Hermann Pfitzner

Phillips' Lemma for L-embedded Banach spaces

Keywords: Mathematics Subject Classification (2000). 46B20 Keywords. Phillips' lemma, L-embedded Banach space, L-projection

In this note the following version of Phillips' lemma is proved. The L-projection of an L-embedded space -that is of a Banach space which is complemented in its bidual such that the norm between the two complementary subspaces is additive -is weak * -weakly sequentially continuous.

Phillips' classical lemma [START_REF] Phillips | On linear transformations[END_REF] refers to a sequence (µ n ) in ba(IN) (the Banach space of finitely bounded measures on the subsets of IN) and states that if µ n (A) → 0 for all A ⊂ IN then k |µ n ({i})| → 0. It is routine to interpret this result as the weak * -weak-sequential continuity of the canonical projection from the second dual of l 1 onto l 1 because this continuity together with l 1 's Schur property gives exactly Phillips' lemma. (Cf., for example, [2, Ch. VII].) Therefore the following theorem generalizes Phillips' lemma (for the definitions see below):

Theorem 0.1. The L-projection of an L-embedded Banach space is weak * -weakly sequentially continuous.

The theorem will be proved at the end of the paper. The theorem has been known in the two particular cases when the L-embedded space in question is the predual of a von Neumann algebra or the dual of an Membedded Banach spece Y . In the first case the result follows from [1, Th. III.1]; in the second case Y has Pe lczyński's property (V) ( [START_REF] Godefroy | Weakly unconditionally convergent series in M -ideals[END_REF] or [START_REF] Harmand | M -ideals in Banach Spaces and Banach Algebras[END_REF]Th. III.3.4]) and has therefore, by [START_REF] Harmand | M -ideals in Banach Spaces and Banach Algebras[END_REF]Prop. III.3.6], what in [6, p. 73] or in [START_REF] Ülger | The weak Phillips property[END_REF] is called the weak Phillips property whence the result by [START_REF] Harmand | M -ideals in Banach Spaces and Banach Algebras[END_REF]Prop. III.2.4].

Preliminaries. By definition a Banach space X is L-embedded (or an L-summand in its bidual) if there is a linear projection P on its bidual X * * with range X such that P x * * + x * * -P x * * = x * * for all x * * ∈ X * * . The projection P is called L-projection. Throughout this note X denotes an L-embedded Banach space with L-projection P . We have the decomposition X * * = X ⊕ 1 X s where X s denotes the kernel of P that is the range of the projection Q = id X * * -P . We recall that a series z j in a Banach space Z is called weakly unconditionally Cauchy (wuC for short) if |z * (z j )| converges for each z * ∈ Z * or, equivalently, if there is a number M such that n j=1 α j z j ≤ M max 1≤j≤n |α j | for all n ∈ IN and all scalars α j . The presence of a non-trivial wuC-series in a dual Banach space is equivalent to the presence of an isomorphic copy of l ∞ . For general Banach space theory and undefined notation we refer to [START_REF] Johnson | Handbook of the Geometry of Banach Spaces[END_REF], [START_REF] Lindenstrauss | Classical Banach Spaces I and II[END_REF], or [START_REF] Diestel | Sequences and Series in Banach Spaces[END_REF]. The standard reference for L-embedded spaces is [START_REF] Harmand | M -ideals in Banach Spaces and Banach Algebras[END_REF]; here we mention only that besides the Hardy space H 1 the preduals of von Neumann algebras -hence in particular L 1 (µ)-spaces and l 1 -are L-embedded. Note in passing that in general an L-embedded Banach space, contrary to l 1 , need not be a dual Banach space.

The proof of the theorem consists of two halves. The first one states that the L-projection sends a weak * -convergent sequence to a relatively weakly sequentially compact set. This has already been proved in [START_REF] Pfitzner | L-embedded Banach spaces and a weak version of Phillips lemma[END_REF]. The second half asserts the existence of the 'right' limit and can be deduced from the corollary below which states that the singular part X s of the bidual is weak * -sequentially closed. Note that X s is weak * -closed if and only if X is the dual of an M-embedded Banach space [4, IV.1.9]. The following lemma contains the two main ingredients for the proof of the theorem namely two wuC-series

x * k and u * k by means of which the theorem above will reduce to Phillips' original lemma. The first one has already been constructed in [START_REF] Pfitzner | L-embedded Banach spaces and a weak version of Phillips lemma[END_REF], the construction of the second one is (somewhat annoyingly) completely analogous, with the rôles of P and Q interchanged, cf. (0.20) and (0.21). (For the proof of the theorem it is not necessary to construct both wuC-series simultanuously but there is no extra effort in doing so and it might be useful elsewhere.) Lemma 0.2. Let X be L-embedded, let (x n ) be a sequence in X and let (t n ) be a sequence in X s . Furthermore, suppose that x + x s is a weak * -cluster point of the x n and that, along the same filter on IN, u + u s is a weak * -cluster point of the t n (with x, u ∈ X, x s , u s ∈ X s ). Let finally x * , u * ∈ X * be normalized elements.

Then there is a sequence (n k ) in IN and there are two wuC-series

x * k and u * k in X * such that t n k (x * k ) = 0 for all k ∈ IN, (0.1) lim k x * k (x n k ) = x s (x * ), (0.2) lim k t n k (u * k ) = u * (u), (0.3) u * k (x n k ) = 0 for all k ∈ IN. (0.4)
Proof. Let 1 > ε > 0 and let (ε j ) be a sequence of numbers decreasing to zero such that 0

< ε j < 1 and ∞ j=1 (1 + ε j ) < 1 + ε. By induction over k ∈ IN 0 = IN ∪ {0} we shall construct four sequences (x * k ) k∈IN0 , (y * k ) k∈IN0 , (u * k ) k∈IN0 and (v * k ) k∈IN0 in X * (
of which the first members x * 0 , y * 0 u * 0 , and v * 0 are auxiliary elements used only for the induction) and an increasing sequence (n k ) of indices such that, for all (real or complex) scalars α j and with β = x s (x * ), γ = u * (u), the following conditions hold for all k ∈ IN 0 :

x * 0 = 0, y * 0 = 1, (0.5)

u * 0 = 0, v * 0 = 1, (0.6) α 0 y * k + k j=1 α j x * j ≤ k j=1 (1 + ε j ) max 0≤j≤k |α j |, if k ≥ 1, (0.7) α 0 v * k + k j=1 α j u * j ≤ k j=1 (1 + ε j ) max 0≤j≤k |α j |, if k ≥ 1, (0.8) t n k (x * k ) = 0, (0.9) u * k (x n k ) = 0, (0.10) y * k (x) = 0, and x s (y * k ) = β, (0.11) u s (v * k ) = 0, and v * k (u) = γ, (0.12) |x * k (x n k ) -β| < ε k , if k ≥ 1, (0.13) |t n k (u * k ) -γ| < ε k , if k ≥ 1. (0.14) We set n 0 = 1, x * 0 = 0, y * 0 = x * , u * 0 = 0 and v * 0 = u * .
For the following it is useful to recall some properties of P : The restriction of P * to X * is an isometric isomorphism from X * onto X ⊥ s with (P * y * ) |X = y * for all * ∈ X * , Q is a contractive projection and X * * * = X ⊥ s ⊕ ∞ X ⊥ (where X ⊥ is the annihilator of X in X * * * ).

For the induction step suppose now that x * 0 , . . . , x * k , y * 0 , . . . , y * k , u * 0 , . . . , u * k , v * 0 , . . . , v * k and n 0 , . . . , n k have been constructed and satisfy conditions (0.5) -(0.14). Since x + x s is a weak * -cluster point of the x n and u + u s is a weak *cluster point of the t n along the same filter there is an index n k+1 such that

|x s (y * k ) -y * k (x n k+1 -x)| < ε k+1 , (0.15) |t n k+1 (v * k ) -(u + u s )(v * k )| < ε k+1 , (0.16) Put E = lin({x * , x * 0 , . . . , x * k , y * k , P * x * 0 , . . . , P * x * k , P * y * k , u * , u * 0 , . . . , u * k , v * k , P * u * 0 , . . . , P * u * k , P * v * k }) ⊂ X * * * , F = lin({x n k+1 , t n k+1 , x, x s , u, u s }) ⊂ X * * . Clearly Q * x * j , Q * y * k , Q * u * j , Q * v * k ∈ E for 0 ≤ j ≤ k.
By the principle of local reflexivity there is an operator R : E → X * such that Re * * * ≤ (1 + ε k+1 ) e * * * , (0.17)

f * * (Re * * * ) = e * * * (f * * ), (0.18) R |E∩X * = id E∩X * (0.19)
for all e * * * ∈ E and f * * ∈ F .

We define

x * k+1 = RP * y * k and y * k+1 = RQ * y * k , (0.20) u * k+1 = RQ * v * k and v * k+1 = RP * v * k . (0.21)
In the following we use the convention

0 j=1 (• • • ) = 0.
Then we have that

α 0 y * k+1 + k+1 j=1 α j x * j = R Q * (α 0 y * k + k j=1 α j x * j ) + P * (α k+1 y * k + k j=1 α j x * j ) , α 0 v * k+1 + k+1 j=1 α j u * j = R P * (α 0 v * k + k j=1 α j u * j ) + Q * (α k+1 v * k + k j=1 α j u * j ) .
Now (0.7) (for k + 1 instead of k) can be seen as follows:

α 0 y * k+1 + k+1 j=1 α j x * j ≤ (0.17) ≤ (1 + ε k+1 ) Q * (α 0 y * k + k j=1 α j x * j ) + P * (α k+1 y * k + k j=1 α j x * j ) = (1 + ε k+1 ) max Q * (α 0 y * k + k j=1 α j x * j ) , P * (α k+1 y * k + k j=1 α j x * j ) ≤ (1 + ε k+1 ) max α 0 y * k + k j=1 α j x * j , α k+1 y * k + k j=1 α j x * j ≤ k+1 j=1 (1 + ε j ) max{ max 0≤j≤k |α j |, max 1≤j≤k+1 |α j |} = k+1 j=1 (1 + ε j ) max 0≤j≤k+1 |α j |
where the last inequality comes from (0.5) if k = 0, and from (0.7), if k ≥ 1. Likewise, (0.8) (for k + 1 instead of k) is proved. The conditions (0.9) and (0.11) (for k + 1 instead of k) are easy to verify because P t n k+1 = 0, Qx = 0 and Qx s = x s thus, by (0.18)

t n k+1 (x * k+1 ) = P t n k+1 (y * k ) = 0, y * k+1 (x) = Qx(y * k ) = 0 and x s (y * k+1 ) = Q * y * k (x s ) = x s (y * k ) = β.
In a similar way we obtain (0.10) and (0.12) (for

k+1 instead of k) by u * k+1 (x n k+1 ) = RQ * v * k (x n k+1 ) = Qx n k+1 (v * k ) = 0, u s (v * k+1 ) = P u s (v * k ) = 0 and v * k+1 (u) = v * k (u) = γ.
Finally, we have

x * k+1 (x n k+1 ) -β = y * k (x n k+1 ) -β = y * k (x n k+1 -x) -x s (y * k )
by (0.11) whence (0.13) for k + 1 by (0.15). Analogously, we get (0.14) for k + 1 via (0.16) and t n k+1 (u * k+1 ) = t n k+1 (v * k ) and (u + u s )(v * k ) = γ by (0.12). This ends the induction and the lemma follows immediately.

Corollary 0.3. The complementary space X s of an L-embedded Banach space X is weak * -sequentially closed.

Proof. Suppose that (s n ) is a sequence in X s that weak * -converges to v + v s . Let u * ∈ X * be normalized, set t n = s nv s . We apply the lemma to (t n ) with u = v, u s = 0 and x n = u and define a sequence (µ n ) of finitely additive measures on the subsets of IN by µ n (A) = (t nu)( k∈A u * k ) for all A ⊂ IN where k∈A u * k ∈ X * is be understood in the weak * -topology of X * and where the u * k are given by the lemma. Then µ n (A) → 0 for all A ⊂ IN and by Phillips' original lemma we get

|t n k (u * k )| (0.4) = |(t n k -u)(u * k )| ≤ j |(t n k -u)(u * j )| = j |µ n k ({j})| → 0.
Thus u * (u) = 0 by (0.3) and u = 0 because u * was arbitrary in the unit sphere of X * . Hence (t n k ) weak * -converges to 0 which is enough to see that (s n ) weak *converges to v s in X s .

Proof. Proof of the theorem: Let X be an L-embedded Banach space with Lprojection P . Suppose that the sequence (x * * n ) is weak * -null and that x * * n = x n +t n with x n = P x * * n . Let x * be a normalized element of X. The sequence (x n ) is bounded and admits a weak * -cluster point x + x s . We use the lemma, this time with the wuC-series

x * k , like in the proof of the corollary and define a sequence Thus x s (x * ) = 0 by (0.2) and x s = 0 because x * was arbitrary in the unit sphere of X * . It follows that each weak * -cluster point of the set consisting of the x n lies in X. Hence this set is relatively weakly sequentially compact by the theorem of Eberlein-Šmulian. If x is the limit of a weakly convergent sequence (x nm ) then (t nm ) weak * -converges to -x. Hence x = 0 by the corollary. This shows that the sequence (x n ) is weakly null and proves the theorem.

  (µ n ) of finitely additive measures on the subsets of IN by µ n (A) = x * * n ( k∈A x * k ) for all A ⊂ IN. Then µ n (A) → 0 for all A ⊂ IN and by (0.1) and Phillips' original lemma we get |x * k (x n k )| = |x * * n k (x * k )| ≤ j |x * * n k (x * j )| = j |µ n k ({j})| → 0.


Hermann Pfitzner

L-embedded Banach spaces and a weak version of Phillips lemma

Keywords: Mathematics Subject Classification (2000). 46B20 Keywords. L-embedded, L-projection, Phillips lemma, Bochner spaces

After a short survey on L-embedded Banach spaces -for which Bochner spaces of the form L 1 (X) with reflexive X serve as examples -we prove that these spaces satisfy a weak form of Phillips' lemma: It is proved that the L-projection of an L-embedded Banach spaces sends relatively w *sequentially compact sets to relatively weakly sequentially compact sets.

Further, if the underlying measure space is finite and if X ⊂ L 1 is (in Godefroy's terminology) nicely placed i.e. L-embedded, for example H 1 0 ⊂ L 1 (I T), then L 1 (X) is L-embedded, too. Finally, if X is reflexive (hence trivially L-embedded) then L 1 (X) is L-embedded.

The class of L-embedded spaces is much larger than the one of duals of Membedded spaces because among L 1 -spaces only l 1 (Γ) (Γ a set) is the dual of an M-embedded space, and, analogously, the only von Neumann predual which is the dual of an M-embedded space is the direct sum in the l 1 -sense of N(H)spaces which in turn is equivalent to the von Neumann predual having RNP [3, Prop. III.2.9, Prop. IV.2.9]. In particular, an L-embedded space need not have RNP whereas the dual of an M-embedded space does. There are dual L-embedded spaces with RNP that are not duals of M-embedded spaces [3, Ex. IV.4.12].

The interest of L-embeddedness lies in the fact that it unifies some Banach space properties which before were known only as special cases. A kind of breakthrough for this general point of view was Godefroy's discovery that L-embedded Banach spaces are weakly sequentially complete. Another example of this unifying point of view is the fact [START_REF] Pfitzner | Separable L-embedded Banach spaces are unique preduals[END_REF] that (at least) separable L-embedded spaces are unique preduals which comprises for example the corresponding result of Dixmier-Sakai for von Neumann preduals.

Talagrand [START_REF] Talagrand | Weak Cauchy sequences in L 1 (E)[END_REF] has shown that the Bochner space L 1 (X) is weakly sequentially complete if and only if X is and Randrianantoanina [START_REF] Randrianantoanina | Complemented copies of ℓ 1 and Peczyski's property (V * ) in Bochner function spaces[END_REF] has shown that L 1 (X) has Pe lczyński's property (V * ) if and only if X does (see also [START_REF] Lin | Köthe-Bochner function spaces[END_REF]). Since L-embedded spaces are weakly sequentially complete and have (V * ) [3, IV.2.7] it seems plausible that L 1 (X) is L-embedded for more Banach spaces X than the ones mentionned above.

A weak version of Phillips' lemma. Phillips' lemma [START_REF] Phillips | On linear transformations[END_REF] states that the canonical projection from the second dual of l 1 onto l 1 is w * -norm-sequentially continuous which if one takes l 1 's Schur property for granted is equivalent to the projection being w * -weak-sequentially continuous. (Cf., for example, [1, Ch. VII].) Therefore the fact that the canonical projection from the third onto the first dual of a Banach space is w * -weak sequentially continuous if the space in consideration has Pe lczyński's property (V) (cf. [START_REF] Harmand | M -ideals in Banach Spaces and Banach Algebras[END_REF]Prop. III.3.6]) can be considered a generalization of Phillips' lemma.

In this note we give a modest -see the remark after the proof -generalization in a similar direction by looking at l 1 as an L-embedded Banach space. Although the result is new the technique of its proof is not. In fact it is a modification of the proof in [START_REF] Pfitzner | Separable L-embedded Banach spaces are unique preduals[END_REF] but holds, contrary to the latter one, for all L-embedded Banach spaces, not only separable ones.

Theorem The L-projection of an L-embedded Banach space sends relatively w *sequentially compact sets into relatively weakly sequentially compact sets.

Proof. We recall that a series z j in a Banach space Z is called weakly unconditionally Cauchy (wuC for short) if |z * (z j )| converges for each z * ∈ Z * or, equivalently, if there is a number M such that n j=1 α j z j ≤ M max 1≤j≤n |α j | for all n ∈ IN and all scalars α j .

Let X be an L-embedded Banach space with L-projection P . We have the decomposition X * * = X ⊕ 1 X s where X s denotes the range of the projection Q = id X * * -P . Let (x n ) be a sequence in X and let (t n ) be a sequence in X s . Further, consider a cluster point x + x s of the x n in the w * -topology of X * * (with x ∈ X, x s ∈ X s ).

Claim Given x * ∈ X * of norm one there are a wuC-series

x * k in X * and a sequence (n k ) in IN such that, for all k ∈ IN,

t n k (x * k ) = 0 (0.1) lim k x * k (x n k ) = x s (x * ). (0.2)
Proof of the claim: Let 1 > ε > 0 and let (ε j ) be a sequence of numbers decreasing to zero such that 0 < ε j < 1 and

∞ j=1 (1 + ε j ) < 1 + ε. By induction over k ∈ IN 0 = IN ∪ {0} we shall construct two sequences (x *
k ) k∈IN0 and (y * k ) k∈IN0 in X * (of which the first members x * 0 and y * 0 are auxiliary elements used only for the induction) and an increasing sequence (n k ) of indices such that, for all (real or complex) scalars α j and with β = x s (x * ), the following hold:

x * 0 = 0, y * 0 = 1, (0.3)
α 0 y * k + k j=1 α j x * j ≤ k j=1 (1 + ε j ) max 0≤j≤k |α j |, if k ≥ 1, (0.4) t n k (x * k ) = 0, (0.5) y * k (x) = 0, and x s (y * k ) = β, (0.6) |x * k (x n k ) -β| < ε k . (0.7)
We set n 0 = 1, x * 0 = 0 and y * 0 = x * . For the following it is useful to recall some properties of P : The restriction of P * to X * is an isometric isomorphism from X * onto X ⊥ s with (P * y * ) |X = y * for all y * ∈ X * , Q is a contractive projection and X * * * = X ⊥ s ⊕ ∞ X ⊥ (where X ⊥ is the annihilator of X in X * * * ). For the induction step suppose now that x * 0 , . . . , x * k , y * 0 , . . . , y * k and n 0 , . . . , n k have been constructed and satisfy conditions (0.3) -(0.7). Since x + x s is a w * -cluster point of the x n there is an index n k+1 such that

|x s (y * k ) -y * k (x n k+1 -x)| < ε k+1 . (0.8) Put E = lin({x * , x * 0 , . . . , x * k , y * k , P * x * 0 , . . . , P * x * k , P * y * k }) ⊂ X * * * , F = lin({x n k+1 , t n k+1 , x, x s }) ⊂ X * * . Clearly Q * x * j , Q * y * k ∈ E for 0 ≤ j ≤ k.
By the principle of local reflexivity there is an operator R : E → X * such that (1 -ε k+1 ) e * * * ≤ Re * * * ≤ (1 + ε k+1 ) e * * * , (0.9) f * * (Re * * * ) = e * * * (f * * ), (0.10)

R |E∩X * = id E∩X * (0.11)
for all e * * * ∈ E and f * * ∈ F . We define

x * k+1 = RP * y * k and y * k+1 = RQ * y * k .
In the following we use the convention

0 j=1 (• • • ) = 0. Then we have that α 0 y * k+1 + k+1 j=1 α j x * j = R Q * (α 0 y * k + k j=1 α j x * j ) + P * (α k+1 y * k + k j=1 α j x * j ) .
Now (0.4) (for k + 1 instead of k) can be seen as follows:

α 0 y * k+1 + k+1 j=1 α j x * j ≤ (0.9) ≤ (1 + ε k+1 ) Q * (α 0 y * k + k j=1 α j x * j ) + P * (α k+1 y * k + k j=1 α j x * j ) = (1 + ε k+1 ) max Q * (α 0 y * k + k j=1 α j x * j ) , P * (α k+1 y * k + k j=1 α j x * j ) ≤ (1 + ε k+1 ) max α 0 y * k + k j=1 α j x * j , α k+1 y * k + k j=1 α j x * j ≤ k+1 j=1 (1 + ε j ) max{ max 0≤j≤k |α j |, max 1≤j≤k+1 |α j |} = k+1 j=1 (1 + ε j ) max 0≤j≤k+1 |α j |
where the last inequality comes from (0.3) if k = 0, and from (0.4), if k ≥ 1.

The conditions (0.5) and (0.6) (for k + 1 instead of k) are easy to verify because P t n k+1 = 0, Qx = 0 and Qx s = 0 thus, by (0.10)

t n k+1 (x * k+1 ) = y * k (P t n k+1 ) = 0, y * k+1 (x) = y * k (Qx) = 0 and x s (y * k+1 ) = Q * y * k (x s ) = x s (y * k ) = β. Finally, we have x * k+1 (x n k+1 ) -β = y * k (x n k+1 ) -β = y * k (x n k+1 -x) -x s (y * k )
by (0.6) whence (0.7) for k + 1 by (0.8). This ends the induction and proves the claim.

Put now x * * n = x n + t n and suppose that (x * * n ) is w * -null. We define a sequence (µ n ) of finitely additive measures on the subsets of IN by µ n (A) = x * * n ( k∈A x * k ) for all A ⊂ IN where k∈A x * k is to be understood in the w * -topology of X * . Then µ n (A) → 0 for all A ⊂ IN and by (0.1) and Phillips' original lemma ( [START_REF] Phillips | On linear transformations[END_REF], [1, p. 83]) we get

|x * k (x n k )| = |x * * n k (x * k )| ≤ j |x * * n k (x * j )| = j |µ n k ({j})| → 0.
Thus x s (x * ) = 0 by (0.2). Since x * was an arbitrary normalized element this means that x s = 0. It follows that the w * -closure in X * * of the set consisting of the x n lies in X. Hence this set is relatively weakly compact (or, equivalently, relatively weakly sequentially compact). This proves the theorem.

It is not clear whether the L-projection is actually w * -weak-continuous. In two cases this happens trivially: firstly if the dual X * is a Grothendick space (which by definition means that in X * * w * -convergent sequences converge weakly) and secondly if X is the dual of an M-embedded Banach space Y (see [3, Ch. III]) because in this case P is the canonical restriction projection from Y * * * onto Y * ([3, Prop. III.2.4]) and the x n tend to 0 on the elements of Y and thus any weak cluster point x of (x n ) is necessarily zero. Alternatively, in the latter case the result [3, Prop. III.3.6], which was mentionned above, applies because M-embedded Banach spaces have property (V) ( [START_REF] Godefroy | Weakly unconditionally convergent series in M -ideals[END_REF] or [START_REF] Harmand | M -ideals in Banach Spaces and Banach Algebras[END_REF]Th. III.3.4]).


Thierry Gousset

Jacob Lamblin

Sandra Valcares

Radioelectric fields from cosmic-ray air showers at large impact parameters

We discuss electric fields generated by cosmic ray air showers at large impact parameters b. An approximation relevant to this situation is given. The formulation makes explicit the relationship between the shower profile and the radio pulse shapes at large b, putting forward one important observational consequence, namely the decrease of the high-frequency cutoff ν c ∝ 1/b 2 when the impact parameter increases. The approximation is also used to give a detailed comparison between two emission models, the geosynchrotron model and the transverse current model.

Introduction

There is nowadays a renewed interest in the extensive air shower radio detection technique [1,2]. Results have been obtained which improved on both the experimental and theoretical exploratory studies that took place during the sixties. An illuminating review of this early phase is given in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF].

From the theoretical point of view, several comprehensive frameworks have been completed or initiated in the last five years [START_REF] Huege | [END_REF]5]. The most advanced approach (geosynchrotron approach) is by Huege and collaborators [START_REF] Huege | [END_REF]6,7]. It gives a wealth of details on the pattern of radio emission with different levels of sophistication for how the shower development is taken into account. More recently, based on the model first proposed by Kahn and Lerche [8], Scholten and Werner developed a different approach (transverse current approach) [5]. The present study can be seen as a follow-up of the latter work as it exploits a limiting behavior, the so-called point-like approximation, motivated in Ref. [5]. We give a somewhat different and complementary motivation for arriving at the same approximation. Our derivation follows a line of reasoning suggested in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. It makes it possible to extend the approximation to the cases of non vertical showers and also to get the corresponding expression for the electric field considered in Ref. [START_REF] Huege | [END_REF].

We believe that, however crude the approximation may be, it might be useful in the present stage of mastering the radio detection technique when progress is made using a 1 delicate mix of physical prejudices and empirical facts. This is provided that the model captures some essential features of the phenomenon. For the approximation at hand a point that can be put forward is the direct relationship at large impact parameter between the time-development of an air shower and that of its radio component. This has two testable consequences. First, the decrease of the high-frequency cutoff of the electric field spectrum when the impact parameter rises: this has never been seen in experiment so far and is a crucial test for our understanding of radio emission. Next, the sensitivity of the radio component to the beginning of the shower development, making radio complementary to traditional techniques for that matter.

The derivation of the electric field expressions in the point-like approximation is given in Sect. 2 for both the transverse current and the geosynchroton approaches. This leads us to discuss the differences between the two approaches concerning on the one hand their phenomenologies and on the other hand their physical motivations. The section ends in a discussion of the domain of impact parameters and frequencies where the approximation makes sense. The relationship between the time pulse and shower evolution is investigated in Sect. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF], showing the importance of the Doppler effect. Sect. 4 is devoted to questions concerning a very large array, a natural topic for an application of large impact parameter calculations. Our main observations are collected in Sect 5.

The point-like model

Electric field expressions

A cosmic-ray air-shower is a charge and current system whose electromagnetic field carries genuine information on some important aspects of the shower. Because the shower front moves at about the speed of light, the electromagnetic signals emitted in the forward direction by the shower core at various stages of the shower history all travel with the particle front. Thus only at observation points located at sufficiently large impact parameters, i.e., away from the forward direction, the sequence of emissions results in a sequence of receptions, and the collected pulse is a reflection of the shower evolution. This is because the time scale at reception associated with the shower development becomes much larger than those coming from the shower extension. The hierarchy of time scales relevant at large impact parameters is thoroughly studied in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF].

This hierarchy suggests a model where all time scales but that due to shower evolution are set to zero. Setting time scales associated with the shower extension to zero corresponds to neglecting the shower extension altogether, hence taking the shower as a point-like system. Of course, this is only meant as a model for the densest part of the shower core where most of the ultra-relativistic charges stand. As discussed in Sect. 2.3, this model makes sense for impact parameters in the km range and at not too large frequencies.

The explicit expression for the electric field depends on both the origin of charge separation and the electric field emission mechanism. For electron-positron separation in the Earth magnetic field, two possibilities have been explored recently, considering that the electric field is generated either by charge acceleration (geosynchrotron emission) [START_REF] Huege | [END_REF] or by charge drift [5]. Both will be considered here in order to show the flexibility of our approach and to gauge the sensitivity of some results. For a given choice of charge separation and emission mechanism the expression of the electric field in the model is essentially an adaptation of the relevant textbook formula to the case where the number of elementary charges N ee evolves with time. Such a scheme was already followed in Ref. [9]. There it is also explained that at a given impact parameter b two effects are in competition for building the electric field. These are the inverse distance 1/R (notations in Fig. 1) between the charge and the observation point and the Doppler factor ∂t ∂t

-1 = 1 1 -nβ cos ϑ (t is the emission time, t is the reception time, ϑ = ( v, -→ QA)
, n is the refraction index and β = v/c). The larger both factors are the larger is the field. For n very close to 1 (air) and v very close to c, the Doppler factor singles out very small angles (for n = 1, ϑ = O(1/γ)). The most favorable situation is therefore at small b. However, if we insist on having a somewhat larger b (so that the point-like model may apply) then the combination N ee /R will single out finite angles1 in contradistinction to the Doppler factor. The competition is illustrated in Fig. 2 as a function of Z (the relation between Z and ϑ is Z = b/ tan ϑ).

As a result of the above competition, the electric field picks up its largest magnitude in a region of small but not too-small angles where the following conditions are fulfilled:

• ϑ 1, the Doppler contraction plays a role, so that relativistic features of radiation are partly kept,

This greatly simplifies the computation of the electric field since all source based time dependences can be expressed as function of t and b, i.e., observer based quantities. The transverse current field reads

E v (t, A) = (N ee (t ) + t Ṅee (t )) e v T 4πε 0 c 1 (ct) 2 = e v T 4πε 0 c 4(ct ) 2 (N ee (t ) + t Ṅee (t )) b 4 . (2)
A complete derivation can be found in Ref. [5]. We give some details in the Appendix and also on how to arrive at the geosynchrotron field that reads

E a (t, A) = N ee (t ) e a T 4πε 0 c 2 b 2 2 (ct) 3 = e a T 4πε 0 c 4(ct ) 2 (-t N ee (t )) b 4 . ( 3
)
a T is the transverse acceleration (transverse with respect to the shower axis) and is a unit vector showing the polarization which is perpendicular to the shower axis and such that (

B T , ) = π 2 + 2 × ( B T , -→ BA). (4)

Z km

Figure 3: d dt (t N ee (t ))/(t N ee (t )) in µs -1 for θ = 0 (right) and 45 • (left). Dashed curves show the respective behaviors of E a (t ), rescaled so that their maxima = 0.1, to help focusing on the regions of interest.

In Eqs. ( 2) and (3), the first expressions make it easy to recognize the one-particle field approximation in the relevant ϑ range, setting N ee (t ) →const., whereas the second expressions are useful for discussing scaling relations with b, see Sect. 2.2.

Comparisons

Eq. (2) shows that the magnitude of the field writes as a function of t divided by b 4 . As noticed in Ref. [5], because of Eq. ( 1) this means that the electric field pulse duration scales as b 2 and its peak value as 1/b 4 . Eq. (3) shows that the same properties hold for the geosynchrotron field.

The functional dependence on N ee is different for E v and E a . E a is proportional to t N ee (t ) whereas E v goes like the derivative with respect to t of this quantity. As a consequence, it is easily checked that the time integral of E v (t, A) is zero, i.e., the E v pulse is strictly bipolar [5]. The E a pulse is predominantly monopolar. Sect. 3 is devoted to the discussion of the shower evolution dependence of the electric pulse.

As for comparison of magnitudes, the fields in the point-like approximation are in proportion of

v T × d dt (t N ee (t )) vs a T × (t N ee (t )). (5)
This amounts to comparing the ratio v T /a T to a shower development quantity. These are genuinely different quantities, which has a consequence on their orders of magnitude, but also on their parametric dependences. Fig. 3 displays d dt (t N ee (t ))/(t N ee (t )) in µs -1 for two zenith angles. (The ratios being time-dependent, E a shapes are also shown to point out the relevant regions.) The estimate given in the Appendix B leads to v T /a T = 0.67 µs. Consequently, the peak value of the E v field is one order of magnitude smaller than the E a field. The parametric dependence of this hierarchy can be discussed by looking at the involved scales. Physically, the shower development is best described in terms of

B T B

Figure 4: E a polarization map for a shower going through the map at point B. Acceleration of positrons is to the right but the polarization is not always opposite to it. (Polarization of E v would be to the left everywhere.) slant depth, X, and a scale ∆X in this development, e.g., the width of N ee (X), can be transformed into a length scale by dividing ∆X by the air density at the point of interest, ∆X/ρ air . The ratio v T /a T ∼ τ is given by the lifetime of a particle (ultra-relativistic electron or positron) in the shower core [8], see also the Appendix B. Since c τ ∼ X 0 /ρ air the relative strength of E v and E a is given by the relative strength of X 0 and ∆X. The above discussed hierarchy can be seen as a consequence of ∆X X 0 for high-energy showers. However, no strong variation of this hierarchy was found when changing shower characteristics, such as inclination or energy.

Eq. (5) suggests that E v and E a are two components that derive from a single quantity. This fact may be connected to the occurrence of angular polarization in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF], which is the single quantity from which the total electric field derives in the latter formalism. In the present formulation, however, the relationship is less apparent when polarization is considered. Polarization of the velocity field is simply given by the direction of v T , i.e., perpendicular to the plan formed by the shower axis and the magnetic field. That of the geosynchrotron field, see Eq. ( 4) and Fig. 4, is more complex: it is still perpendicular to the shower axis, but not always in the direction of a T , hence giving another way to distinguish between E v and E a . The pattern shown in Fig. 4 is a purely relativistic effect. For a single charge this kind of polarization occurs at angle such that cos ϑ < nβ, i.e., precisely the regime where the point-like approximation is formulated (see Sect. 2.1). The polarization pattern obtained in the general geosynchrotron approach is even more complex. In the latter approach, at an observation point located at small b some particles are emitting at very small angles (cos ϑ > nβ) while others are emitting at larger angles resulting in a highly non trivial pattern. However, it is possible to understand qualitatively this pattern as a superposition of a small-angle component, opposite to the acceleration, and a large angle one with the pattern shown in Fig. 4: for antennas located at the north or the south of B (we refer to the cardinal points on the map with north to the top as usual) both fields add and the result is purely EW; for points east or west of B there is a partial cancellation but the superposition remains EW; for direction NE and the like both an EW and a NS components show up. (See Fig. 3 in Ref. [6].)

As a matter of fact E v and E a are genuinely different. The difference has its roots in the underlying physical picture for radio emission. For the E v field the picture is that of drifting charges much as in a conductor in a stationary state: charges are constantly accelerated by the external field but acceleration due to collisions is, on average, opposite to this external field acceleration, and coherence implies that any radiation is that of a system with no net acceleration. In the E a approach, on the contrary, collisions are considered to be inefficient to alter the radiation associated to the external field acceleration and are disregarded (in the conductor picture this would correspond to the transient regime). A dedicated study of shower evolution is necessary in order to see which picture is the right one (or which mix is effectively relevant). Such a study, e.g., based on shower simulations, goes beyond the scope of the present work. Naively, the scales involved have a tendency to promote the drifting regime since the timescale for collision is τ which is smaller than the timescale for evolution of the shower as a whole. In the following, when limiting a study to one field, the E v situation will be considered.

Validity

As was explained in Sect. 2.1, the point-like approximation requires that the shower evolution, as seen by an observer, has a time scale much larger than those due to the shower extension. The relevant time scales were studied in detail in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] and we will essentially reproduce the reasoning here, extending the consideration to somewhat larger b and also to inclined showers.

Allan makes his estimate for a vertical shower with energy around 10 17 eV. He first estimates a rise time for the pulse associated with the shower evolution (the so-called obliquity effect in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]) by computing the time elapsed between the reception of a signal emitted at a height 10 km (shower "start") and that at a height 5 km (shower maximum). At small b, the shower evolution alone, i.e., the function f 1 (t ) = N ee (t ), leads to the above choices for 'start' and 'max'. At larger b, Eqs. ( 2) and [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]

show that it is rather f 2 (t ) = t 2 |N ee + t Ṅee | or f 3 (t ) = |t | 3 N ee (t )
that drive the pulse behaviors. Values of Z = -c t at which f 1 , f 2 , and f 3 go through their first maxima are given in Table 1, for various inclinations and energies E p = 10 19 and 10 17 eV.

The rise time (multiplied by c) is, using Eq. ( 1), Consideration of length scales is also important in the frequency domain. When ∆ g ∆ l and ∆ g ∆ r the loss of coherence is set by obliquity at a frequency ν c ∼ c/(2π∆ g ). Since ∆ g ∝ b 2 , an important signature of shower radio emission is provided by the observation of the scaling law ν c ∝ 1/b 2 at large b. There is also an upper bound for application of the point-like approximation in the frequency domain given by ν c ∼ c/(2π∆ l ). Taking ∆ l = 3 -10 m gives ν c = 5 -15 MHz. These figures are also put forward in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. It should be noted that the experimental situation is not clear concerning the loss of coherence at small b where the scales ∆ l and ∆ r are still relevant (∆ g → 0). It seems that a more or less flat spectrum has been observed around 50 MHz [10]. Such results call for confirmation but it may well be that the above figures for ν c are underestimated making the range of applicability of the point-like model extending somewhat beyond 5 -15 MHz. In any case, observation of loss of coherence is one of the major goal to be achieved by the present radio detection experiments.

∆ g = b 2 2Z max - b 2 2Z start , θ max f 1 max f 2 max f 3 0 2.
∆ r = b r Z , ∆ g ∆ l = l,
3 Deciphering the shower profile

Pulse shapes and spectra

A selection of pulse shapes and spectra is shown in Figs. 5 and6, respectively for E v and E a , for showers of energy E p = 10 19 eV and X max = 770 g/cm 2 and an observer at b = 1 km.

For E v the beginning of the electric field pulse is polarized along t Ṅee (t ) v T , i.e., opposite to v T ∝ v × B with v the unit vector along the shower axis and B the Earth magnetic field. For the latter we consider the situation in central Europe: the zenith angle is θ B = 153 • and the horizontal component goes from south to north. Consequently, the polarization is east-west for showers coming from north or south. Furthermore, the initial polarization is westward for a vertical shower while it is eastward for showers coming from south with θ ≥ 27 • . This explains the various pulse signs in Fig. 5. For other shower directions, the study of polarization can be made along the same lines.

For E a , the electric field polarization depends on the observer location with respect to the shower (see Fig. 4). Here an observer standing east or west of the shower is considered. As in the E v case there is a flip of sign when crossing the magnetic lines at θ = 27 • .

The explicit relationships between pulse shape and time evolution of the shower charge number are given in Eqs. (2,[START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. In both cases, the Doppler-like effect, t → t, distorts the time dependence. From Eq. (1):

∂t ∂t = 1 2 b c t 2 .
Early times (corresponding to t negative and large) getting more contracted than later times. This explains the sharp rise, the slower decay, and in the E v case the even slower undershoot of the pulses shown in Figs. 5 and6. The variation of rise time with θ was already discussed in Sect. 2.3. The probing of earlier shower stages when θ increases affects pulse shape and has the tendency to give larger fields. To get the complete change, this increase has to be combined with the overall amplitude modification coming from the variation of sin α(θ, ϕ), where α is the angle between the shower axis and the magnetic field. At fixed θ, the azimuth dependence is only through the latter overall factor. Also remember that the b dependence is simple in the approximation: pulses for another b can be deduced from Figs. simple form f (t )/b 4 for both fields Eqs. (2,3) together with Eq. ( 1) leads to

F (ν, b) = dt f (t ) b 4 e 2iπνt = 1 b 2 dt f (t ) 2(ct ) 2 e -iπν(b/c) 2 /t
which implies that

F (ν, b) = b 0 b 2 F (νb 2 /b 2 0 , b 0 ).
As for the time pulse, it is thus possible to obtain a spectrum at another b from Figs. 5 or 6 taking into account that frequencies and spectrum magnitude are both ∝ 1/b 2 . A specific application of the rescaling law for frequencies is the scaling behavior of the high-frequency cutoff, ν c ∝ 1/b 2 , already discussed in Sect. 2.3.

For the E v field the spectrum is 0 at ν = 0 (bipolarity), goes through a maximum in the MHz range for b = 1 km and then falls off almost as a pure exponential (loss of coherence) at least at not-too-large θ. At θ = 60 • the loss of coherence starts beyond 10 MHz, i.e., a value comparable to the estimate given for ν c , reminding us that the pointlike approximation is marginally applicable at b = 1 km for this large inclination (see Sect. 2.3). The loss of coherence for the E a spectrum follows the same trend. The main difference is at small frequencies, the E a spectrum being finite at ν = 0.

Sensitivity to shower rise

An interesting consequence of the Doppler distortion lies in the greater sensitivity of the radio signal to the beginning of the shower. An illustration of this feature is shown in Fig. 7. The pulse height difference is a signal of the difference between the two developments (here around 8 km). The region of greater sensitivity is thus different in radio detection and in fluorescence. Note that this property is generic in radio emission at large impact parameters [9]: in the present study it was observed for both the E v (Fig. 7) and the E a (not shown) emission models.

The effect is more visible in the frequency domain (right plot in Fig. 7). The shower growth being pushed at larger frequencies than the shower decay by the Doppler effect, the stiffening of the time pulse for the Gaisser-Hillas profile stands out in the spectrum at large frequencies. This may be important in practice, since large radio frequency interferences often render radio observation difficult at frequencies below 20 MHz. In such circumstances the greater sensitivity of radio detection to the first stage of shower evolution may be even more pronounced.

Profile extraction

From Sect. 2.1, the electric field magnitude observed at the location of antenna i, A i , reads for the transverse current component, and

E v (t, A i ) = K v (ct) 2 (N ee (t ) + t Ṅee (t )), t = -b 2 i 2c 2 t ,
E a (t, A i ) = K a b 2 i (ct) 3 N ee (t ),
for the synchrotron one. K v and K a are both independent from the observation point of the event and are thus irrelevant. They can be set to 1 for the reasoning. Such simple expressions for electric fields make it possible to illustrate what knowledge can be gained from radio measurements. Imagine the full geometry of the shower is known, i.e., both the shower incidence and the impact point location, either by some joint particle detector array or by an analysis based on radio alone [START_REF] Gousset | contribution to the 30th International Cosmic Ray Conference[END_REF]. Then it is possible to obtain the full time evolution of N ee from the time pulse collected on antenna say i = 1 by looking at either (ct

) 2 × E v (t, A 1 ) = g 1 (-b 2 1 /(2ct)
), for the transverse current field, or

(ct) 3 × E a (t, A 1 ) = g 2 (-b 2 1 /(2ct)),
for the synchrotron field. In the latter case, the function g 2 (t ) is directly proportional to N ee (t ). In the former g 1 (t ) = N ee (t ) + t Ṅee (t ).

g 1 (t ) is the time derivative of t N ee (t ), so a time integration, followed by a division by t , is necessary to obtain N ee (t ). In any case, the unfolding of the shower profile is quite simple in the model, in spite of the presence of the Doppler distortion.

In principle, the signal collected on one single antenna is enough in order to make the aforementioned extraction, but several antennas may be used to enhance the signal over background ratio. In addition, the signal can only be extracted in practice in a limited frequency-range. For sure this is the more severe limitation to the suggested analysis. However, the shower evolution is not a complete unknown and various tools of signal analysis can be considered to obtain pieces of information on the N ee evolution. Having one such tool in mind, the point-like model expressions are sufficiently easy to manipulate in order to test the relevance of the foreseen method. It is not the purpose of the present study to elaborate on those methods which depend to a certain extent on the specifics of experiments and the noise situation at their locations.

Towards a large array

The radio detection technique is nowadays considered as a possible surrogate to more traditional techniques for future giant array, such as Auger north. One elementary detector per km 2 is the typical scale for such an array, setting the scale of closest approach to 1 km. Since the phenomenology discussed above applies just to this regime, it may have consequences on the exploitation of a large array which we examine here. Given a detection criterion it is straightforward to count the number of antennas that see a given cosmic ray shower event, once the distribution of antennas is fixed. The geometry considered is that of an hexagonal array on a horizontal ground located at sea level and the detection efficiency is studied as a function of the antenna spacing. Our detection criterion is that the electric field magnitude is above a threshold to be defined on at least three antennas belonging to the same elementary triangle. Following Ref. [START_REF] Falcke | [END_REF] the threshold is first fixed to the root mean square of the galactic noise below 100 MHz (see App. D):

2 E T = σ sky = 30 µV/m.
This is a very crude model for the detection problem but it is sufficient to demonstrate some qualitative trends. The main drawback is the assumption that the full bandwidth, and especially the low frequency range 0 -20 MHz, might be used for radio detection at the location of the experiment. We come back to this question at the end of the section.

A look at Fig. 5 indicates that an antenna located at 1 km receives an electric field magnitude above this threshold for a vertical shower. For such vertical events and an antenna spacing d = 1.5 km, Fig. 8 depicts the area spanned by impact parameters of E p = 10 19 eV events seen by 3 antennas. The proportion of detected events is simply given by the ratio of the area of one filled region to that of a basis triangle. For 10 19 eV vertical showers and d = 1.5 km this is about 35%.

The behavior of this ratio as a function of antenna spacing is given in Fig. 9, left panel, for 10 19 eV vertical showers. The efficiency τ is a function of the impact parameter at which the threshold is reached, b T , and the antenna spacing, d, only through the ratio of these quantities:

τ ≡ τ (d/b T ).
b T appears as a natural unit for d and is in addition the value of d at which τ departs from 1.

From this it is easy to see the effect of a change in the threshold level: for E T = 30 µV/m, b T = 1.15 km; doubling E T leads to (using E max ∝ b -4 ) b T = 1.15/2 1/4 = 0.97 km. It is also easy to obtain the efficiency if geosynchrotron emission is assumed. Using the figures given in Sect. 2.2 we get b T = 2.2 km for E T = 30 µV/m. This is a large increase for the detection capability which is of course a direct consequence of the hierarchy between E v and E a discussed in Sect. 2.2.

The right panel in Fig. 9 displays efficiency integrated over arrival directions from vertical to 60 • -inclined showers with a weight sin θ cos θ, in order to incorporate changes both in solid angle and in the effective area of the antenna array. As already explained in Sect 3 (see also [9]) inclined showers are in general more effective at large impact parameters, making them easier to detect than vertical ones: for d = 1.5 km, the efficiency is now 90 %. This is illustrated in Fig. 10 (left panel) where the detection efficiency as a function of zenith angle in the north direction for an antenna spacing of 2 km and an energy of 10 19 eV is presented. The effect of the zenith angle is strong. Vertical showers are never detected whereas all showers with inclination bigger than 40 degrees are detected. Of course, in practice, this would be moderated for antennas with maximum directivity pointing towards the vertical direction.

In Fig. 10, the efficiency versus azimuth for a zenith angle of 30 degrees (same antenna spacing and energy) is also shown. Fixing θ singles out the geomagnetic field effect, otherwise entangled with the zenith angle effect discussed above, since the change of efficiency in this case can only be attributed to the variation of the field strength with α(θ, ϕ). The efficiency falls down when approaching the geomagnetic field direction towards the south. Here the effect is large because the choice θ = 30 • is close to the magnetic lines at 27 • .

Obviously, all these results depend on shower energy. Fig. 11 shows the detection efficiency as a function of energy for an antenna spacing of 2 km, integrated over all arrival directions as in Fig. 9. The variation is not very strong as it takes three energy decades to reach 100 % efficiency. The above detection condition requires very favorable conditions which may be approached only at exceptionally quiet location. Seemingly the site of the Pierre Auger Observatory in Malargue is of this kind, as the study of Ref. [13] shows. Even there, however, the atmospheric noise is likely to take over the galactic noise at low frequency [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. This is not included in the computation of σ sky . A dedicated study of the noise at the experimental site and as a function of time (day-night variation) would be mandatory to obtain a more realistic estimate of the background. More often, radio frequency interferences show strong variations with frequency. A detection criterion is thus best formulated in the frequency domain. In a rather quiet radio environment, a detection threshold of 1 µV/m/MHz above 20 MHz in a 1 µs time window can be achieved [10]. Such a detection condition leads to similar results with typically a factor 0.5 on the antenna spacing. The antenna spacing for a large array should be around 1 km rather than 2 km.

Outlook

In the present study we showed how extensive air shower radio electric fields can be evaluated at large impact parameter with an analytical formulation. Expressions were derived for two geomagnetic effects currently under investigations in the literature, thus helping pointing out aspects for which both approaches are at variance. The relationship between radio pulse shape and shower profile, and some considerations on the sizing of a giant array could be easily discussed.

The specific figures are certainly no more than illustration, not only because the pointlike approximation is too crude, but also because of their sensitivities on some shower and emission mechanism parameters. However, the approximation may be sufficient to describe tendencies and we think to make a point on some issues in the field.

As it has been understood so far in every approach to radio emission, the radio component of an air shower is driven by its time-development at large impact parameter. This is a simple consequence of propagation laws once it is assumed that the radio component is generated by the bulk of relativistic electrons and positrons in the shower. Let us collect two essential predictions that result from this fact.

The high-frequency cut in the spectrum of the shower radio-component decreases when the impact parameter increases. This prediction is not in any sense new as it was already at the heart of the discussion in Ref. [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF]. Recent studies [START_REF] Huege | [END_REF]5] show specific realizations of this universal phenomenon. Only in the point-like model developed here, it is easy to isolate the origin of the phenomenon and because of its universal nature to emphasize the need to look for it. As a byproduct, we stressed that triggering at high frequency has the consequence of biasing radio detection towards small impact parameter and thus making the corresponding device blind to the phenomenon. We also emphasized that the interrelation between impact parameters and accessible frequency range is one of the main topic to be addressed when designing a large radio detection array.

The radio signal is sensitive to the beginning of the shower development, much more so than the fluorescence signal is whose sensitivity is directly proportional to N ee . Thus radio is complementary to the fluorescence technique for the purpose of measuring the shower development. Such a sensitivity may help to discriminate between various high-energy hadronic scenarios, hence providing a rich potential for the radio technique.

A Derivation of electric fields in the point-like approximation

A.1 Point-Like expressions

For a point-like charge, ρ(t, x) = qδ (3) ( x -vt),  = ρ v a direct application of Maxwell equations in a medium of refractive index n leads for the scalar and vector potentials in Lorentz gauge to

V (t, x) = 1 4πε 0 n 2 q R|1 -n v • n/c| , A(t, x) = 1 4πε 0 c 2 q v R|1 -n v • n/c| ,
with n = R/R. For an accelerating charge, the derivation of the acceleration field is in textbooks. A positron and an electron at the same location and with the same speed have opposite accelerations and therefore produce the same electric field. Then, in the point-like approximation, N ee positrons and electrons generate

E a = N ee e 4πε 0 c 2 n ∧ ( n -n v/c) ∧ a R|1 -n v • n/c| 3 ,
where a is the positron acceleration. For E v , the derivation from the potentials with a time varying charge q → N ee (t ) × e leads to (see also [5])

E v = N ee e v T 4πε 0 nc n 2 v 2 /c 2 -v • n/c R 2 |1 -n v • n/c| 3 -Ṅee e v T 4πε 0 c 2 1 -n v • n/c R|1 -n v • n/c| 3 ,
where v T is the positron drift velocity.

A.2 Small angle approximation

A further simplification is motivated in Sect. 2.1, relevant in the region of small angles such that |1 -n v/c| ϑ 1.

Applying this simplification to the above electric field expressions makes use of the following set of rules

R → b/ϑ, 1 -n v • n/c → ϑ 2 /2, ϑ → 2 ct/b.
The numerator of E a is given by

n ∧ ( n -nR v/c) ∧ ˙ v ≈ ϑ 2 2 2( b • a T ) b -a T ≡ ϑ 2 2 a T ,
where a T is the acceleration transverse to v, or equivalently to n since this is the same at small ϑ. These replacements (together with n v/c → 1) lead to Eqs. ( 2) and (3).

B Acceleration and drift velocity

In order to get quantitative values for the electric fields we need estimates for a T and v T which we collect from various sources in the literature. For a shower whose axis makes an angle α with the geomagnetic field B, the acceleration of an electron or a positron in the shower core is ecB sin α/(γm e ). Averaging this over γ (1/γ = 1/γ ) leads to an estimate for a T in Eq. ( 3) a T = ecB sin α γm e .

Taking γ = 60, which corresponds to the median value [START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] and to a mono energetic approximation in Ref. [START_REF] Huege | [END_REF], thus enabling us to make contact with computation there, together with B sin α = 20 µT we get a T /c = 0.06 µs -1 .

After time t the electron or positron has a transverse velocity (1/γm e )ecB sin α t. Kahn and Lerche [8] use this with replacements t → τ = 1 µs (τ is the lifetime of the particle in the shower) and γ → γ. With γ = 60, this gives v T /c = 0.06. 3 A detailed approach to more properly estimate the drift velocity taking into account both energy loss and energy dependence of the lifetime τ is given in Ref. [5]. It leads to v T /c = 0.04, and this is the value which is used in the present paper for vertical showers.

Figure 1 :

 1 Figure 1: Geometry and notations: R = -→ QA and ϑ = ( v, -→ QA).

Figure 2 :

 2 Figure 2: Shape of N ee /R (θ = 45 • ) [thick line] and that of 1/(1 -nβ cos ϑ) (b = 1.5 km, n = 1, β = 1 [solid], n = 1, γ = 60 [long dashes], n = 1.0003, β = 1 [short dashes]).

4 Z

 4 where r and l are characteristic lateral and longitudinal displacements in the shower. For electrons and positrons of critical energy[START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] The criterion ∆ g ∆ r gives b 4 r. For a 10 19 eV shower, the maximum of f 2 or f 3 occurs around a 10 km height where p = 0.3 atm, leading to 4 r ≈ 500 m. The second criterion, ∆ g ∆ l , is equivalent to b √ max l. Plugging f 2 maxima for the 10 19 eV case gives b 2 Z max l = 0.6, 0.7, 0.8, 1 km, respectively, for inclinations 0, 30, 45 et 60 • . The figures show that it is the second criterion that is the most relevant.

  5 or 6 knowing that times are ∝ b 2 and amplitudes are ∝ 1/b 4 .Most of these features have of course counterparts in spectra. For the b dependence the

Figure 5 :

 5 Figure 5: Transverse current electric fields at b = 1 km (component towards east) for showers coming from south at various zenith angles: θ = 0 • , 30 • , 45 • , 60 • .

Figure 6 :

 6 Figure 6: Geosynchrotron fields for the same configurations as in Fig. 5. The observer location is east (or west) of the shower.

12

 12

Figure 7 :

 7 Figure 7: Sensitivity to shower rise: left plot shows two 10 19 eV vertical-shower profiles (Gaisser-Hillas and Greisen) as a function of altitude; the corresponding E v field pulses and spectra (at b = 1 km) are shown respectively on the right and bottom plots. The effect is similar for E a (not shown).

Figure 8 :

 8 Figure 8: Hexagonal array and area of detection.

Figure 9 :

 9 Figure 9: Detection efficiency vs antenna spacing for 10 19 eV showers assuming transverse current emission: vertical configuration (left) and integrated over inclination (right).

Figure 10 :Figure 11 :

 1011 Figure 10: Detection efficiency assuming transverse current emission as a function of arrival direction for E p = 10 19 eV and antenna spacing d = 2 km. Left: as a function of θ towards the north. Right: as a function of ϕ at θ = 30 • .

Table 1 :

 1 Z max in km: E p = 10 19 eV (left) and E p = 10 17 eV (right).where Z start gives an estimate for the signal start. In Allan's estimate, Z start = 2Z max and ∆ g ≈ b 2 /(4 Z max ). This gives the main scaling behavior of ∆ g . More precisely, defining a starting point as the instant at which the relevant function (f 2 or f 3 ) reaches 10% of its maximal value, ∆ g is 90% of b 2 /(4 Z max ) for vertical showers and 2/3 of b 2 /(4 Z max ) for 60 • -inclined showers.Following estimates of Ref.[START_REF] Allan | Progress in elementary particle and cosmic ray physics[END_REF] the timescale criterion for the point-like approximation reads ∆ g

					θ max f 1 max f 2 max f 3
		5	7.5	6	0	4	9	8
	30	4.5	9.5	8	30	6.5	12	10
	45	8	13.5	11.5	45	10.5	16.5	14
	60	17	24.5	21	60	20.5	28.5	24.5

This is not true for very inclined showers if b is kept in the km range. For this reason inclinations were limited to θ ≤ 60 • in the present study.

The assumed bandwidth is fixed by the radio experiment, i.e., not necessarily restricted to observations at large impact parameters for which the effective bandwidth is much more limited (see Sect.

3.1).

In Ref.[8], γ = 200 is quoted, which would lead to v T /c = 0.02.

Acknowledgments

This work was supported in part by grant ANR-NT05-2 42808 (CODALEMA) from the French Agence Nationale de la Recherche.

C Shower characteristics

For shower evolution we primarily used the Greisen parametrization [START_REF] Stanev | High energy cosmic rays[END_REF] N (X) = 0.31

with E C = 89 MeV and X 0 = 36.7 g/cm 2 . Throughout, X max was chosen as 770 g/cm 2 (respectively 700, 630) for E p = 10 19 eV (respectively 10 18 , 10 17 ) [5].

For comparison, the Gaisser-Hillas parametrization [START_REF] Gaisser | Cosmic rays and particle physics[END_REF] was also considered

In addition, the observation level was always chosen as that of sea-level and the atmosphere thickness as a function of altitude, for a shower coming from the direction of zenith angle θ, was taken as

D Detection threshold

To be visible a shower radio signal should be above the contribution from the galactic noise, whose power spectral density (for one polarization) is given by [START_REF] Kraus | Antennas[END_REF]]

This can be related to the electric field ν to which an antenna with gain G (effective area A e = Gλ 2 /(4π), λ = c/ν) is exposed for a time ∆t

where Z 0 = 1/(ε 0 c) is the intrinsic impedance of free space. The variance of an electric field can be computed from its spectrum by

where E ν is the Fourier transform of E(t) restricted to the time window [t 0 , t 0 + ∆t]. E ν is defined for positive and negative frequencies, while ν is a positive-frequency quantity such that 2 ν = E 2 ν + E 2 -ν . For the sky noise, this gives

where the sky temperature quoted in Ref. [START_REF] Falcke | [END_REF] (T sky ∝ ν a-2 , a = -0.5 and T sky (55 MHz) = 4800 K) is used and an upper bound ν max is set to the ν integral. For G = 1 and assuming an observational upper bound ν max = 80 MHz we get σ sky ≈ 30 µV/m.


Christian Meyer

F Alberto Gallo

Meyer

Elzbieta Witkowska-Zaremba

Dans son commentaire de la Physique, Thomas d'Aquin avait esquissé les fondements d'une théorie de la musique, à la croisée de la physique et de l'arithmétique : la musique comme science médiane du nombre rapportée au son 1 . On sait aussi combien la conception aristotélicienne du temps comme succession d'instants a favorisé la mathématisation de la notation des chants polyphoniques au cours de la seconde moitié du XIII e siècle et, in fine, l'essor des techniques de composition. En dehors de ces acquis déterminants pour l'essor de la musique polyphonique au siècle de l'Ars nova, la réception de la physique aristotélicienne, et plus généralement des diverses sciences qui s'épanouissent dans le cadre de ce qu'il est convenu d'appeler la philosophie de la nature, n'a toutefois guère laissé de traces dans les écrits sur la musique de la fin du Moyen Age. C'est du moins le constat sommaire qu'autorise une investigation au sein du vaste corpus de traités de musique dont nous disposons aujourd'hui dans de bonnes éditions, et, sans nul doute, tout à fait représentatif de la littérature musicale du Moyen Age.

Pour durcir le propos, on pourrait même avancer que la théorie de la musique, dans sa dimension la plus spéculative, et telle qu'elle était enseignée dans les facultés des Arts au cours des XIV e et XV e siècles, se trouvait alignée sur l'autorité de Boèce, en particulier grâce à l'extraordinaire diffusion de la Musica dite speculativa de Jean de Murs 2 . Ce traité, rédigé en 1323 à l'université de Paris, qui nous est parvenu dans deux versions et par une soixantaine de copies, livre en effet un exposé axiomatisé et sommairement commenté, des fondements mathématiques de l'échelle des sons selon l'exposé du De institutione musica de Boèce, sur lequel précisément, la théorie de la musique occidentale s'était édifiée depuis le IX e siècle 3 . Cet alignement sur Boèce consolidait sans nul doute une approche mathématique de la musique illustrée par la construction du monocorde dont l'étude constituait, depuis le début du XI e siècle, le socle de la théorie du chant liturgique.

Pour le reste, la théorie de la musique, dans son versant plus pratique, tourné vers le chant liturgique mais aussi vers la composition et la notation de chants monodiques ou polyphoniques, développait un discours essentiellement pragmatique et normatif. La théorie du chant liturgique s'attachait, dans le prolongement d'une tradition pluriséculaire, à régler les problèmes du chant antiphoné : il s'agissait pour l'essentiel d'affiner les critères d'identification du mode des antiennes de l'office ou des chants responsoriaux de la messe. Du côté de la polyphonie, les traités de déchant, puis de contrepoint, s'appliquent à fixer les règles de succession des intervalles en consonance ; enfin les règles formulées dans les traités de la notation garantissent un mode de figuration univoque des rapports de durées aux structures puissamment mathématisées. La dimension pragmatique de ces textes -et de la préoccupation des musiciens -est évidente ; l'outil conceptuel demeure à bien des égards un appareil mathématique rudimentaire et une pensée rivée à l'horizon du pythagorisme diffus des nombres harmoniques.

La théorie de la musique, rapportée à une pratique musicale refermée sur elle-même et à l'exercice du chant liturgique, formalisée pour l'essentiel par des chantres ou des maîtres de grammaire des écoles épiscopales ou monastiques, livre l'image d'un monde clos, imperméable, voire inaccessible, à des cultures musicales ou à des théories d'autres cultures -on pense notamment à celles du monde arabe, qui a tant, par ailleurs, sollicité la curiosité des gens de savoir. La section XIX des Problemata aristotéliciens appartient ainsi à ce monde insaisissable, à l'égard duquel -on le verra -les musiciens observent une relative indifférence.

Les cinquante questions réunies dans la XIX e section des Problemata sollicitent en effet d'obscurs horizons théoriques relatifs à l'émission, à la diffusion et à la perception des sons, aux effets de la musique sur l'homme et son comportement, ou encore à la structure des échelles musicales. De plus, ces problèmes sont parsemés d'allusions aux pratiques musicales, vocales ou instrumentales, de l'Antiquité, et leur lecture est entravée par les nombreuses difficultés sémantiques de la traduction latine. C'est dire que l'éloignement dans le temps, à la fois des cultures musicales impliquées dans ces problèmes, mais aussi et surtout de la théorie de la musique antiqueque Boèce, Cassiodore, Martianus Cappella ou Isidore de Séville n'avaient transmis que par bribes -devait poser au lecteur médiéval de redoutables difficultés. Pierre d'Abano en témoigne lui-même lorsqu'il déplore, vers la fin du commentaire de cette section, la « profonde obscurité, embrouillée par la corruption du texte » de l'ensemble des problèmes de mathématiques auxquels se rattache cette section des Problemata4 .

Il est évidemment impossible, dans le cadre de cette présentation sommaire, de décrire les difficultés auxquelles le commentateur médiéval pouvait être confronté dans l'étude de cette section. Mais on peut tenter de porter un jugement d'ensemble sur les caractéristiques majeures -et les qualités -du commentaire de Pierre d'Abano. Pour commenter cette XIX e section des Problemata, Pierre d'Abano sollicite de multiples autorités. Ce sont tout d'abord les textes de la médecine grecque et de ses commentateurs arabes. Ce sont aussi, bien sûr, les écrits d'Aristote dans lesquels le commentateur puise très largement : les écrits de logique et de métaphysique, de philosophie morale et de politique, enfin les traités de philosophie de la nature. Sur la théorie de la musique, Pierre d'Abano convoque Boèce et, secondairement, Macrobe ou Isidore. Mais on notera qu'en dépit de son voyage à Constantinople, Pierre d'Abano n'a, à l'évidence, aucune connaissance des écrits d'Aristoxène, de Cléonide ou d'Aristide Quintilien, par exemple, qui lui auraient permis de commenter plus précisément les points de théorie de la musique ignorés par Boèce et la tradition médiévale, comme par exemple la notion de son "antiphone", qui revient à de multiples reprises au fil de cette section, et qui caractérise cette résonance spécifique créée par la fusion de deux sons en rapport d'octave. Sur cette notion et sur bien d'autres, nécessairement obscures pour n'importe quel lecteur de ce temps, Pierre d'Abano se contente de tisser un réseau de conjectures (souvent fondées sur des lieux parallèles disséminés dans le corpus des Problemata) ou sur des remarques plus empiriques reposant à l'évidence sur une expérience et un vécu personnel. On notera à cet égard que le commentaire ne fait, et à juste titre, aucune référence aux structures ou à l'exécution du chant liturgique (ni d'ailleurs à la théorie du plain-chant), alors qu'il cède volontiers à des allusionsmais souvent difficiles à apprécier -à des pratiques de chants en langue vernaculaire ou à des observations organologiques. Mais audelà de la fragilité du commentaire, il semble bien qu'il y ait eu, chez Pierre d'Abano, l'idée implicite de construire à partir de ces problèmes une sorte de théorie de la musique aux confins de la philosophie de la nature, un projet rappellant singulièrement celui de cette science médiane, à l'intersection de la théorie arithmétique héritée de Boèce et des sciences de la nature et du vivant qui se consolide au cours de la seconde moitié du XIII e siècle.

Ce projet est évidemment passé inaperçu, même dans les écrits les plus savants et les plus prolixes des théoriciens de la musique des XIV e et XV e siècles, dans le Lucidarium rédigé par Marchettus de Padoue au cours des années 1310 5 , dans l'immense Speculum Musicae compilé dans les années 1320 par Jacques de Liège 6 , ou encore et même, au siècle suivant, dans les sept livres de la Declaratio musicae composée vers 1430 par Ugolino d'Orvieto 7 -lequel cependant fonde le concept d'une subordination de la musique à la physique 8 .

5

. Jan W. Herlinger, The Lucidarium of Marchetto of Padua. A Critical Edition, Translation, and Commentary (Chicago-London, 1985) 6 . Roger Bragard (éd.), Jacobi Leodiensis speculum musicae (Roma, 1955(Roma, -1973 ; ; Corpus scriptorum de musica, 3), 7 vol.

7

. Albert Seay (éd.), Ugolini Urbevetani declaratio musicae disciplinae (Roma, 1959(Roma, -1962 ; Corpus scriptorum de musica, 7), 3 vol.

8

. « Praeterea illa dicitur esse scientia quae sibi superiori scientiae subalternatur et principia sua in ea superiore probantur. Sed musica subalternatur philosophiae naturali sumendo philosophiam in communi, nam primo posteriorum dicit Aristoteles quod illa scientia subalternatur alteri quae habet principia probata in illa sibi superiore, ut perspectiva geometriae supponitur. Sed musica habet principia sua probata in philosophia naturali sumendo eam in communi, igitur subalternatur illi. Musica enim considerat tempus, motum, sonum, quantitatem, numerum, voces graves et acutas, intensionem et remissionem vocum, similiter earum velocitatem et tarditatem, intervallum unius vocis ad aliam. Considerat etiam consonantias vocum secundum varias proportiones unius vocis ad aliam in acuitate et gravitate consideratarum, quorum omnium multa probata sunt in philosophia naturali quae tamen praesupponuntur in musica. Igitur musica est scientia. » (Lib. I, ch. 2, « Quod musica sit scientia », éd. cit. p. 19-20). Sur ce point, voir aussi Fr. Hentschel, op. cit., p. 153 n. 257.

L'autorité de la section XIX des Problemata, et l'idée que cette section livre de manière diffuse les théories d'Aristote en matière de musique ne perce cependant qu'au cours du dernier quart du XV e siècle. Cette reconnaissance tardive a sans doute été largement favorisée par la curiosité que manifestent, depuis plusieurs décennies, les érudits italiens pour la théorie de la musique grecque. Le premier témoignage est celui de l'historien Nicolò Burzio, né à Parme vers 1453, qui, dans un traité de musique -son Florum libellus imprimé en 1487 à Bologne 9 -, instaure en ces termes l'autorité d'Aristote :

« Secundum quam rationem speculamur sonorum varias naturas ac potestates, et ex quibus invicem proportionibus consonantias dissonantiasque causari contingat. Aristoteles idcirco in problematibus, ubi de hoc satisabunde, musicen naturalem habere delectationem omnibusque aetatibus ac singulis moribus amicam esse docuit. » 10 Quelques années plus tard, dans sa Theorica musice publiée à Milan en 1492, Franchinus Gafurius, maître de chapelle à la cathédrale de Milan, réservera une place prédominante à Aristote dans son catalogue des autorités en matière de musique : « Aristoteles librum unum musices unumque pythionice musices atque problemata et multis in libris musica elementa commiscuit. » 11 Gafurius est aussi le premier auteur de musique à signaler le commentaire de Pierre d'Abano. C'est d'ailleurs principalement à travers ce commentaire qu'il semble avoir lu et tenté de comprendre le texte des Problemata. * Sur cette question de la réception des Problemata aristotéliciens chez les théoriciens de la musique, et pour être aussi complet que possible, dans l'état actuel de nos connaissances, on ne peut faire l'économie de deux témoignages ou témoins, sensiblement contemporains de Pierre d'Abano.

Pour le premier il s'agit d'un recueil de traités de musique conservé aujourd'hui à Pesaro, qui constitue l'un des témoignages les plus anciens de l'intérêt qu'un théoricien de la musique pouvait porter aux Problemata aristotéliciens 12 . Ce volume daté de 1291 contient un florilège de traités de musique compilé à Bologne par un certain Sinibaldo da Volterra. Il s'agit, en l'occurrence d'extraits de traités de musique bien connus (Guy d'Arezzo, le petit Dialogue attribué au Moyen Age à un certain Odon, Cassiodore, ou les Etymologies d'Isidore), mais aussi des écrits plus singuliers comme l'Ars musica . Giuseppe Vecchi (éd.), Nicolo Burzio. Musices opusculum (Facs.) (Bologna, 1969 ; Bibliotheca musica Bononiensis II/4).

10

. Op. cit, f. a5v.

11

. Réimpr. facs. New York, 1967, f. a6v. 12 . Pesaro, Biblioteca Comunale Oliveriana, Ms. 1336. Sur ce manuscrit, voir Répertoire international des Sources Musicales, B III,6 (2003), p. 557-562. du maître Guido Faba, actif à Bologne dans les années 1220-1230, et quelques petits traités de plain chant inconnus par ailleurs. Cette collection intègre également, assez curieusement, une copie de la section XIX des Problemata dans la traduction de Barthélemy de Messine et dont les sept premiers problèmes sont perdus par suite d'une lacune matérielle. Le cinquantième et dernier problème de la section est suivi, sans explicit ni transition, d'une série de définitions des arts libéraux et des chants liturgiques extraits des Etymologies d'Isidore de Séville. Hélas, rien dans contexte proche ou lointain ne permet aujourd'hui de préciser l'intérêt du compilateur pour ce texte, ni l'usage auquel il le destinait, moins encore le projet intellectuel qui l'avait conduit à le recueillir. Nul doute cependant que l'écriture textuelle et appliquée du copiste conférait à ce volume une certaine autorité et le destinait sans doute à un usage public.

Le second est un témoin moins anecdotique, car il s'agit bien de l'un des très rares lecteurs avérés de la section XIX des Problemata. Il s'agit d'Engelbert, dit d'Admont, né vers 1250 et décédé en 1331 en l'abbaye d'Admont en Styrie dont il fut l'abbé à partir de 1297 13 . On sait qu'il découvrit les libri naturales d'Aristote lors de ses études à l'école de la cathédrale de Prague entre 1271 et 1274. Au cours des années 1276/77 -1285/87 il séjourne à Padoue où il étudie, selon son propre témoignage, la théologie auprès des Dominicains de la ville, et la logique et la philosophie à l'université. Mais c'est en l'abbaye d'Admont qu'Engelbert rédige son traité sur la musique, et c'est dans le fonds de la bibliothèque de cette abbaye qu'il puise ses références aux autorités de la science de la musique : un Boèce du XII e siècle 14 , et un florilège de traités de musique formé au XII e siècle, aujourd'hui conservé à Rochester 15 . Le traité d'Engelbert tient d'ailleurs une place singulière dans la littérature musicale des XIV e et XV e siècles non seulement par l'abondance et la diversité des références aux écrits d'Aristote et à certains de ses commentateurs dont Averroes 16 , mais aussi par le fait qu'Engelbert a lui-même rangé son traité au nombre de ses écrits relevant de la philosophia naturalis 17 . Le projet d'Engelbert semble donc bien avoir été de construire un discours sur la musique à l'intersection d'une part de la théorie du plain-chant formée aux XI e et XII e siècles et de la philosophie de la nature de l'autre, inaugurant ainsi une sorte d'autre science médiane. Hélas, le traité Ce long problème était en effet bien à l'esprit de l'auteur puisqu'il s'y réfère explicitement pour affirmer ailleurs « que la nature se réjouit de tout ce qui est conforme à la nature : or le mouvement est ordonné selon la nature » 23 .

Que faut-il entendre par mouvement ? Engelbert consacre un chapitre tout entier à sous l'égide des Problemata, cette notion 24 qui ne s'éclaire toutefois véritablement qu'à la lecture du cinquième livre de la Physique. Engelbert décrit en effet trois types de mouvements : un mouvement naturel (motus naturalis), un mouvement forcé (motus violentus), et un troisième mouvement associant les deux mouvements précédents. Le mouvement naturel, explique-t-il, est relâché en son début (in principio remissus) et tendu en sa fin (in fine intensus). A l'inverse, le mouvement forcé est d'abord tendu, puis relâché. Le mouvement combiné, associant mouvement naturel et mouvement forcé, est tantôt tendu, tantôt relâché, au début, au milieu ou à la fin 25 .

Si ces notions physiques semblent bien pouvoir être interprétées en terme d'agogique musicale, c'est-à-dire d'accelerando ou de ritenuto 26 , il faut sans doute, comme le suggère Engelbert lui-même, étendre leur domaine d'application à d'autres paramètres du son ou du chant. Elles peuvent être appliquées au mode de production du son : la tension de la corde ou la pression du souffle agissent en effet sur la hauteur ou sur l'intensité du son. De même, rapportés à la ligne mélodique du chant, le relâchement et la tension peuvent être interprétés en terme de progression descendante ou, au contraire, ascendante, par degrés conjoints, ou au contraire, par intervalles disjoints. Enfin, rapportées à telle ou telle note d'un chant, ces notions s'appliquent enfin à la possibilité de hausser ou d'abaisser un son d'un demi-ton -comme le suggère Engelbert lui-même, lorsqu'il explique que les demi-tons fonctionnent comme un moyen d'infléchir un son vers le grave ou, au contraire, vers l'aigu 27

26

. P. Ernsbrunner,op. cit., . voir la note suivante. l'orienter vers la pratique musicale, Engelbert fait d'ailleurs explicitement référence aux chants tragiques évoqués dans le sixième problème qu'il paraphrase avec une certaine emphase en insistant sur le caractère extrême de ces chants toujours animés soit par la louange, soit par l'imprécation 28 . Enfin, pour Engelbert, seul précisément le cantus naturalis est animé de ce mouvement complexe combinant mouvement naturel et mouvement forcé -et c'est à ce titre que le chant peut provoquer chez l'auditeur des sentiments d'allégresse ou de tristesse, le plonger dans un état de jouissance ou au contraire d'effroi 29 .

Voici donc, in fine, une réponse "physicienne" à la question de la puissance de la musique qui revient avec insistance, depuis Guy d'Arezzo au moins, dans la théorie du chant liturgique : pourquoi les uns prennent-ils plaisir à telle mélodie et les autres à telle autre ? ou comme le répète cette longue tradition pythagorico-platonicienne récapitulée par Boèce : pourquoi tel mode ou telle mélodie a-t-elle le pouvoir de calmer tel esprit, ou de mettre en transe tel autre ? Pour Engelbert d'Admont, la théorie de la musique désormais armée de la science de la nature possède les moyens d'ordonner scientifiquement ce topos, mais aussi de proposer des modèles opérationnels susceptibles de guider les chantres dans la composition de mélodies ou la révision des livres liturgiques notés.

Tout cela, bien entendu, n'a pas eu lieu, et les Problemata ne connaîtront une réelle fortune que vers la fin du XVI e siècle lorsque les intellectuels et les musiciens italiens s'empareront de ce corpus pour conceptualiser les structures de l'opéra naissant -mais cela est une autre histoire. En revanche, on peut s'interroger sur les raisons profondes de cette résistance de la théorie de la musique du Moyen Age à l'appareil conceptuel et aux théories de la philosophie de la nature. On peut avancer sur ce point un exemple emblématique qui rend parfaitement compte de l'attachement des théoriciens de la musique aux modèles de la tradition mathématique pythagoricoplatonicienne héritée de la lecture des Institutions musicales de Boèce. Il s'agit en l'occurrence de la persistance du modèle intellectuel de l'Âme du monde, hérité du Timée et du commentaire qu'en a donné Calcidius. Or il s'agit là précisément du modèle mathématique bien connu des nombres harmoniques -des puissances de 2 et de 3 et de leurs produits -qui fonde non seulement la gamme pythagoricienne que les savants et les musiciens du Moyen Age avaient reconstruite sur le monocorde, mais aussi l'architecture mathématique sur laquelle repose la notation mesurée qui s'est imposée au cours du XIV e siècle. Mais plus encore, ce modèle avait permis une classification des sons en concordance, là où précisément le classement sub auditu, qui distingue les consonances en fonction de leur tolérance à l'oreille, ne pouvait plus rien. Cette classification qui apparaît au cours de la première moitié du XIII e s. et dont Jean de Garlande est l'un des premiers témoins, peut être commodément représentée en superposant au lambda pythagoricien présentant la suite des rapports issus des nombres harmoniques la classification des consonances en vigueur au XIII e s. Le diagramme ainsi formé représente cet état de la théorie des sons en concordance et leur classification en consonances concordantes, discordantes, parfaites, moyennes ou imparfaites, dont le critère ultime de discordance est le degré d'éloignement de l'unité. Ces principes philosophico-mathématiques hérités de l'Antiquité fondent ainsi une hiérarchie des consonances sur laquelle reposent à leur tour les principes d'organisation des polyphonies de la fin du Moyen Age. La prégnance de ce modèle mathématique -qui a modelé en profondeur des segments entiers de la pratique musicale -et la diffusion considérable de la musique mathématisée pythagoricoplatonicienne dans le prolongement de l'enseignement de Jean de Murs expliquent sans doute pour une bonne part la résistance des théoriciens de la musique aux outils conceptuels des sciences physiques.

  30

30.

  Michel Huglo, « La notation franconienne. Antécédents et devenir », Cahiers de civilisationmédiévale, 31 (1988), p. 123-132, voir p. 125 Tabl. I : Extension des consonances à la fin du XII e s.) ; Chr. Meyer, « Mathématique et musique au Moyen Age », in : Quadrivium: Musique et sciences (Paris, 1992), p. 107-121 ; Id. et Jean-Francois Wicker, « Musique et mathématique au XIV e siècle. Le De numeris harmonicis de Leo Hebraeus », Archives internationales d'histoire des sciences, 50 (2000), p. 30-67. Sur la fortune du lambda pythagoricien au Moyen Age : Chr. Meyer, « Le diagramme lambdoïde du ms. Oxford Bodleian Library Auct. F.3.15 (3511) », Scriptorium, 49 (1995), p. 228-237 ; Id., « L'Âme du monde dans la rationalité musicale : ou l'expérience sensible d'un ordre intelligible », in : Harmonia mundi. Musica mondana e musica celeste fra Antichità e Medioevo, a cura di Marta Cristiani, Cecilia Panti e Graziano Perillo (Firenze, 2007), p. 57-75.

  Aujourd'hui conservé à la Newberry Library de Chicago, Case Ms. f. 9. Voir Répertoire international...,op. cit., B III 4 , p. 151-152 (notice de Michel Huglo). Eastman School of music, Ms. 92 1200. Sur ce ms. voir Répertoire international..., op. cit. B III4 , p. 183-186 (notice de Michel Huglo). 'est conservé que dans une seule copie et ne semble avoir eu aucune descendance intellectuelle.Retenons cependant pour notre propos qu'au-delà des références au corpus aristotélicien, les nombreuses références aux Problemata sont remarquables et tout à fait singulières dans la littérature musicale de la fin du Moyen Age. On peut d'ailleurs supposer avec Pia Ernsbrunner qu'Engelbert s'est servi du Liber problematum aristotelis secundum speciem compilationis du manuscrit conservé aujourd'hui à laBibliothèque d'Admont (cod. 345) 18 . Le traité d'Engelbert observe un plan en quatre parties. La première expose les fondements de l'ars musica : définition et classification de la musique, nomenclature et structure de l'échelle des sons. La seconde traite des intervalles entre les sons, des consonances et de leurs proportions. Les troisième et quatrième abordent des questions plus pratiques. La troisième expose les principes de la solmisation. La quatrième est la partie la plus originale du traité. L'auteur y tente une fusion inédite entre, d'une part, la théorie classique des huit tons ecclésiastiques telle qu'elle a été formalisée dans une tradition allemande de la seconde moitié du XI e s., et, d'autre part, une approche psycho-acoustique du son et une théorie du mouvement d'inspiration aristotélicienne. C'est là qu'il développe sa conception tout à fait singulière d'un cantus naturalis. .. ergo voces et soni debent de congruitate ordinis sui ad illum finem taliter in se et ad se invicem proporcionari et ordinari, quod demulceant auditum et moveant animum et affectum audiencium. Unde multa et mira refert Boecius in prologo Musice, de natura armonie musicalis, qualiter diversitates tonorum musicorum diversas passiones et affectiones gaudii et tristicie et audacie, et desperacionis causaverint in quibusdam. » (IV, 3 ; éd. cit. p. 286-287). de l'ars musica -« propter radicalem intelligenciam » 20 . L'idée fondamentale est celle que la nature humaine, dans sa complexion ou sa constitution naturelle, prend naturellement plaisir à ce qui est constitué comme elle, c'est-à-dire selon les proportions harmoniques. Voilà qui explique, selon Engelbert, que l'âme est davantage touchée par les sons que par les paroles et qu'en dans le chant, c'est davantage la mélodie que les mots qui affecte l'âme -« quod cantu musico magis inmutantur animi quam verbis » 21 . Engelbert rapporte explicitement cette idée au texte des Problemata et il s'agit peut-être bien d'une réminiscence d'une lecture du Problème 38 qui est ainsi formulée : « Pourquoi tout le monde prend-il plaisir au rythme et au chant ? Les mouvements (ou changement, motus) ordonnés selon la nature nous invitent à nous réjouir selon la nature. Pour preuve, les enfants y prennent spontanément plaisir. Par habitude (propter consuetudinem dans la traduction de Barthélemy de Messine) nous prenons plaisir aux modes des chants (modis cantuum -c'est peut-être ici qu'il faut sous-entendre avec Engelbert d'Admont « aux mélodies et non aux paroles »). Quant au rythme, nous l'aimons dès lors qu'il repose sur un nombre connu et structuré et qu'il nous anime de façon ordonnée. Car le mouvement ordonné par la nature, nous est plus familier que celui qui ne l'est pas, c'est pourquoi il est plus conforme à (notre) nature. La preuve en est que si le travail, le boire et le manger sont réglés, nous conservons et nous augmentons notre nature et nos capacités ; lorsqu'ils désordonnés, nous les altérons et les faisons dévier. Les maladies sont des changements (motus) de l'ordonnancement du corps non conformes à la nature. Nous prenons plaisir à la consonance puisqu'elle est un mélange (complexio) de contraires possédant un rapport (proportio) entre eux. Le rapport est ordonnancement (ordinatio) qui (-nous l'avons dit -) est chose agréable à notre nature : ce qui est mélangé (complexionatum) est plus agréable que ce qui ne l'est pas. D'autant plus que s'agissant d'un objet sensible, dans la consonance, le rapport procède des deux extrêmes à puissance égale. » 22 20 . « Ideoque ad principem philosophorum ipsius nature praecipuum scrutatorem Aristotelem veniamus, et ipsius verba et raciones de natura musice harmonie in libro Problematum parte XVIIII a subsequenti doctrine de proprietatibus et differentiis et convenientiis musicorum tonorum propter radicalem intelligentiam premittamus. » (IV,4 ; éd. cit. p. 290-291.) Propter quid et in rithmo et cantu et universaliter omnino consonantiis gaudent omnes ? Quia motibus, qui sunt secundum naturam, gaudemus secundum naturam. Signum autem pueros confestim natos gaudere ipsis. Propter consuetudinem autem modis cantuum gaudemus. Rithmo autem gaudemus propter notum et ordinatum numerum habere, et movere nos ordinate magis. Prius enim ordinatus motus natura inordinato, quare et secundum naturam magis. Signum autem, laborantes enim et bibentes et comedentes ordinate, salvamus et augemus naturam et virtutem, inordinate autem corrumpimus et stupefacimus ipsam. Egritudines enim ordinationis corporis non secundum naturam motus sunt. Consonantia gaudemus quoniam complexio est contrariorum habentium proportionem adinvicem. Proportio autem ordinatio, quod erat delectabile natura. Complexionatum autem omne incomplexionato omni delectabilius, aliter quamvis sensibile sit utrisque extremis ex equali potentia habet in consonantia proportio. » Problematum Aristotelis translatio duplex antiqua s. & ea quam Theodorus Gaza ed. Cum Petri Apponensis Expositionibus nuper corr. Adiuctis his preterae

	Engelbert retient tout d'abord l'idée que les sons, par essence
	différents et contraires, peuvent fusionner dans un moyen terme, une
	moyenne ou un milieu, ou résister à cette fusion et se maintenir dans
	une sorte d'altérité plus ou moins irréductible. La fusion harmonique
	ou, au contraire, la résistance ou la répugnance des sons à fusionner
	provoquent ainsi chez l'auditeur soit un sentiment de plaisir, soit, au
	contraire, un sentiment de répulsion, de tristesse ou de mélancolie 19 .
	2. Moins anecdotique, la référence aux Problemata est bien au
	contraire fondamentale pour les développements de la quatrième et
	dernière partie du traité dans laquelle il est abondamment question de 21 . Ibid., p. 291.
	ce chant qu'Engelbert d'Admont qualifie de naturel, ce cantus
	naturalis dont les caractéristiques formelles ont été analysées dans la
	section précédente. Dans cette dernière partie, la section XIX des
	Problemata est expressément convoquée pour assurer les fondements
	13 . Pia Ernsbrunner, Der Musiktraktat des Engelbert von Admont (ca. 1250-1331)
	(Tutzing, 1998 ; Musica mediaevalis Europae occidentalis, 2). 18 . Ibid., p. 290-291 (note).

14 . 15 . Rochester, 16 . P. Ernsbrunner, op. cit., p. 200. . Ibid., p. 23, 38 (« In philosophya naturali »).

n19 . « .

22 . «

  . Pour appuyer le propos et Motus enim naturalis est in principio remissus, in fine intensus, motus vero violentus e converso est in principio intensus, in fine remissus. Motus autem compositus ex naturali et violento nunc intenditur, nunc remittitur in principio et in medio et in fine. Cantus autem naturalis non semper uniformiter remittitur et intenditur in principio, et in medio, et in fine, sed nunc sic nunc sic, per singulos tonos musicos secundum singulorum differentias et proprietates : ergo sequitur modum et naturam motus compositi ex naturali et violento. » (IIII, 7 ; éd. p. 295-297).

	Alexandri Aphrodisii & Plutarchi Problematis (Venetiis : Octavianus Scotus, 1519),
	f. 177.
	23 . « ... qud natura gaudet in omni eo quod est secundum naturam : Motus autem
	ordinatus est secundum naturam... » (IIII, 4 ; éd. cit. p. 291).
	24 . IIII, 7 : « Secundum quem speciem motus artificialiter reguletur cantus
	naturalis », éd. cit. p. 295-300.
	25 . «

  Unde dicit Aristoteles in libro predicto, quod in subita elevacione et pressione sonorum et vocum est labor et violencia ex eo quod sunt motus contrarii elevacio et depressio qui sine quietate non possunt sibi continuari. Une semitonia interveniunt <tamquam> media deflexionis acute vocis in gravem vel e converso, ubi ipsa deflexio est loco quietis. Et subiungit ibidem, quod cum in tragicis cantibus qui continent vel magnam laudem vel magnam vituperacionem, que sunt duo extrema, semper medium fugiatur, ideo cantantur in yppaton et parypaton i. e. in principalibus seu gravibus, quorum cantus habent se ad gravius extremorum in vocibus et ad forcius : quia gravior vox tanquam habens angulum obtusum forcius inpellit et facilius cantatur et sic etiam forcius inprimit et inprimitur. »(IIII, 7 ;. Naturalis itaque regulacio cantus consistit in modulacione ipsius secundum modum et naturam motus quem sequitur cantus naturalis. Sed secundum Aristotelem in problematibus, parte XVIIII a modus quo cantus movet animum et affectum cantancium et audiencium ad leticiam et tristiciam, et ad delectacionem vel horrorem, sequitur naturam modus conpositi ex motu naturali et violento. »(IIII, 7 ;

28

. « 29 . «

. « Ex hiis utique quam brevius prodierunt in lucem utcumque circa liberales artes et maxime mathematicam problemata grandi textuali caligine cum eius corruptione involuta quandoque in quibus etiam quia plurimis extranea non grandis utilitas, que hodie appetitur summe reperitur. Quare ipsa celerius percurri, nunc autem deinceps circa reliquas partes philosophie naturalemque maxime problemata tam circa composita quam simplicia existentia tractabuntur. »