Michel Berthier
email: michel.berthier@univ-lr.fr

Edoardo Provenzi
email: edoardo.provenzi@math.u-bordeaux.fr

From Riemannian trichromacy to quantum color opponency via hyperbolicity

We propose a mathematical description of human color perception that relies on a hyperbolic structure of the space P of perceived colors. We show that hyperbolicity allows us to reconcile both trichromaticity, from a Riemannian point of view, and color opponency, from a quantum view-point. In particular, we will underline how the opponent behavior can be represented by a rebit, a real analog of a qubit, whose state space is endowed with the Hilbert metric.

Introduction

The modern description of color perception involves two complementary processes. The first one is based on the existence of the LM S cones with sensitivity pick in the long, medium and short wavelength range of the visible spectrum, respectively. This physiological fact is responsible for the trichromatic character of human color perception. The second process is related to the opponent color mechanism first described in [START_REF] Hering | Zur lehre vom lichtsinne[END_REF] and then proven to be physiologically related to the action of retinal neurons, see e.g. [START_REF] Shevell | Color opponency: tutorial[END_REF].

Classically, the opponent mechanism is described as follows: for simplicity, let us denote again with L, M, S the output of retinal cones, then the opposition L -M or M -L gives the red-green axis, the opposition S -(L + M ) or (L + M ) -S gives the blue-yellow axis, whereas the sum L + M + S, or the difference -L -M -S, gives the achromatic axis, i.e. the gray intensity. The Hering disk illustrates the oppositions by identifying L -M with the point of the unit circle of angular coordinate θ = 0, S with the point given by θ = π/2, M -L with θ = π and -S with θ = 3π/2. The center of the disk represents achromatic values.

Natural image statistics has systematically investigated the problem of why humans have evolved to develop such a peculiar codification system. To the best of our knowledge, the first description of chromatic opponency in this field has been provided by [START_REF] Ohta | Color information for region segmentation[END_REF] via a principal component analysis (PCA) on a database of natural RGB images. The authors interpreted the ensemble of RGB image values as random variables and studied the covariance matrix of this statistical ensemble. If v 1 , v 2 and v 3 are the eigenvectors of the covariance matrix relative to the eigenvalues λ 1 , λ 2 and λ 3 such that |λ 1 | > |λ 2 | > |λ 3 |, then their empirical results showed that v 1 is obtained via a linear combination of the R, G, B axes with all positive coefficients, while v 2 and v 3 are built via linear combinations of R, G and B with coefficients of opposite sign. As a results, Ohta et al.'s PCA showed that the most efficient way to codify the chromatic content of the ensemble of RGB images was via the achromatic plus opponent channel representation foreseen by Hering.

The elegant work of [START_REF] Buchsbaum | Trichromacy, opponent colours coding and optimum colour information transmission in the retina[END_REF] gave a rigorous explanation of the reason underlying the results just described. Due to the strong correlation between either the L, M, S sensitivity curves, or the R, G, B sensors of a camera, the chromatic covariance results in a positive matrix, i.e. a matrix with all positive coefficients, not to be confused with a positive-definite matrix. For positive matrices, the Perron-Frobenius theorem, see e.g. [START_REF] Berman | Nonnegative Matrices in the Mathematical Sciences[END_REF], guarantees that opponency must appear in the second and the third eigenvector.

[9] confirmed experimentally the predictions of Buchsbaum and Gottshalk by using a multispectral database. A more modern overview and further information about this topic can be found e.g. in [START_REF] Hyvärinen | Natural Image Statistics: A Probabilistic Approach to Early Computational Vision[END_REF] or [START_REF] Provenzi | On the second order spatiochromatic structure of natural images[END_REF].

It is nonetheless important to stress that what we have described above is a posteriori description of the phenomenon of chromatic opponency and not a mathematical model able to intrinsically account for this phenomenon. This remark applies also to the color description performed by the CIE color spaces as e.g. the CIELab space.

The main aim of this paper is to fill this gap by providing a novel model with respect to the state of the art in which color opponency appears naturally without the need of a posteriori analysis.

To do this, in section 2, we describe the possible Riemannian geometries of the space P of perceived colors viewed as a chromatic space. Classical historical axioms say that this chromatic space is a regular convex cone of dimension 3. Following the seminal work of [START_REF] Resnikoff | Differential geometry and color perception[END_REF] and under a supplementary axiom that is discussed in the sequel, we show that P must be isometric to either the flat space R + × R + × R + endowed with the Helmhotz-Stiles metric, or to the cone H + (2, R) of positive definite 2 × 2 real symmetric matrices endowed with the Rao-Siegel metric. This latter is shown to be hyperbolic in the sense that it is foliated with leaves isometric to the Poincaré disk.

The fundamental remark that we develop in section 3 is that H(2, R), equipped with the Jordan matrix product, is the algebra of observables of a real quantum system with Hilbert space R 2 . This quantum system, called rebit, is a real analog of the qubit associated to 1/2-spin particles. Then H + (2, R) is the cone of a state space S that is isometric to the Klein disk, i.e. the unit disk equipped with the Hilbert metric.

As a consequence, we obtain the following mathematical description of human color perception: the perceived color space can be represented as the cone of positive elements of the Jordan algebra H(2, R) and it encodes both the Riemannian geometry of the trichromacy and the quantum opponent mechanism with the rebit, the Bloch disk of which coincides with the Hering disk. This proves that the missing mathematical link between the chromatic and color opponency stages is nothing but the hyperbolic structure of the space P of perceived colors.

We kept this paper intentionally short because we prefer to focus on the communication of the new ideas underlying our discoveries. All the many technical mathematical aspects will be detailed in future works.

Riemannian trichromacy

Our starting point is that the trichromatic space of perceived colors P is a regular convex cone of dimension 3. In order to classify the possible geometric structures of this space, in [START_REF] Resnikoff | Differential geometry and color perception[END_REF] a supplementary axiom is added. Starting from the observation that no color can be considered special with respect to another one, Resnikoff is led to postulate that P is a G-homogeneous space, where of course the group G must represent the transformations on P that permit to pass from the perception of a color to that of any other color.

With a choice that is still debated today, Resnikoff selected for G the following group: if V =span(P), then G ≡ GL + (P) = {B ∈ GL(V ) : det(B) > 0, Bx ∈ P ∀x ∈ P}. A transformation B ∈ GL + (P), i.e. an orientation-preserving linear map that preserves P, is called a change of background illumination. Resnikoff noticed that by suitably changing the background illumination of a color stimulus, as in Fig. 1, we can modify its perception to any another color of P sufficiently close to it, i.e. P is a G-locally homogeneous space. Since P is a convex cone, local homogeneity implies that G acts transitively on the whole P, which then becomes a globally G-homogeneous cone. By using the dimension axiom, i.e. that dim(V ) ≤ 3, it is possible to prove that P can be isomorphic to either the classical to colorimetric trichromatic space P 1 = R + × R + × R + or to a novel space for colorimetry, namely P 2 = H + (2, R), the space of positive definite 2 × 2 real symmetric matrices.

Due to the essential role played by the group G for his theory, Resnikoff searched for Ginvariant Riemannian metrics on the spaces P 1 and P 2 . It turned out that the only G-invariant Riemannian metric on the classical space P 1 is the equally classical Helmholtz-Stiles metric

ds 2 = α 1 ( dx1 x1 ) 2 + α 2 ( dx2 x2 ) 2 + α 3 ( dx3 x3 ) 2 , α i > 0, i = 1, 2, 3.
Remarkably, the only G-invariant Riemannian metric on the space P 2 is the Rao-Siegel metric

ds 2 = 1 2 Trace[(x -1 dx) 2 ]. If H + 1 (2, R
) denotes the subset of matrices belonging to H + (2, R) with unit determinant, then we have that H + (2, R) R + × H + 1 (2, R). One can check that, when endowed with the Rao-Siegel metric, this space is isometric to R + × D, where D is the Poincaré disk. An explicit isometry between the Poincaré disk D and H + 1 (2, R) is given by the following transformation:

ϕ : D -→ H + 1 (2, R) (w 1 , w 2 ) -→ ϕ(w 1 , w 2 ) =     1 + 2w 1 + (w 2 1 + w 2 2 ) 1 -(w 2 1 + w 2 2 ) 2w 2 1 -(w 2 1 + w 2 2 ) 2w 2 1 -(w 2 1 + w 2 2 ) 1 -2w 1 + (w 2 1 + w 2 2 ) 1 -(w 2 1 + w 2 2 )     , (1)
so that we have:

1 2 Trace (ϕ(w 1 , w 2 )) -1 (dϕ(w 1 , w 2 )) 2 = 4 (dw 1 ) 2 + (dw 2 ) 2 (1 -(w 2 1 + w 2 2 ) 2 = ds 2 D . (2)
At this stage, it is not clear how to recover the colorimetric attributes of perceived colors nor to give a meaningful description of the color opponency mechanism. In fact, it turns out that the missing link to obtain these results is given by Jordan algebras formalism. This not only permits to reformulate Resnikoff's description of P in a much more concise way, but, more importantly, it also opens a whole new perspective about color perception, formalizing a link with the algebraic theory of quantum physics.

According to the classification theorem of Jordan, von Neumann and Wigner, see e.g. [START_REF] John | Division algebras and quantum theory[END_REF], a formally real Jordan algebra of real dimension 3 is isomorphic to either R ⊕ R ⊕ R or H(2, R). The positive elements of these two Jordan algebras are the symmetric cones R + × R + × R + and H + (2, R). This fact leads us to postulate only the following axiom for describing human color perception from a mathematical point of a view.

The trichromatic axiom: the space P of perceived colors is the symmetric cone of positive elements of a formally real Jordan algebra of real dimension 3.

Remark: we notice that, since symmetric cones are intrinsically homogeneous, this axiom has the advantage of circumventing the need of an explicit specification of a transitive action on P, which is the most debated and controversial issue of Resnikoff's theory.

The main contribution of this paper is to show that, contrary to the flat space P 1 = R + × R + × R + , the hyperbolic space P 2 = H + (2, R) is suitable to describe the dynamic of the chromatic opponent mechanism. One simple, but non trivial, result asserts that H(2, R) is isomorphic, as a Jordan algebra, to the spin factor R ⊕ R 2 equipped with the Jordan product • define by:

(α + u) • (β + v) = ( u, v + αβ + αv + βu), ∀α, β ∈ R, u, v ∈ R 2 .
(

) 3
The explicit isomorphism of Jordan algebras between R ⊕ R 2 and H(2, R) is given by:

(α + u) -→ α + u 1 u 2 u 2 α -u 1 . (4)
If we consider the Minkowski metric induced by:

(α + v) • (β + w) = αβ -v, w , (5)
on R ⊕ R 2 , the hyperbolic space H + (2, R) space corresponds isometrically to the future lightcone:

L + = {a = (α + v), α > 0, a • a > 0} . (6)
and H + 1 (2, R) to the level set:

L 1 = {a = (α + v) ∈ L + , a • a = 1}. (7)
This precisely means that H + 1 (2, R) equipped with the Rao-Siegel metric is isomorphic to the Klein disk

K = {(α + v) ∈ L + , α = 1}, (8)
where L + is the topological closure of L + , or to the Klein disk

K 1/2 = {(α + v) ∈ L + , α = 1/2}, (9)
with Klein metric given by the Hilbert metric. This latter disk provides a very simple vizualisation of the states of the quantum system described below. The fact that the distance on K 1/2 is the Hilbert metric is a crucial fact that we will discuss in the conclusion.

Quantum color opponency

The algebra H(2, R) is the observable algebra of a real quantum system with Hilbert space R 2 . Such a system is called a rebit and it is a real analog of a qubit, i.e. a two-level complex quantum system of a 1/2 spin particule. The state space S is the set of density matrices, i.e. the set of non-negative matrices of H(2, R) with trace equal to 1. Such a density matrix can be written as:

ρ(v 1 , v 2 ) = 1 2 1 + v 1 v 2 v 2 1 -v 1 , ( 10
)
where v = (v 1 , v 2 ) ∈ R 2 with v ≤ 1.
We can considere S as the unit disk {v ∈ R 2 , v ≤ 1} embedded in the Klein disk K 1/2 via the map

s = (v 1 , v 2 ) -→ 1 2 (1 + v) = 1 2 + v 1 2 , v 2 2 . ( 11
)
Now, we explain the opponency mechanism of the rebit. First of all notice that, by direct computation, ρ ∈ H(2, R) is a state density matrix if and only if it can be written as:

ρ(v 1 , v 2 ) = 1 2 (Id 2 + v • σ) = 1 2 (Id 2 + v 1 σ 1 + v 2 σ 2 ), ( 12
)
where σ = (σ 1 , σ 2 ) with:

σ 1 = 1 0 0 -1 , σ 2 = 0 1 1 0 , (13)
and, again, v = (v 1 , v 2 ) ∈ R 2 with v ≤ 1. The matrices σ 1 and σ 2 are Pauli-like matrices. They give a two direction opponency mechanism in the sens that we are going to explain. By using the typical bra and ket Dirac notation of quantum mechanics, let us denote with |u 1 , |d 1 , |u 2 and |d 2 the four state vectors defined by:

|u 1 = 1 0 , |d 1 = 0 1 , ( 14
)
|u 2 = 1 √ 2 1 1 , |d 2 = 1 √ 2 -1 1 . ( 15
)
With a straightforward calculation, we can express σ 1 and σ 2 as a combination of projectors on the state vectors defined above, more precisely:

σ 1 = |u 1 u 1 | -|d 1 d 1 | , σ 2 = |u 2 u 2 | -|d 2 d 2 |. (16)
The state vectors |u 1 and |d 1 , resp. |u 2 and |d 2 , are eigenstates of σ 1 , resp. σ 2 , with eigenvalues 1 and -1. By using the polar coordinates reparameterization given by v 1 = r cos θ, v 2 = r sin θ, r ∈ [0, 1] and θ ∈ [0, 2π), we can write ρ(v 1 , v 2 ) as:

ρ(r, θ) = 1 2 1 + r cos θ r sin θ r sin θ 1 -r cos θ = 1 2 [(1 + r cos θ)|u 1 u 1 | + (1 -r cos θ)|d 1 d 1 | + (r sin θ)|u 2 u 2 | -(r sin θ)|d 2 d 2 |] . (17)
This gives, for instance1 :

L -M := ρ(1, 0) = |u 1 u 1 | = 1 0 0 0 , (18)
M -L := ρ(1, π) = |d 1 d 1 | = 0 0 0 1 , (19)
S := ρ(1, π/2) = |u 2 u 2 | = 1 2 1 1 1 1 , (20)
-S := ρ(1, 3π/2) = |d 2 d 2 | = 1 2 1 -1 -1 1 . (21)
More generally, for all θ ∈ [0, 2π), we have

ρ(1, θ) = |(1, θ) (1, θ)|, with: |(1, θ) = cos(θ/2)|u 1 + sin(θ/2)|d 1 . (22)
Let us recall that the pure state density matrices are state density matrices ρ that are also projectors, i.e. which satisfy ρ • ρ = ρ. An equivalent characterization can be given in terms of the von Neumann entropy of the state density matrix ρ, i.e. S(ρ) = -Trace(ρ log ρ): ρ defines a pure state if and only if S(ρ) = 0.

The pure state density matrices correspond exactly to the points (1+v)/2 of K 1/2 with v = 1, i.e. to the boundary points of K 1/2 . In fact, K 1/2 is the Bloch disk of the rebit, a real analog of the Bloch ball of the qubit, embedded in L + .

The opposite case is that of the maximal entropy state, that, for a general formally real Jordan algebra, corresponds to ρ 0 ≡ u/Trace(u), where u is its unit element. In particular, for the Jordan algebra H(2, R), ρ 0 = Id 2 /2.

Consequences on the mathematical description of color perception

The previous mathematical discussion can be translated in colorimetric terms as follows. A perceived color c = (α + v) is an element of L + , viewed as the Riemannian space of trichromacy. The real value α is the magnitude of the c. The element c/(2α) = (1/2 + v/(2α)) belongs to the Klein disk, or the Bloch disk, K 1/2 . The chromatic state of c is identified with the quantum state s c = v/α of S, the quantum chromatic state space.

A perceived color relative to a pure chromatic state, i.e. with zero von Neumann entropy, is a pure perceived color. On the contrary, a perceived color c associated to a chromatic state with maximal von Neumann entropy, i.e. ρ 0 = Id 2 /2 = (1 + 0)/2, is an achromatic perceived color. In fact, in the proposed framework, the classical attribute of saturation is nothing else than a mesure of the von Neumann entropy.

Consider now a chromatic state s = (v 1 , v 2 ) and the corresponding density matrix ρ s (v 1 , v 2 ) = ρ s (r, θ). In order to evaluate the chromatic state s with regard to the opposition "red-green" given by σ 1 , we compute the value:

σ 1 ρs = Trace(ρ s • σ 1 ), (23)
which gives the value of the observable σ 1 in the state s. We obtain:

σ 1 ρs = r cos θ. (24)
If -π/2 < θ < π/2, then s is a reddish chromatic state, whereas it is greenish when π/2 < θ < 3π/2. If θ = π/2 or θ = -π/2, then s is achromatic in the opposition red-green. Note that Haring's statement: "greenness and redness can not be simultaneously attributes of a color" in this context is simply equivalent to the fact that cos θ cannot be simultaneously non-negative and non-positive. We obtain similar conclusions for the "yellow-blue" opposition given by σ 2 because:

σ 2 ρs = r sin θ. (25)
Finally, it easy to check that every density matrix ρ(r, θ) can be written as:

ρ(r, θ) = ρ 0 + r cos θ 2 (ρ(1, 0) -ρ(1, π)) + r sin θ 2 (ρ(1, π/2) -ρ(1, 3π/2)) , (26)
where the achromatic state density matrix is the mixture with equal probability:

ρ 0 = 1 4 ρ(1, 0) + 1 4 ρ(1, π) + 1 4 ρ(1, π/2) + 1 4 ρ(1, 3π/2). (27)
The difference (ρ(1, 0)-ρ(1, π)) encodes the red-green opposition whereas the difference (ρ(1, π/2)ρ(1, 3π/2)) encodes the blue-yellow one.

To conclude, the quantum-like framework provided by the Jordan algebra formalism accounts for an intrinsic description of the achromatic plus opponent color phenomenon that does not require any a posteriori analysis.

Conclusions and perspectives for future works

We have shown that the human color perception can be mathematically described from the sole axiom that the trichromatic space is given by the positive elements of a Jordan algebra of real dimension 3. The hyperbolic Riemannian chromatic space is the future lightcone L + of the 3dimensional Minkoswki space. The Klein disk K 1/2 of L + appears to be the embedding of the chromatic quantum state space S of a rebit whose Bloch disk coincides with the Hering disk of color opponency.

One of the key points of this description of human color perception is that the hyperbolic Klein metric on K 1/2 is given by the Hilbert distance. Hilbert's distance is defined on every convex set Ω and is always a Finsler metric whose asymmetric norm is given by:

v p = 1 2 1 |p -p -| + 1 |p -p + | |v| ( 28
)
where p ∈ Ω and p ± are the intersection points with the boundary ∂Ω of the oriented line in Ω defined by the vector v with Euclidean norm |v| based at the point p. It is well known that this Finsler metric is Riemannian if and only if the boundary ∂Ω is an ellipse. Now, both the Riemannian trichromatic and quantum chromatic spaces that we have described are ideal spaces that do not involve specific characteristics of an observer. In particular, the rebit is a so-called closed quantum system. It is natural to envisage to characterize every observer's capability regarding color perception by a convex subset Ω of the state space S, or equivalently of the Klein disk K 1/2 , endowed with the Finsler metric given by the Hilbert distance. This convex subset Ω is in some sense the restriction of the ideal chromatic state space due to the limitation of the observer perception. Work in progress is devoted to identify, for each observer, the convex subset Ω by comparing the balls of the Finsler metric and the MacAdam ellipses drawn by the observer.

Finally, let us mention that the chromatic quantum system that we have described is, to the best of our knowledge, the first example of a meaningful real quantum system in nature, by opposition to the complex systems of quantum physics.

Figure 1 :

 1 Figure 1: The observation arrangement considered by Resnikoff: a simple color stimulus, typically spanning a foveal aperture, embedded in a uniform background.

We stress that the notations that we used here do not mean that M -L is the opposite of L -M and that -S is the opposite of S. We adopt these notations according to the color opponency mechanism explained before.


Armando Alaminos Bouza

A Fast and compact approximation of energy loss fluctuation for Monte Carlo simulation of charged particles transport

A simple and fast functional model is proposed to approximate energy loss distributions of charged particles crossing slabs of matter. The most accepted physical models for treating this problem was created by Landau and later improved by Vavilov. Both models depend on complex functional forms with exact solutions that are, by far, too CPU intensive to be directly included in existing Monte Carlo codes. Several authors have proposed approximations with varying degree of accuracy and performance. This paper presents a compact and efficient form that approximates with enough accuracy the Vavilov distribution and its extreme cases of Landau and Gaussian shapes. Our functional form could be interpreted as a generalization of the basic Gaussian distribution. Some parameter fits are illustrated with various test cases. Our model also represents a simple functional form to use for regression analysis with experimental energy loss data.

Introduction: While crossing a slab of matter a charged particle suffers inelastic interactions and releases energy to the medium. Due to the random nature of those interactions, the resulting energy of the particle emerging at the end of the slab exhibits a probability distribution. Two basic models representing such distributions are in common use inside Monte Carlo (MC) codes: Landau [START_REF] Landau | On the energy loss of fast particles by ionization[END_REF] and Vavilov [START_REF] Vavilov | Ionization losses of high-energy heavy particles[END_REF].

For a MC code to be used in radiotherapy planning, the need of fast convergence is paramount. In order to be fast, electron transport codes implement condensed history steps to avoid the simulation of billions of interactions per particle history. On the other hand, condensed history simulations alone is less realistic than individual simulation of interactions. As an alternative, most MC uses condensed history steps representing the vast majority of the events but interactions involving high energy transfer are treated individually. Those events over a defined energy threshold are usually called "catastrophic" and its physics is simulated explicitly. Events considered as catastrophic are knocked-on electrons and bremsstrahlung. The introduction of catastrophic events in a code generates some degree of energy loss straggling for the particle, but this fluctuation represents only the extreme cases of fluctuations. All other energy loss fluctuations below the energy threshold are not treated individually and should be included with one of the models by Landau or Vavilov.

The Landau distribution depends on the universal Landau function:

The validity of Landau's theory depends on some restrictions. One of its restrictions conflicts with MC codes including catastrophic events over a cut-off energy: there is no upper limit for the maximum energy loss for a single interaction in Landau's model (ES = ∞).

The Vavilov distribution is more realistic because it contains a maximum limit to the energy transfer in a single interaction (ES).

Where x is the path length of the particle and ∆ is the actual energy loss over the condensed history step.

It is useful to describe the energy loss fluctuation as a function of a parameter K defined as:

K = ξ / ES
K is proportional to the ratio of mean energy loss to the maximum allowed energy transfer in a single interaction.

The parameter K could take values in the interval (0,.., ∞) . If K → 0, it means that ES is very high compared to the mean energy loss. If K → ∞, it means that ES is very low compared to the mean energy loss. For K → 0, the Vavilov distribution should reproduces Landau's shape. On the other hand, if for K → ∞, the distribution became symmetric and identical to a Normal distribution.

Analytic solutions for Landau and Vavilov are too complex and time-consuming for use in MC simulation [3]. Several approaches have being proposed to alleviate the time-consumption impact on MC codes [START_REF]Theory of ionization fluctuations[END_REF][START_REF] Tabata | Approximations to Landau's distribution functions for the ionization energy loss of fast electrons[END_REF][START_REF] Chibani | New Algorithms for Vavilov distribution calculation and the corresponding energy loss sampling[END_REF]7].

A possible approach is the use of pre-generated tabulated distributions. This approach has several drawbacks. While it is a reasonable solution for Landau's distribution function, it is too costly to implement for all possible variation of the K parameter in Vavilov's distribution. An additional performance consideration is that searching into many tables at different address into RAM forces the continuous flush of processor's internal cache, which translate into serious performance penalization in modern superscalar CPUs. Some authors have used rational functions approximations dividing the distributions in subintervals for better agreement [START_REF] Tabata | Approximations to Landau's distribution functions for the ionization energy loss of fast electrons[END_REF]. A clever approach by Chibani proposed the use of the lognormal distribution [START_REF] Chibani | New Algorithms for Vavilov distribution calculation and the corresponding energy loss sampling[END_REF]7]. The use of Edgeworth expansion is a well-known method to correct a Gaussian distribution and has also being investigated [START_REF] Vanginneken | Edgeworth series for collision energy loss and multiple scattering[END_REF].

METHOD

Both, Landau and Vavilov, are skewed distributions, but there is a resemblance between those distributions and the normal distribution. Modifications of the normal distribution have been used to skew the original shape of the function, as in the case of the human longevity distribution [7]. The proposed model has five parameters for fitting the shape of any energy fluctuation distribution, from the ideal Landau case to Vavilov, including the extreme case of a "pure" Gaussian form.

The parameter u is a translation governing the distribution mean, N is a normalization or scale factor and the other three parameters adjust the variance and skewness of the resulting distribution.

The parameters of the proposed model were fitted to analytical solutions of Landau and Vavilov. A non-linear regression method was implemented for fitting the parameters. A different set of the five parameters should be fitted for any value of K and particle speed in the Vavilov distribution function. The Landau's distribution was generated implementing the solutions by Börsch-Supan [3]. Our results were compared with those published [3] before its use. The exact implementation of Landau demonstrated that was too time consuming for inclusion in MC codes.

Our Vavilov exact solution was based on the original paper [START_REF] Vavilov | Ionization losses of high-energy heavy particles[END_REF], complemented with some ideas presented by Seltzer and Berger [7]. Our implementation is improved because we used a better approximations of the sine and cosine integrals, based in the Padé approximants for the argument in the range [0..4] and the use of Chebyshev-Padé expansions of the auxiliary functions for arguments x > 4.0 [START_REF] Temme | Exponential, Logarithmic, Sine, and Cosine Integrals[END_REF][START_REF] Abramowitz | Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables[END_REF]. Our sine and cosine integral are accurate to better than 10 -16 for the whole range. As expected, the computation time for exact Vavilov is exceedingly large for direct inclusion in MC codes.

Results and discussion.

Figure 1 presents a plot of Landau distribution based on the evaluation method by Börsch-Supan [3]. Fig. 1 Figure 2 presents the overlaid plot of Börsch-Supan and the fitting model evaluation. The blue curve is Börsch-Supan while the fitting model is red. The optimum parameters were: N=0.179159, u = 0.390788, A = 2.202571, B = -0.223021, C=-0.000175. Only a few traces of the red curve are visible because of the remarkable proximity between the exact distribution and the proposed approximation. Fig. 2 Figure 2 shows that the correspondence between a rigorous evaluation of Landau's distribution and our fitting model is good for most simulations purposes.

Figure 3 shows the Vavilov distribution for electrons in Water. The initial energy is 10 MeV, Es is 0.2 Mev and K 0.128. The curve in blue is Vavilov, the red is our model. The optimum parameters were: N=0.98735, u = 0.4713, A = 0.04898, B = 0.32578, C=-0.282869. Fig. 3 Figure 4 shows Vavilov exact distribution overlapped with the implementation of Rotondi and Montagna [START_REF] Rotondi | Fast calculation of Vavilov distribution[END_REF] model. The implementation of Rotondi and Montagna was taken from CERNlib and translated into C99. The physical conditions are the same as those in figure 3. The blue trace is Vavilov and the red is Rotondi and Montagna. The model reproduces very well the Vavilov distribution at its central region. At the right tail (region of high energy loss) there is a step that almost ignores the possibility of energy losses with probabilities bellow 1%. As the documentation said, the error increases toward both tail, but in our experiments bigger errors were found on the high energy tail. Fig. 4 Figure 5 shows the same test case from figure 3, but it compares Vavilov with Log-normal and our model. Vavilov curve is blue, Log-Normal is red and our model in yellow. This test case has the same conditions as one from Chibani work [7] using the log-normal distribution. Fig. 5 Figure 6 shows the Vavilov distribution for electrons in Water. The initial energy is 10 MeV, Es is 0.1 Mev and K 0.266. This test case has the same conditions as one from Chibani work [7] using the log-normal distribution. The curve in blue is Vavilov, the red is our model, it is almost a perfect superposition. The optimum parameters were: N=99762, u = 0.48339, A = 0.043564, B = 0.250287, C=-0.415041.

Fig 6

Figure 7 shows the fit of the model to experimental data for protons by Hancock et. al. [8]. Curve in blue follows the experimental data (range of error not shown) and the red curve is our model. We could not access the original data by Hancock et. al., the blue curve is the product of a digitalization from figure 2a at referenced paper [8]. The idea proposed by Tabata et. al. [START_REF] Tabata | Approximations to Landau's distribution functions for the ionization energy loss of fast electrons[END_REF], of using rational function approximation, was also tested. The original paper investigated only the use of rational fractions for Landau distributions, but we try the same method with some Vavilov cases. Figure 8 shows a Vavilov distribution for electrons in Water. The blue curve is the exact Vavilov distribution, the yellow curve is our model and the red curve is the rational function approximation. The rational function used had nine coefficients, four for the numerator and five for the polynomial denominator. It is necessary to split the range of energy loss, at less, into two subintervals; the position of the maximum of the distribution was the division of the domain. Therefore, this approximation needs 18 parameters for any case of the Vavilov distribution. As figure 8 shows, care must be taken with both ends of the range because oscillations of the distribution are common, even taking negative values. Our model (yellow) follows closely the exact Vavilov distribution, the only region with apparent differentiation between our model and Vavilov is close to the maximum. Fig. 8 The execution time needed to compute a Vavilov distribution was the main motivation of this and other works, so let us illustrate the issue with some results. All our numerical experiments used an Intel i7-4930K CPU at The implementation of all the models is in C99. The compiler used on the experiments was Intel C++ Composer XE 2013, SP1 [START_REF]Intel Parallel Studio, C++[END_REF].

Landau and Vavilov exact implementation had the Simpson's integration parallelized using OpenMP, but no other optimization technique was employed. Our Generalized Normal model, the Log-normal model and the rational functions did not gain with parallelization.

As the figures demonstrate, a direct implementation of either Landau or Vavilov are not feasible for most applications. A typical MC code with one million initial electron tracks in water may evaluate 100 million times the energy loss distribution. Assuming that the time for evaluation is equal to the number in the table it will take (it is more than that) 471 hours of CPU time only for Vavilov computation. The same task could take only 2 seconds using our model.

The Log-normal implementation takes 2.8 more CPU time than our generalized normal distribution. This should be the result of a logarithm not present in our model.

The rational function and our model shows equivalent execution time, but the need of 18 parameters and the risk of having oscillatory values at both ends renders the method as not recommended.

The Rotondi and Montagna method has equivalent execution time with our model, but our model has a much simpler functional form and is better for experimental data fitting.

As several of these models use transcendental functions, the resulting CPU times depends on the implementation of those transcendental functions in the compiler library. Some modern implementation of the exponential function, based in SSEx/AVX technologies, are very efficient, as it is the case with Intel compilers (C, C++ and FORTRAN).

  Our proposed model for the probability distribution function, representing Landau and Vavilov is: equ.1 where equ.2

Fig. 7 .

 7 Fig. 7.

  3.40 GHz. The next table presents the results.

	Function Model	Time to compute 1 point in
		distribution [s]
	Landau exact (estimated better	2.983E-3
	than 10 -5 )
	Vavilov exact (estimated better	1.696E-2
	than 10 -5 )
	Log-Normal	5.911E-8
	Rational function	2.157E-8
	Our model (GenNormal)	2.092E-8
	Rotondi/Montagna	2.105E-8
	From CERNlib

Conclusion:

A functional form, simple and easy to evaluate, is presented to approximate the behavior of energy loss fluctuation of charged particles crossing slabs of matter. The model could be interpreted as a generalization of the original Gaussian distribution, a generalization that introduces and control skewness. We have experimented with several test cases, including theoretical distributions from Vavilov and Landau's model as well as measurements results published by other authors. Due to its simplicity the model is a good candidate for use as a regression function with experimental measurements of energy loss. NOTE: Source code is available upon request from the author, Including Landau and Vavilov exact computations and the regression method to fit our model parameters.


John Mullen

Chapter 1: What Voices Can Be Heard in British Music Hall Songs of the First World War?

Following on from my book, 'The Show Must Go On': Popular Song in Britain during the First World War published in 2015 by Routledge, this chapter will look at two aspects of wartime song, both included in the term 'voices'. Firstly, it will look at the interests and priorities defended in the repertoire: can we speak of 'a voice of the people' or 'a voice of the elite' or both? Similarly, the question of gendered priorities expressed in the song repertoire will be examined. The second part of the chapter will look at the sound material of the World War one singers: what different voices did they use on stage (operatic, stage cockney, feminine, regional, working-class and so on), why did they make these choices and what did they mean? The voice is one of the few musical tools in First World War music hall to remain under the comprehensive control of the artiste (since generally the singers were accompanied by a different house orchestra in a different town each week), and its theatrical usage and technological constraints (no microphone in a two or three thousand seat theatre) can tell us much about the meaning of music hall in people's lives during the war.

What Voices Can Be Heard in British Music Hall Songs of the First World War?

Popular music of many genres has been claimed as expressing a particular social voice.

For Cecil Sharp and many others, folk music was or is the voice of the people; 1 rap or punk rock have been presented as the voice of marginalized, dominated youth. [START_REF] Michael | The Sociology of Youth Culture and Youth Subcultures: Sex and Drugs and Rock'n'Roll?[END_REF] What can we say about British wartime music hall of a century ago? What voices could be heard? The title of my chapter is deliberately ambiguous. It would normally be taken to mean 'Whose interests and priorities are expressed in music-hall songs of the war period?' My (tentative) answers to this question will constitute the first part of my chapter, during which I will look at elite voices, working class voices, and gendered voices in music-hall repertoires and practices.

However, in the second part, I will look at the question 'How was voice used by music-hall singers during the Great War?' with the help of a few examples of hit songs. Finally, I will try to sketch out some links between the two sets of conclusions.

The question of music hall as the voice of the people or of the elite has been posed by others for earlier periods, notably Victorian Britain, the debate occasionally simplified as 'class expression versus social control'. [START_REF]Gareth Stedman Jones criticizes simplistic views in his article 'Class Expression versus Social Control?[END_REF] Some writers have emphasized the conservative and jingoistic politics often to be heard in music hall lyrics, and have linked this to the social position of theatre owners and managers, newcomers to local elites. Others have underlined the usefulness to the urban working class of songs which put their lived experience at the centre of an evening's entertainment, and allowed them to laugh and sing along with stars who came from their own milieu and sang in their common language about the hardships, satisfactions and dreams which made up their ordinary life. In this way, they may have validated urban working-class experience in a way that other repertoiresfolk or operetta, for examplecould not.

Before we examine the wartime repertoire with this debate in mind, it is worth thinking about the context in which the voices were heard: the music hall itself. Who was at home in the music hall? The furnishingsred plush and marble pillarsimitated elite tastes to the point of caricature. The appearance of luxury was essential, and this is suggested by the very names of the theatres: the Empire, the Palace, the Coliseum or the Alhambra. At the Oxford theatre in London one moved among 'Corinthian columns and bars smothered in flowers and glittering with mirrors'. 4 At the Palace, a 'grand staircase columned with green marble and gilt capitals' led up to the royal box. 5 The Nottingham Empire boasted smiling idols representing Krishna on either side of the stage and four giant gilt elephant heads in the four corners of the auditorium. [START_REF] Russell | Popular Music in England[END_REF] Yet the mass of the audience was working-class, whether traditional or (more and more) 'white-collar': aspirational office clerks, minor civil servants and shop workers.

These growing groups had clean hands and often felt themselves a cut above the manual workers, but were frequently poor and over-worked. 7 The music hall was a place where, for once, the luxury was for them. And yet, unlike the also popular municipal concert halls, where classical concerts were presented at very low prices for the masses, the music hall was not obsessed with 'elevating'. There was no need to sit in silence and worry about clapping in the wrong places. In the music hall the worker felt at home, with singalong and neighbourly bonhomie for a 'Sweet Saturday night' [START_REF] Macinnes | Sweet Saturday night -Pop song 1840 to 1920[END_REF] . The term 'people's palace' was not used without a reason. But within this pretence of a home, what voices were dominant?

Elite voices

The most important way in which wartime music hall might be seen to be acting as an elite voice is that it was a tool building support for the war drive, for the empire's cause. This role was particularly important in Britain, where conscription was not used during the first years of the conflict, (whereas Germany and France mobilized millions of men by conscription in the first weeks of the war). In Britain, political leaders, trade union leaders, feminist leaders and church leaders almost unanimously called for men to join up and fight, and music-hall stars joined in the chorus. Certainly, writers such as Siegfried Sassoon [START_REF] Sassoon | The Old Huntsman and Other Poems[END_REF] or J B Priestley [START_REF] Priestley | Margin Released --a Writer's Reminiscences and Reflections[END_REF] tended to dismiss the wartime music-hall repertoire as mainly jingoistic and wholly subordinate to the establishment. They based their complaints on a number of elements. First and foremost, there was the wave of songs of recruitment and enthusiasm which flooded into the music hall in 1914, with titles such as the following:

We Don't Want to Lose You but We Think You Ought to Go

The Army of Today's All Right. The Baby Tank 1916 These songs, along with the reports of rousing speeches on the stage of the music hall (sometimes in the tone 'Two shillings for the first man to join the army here tonight!'), and the leading role of a small number of music-hall stars (such as Harry Lauder and Vesta Tilley) in the recruitment campaigns, have led to conclusions which echo the words of Lawrence Senelick, who wrote, of an earlier period: Much of the energy the working man might have directed to ameliorating his own situation was rechannelled by the music hall to the advancement of [the British] empire. [START_REF] August | A Culture of Consolation?[END_REF] There are, however, grave problems with this conception. If the music hall artistes and music hall songs which campaigned for the empire may have been particularly effective (working-class singers in the workers' favourite place for a Saturday night out are more convincing than 'posh' politicians or bishops), they were by no means exceptional. Almost all religious leaders, feminist leaders, union leaders, politicians and intellectuals went at least as far.

My

More importantly, there has been, until recently, no attempt to judge how representative the enthusiastic war songs were in the song repertoire. The same dozen songs are always quoted, among the thousands produced at the time which lie forgotten in the archives of the British Library. When I was able to analyse a full corpus of over a thousand wartime music hall songs, the conclusions were somewhat different. [START_REF] Mullen | The Show Must Go On! Popular Song in Britain during the First World War[END_REF] Firstly, recruitment songs, and heroic songs about the war, all but disappeared after only a few months of the conflict. The recruiting campaign was at its fiercest in the second half of 1915, but the collection of 'Greatest Hits of 1915' published by the biggest of the sheet music companies, Francis and Day, [START_REF]Francis and Day Annual[END_REF] contains no recruitment songs at all. Secondly, only a third of wartime songs speak of the war or the changes in life because of the war, and enthusiastic songs make up a small minority of this third. Songs dreaming of the end of the war rapidly take over from heroic songs. Indeed 'home' is the word most frequently encountered in song titles. The seven most common words in titles (in a corpus of Hardly a glossary of jingoism. For every 1918 song with the word 'victory' in the title there are ten with the word 'home', and the same is true the following year. 14 Finally, even at the start of the war there are no songs about hating Germans. In contrast with the French repertoire, which benefitted from an already established stock of anti-German songs (written in the period after France lost Alsace and Lorraine to the new German Empire in 1871), British songs about hating Germans are completely absent. There are a number of reasons for this: the German people were close to the British before 1914, and particularly in the music industry. The 'German band' was an important feature of entertainment, and the best pianos were imported from Germany [START_REF]Excellent German overstrung instruments cost about £30[END_REF] . The music-hall song experience depends on the audience singing in unison, and it seems likely that artistes understood that singing about hating Germans was not going to be possible. In addition, songs of hatred sat uneasily with the almost obligatory jauntiness which was part of the music hall ethos.

In other domains of public discourse, such limitations did not exist. The Bishop of London, Arthur Winnington-Ingram, became known as a fiery orator for the cause, declaring:

[Let us] band in a great crusade… to kill Germans. To kill them, not for the sake of killing, but to save the world; to kill the good as well as the bad; to kill the young men as well as the old, to kill those who have shown kindness to our wounded as well as those fiends who crucified the Canadian sergeant…to kill them lest the civilization of the world should itself be killed. [START_REF] Moynihan | God on Our Side -the British Padre in World War One[END_REF] Winnington-Ingram liked to repeat that the war was a combat 'between nailed hand and mailed fist', that is, between the suffering Jesus, dying for man's sins, and brute violence. Such excesses were entirely absent from the music hall song repertoire.

We may conclude, from these diverse elements, that the music hall cannot be considered in any straightforward way a jingoistic channel by which elite priorities concerning the war were imposed on working class audiences.

If jingoistic warmongering was generally absent, so too was opposition to war. The tone was very much of acceptance, even after the enthusiastic recruitment songs had disappeared after the first few months. Songs such as 'Lloyd George's Beer' which humorously complained about government policy of reducing the alcohol content in beer, and which could stand in for more general discontent with inequality in wartime, felt obliged to begin by underlining faith in victory:

We shall win the war, we shall win the war As I've said before, we shall win the war Before going on with the theme of the song Oh, they say it's a terrible war, oh law, And there never was a war like this before, But the worst thing that ever happened in this war Is Lloyd George's Beer.

Buy a lot of it, all they've got of it.

Dip your bread in it, shove your head in it

From January to October, And I'll bet a penny that you'll still be sober. 17 The war was, though, not the only priority of the elite. One long term priority which continued to be pushed throughout the conflict was the campaign against 'vulgarity' and in favour of 'respectability', this last being a tremendously powerful cement of conservative social consensus. [START_REF]I analyse the social practice and ideology of music hall respectability more fully in John Mullen[END_REF] For the elite in general, this helped mobilize large sections of the working class behind ideological priorities of the elite. For if being 'respectable' included respecting yourself and your family (no excessive drunkenness, no interpersonal violence between adults within the family), it also includes respecting 'one's betters', sending the children to Sunday school, and not banding together to resist the status quo.

The music hall trade newspaper, The Encore, devoted more editorials between 1914 and 1918 to the defence of respectability than it did to questions connected with the war, and theatre managers were eternally eager to stop artistes overstepping the bounds, banning all adlibbing and sacking artistes who were too rude. Newspaper review of shows habitually reassured readers that there was 'no vulgarity' involved in the evening's entertainment. The great Moss Empires theatre chain advertised in the trade press insisting on strict rules: all scripts to be submitted three weeks before appearance artistes' business must be absolutely free from vulgarity artistes must attend rehearsals, no deputies being accepted 19 In April 1916, when one comedian wrote a letter to the newspaper of the performers' trade union, saying that vulgarity was not such a bad thing ('Who needs all this refinement?'), the paper received an avalanche of letters disagreeing. 20 The same year the Bishop of London brought his very considerable influence to bear, denouncing indecency or improper behaviour in music hall lounges. 21 In summary, one might say that more complex elite priorities, which had had more success implicating the general population, were very much carried in music hall practices. Since they [music-hall songs] were chiefly written by, and sung by, working-class men and women for working-class audiences, we may hear in them a vox populi which is not to be found in Victorian and Edwardian literature [START_REF] Macinnes | Sweet Saturday night -Pop song 1840 to 1920[END_REF] .

How working-class voices were present

What does this vox populi mean when we look at the wartime songs? Firstly that 'life goes on'. Life was hard before the war, life is hard during the war: many themes sung about do not change, and the valorisation of ordinary working-class life continues as a main plank of music hall content. Many pieces have as their subject good food and drink, in particular good working-class food and drink, conjured up in song to exorcise memories of hunger, and defended against bourgeois pretention, as for example in a series of hits by Harry Champion including 'Home-made Sausages' (1912) 'A Little Bit of Cucumber' (1915) 23 and 'Pudding, Pudding, Pudding (1917). There are plenty of songs about working class courtship and marriage, [START_REF]See for example George Formby Senior 'since I a Row with the Wife[END_REF] and about the ups and downs of working class friendship. The songs about the war are about Tommy, not about the empire. And there are very large numbers of songs dreaming of a distant paradise, far from the war (generally rural Ireland or 'Dixieland', both places which most of the audience would never go to, and which seemed to them to be vaguely similar).

The singers are from working class backgrounds, with working class stage names (Harry, Gus, Lily, Fred, Kitty, Sam, Jock, Florrie, Daisy, Ernie…) and working-class accents.

They sing in a tone of down to earth neighbourliness, as can be seen in these opening lines from Marie Lloyd's song 'Now You've Got Yer Khaki on': P'r'aps you don't know Johnny Brown who sold the flowers and fruit? I stood 'im 'cos he used to make me laugh The other day he rolled up in Khaki, if you please I says, 'Lumme! don't it suit yer, Jack. Not 'arf!' There is, then, a working-class voice in the repertoire, which is generally (but not completely) reserved for the expression of everyday individual joys and difficulties: a political voice is all but absent. So, when there are waves of strikes before, during or after the war (the Great Unrest of 1910-14, the enormous strike wave of 1919, or the important industrial conflicts of 1917) the strikes may be popular among working class audiences, but we see almost no songs speaking of them. Music hall remains a consensual genre, principally because of the combination of sing-along practices which oblige singers to present choruses that the entire audience will like, and the power of generally conservative theatre managers.

Many maintain that the war was not in the interests of working-class people, yet there was little on the music-hall stage to reflect this. Though the anti-war movement was very much marginalized (except in Ireland) in 1914, in 1916 it was strong enough to hold 'Stop the War' meetings in every significant town and a 1500-strong delegate conference in London. [START_REF] Cummins | The Hound of Conscience: a History of the No-conscription Fellowship[END_REF] The echoes of this dissent on the music hall stage were more than muted (since consensus was so important), but did exist. A song like 'The Military representative' sarcastically mocking the committees who refused exemption from military conscription on any grounds whatsoever, was a smash hit. [START_REF][END_REF]

Gendered voices

Men, women and love are major themes in the repertoire, and the authorial voice of the songwriter in music hall with regard to gender should be looked at. The authorial voice is generally little developed in the music hall song compared to later genres of popular music such as rock. The showman was more present than the auteur in song writing in this period. This can be shown by the fact that one of the most prolific songwriters, Fred Godfrey, wrote thirty or so popular 'Scottish songs' and at least as many 'Irish songs', with titles such as 'It Takes an Irish Heart to Sing an Irish Song' [START_REF][END_REF] and 'The Mother Old Ireland Gave Me'. Godfrey claimed that it was because of his Irish mother that his Irish songs came straight from the heart. In fact, his father was Welsh and his mother was English; he had no Irish roots at all.

The auteur was even more clearly a fabricated construct than they are today.

Nevertheless, one might think that the fact that 90% of the songwriters were men, [START_REF]A quick count of the first three sections (A-C) of Kilgarriff's list of lyricists and composers 1860-1920 gives a list of 251 men and 28 women. Michael Kilgarriff, Sing Us One of the Old Songsa Guide to Popular Song 1860 -1920[END_REF] and none of the most well-known and prolific were women. (whereas at least a third of the singers were women) affected the content of the songs. It is certain that women do not get treated well in the repertoire. A very large number of songs complain, with hyperbole, about the position of the husband in marriage (and indeed, in most working-class families, the woman was in charge of the finances: most men brought their wage packets home and were given pocket money). [START_REF] Roberts | A Woman's Place: an Oral History of Working-class Women 1890-1940[END_REF] Many of these pieces wax lyrical about domineering or even violent wives, whereas there are no equivalent songs complaining about bossy or violent husbands. Billy Merson's 1911 song 'Take an Old Woman's Advice' (which he sings in drag) includes the following sing-along chorus: Girls, girls, when you are married Take a tip from one who knows Treat your old man like you would a pet dog Kind words are better than blows! A 1914 song 'She's So Jealous' portrays a wife who gives very little pocket money to her husband, for fear he should spend it on drink and women; to guard against any risk of infidelity, she cuts off his ears. George Formby Senior sang in 1917 his hit 'Since I've had a row with the wife' in which his wife made him sleep in the kennel with the dog, after having beaten him. [START_REF]These songs and others by George Formby Senior can be heard[END_REF] The theme of the dominating wife could, furthermore, always be brought up to date with current events: in 1918, the piece 'Since My Wife Joined the WAAC [START_REF]The Women's Army Auxiliary Corps[END_REF] ' is billed in The Performer of 25 April 1918 as a 'bang up to date henpecked husband song'.

Of course, popular comic songs have always been very much marked by hyperbole, but the fact that these pieces are not ever balanced by narratives about dominated wives shows that they are part of an ideological structure validating traditional roles. The bullying wife is sung about and ridiculed because she is considered unnatural, whereas the bullying husband is seen as a regrettably normal part of social life, nothing to make a song and dance about, so to speak. One of the important functions of music hall was to bring into the public sphere for the purpose of collective singing and laughing, the everyday difficulties of working class life.

These could be joys and problems with friends, difficulties in courtship, difficulties in marriage. Men's difficulties were explored far more than women's. The women characters in music hall lyrical content are more limited in scope, and often examined only in their relationships to men.

Further, fantasies of perfect partners, very common in music hall songs, are in their vast majority male fantasies, such as this one:

Off in the gloaming when the lights are low Comes a picture in fancy of one girl I know;

A maiden so sweet with the charms of a rose,

Though of ways of the world tis but little she knows

In all Erin's land there is none fairer than she

The dearest of all in this wide world to me! The link with the absolute male domination of the songwriting profession is doubtless not the only reason for this, but is part of it. This is not to say that all discourses about men's and women's roles can be categorized as gendered voices. Some types of song are more complex. There are dozens of songs, for example, about women's new roles in wartime, with titles such as: Many of these songs do not pronounce themselves in favour nor opposed to women's new roles, but intrigued or anxious about them, and this cannot be necessarily categorized as a masculine voice of anxiety: there is ample evidence that women too were worried about the changes. In a number of situations where women took over commercial roles, they could find that customers, both women and men, simply refused to accept them. [START_REF]The example of a woman who took over her father's role delivering meat can be found[END_REF]

How is voice used in the music hall?

This overview has shown some key elements of social voices present in the halls. How did this affect the literal use of voice by the singers? I am not intending to carry out a technical, musicological analysis of voice in music hall, but to look at how different aspects fit in to the entertainment experience and are used to carry meaning. Using some of Philip Tagg's ideas about understanding popular music, I aim to see what the use of voice was, not from the point of view of the singer, but from the point of view of the listener. [START_REF] Tagg | Music's Meanings[END_REF] By way of introduction, three basic points must be made about the 'consumption' of voice in music hall a century ago. Firstly, there are no microphones on the music hall stage until at least the mid-1920s, yet the variety theatres, especially the newer ones, seat two or three thousand people. 34 This meant that 'intimate' voices could not be used, projection was essential, and indeed many artistes would use a mix of reciting and singing. The practice of audience singalong allowed to inject some much-needed vocal volume into the show.

Secondly, the star's voice was a much rarer commodity than it is today. A fan who goes to a concert in the twenty first century has heard the artistes sing many times on CD, on the radio, on the television or on Mp3. The audience a century ago might know the song well, having bought the sheet music and sung it around the piano, or listened to street singers doing their versions (hits sold hundreds of thousands of copies in sheet music, and there were three or four million pianos in Britain at the time). They would not, however, have much heard the voice of the artiste: the gramophone was still reserved to a small minority of the population.

For the price of one record containing two songs, one could buy, in 1916, six tickets for a music hall evening; for the price of a cheap gramophone, 220 tickets. Many soldiers heard a gramophone record for the very first time during their years in the army.

Finally, the centrality of sing-along needs to be again underlined. The artiste proposes a position, a fantasy, an emotion etc. which he or she invites the whole audience to project together. As Philip Tagg has pointed out, sing-along (or singing in unison) has a social meaning: it constitutes 'a rather obvious aural icon of individuals sharing (supposedly or actually) a common cause, or identity, or set of values and beliefs.' [START_REF] Tagg | [END_REF] Let us turn to our sample of wartime music hall songs. The recordings used for this analysis are not live recordings, which contemporary technology could not provide; nevertheless, the artistes at the time tried to record in the same way as they sang on stage, since the stage show was considered primary. 36 The first song 'Your King and Country Want You' performed by Helen Clarke, 37 is one of the best known of the 1914 recruitment songs. It was taken up by a number of artistes at the time. The operatic style of voice (received pronunciation, accentuated vibrato, rolled R's) represents 'high culture' and communicates not just respectability but also prestige. The audience singing along feels it absorbs some of the prestige. This use of voice contributes to the production of an anthem giving an establishment point of view. The seductive voice fits with the somewhat daring end to the song (indeed some sang 'bless you' instead of 'kiss you' in the last line of the chorus, in order to maintain an air of respectability). Nevertheless, the opera voice contrasts a little incongruously with the jolly sing-along tune. The use of voices taken from opera or operetta was common on the music hall stage, and stars such as Gertie Gitana and Zona Vevey made much use of it. It was of course quite suited to large halls without microphones.

The second piece, sung by Ernie Mayne in 1917, is 'Lloyd George's Beer'. 38 The lyrics of the song complain sarcastically about the new regulations reducing the strength of beer in licensed premises, introduced as part of the government's vision that alcohol consumption was putting the war drive in danger. Mayne sings in a working-class voice, coming over as 'one of the lads'. Slightly out of tune at times, his untrained voice might give an impression, along with the lyrics, of 'a favourite uncle, a bit rough around the edges but with his head screwed on'. He sounds as if he has been drinking, but this may well be projected as a positive characteristic ('A man who likes a drink'). His voice is that of the sardonic workingclass civilian, an 'ordinary bloke' who accepts the need to win the war, but resents the restrictions imposed, he feels, solely on ordinary people. The naming of Lloyd George as personally responsible allows the song to express (in the sing-along chorus) bitterness against the Prime Minister, but on a topic which does not seriously threaten patriotic ideology.

In fact dissent, by 1917, was widespread. Large waves of strikes [START_REF] Horne | A Companion to World War I[END_REF] were taking place, and anger about the conscription process was very common. 40 This song probably expressed the dissent, but attached it to a subject which was relatively harmless, and which fit a music hall tradition of singing about drink.

Both of these songs, then, use clear class markers in their voices. The opera voice represents both virtuosity and bourgeois respectability, this last being at the time a crucial plank of society. 41 The implication is that being respectable includes actively supporting the war drive. The working-class voice underlines that some popular priorities are despised by the powers that be, but it can be taken in two different ways. Much of the audience will identify with the sardonic narrator, whereas other, perhaps more expensive, parts of the theatre might see him as an object of fun. Ambiguous presentations of this sort were common in music hall.

The third song we will discuss is 'Goodbyee' by Courtland and Jefferies, from 1918. 42 A received pronunciation 'posh' voice is used in the first verse, which goes along with the use of some elements of bourgeois slang in the chorus. In the music hall, such bourgeois slang will be being mocked. The voice is clipped, that of someone who has difficulty expressing their feelings, perhaps. The title word 'Goodbyee' and several rhyming words have a distorted pronunciation which can be interpreted in different ways. Might it be baby-talk, the tragedies being lived through making adult reactions no longer possible? The power of the song is to reproduce in light-hearted tone the dreadful and sometimes tragic goodbyes which are part of the audience's wartime experience. The laughing, indirect references to death, the obsessive repetition of 'goodbyee' in the chorus sung by the whole audience, give an impression of bitterness and inexpressible pain.

Good Byee! Good Byee! Wipe the tear, baby dear, from your eyee Tho' it's hard to part, I know, I'll be tickled to death to go.

Don't cryee! don't sighee!

There's a silver lining in the skyee Bonsoir, old thing! cheerio! chin-chin! Nah poo! Toodleoo! Goodbyee! There are no tearful goodbyes on the music hall stage during the war. The voice in this song plays a role in the communication of ambiguous psychology, rather than being an elite voice or a working-class voice.

Our fourth example is the hit song 'When I take my Morning Promenade' by Marie Lloyd in 1916. 43 Marie Lloyd was tremendously popular, and her nickname of 'Our Marie' symbolized her status as a perceived working-class woman's voice on stage, ready to eagerly mock prudery and excessive respectability while never going 'too far', never being clearly vulgar. In this she is part of a popular music tradition of 'moderate transgression which continues to this day. [START_REF] Perry | 's song 'I kissed a Girl' which both described this supposedly daring experience while reassuring the audience of the narrator's heterosexual credentials is a twenty first century example of this moderate transgression[END_REF] Her working-class voice was central to her stage personality. In this song, she uses 'bourgeois' vocabulary like 'promenade', and a 'flirty' tone of voice, using almost a high yodel to sound cheeky. She represents a woman in control as an independent individual, through the seduction game, though in no way undermining the separation of gender roles, which are felt and presented as 'natural' in this and other songs of hers. Indeed her 1915 song 'Woman's Opinion of Man' whose title might lead one to expect some critique of masculine domination, is on the contrary a text which mocks women's attempt to critique men, and insists that traditional roles are wholesome and desirable. She understands man, or she thinks so at least She's read the six novels, and calls him a beast She vows she won't marry, but when she is tough Oh, she's fearfully certain she won't get enough Dear, dear, hum, hum, don't fret If a fellow says 'Wilt? will she wilt? You bet To return to 'When I take my Morning Promenade', her use of voice includes both class elements: accent and vocabulary, and gendered elements: seduction and flirting. The whole remains deliberately ambiguous: respectable but mocking bourgeois pretention, seductive but not vulgar.

Our final song is 'Jolly Good Luck to the Girl who loves a soldier!' 45 , a 1914 piece by the male impersonator Vesta Tilley, which mixes an expression of support for the war drive with gentle mocking of the young officer narrator, who is something of a 'ladies' man'. The most notable feature of her use of voice is the question of gendering. She is a male impersonator, well-known for precise imitation of her male characters' dress, gait and gestures, yet there is no attempt to imitate a male voice; on the contrary she uses a semioperatic voice. At the time, male impersonation was still considered a little 'risqué': though Vesta Tilley was allowed to perform her turn in front of the royal family at the Royal Variety Performance in 1912, it was said that the queen was shocked by the sight of a woman in trousers. This is why the imitation of masculinity must not go too far. Indeed, off stage, Vesta Tilley was careful to look as 'feminine' as possible, in jewels and fur, and, having no children of her own, was very visibly involved in children's charities. 46 The feminine voice she uses in this song is a sign of the tension around gender roles which will become even sharper as the war wears on and women are able to take on jobs previously closed to them. This is also shown in other aspects of her stage showfor example she never wore false beard or moustache and generally played androgynous young men. The clear, almost exaggerated diction underlines the theatrical atmosphere. We are not supposed to be fooled by her impersonation.

Conclusions

Music hall could not be an unambiguous voice of the elite, nor an unambiguous voice of the people, since it relied on consensus, customer demand and sing-along, but also on respectability and establishment support. The actual use of voice by the showmen and showwomen who were singing reflects their communicative aims quite deliberately. They could valorize working-class experience, 'common-sense' or scepticism, or they could symbolize integration into respectable society with its establishment priorities. It has only been possible to analyse here a small sample of voice use in wartime music hall, but we hope it has enabled an exposition of what is at stake, and shown the importance of a close-textured analysis of popular songs which is able to bring about some of the hidden communication and deliberate ambiguities inherent in twentieth century popular music.

  Volunteer (he's come from his desk in the city) Our brave Colonials Sandy boy, my soldier laddy Men of England, You Have Got to Go! Be a Soldier, Be a Man! For the Honour of Dear Old England I'd Like to Be a Hero, Too! March on to Berlin! It's a Grand Sight to See Them Going Away! Tommy is as Good a Man as Any Knight of Old And other, later songs showing a light hearted and enthusiastic view of the war: Cheer Up Little Soldier Man 1915 God Give Us Victory! 1915 Our Whistling Tommies 1915 We'll Never Let the Old Flag Fall 1915 The Tanks that Broke the Ranks Out in Picardy 1916

  Apart from elite voices, what other voices might we hear? Examining the repertoire, the important presence of questions of everyday working-class life and fantasy are perhaps what mark it most clearly as a working-class voice. Colin MacInnes pointed this out almost fifty years ago:

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  Kitty, the Telephone Girl 1914 -The Ladies Football Club 1915 -The Editress 1915 -Which Switch Is the Switch, Miss, for Ipswich? 1915 -If the Girlies Could Be Soldiers 1915 -The Lady Bus Conductor 1916 -Tilly the Typist 1916 -Dance of the Fire Brigade girls 1915 -Women's Work 1917 -Where Are the Girls of the Old Brigade? 1917 -Polly from the GPO 1918

Can be heard online at https://www.youtube.com/watch?v=lRWSng0bSYA Accessed November 12, 2017.

Sara Maitland, Vesta Tilley (London: Virago, 1986), p. 58.