

• aba'c (c can be equal to b (or to a))

• aa'bc (c can be equal to b (or to a))

In all 3 cases above, elements a' and/or a" are considered as transformations or variants of element a (including identity).

When some well-defined property of a musical information layer follows one of these patterns at a certain time scale, it constitutes a structuring property, i.e. it induces locally the existence of a syntagmatic cohesion at that scale. The set of musical information layers which follow one or another of these morphological 6 patterns form the set of structuring properties (one layer may follow a particular pattern and simultaneously another layer follow another pattern).

Note that a constant layer (i.e. a layer for which the behavior is identical on all the sub-blocks of the carrier system, i.e. a pattern aaaa) is not necessarily a structuring property. However, it is bound to acquire the status of structuring property if its apparition and/or disappearance is synchronized with the reactualization of other morpho-syntagmatic patterns.

4.b -Localization of structural blocks (segmentation)

Similarity relationships also occur at a longer distance in the piece, but in general it is possible to identify, at the level of the entire piece, a typical (time-)radius above which the density of relations between morphological units becomes irregular and decreases rapidly. This induces a structural pulsation and is the basis for the definition of segment boundaries, beyond which similarities can be somehow "ignored" without hurting the local musical consistency.

As a consequence, structural blocks appear as self-sufficient musical units, i.e. they possess the property of iterability (see former work [1] [2]). They also can be removed from the piece with no severe effect on the rest of the piece, hence the property of suppressibility.

A key element to detect block boundaries is the presence of a contrast, i.e. an element "c" in the morphosyntagmatic system which departs from the logical sequence induced by the first three elements of the carrier system. This acts simultaneously as a logical modulation and as some sort of punctuation mark which announces an imminent border (located after the contrast)6 .

4.c -Characterization of structural blocks (semiotic properties)

Internal systems forming blocks are refreshed regularly in the following manners :

1. by presenting a new occurrence of a previously observed morphological unit a. either for introducing what will appear to be a variant of an already known system b. or to expose a new system which partly re-uses material from a previous system 2. by introducing significantly new musical material (which means, the beginning of a totally new system)

For each musical layer, a structural block can be described, by :

1) The structuring property that participates to the system 2) The network of relations between the 4 sub-blocks composing the stem Some of these attributes constitute semiotic properties.

A semiotic property is defined as a property which contributes to the description of the equivalence classes on which the semiotic alphabet is based. In general, semiotic properties are a subset of the structuring properties but they also happen to be defined as meta-properties of the structuring properties.

As an example, here is a non-exhaustive list of possible semiotic properties of the layer "melody" :

-the melody itselfs -the support notes of the melody -the rhythm of the melodic line -the variations of the melody -the energy fluctuations caused by the melody -the placement of the melody w.r.t. downbeats -etc… -These various properties of the melody may be used to explain latent similarities despite surface differences between two segments : for instance, the melody can be distinct but the support notes may turn out to be identical.

Strong and accurate formulation of semiotic properties will be preferred (namely because they convey more information on the actual behavior of the musical layer). But sometimes a weaker formulation will be relevant, when seeking for common properties which are bound to explain the affiliation of two blocks to a same class (or a same root).

Determining block labels can be approached as a standard task of semiotic analysis which can be addressed by the commutation test which involves substituting or exchanging blocks with one another and decide whether this substitution changes the perception of the overall "narrative" organisation of the song.

However, the approach based on commutation test is prone to divergences of interpretation between listeners and we prefer to formulate the task of assigning semiotic labels to structural blocks as a matter of comparing morpho-syntagmatic systems by analyzing what makes them different but also what they have in common, beyond their immediate surface properties (paradigmatic analysis), but also by taking into account their context in the song (syntagmatic analysis).

This is detailed in the forthcoming two sections.

4.d -Paradigmatic comparison of structural blocks

The semiotic comparison of blocks is based primarily on the comparison of their internal morphosyntagmatic carrier system. For the purpose of explaining this point, let's denote as X and Y two instances of structural blocks.

For a given music information layer, the carrier systems of X and Y are considered as equivalent if and only if, there exist a property of the considered layer for which the corresponding network of internal relationships of X and Y are homologous.

This comparison can then be carried out for all structuring properties and, if the two blocks are somehow similar, a subset of common properties will emerge as being potentially characteristic of a class grouping X and Y.

At the same time, some of these common properties may also be common to other blocks which however are unlikely to belong to the same class because they strongly differ from X and Y with respect to other structuring layers.

The determination of the subset of structuring properties (and the corresponding network of internal relationships) which are strictly relevant to the semiotic classification of structural blocks is called the paradigmatic analysis and it aims at defining the distinctive properties of the semiotic structure.

4.e -Syntagmatic comparison of structural blocks

The position and context of the structural blocks in the transcription are also taken into account in order to guide their membership to a given class. In other words, two blocks are more likely to belong to the same equivalence class if they appear in similar contexts in the piece.

Blocks are considered to occur in a similar context if they are located beside similar left and right segments within a recurrent sequence. Context definition can be relaxed to left-only or right-only situations and similarity can be understood more or less strictly (for instance, be extended to "functional" similarity).

Therefore, structural blocks which have a similar neighborhood are more likely to be considered from a same class (or at least, one of them to be rooted on the other -see section 7), even if they have quite different surface properties.

It is also worth considering that differences between blocks should not be appreciated in the same way if the two blocks are immediately next to each other or at some distance in the piece : a slight difference observed between 2 successive similar blocks may be distinctive (especially if this opposition is recurrent in the piece) whereas a stronger difference at a long distance may be just an expressive variation, especially if the two blocks are surrounded by a similar context.

Therefore, the guidance of a structural pattern (see section 6) is considered as essential to weigh similarities and differences between and across blocks and interpret those with respect to the global organization of the piece.

However, while the semiotic structure of music tends to be based on recurrent patterns, occasional irregularities and singularities are commonly observed in the realization of music structure. Those tend to be more frequent and more spectacular towards the end of the piece.

Notations for semiotic labels

5.a -Primary notation of semiotic labels (transcription)

The choice of the semiotic label of a block is primarily guided by the following considerations :

 When two blocks are from two distinct classes (i .e. when they are considered to be built on 2 distinct carrier systems) they are denoted by 2 distinct alphabetic capital letters : A vs B.

 When two blocks show equivalent carrier systems but different contrasts, this difference is noted as a subscript : A 1 vs A 2 . Note that suffixes and infixes within the last sub-blocks are treated as specific forms of contrasts. Blocks showing no (or extremely weak) contrasts on any layer will be denoted A 0 .

Blocks showing exceptional contrast (usually resulting of combinations of several factors) will be denoted as A * (they usually tend to occur at the end of the piece).

 When two blocks are considered as equivalent, i.e. when their system+contrast are equivalent but they differ in their (surface) realizations, the variants are denoted with a superscript, for instance : A' vs A''. Optionally, the superscript may be chosen specifically to indicate the nature of the variant (see extended conventions in section 8.a). The notation A 0 and A * can be used to denote exceptionally poor or exceptionally rich variants of A.

 When a property or a set of properties disappear gradually or degrades progressively, this will be denoted as a fade-in or a fade-out. This may apply to surface properties such as the intensity < A, A > , the instrumentation support, but also to strengthening or vanishing properties of the carrier system, then denoted < A, A > .

 If a block is realized only in a half form, specific notations are used : A/2 for a block which is half the size of the structural pulsation period but which forms a complete, autonomous system, |A and A| for truncated half blocks corresponding to the right (resp. left) part an entire block A, (usually realized in full form somewhere else in the piece). More general truncations will be denoted as … A or A …, depending on the side of the truncation, and if necessary, accompanied by an indication of the fractional size of the segment, for instance : (3/4) A … which indicates a block based on a square system which has been truncated after the third morphological unit.

 Inevitably, some situations may reveal ambiguities 7 . Therefore, a set of additional notations are designed to render these ambiguities with composed labels : AB, B/A, B|A, B>A, A&B, etc… They are described in details in section 7.

Summary of main notation conventions

Regular

5.b -Hypermetric notations (size)

Semiotic labels can be combined with size (hypermetric) information in a second field, in brackets. For instance [16+4], [-2+16], [16-4], [16&2], etc… following conventions previously introduced to denote block sizes [1].

It is also admissible to describe the hypermetric information as an operation reflecting the internal organization of the morphological elements, for instance :

[4x4], [2x4x2], [6x4],[3x4(+4)+4], etc 8 …
By default, the block size will be considered as equal to 16 snaps.

7 This is indeed a natural outcome of the semiotic strategy : creating an alphabet of elements, getting the listener to learn them and ultimately combining these elements to create hybrid objects (some sort of chord, at the level of the semiotic structure).

8 These conventions will be detailed in a future document

5.c -Internal morpho-syntagmatic system

A third field, between parenthesis, can be used to denote the internal morphology of the block (usually referring to the semiotically dominant musical information layer). For instance (abac), (aabc), (aba'c), (aa'a"c), etc9 …

5.d -Comprehensive description

In minimalistic situations, the semiotic annotation will therefore consist of a capital letter and a simple subscript and/or a simple superscript, or both, for instance : A 2 , A', A" 1 , A°> , etc… Semiotic symbols can be put into brackets to facilitate visual parsing, especially when they are complex :

[A 0 ] [A 1 ] [A 2 ] [B] [C] [A' 1 ] [A' 2 /2] [B'] [C'] [|C'] [D] [D* > ]
In more extensive descriptions, a semiotic label may look like this : A 2 [4x4+4] (abacc) and the description of the piece will preferably be presented vertically.

Structural patterns

As part of the approach, the sequence of semiotic labels is assumed to derive from an underlying structural pattern, namely a regular sequence of symbols governed by some systematic organization. In practice, the actual sequence of semiotic labels is less regular than its underlying pattern, but it can be understood as an more or less irregular realization of the structural pattern.

Denomination Illustration Codification Trivial A A A A A A A A … (A) Binary A B A B A B A B A B … (AB) Ternary A B C A B C A B C A B C … (ABC) Quaternary A B C D A B C D A B C D … (ABCD) Double binary A A B B A A B B A A B B … (2A,2B) Double ternary

A A B B C C A A B B C C … (2A,2B,2C) Alternate A A B C C D A A B C C D … (2A,B,2C,D) Cyclic A B B C C D D D A B B C C D D D … (A,2B,2C,3D) Acyclic A B C D E F G H … "A,B,C,D, … H, …" Double acyclic A A B B C C D D … H H "2A,2B,2C,2D, …, 2H" Ergodic A C D C B C D A A A B C C A … {ABCD} Extensive A B B C C D D D D E C C C D F … "A,2B,2C,4D,E,3C,D,F"
As can be understood from this table, the use of parentheses indicates that a pattern is cyclic ; the use of inverted commas denotes an enumeration and the use of brackets denotes randomness in the order of elements (the term "ergodic" is used in the sense that the probability of a symbol does not depend on its position in the sequence).

Assuming that the structure of the piece derives from "double" (or "triple") structural patterns (i.e. using symbols recurrently repeated twice or three times) should encourage the annotator to consider the possibility of upscaling the structural pulsation period, or of investigating on the existence of semiotically distinctive properties between successive similar elements. However, such adjustments should not be systematic, as there do exist situations for which double / triple structural patterns are perfectly relevant.

As mentioned in the introduction, the actual realization of a structural pattern in a music piece generally shows irregularities (which tend to increase when getting towards the end of the piece). As already mentioned previously, more irregularities tend to occur as the piece progresses. It is conjectured that the distribution of labels in the realization of musical structures could tend to follow some sort of Zipf law.

Handling ambiguities

In this section, we address a number of situations where ambiguities may occur and we detail criteria and solutions for their consistent identification and description. It is worth underlining that the concepts defined so far make it easy to handle very generically a variety of ambiguous situations and to characterize them quite clearly.

In fact, the possibility for the annotation to render ambiguities is essential. As already mentioned in a footnote, ambiguities correspond to existing situations which are inherent to the high-level semiotic process that develops along a music piece ; namely the possibility of creating elements which result not only from simple variations within a given semiotic class, but also from combinations, offsprings, crossbreeds or strains of previously (or, sometimes subsequently) introduced material 10 .

Therefore, it is essential that the annotator has hybrid symbols at his/her disposal to recognize ambiguous situations and annotate them as such, rather than being forced to assign a segment to one semiotic category or the other, when in fact this is inherently not decidable.

We consider successively the following situations : We denote as X (and Y) unlabeled blocks and by A, B (and C) class labels from the semiotic alphabet.

10 An analogy can be made using concepts stemming from genetics : each main semiotic class possesses its own genotype, shared by all members of that class (beyond surface variations). However, some structural elements result from the hybridation, mutation or transposition of the genotype of other classes and should be identified (and annotated) as such.

7.a -Notations for mixed labels

Before giving indications on how to address the various ambiguities, we introduce the following notations :

AB : block showing undecided prevalence of properties of classes A and B (vertical hybridation)

A&B : block showing intertwined properties of classes A and B (alternation). A&B has usually a size double than that of A (and B) alone.

B|A : new system (B) in the first half of X but recall of morphological units of A in its second half (horizontal hybridation)

B>A : block (B) acceptable as autonomous at the current scale but showing strong cohesion with the previous block (A). The sequence [A]

[B>A] could be described as a single system+contrast at the upper scale (kinship).

B/A : system similar to A (rooted in A) but with properties strongly departing from those of the other elements of class A on a subset of layers (mutation). This includes cases when the deviating property is purely missing.

We also introduce the following ways of denoting short segments that occasionally intervene in between regular ones : A_B : tiling segment, i.e. segment corresponding to a partial superposition between A and B (overlap)

_AB_ : common affix located between A and B which can not decidedly be related primarily to A or B (connexion)

In a future version of this document, a diagram will illustrate these various situations.

In AB blocks, the hybrid nature is vertical : the corresponding segment clearly results from the superposition of a segment of class A played simultaneously with a segment of class B. BA should be used instead of AB when it is felt that B is slightly prevalent over A.

In A&B blocks, properties of A and B are alternating, resulting in a block whose size is the total of that of A and B. Here too, B&A should be used instead of A&B, if B starts first.

In B|A blocks, the hybrid nature is horizontal : the system of the first half of X may be specific (hence the B label) but the system of the second half of X obviously relates to a block of class A (sometimes, with a different contrast, though, then noted B|A i ). Block B|A is considered as a new object at the scale of the semiotic structure, but its strong and obvious similarities with class A reflects in the choice of the hybrid label. At the immediately lower scale, the sequence Notation B>A covers situations where the current block (B) and its predecessor (A) would exhibit strong internal relations if considered as a single segment at a scale twice that of the block size. In other words, the sequence [A] [B>A] indicates that the structural pulsation period could have locally be chosen twice larger. This typically happens (but not exclusively) when, at the immediately lower scale, the sequence

[A] [B>A] appears as [A A] [B C] or [A B] [A' C],
with a significant number of properties common to these 4 elements.

In B/A segments, the current segment X can be viewed as some sort of "mutation" of segment A : some musical layers continue to reflect class A but some others are showing major differences with those of a regular block from class A, which are judged to go well beyond a simple variation. However, it usually remains possible to describe these differences through relatively simple transformations of the characters of class A, hence justifying the fact to consider X as "rooted on" A.

 A typical such situation happens when X is a solo (usually towards the end of the piece) where an instrument takes over the vocal lead and revisits a verse or a chorus with strong modifications of the melody supported by the very same harmony as that of block A. The strong change in the melody information layer is considered to create a new semiotic elements B, but it derives so clearly from A that it can be denoted as B/A.

 In many cases, a segment X functioning as an intro or an instrumental passage (with no melody layer) turns out to be unambiguously related to another segment of the piece (assumed to be labeled A) where a clear melodic lead develops. In that case, segment X should be considered as a particular case of mutation of A (by the loss of a character) and denoted B/A. Note however that constant layers are not covered by this situation as they are not considered as structuring layers. It is also important to distinguish this configuration from the observation of a "minimal variant", i.e. when the the musical content in the block just preserves sufficient information to maintain all distinctive semiotic systems alive.

7.b -Label ambiguities

Class ambiguities happen when the annotator is facing the following question : should the current block X be labeled as belonging to class A or to class B ?

In this case, the annotator should first identify :

a. The set of common properties and systems for all blocks in classes A and B b. The set of distinctive properties and systems between blocks of classes A versus B c. The set of irrelevant properties and systems across blocks of classes A or B indistinctively On this basis, the annotator should then choose the label corresponding to the characters most in adequation with X. If this is impossible to decide, an option is to use a hybrid notation and/or to consider creating a new symbol -see next point.

Alphabet ambiguities occur if the annotator is wondering : should a new class B be created to account for current block X or should it be considered as a variant of class A ?

In this case, the annotation should be guided by the following analysis steps : a. Identify the set of common properties and systems for all blocks in class A and see if they are compatible with those of X b. Determine the nature of distinctive properties between the various classes across the piece and see if one of them is applicable to X c. Take into account the position of X in the sequence of labels within the piece (index and context) d. Avoid inflation of the number of semiotic labels used over the whole piece e. Consider the option of using a mixed label, especially a "mutation" Indeed, ambiguities usually arise from the fact that some properties of X deviate from those of class A, but it is difficult to evaluate whether this deviation is (only) a strong variation of A or a new semiotic element. Here are some guidelines which should be considered to resolve the situation :

• If the deviation observed is concerning properties which have been used in other parts of the piece as distinctive properties between classes, this reinforces the plausibility of a new label B.

• On the other hand, if the block is occurring in a context (neighbouring labels) where class A is usually observed in other parts of the piece, this reinforces the plausibility of a variant of A.

• The absolute position of block X in the sequence of blocks should also be taken into account as deviations and/or irregularities tend to increase as the piece progresses. Therefore, more deviant variants of A can be tolerated at the end of the piece as opposed to the beginning.

• Note that reusing existing symbols leads to a more economical decomposition (pleading in favour of A) and that semiotic elements tend to be more obviously distinct from others when they are rare.

After having analyzed all these factors, the annotator must either arbitrate between A or B on the basis of the decisive prevalence of one of the above considerations, or opt for a hybrid notation, for instance B/A, thus describing a partly new semiotic element strongly rooted on an existing one. In case of uncertainty, the second solution is preferable.

7.c -Segmentation ambiguities

In this subsection, we briefly review a few situations which may trigger questions during semiotic labeling about the segmentation or the segmentation scale used to define the boundaries of the semiotic blocks.

Boundary ambiguities: this situation corresponds to the question that can be stated as : should the border of segments labeled A and B be reconsidered ?

Here, we shall distinguish 4 cases :

• The last sub-block of the stem of a block (labeled A) also acts as the first block of the stem of the next block (labeled B). This is a situation of tiling and this requires the creation of a specific segment where the overlap takes places, with label A_B, yielding a sequence :

[A…] [A_B] […B]

• A sub-block located immediately after the end of the stem of block A (called suffix) acts as the first subblock of block B but it is also showing properties which relates strongly to the system of block A.

Here, the annotator may also decide to extend the block into a tiling with the next block (with label B) and label the overlapping part as A_B, yielding again :

[A…] [A_B] […B]

• A small sub-block may be found in between blocks A and B, but it is not part of the stem of A nor of that of B and its properties partially relates and partially departs both from those of class A and B. In that case, the label _AB_ shall be used. Here, this notation denotes some sort of connection unit, yielding the sequence :

[A] [_AB_] [B]
• Finally, some sub-blocks may occasionally function both as the suffix (or prefix) of a block and a semiotic unit of their own (for instance a break that is used both as a contrast and as a re-intro). In that case, it is acceptable to denote this unit as a tiling A_B, where the right boundaries of A and B coïncide :

[A…] [A_B]
In general, however, blocks shorter than half of the typical block size should be avoided. Blocks of a quarter size should preferably be merged to neighboring blocks as suffixes, prefixes, etc… unless they clearly act a functional units, especially intros and re-intros. Conversely, an individual affix should not exceed the quarter of a block (but several consecutive affixes may extend the size of a block beyond this recommendation).

Scale ambiguities : a number of situations may raise questions concerning the scale chosen for the structural segmentation, as of whether the local/global scale of the segmentation should be reconsidered, in particular by merging successive symbols :

• Large/complex structural patterns ; for example (ABCDEFGH)

• Recurring multiplicity of similar symbols in sequence; for example (nA,nB,nC,nD)

In both cases, the grouping of segments 2 by 2 should be considered, privileging simple n-ary structural patterns with n preferably around 4 (quaternary structural pattern).

However, this trend should be balanced with the objective of describing the structure around a scale which is comparable across pieces, namely segments with stems of 16 snaps, and a typical snap duration around 1 second per snap. Ultimately, for some (rare) cases, two scales may be equally admissible and in that case, the lowest one should be preferred. Appropriate symbols should be carefully chosen, in order to preserve as much information as possible on the relationships between successive blocks at the upper scale. This should result in a transcription with many occurrences of :

Note that repeated successions of [

[A i ] [A j ], [A] [B>A] and/or [A] [B|A].

7.d -Ultimate ambiguities / impossibility to annotate

In (hopefully) rare situations when the annotator is unable to decide on the label of a segment, he/she can use the following notations

• X, Y, Z = multiple possible interpretations between several labels • X ? Y ? Z = hesitation of the annotator between several incompatible possibilities • ? : complete inability to label the segment The choice of "multiple interpretations" corresponds to cases when the annotator is convinced that the ambiguity between several options is somehow intentional and unresolvable. Choosing "hesitation" means rather that the annotator is not able to decide which of the various options is the most likely.

Extended conventions

The additional conventions proposed in this section offer a possibility to refine the proposed basic notation, with the two following goals :

1) to offer a finer description of the specific properties of observed variants of the semiotic elements 2) to provide an indication of some (proto-)functional status of the blocks within the piece These conventions should however be used by experienced annotators only, once the basic notation is well understood and has become familiar. However, their knowledge should be useful to help the annotator identifying the type of variation that is observed in a given situation.

8.a -Expressive variants

The annotator may use a variety of superscripts to make more explicit the nature of observed variants of a same semiotic label A :

A f , A ff : increased loudness A p , A pp : decreased loudness A » , A »» : accelerated tempo A « , A «« : slowed-down tempo A # , A ## : upwards transposition A b , A bb : downwards transposition A M , A m : variants in major / minor modes A t+ , A t-: upwards / downwards translation (inducing a change of modality) A i+ , A i-: inclusion / deletion of an infix (can be used for prefixes and suffixes, with letters p and s) A + , A ++ : increased support (more sources participate to the carrier system) A -, A --: decreased support (less sources participate to the carrier system) A x , A xx : reinforced properties (carrier systems applies to stronger or more well-defined properties) A ~, A ~~ : weakened properties (carrier systems applies to weaker or less well-defined properties) A w1 , A w2 : variations of lyrics (can be neglected, except if considered as significantly expressive)

These various conventions should be understood as special cases which replace the standard superscript notation (A', A", A (3) , … ) in situations when it is possible to pinpoint the precise nature of the corresponding variant. However, before using these notations as expressive variations, the annotator must ascertain that the observed variations are not of a semiotic nature. This list is probably not exhaustive and additional notations on this basis may be designed to account for unpredicted situations.

8.b -Functional indications

Even though latent semiotic structure description is distinct from functional structure description, it can be informative to choose the semiotic labels in such a way that they somehow reflect the functional status of the block within the piece.

We therefore propose to use, as much as possible, the alphabetic letters with the following correspondence :

Summary of proto-functional conventions

Intro

Pre

Future work

On the basis of the elements presented in this document, future work will be directed towards :

-the further formalization of the concepts and methodology established in this report -the annotation and distribution of 500 songs annotated with the proposed conventions -the expression of optimization criteria for structural description, derived from information theory -the modeling of semiotic structure in terms of adequate hidden state automata -the design of new algorithms aiming at inferring automatically the semiotic structure

  For instance the structural description A B C D A B C D B C D E C D C D D can be viewed as a realization of 5 cycles of structural pattern (ABCD) with increasing irregularities towards the end of the piece. It could indeed be written as follows : A B C D A B C D A B C D A B E C D A B C D D, where double-barred characters are suppressions and underlined ones are insertions.

  [A] [B|A] would generally appear as [A A] [B A] or, more generally, as [A B] [C B].

  Main « set » : ABCDEF-IJKLMN-XYZ • C : Main central element (C for Central / Chorus) • D : Secondary central element (may or may not be situated just after C) • A,B : Elements that precede the main central element • E,F : Elements that follow the main central elements • I, J : Introductive / re-introductive elements (locally or globally) • K,L : Conclusive / pre-conclusive elements (locally or globally) • M,N : Other recurrent non-central elements • X,Y,Z : Singular elements (occurring only once or spuriously) Secondary « set » : GH-PQRSTUVW • R,S : Other central elements • P,Q : Elements which precede central elements • T,U : Elements which follow central elements • G,H : Introductive, conclusive, bridging elements • V,W : Any use A central element is defined as an element that is representative of the piece in terms of musical content and of frequency of occurrence.

The fundamentals behind these principles will be subject to a separate document.

A future version of this document will contain a brief survey of previous attempts to define music structure

The distinction between structuring and semiotic properties is essential.

Conceptual generalization to non-square stems is rather straightforward.

Corresponding publication is in preparation

This will be detailed in a separate publication.

This notation is currently under consolidation and is therefore subject to future evolutions.

Acknowledgements

Part of this work was achieved in the context of the QUAERO project, funded by OSEO.

We would like to thank our colleague Stanislaw Raczinsky who read, commented and made very useful suggestions to improve the pre-final version of this document.