Alain De Cheveigné
email: alain.de.cheveigne@ens.fr

Why is the perceptual octave stretched? An account based on mismatched time constants within the auditory brainstem

This paper suggests an explanation for listeners' greater tolerance to positive than negative mistuning of the higher tone within an octave pair. It hypothesizes a neural circuit tuned to cancel the lower tone, that also cancels the higher tone if that tone is in tune. Imperfect cancellation is the cue to mistuning of the octave. The circuit involves two neural pathways, one delayed with respect to the other, that feed a coincidence-sensitive neuron via excitatory and inhibitory synapses. A mismatch between the time constants of these two synapses results in an asymmetry in sensitivity to mismatch. Specifically, if the time constant of the delayed pathway is greater than that of the direct pathway, there is a greater tolerance to positive mistuning than to negative mistuning. The model is directly applicable to the harmonic octave (concurrent tones), but extending it to the melodic octave (successive tones) requires additional assumptions that are discussed. The paper reviews evidence from auditory psychophysics and physiology in favor -or against -this explanation.

I. INTRODUCTION

It sometimes happens that competing theories differ on subtle effects that allow us to decide between them. A famous example is the precession of the perihelion of Mercury, a tiny effect (43 arc-seconds per century) that provided support for General Relativity over Newtonian physics. The asymmetry of mistuning for the octave interval [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF][START_REF] Demany | On the perceptual limits of octave harmony and their origin[END_REF] might play a similar role in hearing.

The octave (frequency ratio 2:1) is one of a series of small integer ratios that appear naturally between partials of a harmonic complex sound, or between the subharmonics of any periodic tone, complex or pure. Tones with frequencies in this ratio display a perceptual affinity when they are played together (harmonic octave) or in succession (melodic octave).

The octave plays a special role in music of many cultures and genres, and there is some evidence that melodic octave similarity (if not "equivalence") is innate [START_REF] Demany | The perceptual reality of tone chroma in early infancy[END_REF]. Listeners, particularly musicians, are sensitive to small mistunings from the exact octave. "Asymmetry" refers to the fact that a compressed octave (ratio smaller than 2) is better detected than a stretched octave (ratio larger than 2).

Explanations of octave affinity have been imagined for both spectral and temporal theories of auditory processing. For example [START_REF] Terhardt | Pitch, consonance, and harmony[END_REF] proposed that listeners acquire harmonic templates from exposure to harmonic sounds, whereas [START_REF] Ohgushi | The origin of tonality and a possible explanation of the octave enlargement phenomenon[END_REF] remarked that intervals between auditory nerve spikes (action potentials) evoked by the lower tone are also found between spikes evoked by the octave. Both ideas can be developed into models to account for harmonic and melodic octave affinity [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF].

Why is the perceptual octave stretched?

To the extent that those explanations involve integer ratios, it seems odd that the octave might be perceptually stretched. Deciding if the pitches of two consecutive pure tones form an octave requires enough musical skill to at least understand instructions, and a common outcome is that the frequency of the upper tone (f H ) is matched higher than twice the lower (2f L ) by a few percent [START_REF] Ward | Subjective Musical Pitch[END_REF]. This is known as melodic "octave stretching" or "enlargement". Mistuning from an octave relationship between two concurrent pure tones can likewise be detected. Here, it is observed that tolerance is greater for positive than negative mistuning [START_REF] Demany | On the perceptual limits of octave harmony and their origin[END_REF]. It is natural to search for a common explanation for melodic stretch and harmonic mistuning asymmetry, but experimental support has been elusive, some studies favoring distinct mechanisms [START_REF] Bonnard | Harmonic fusion and pitch affinity: Is there a direct link?[END_REF][START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF], others a common mechanism [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF].

The aim of this paper is to explore an explanation of mistuning asymmetry according to which the asymmetry arises from a mismatch in time constants of neural signal processing networks within the auditory brainstem. This explanation applies to the harmonic octave, but extending it to the melodic octave requires additional assumptions that are considered.

The psychophysics of octave tuning are further reviewed in the Discussion, together with relevant evidence from auditory physiology.

II. METHODS AND RESULTS

This section introduces the required concepts and tools, and describes the main result. a. Stimuli. In contrast to the harmonics-rich sounds of many musical instruments, most studies employ pure tones, so as to avoid the confound of beats between partials. A Why is the perceptual octave stretched? pure tone of frequency f can, however, be treated as a harmonic complex with only one harmonic, with "fundamental frequency" f and period T = 1/f . b. Cochlear frequency analysis. As a first step of auditory processing, stimuli undergo filtering in the cochlea, modeled here as a gammatone filter bank with bandwidths from [START_REF] Moore | Suggested formulae for calculating auditory-filter bandwidths and excitation patterns[END_REF]. Those bandwiths are roughly 1/3 octave (except in the low frequency region where filters are wider), so pure tones spaced one octave apart are well resolved, as illustrated in Fig. 1. Channels with characteristic frequencies (CFs) close to a tone's frequency are dominated by that tone by up to 50 dB or more, whereas some intermediate channels respond to both. an octave (black) and for each tone alone (blue and red, offset vertically for clarity). (b) Power of higher tone divided by power of lower tone (blue: lower dominates, red: upper dominates). These patterns were derived from a gammatone filterbank implementation [START_REF] Slaney | An efficient implementation of the patterson-holdsworth auditory filter bank technical report 35[END_REF] with bandwidth parameters from [START_REF] Moore | Suggested formulae for calculating auditory-filter bandwidths and excitation patterns[END_REF].

Figure 1 gives us an idea of the information available for the auditory brain to detect the octave relationship between two pure tones. The profile of excitation across filter CF (panel a) shows a clear peak at the frequencies of both tones, which could thus be extracted from a tonotopic pattern. Multiple channels are dominated by one tone or the other (panel b),

Why is the perceptual octave stretched? so information about those frequencies is also available from the temporal fine structure of auditory nerve discharges. Finally, a smaller set of intermediate channels responds to both tones (circa 700 Hz) possibly offering a third temporal cue in the form of their interaction.

It is worth noting that these patterns depend largely on the shape of the skirts of cochlear filter transfer functions, whereas most filter models are optimized for their shape near the peak [START_REF] Patterson | Auditory filter shapes derived with noise stimuli[END_REF].

c. Period estimation and pitch. A popular model of pitch perception suggests that pitch is derived from the temporal fine structure of auditory nerve discharges, for example according to an autocorrelation-like mechanism [START_REF] Cariani | Neural correlates of the pitch of complex tones. i. pitch and pitch salience[END_REF][START_REF] Licklider | A duplex theory of pitch perception[END_REF][START_REF] Meddis | Virtual pitch and phase sensitivity of a computer model of the auditory periphery. I: Pitch identification[END_REF]. Using a sampled-signal notation, the autocorrelation function (ACF) can be defined as:

r t (τ ) = t+W θ=t+1 s(θ)s(θ + τ ) (1)
where s(t) is the acoustic signal (or some transform of it), W is an integration window, t is the time at which the function is calculated, and τ is the lag parameter (Fig. 2, left). Applied to a periodic stimulus, such as a pure tone, the function shows a peak at all multiples of T , where T = 1/f 0 is the period. An alternative to the ACF is the the "Squared Difference

Function" (SDF):

d t (τ ) = t+W θ=t+1 [s(θ) -s(θ + τ )] 2 .
(2) calculated as the Euclidean distance between two windows, one fixed the other shifted by τ . These functions are closely related: d t (τ ) = p t (τ ) -2r t (τ ), where p t (τ ) = r t (0) + r t+τ (0)

Why is the perceptual octave stretched?

reflects signal power circa time t. Applied to a periodic stimulus, such as pure tone, the SDF takes its smallest possible value (zero) at T and all its multiples (Fig. 2, right). to take values between -1 and 1 (ACF) or 0 and 1 (SDF).

In the autocorrelation model of pitch, a neural approximation of the ACF is calculated within every channel of a peripheral filter bank [START_REF] Licklider | A duplex theory of pitch perception[END_REF]. In response to a stimulus of period T , all channels show a peak at T , and the position of this common peak is the cue to the period of a sound, and thus its pitch. The CF × lag pattern can also be summarized by averaging it over all channels to form a summary autocorrelation function (SACF), that typically also shows a peak at T [START_REF] Meddis | Virtual pitch and phase sensitivity of a computer model of the auditory periphery. I: Pitch identification[END_REF]. Alternatively, in some tasks it might be beneficial to restrict the pattern to a subset of channels. For example, looking at Fig. 1, it would seem best to estimate the period of the lower tone from channels with CF below ∼700 Hz, and the period of the higher tone from channels with CF above that limit.

Replacing the ACF by the SDF yields a pitch model with properties very similar to the autocorrelation model, the cue to the period being the position of a dip (rather than a peak) in the CF × lag pattern (Fig. 2) [START_REF] De Cheveigné | Cancellation model of pitch perception[END_REF]. A reason for considering this variant is that it can easily be extended to detect mistuning.

d. Cancellation and mistuning. The SDF can be understood as quantifying variance of the output of a time-domain "harmonic cancellation" filter

h(τ ) = δ 0 (t) -δ τ (t).
(3) calculated over a window W , as a function of the delay parameter τ , where δ τ is the Kronecker delta function translated to τ . Figure 3 (left) shows such a filter with lag parameter T , together with its transfer function (Fig. 3 right). The transfer function of this filter shows deep dips at 1/T and its multiples, the output being zero for any periodic input signal with period T /k. Mistuning of a signal of frequency f away from k/T can be detected as an increase in power at the output of the filter.

More generally, harmonic cancellation is a hypothetical process according to which the auditory system exploits harmonicity of a background sound to suppress it (de Cheveigné, 2021), for example using a harmonic cancellation filter (Eqn. 3). This process might explain why mistuning a component within a harmonic complex makes it more salient [START_REF] Hartmann | Hearing a mistuned harmonic in an otherwise periodic complex tone[END_REF][START_REF] Moore | Thresholds for the detection of inharmonicity in complex tones[END_REF] and its pitch easier to match [START_REF] Hartmann | On the pitches of the components of a complex tone[END_REF].

Octave-spaced tones are a special case of this situation.

e. Octave mistuning. Harmonic octave mistuning can be detected with the following steps: (1) estimate the period of the lower tone, (2) use it as the parameter of a cancellation filter applied to the higher tone, (3) decide that the octave is mistuned if the cancellation residual exceeds a threshold, and in-tune otherwise.

Why is the perceptual octave stretched? Referring to Fig. 2 (right), it appears that low CF channels represent well the lower tone, high CF channels the higher tone, and a smaller set of intermediate channels represent both. There are at least three plausible scenarios. A first is that the period is estimated in low-CF channels and used to parametrize a harmonic cancellation filter applied to high-CF channels. A second is that both operations are applied to the set of intermediate channels (avoiding the need to carry the estimate across channels). A third is that both are applied after cross-channel convergence (see section on physiology in the Discussion).

f. Neural implementation. Processing within the auditory brain must rely on the spikebased neural representation created by transduction in the cochlea. [START_REF] Licklider | A duplex theory of pitch perception[END_REF] imagined that the ACF might be approximated by an array of elementary neural circuits, each comprising a neuron fed from direct and delayed pathways via two excitatory synapses (Fig. 4, top left). The neuron fires only if spikes arrive together (within some short window) on both pathways. For example, short excitatory post synaptic potentials (EPSPs) evoked by spikes might interact additively [START_REF] Van Der Heijden | Directional Hearing by Linear Summation of Binaural Inputs at the Medial Superior Olive[END_REF], reaching threshold only if they overlap in time (Fig. 4, top right). Instantaneous output spike probability then approximates the product of input spike probabilities, as required to approximate Eq. 1. Such

Why is the perceptual octave stretched? Bottom: a neuron receives direct and delayed input via excitatory and inhibitory synapses and fires whenever a spike arrives on the excitatory input, unless a spike arrives simultaneously on the inhibitory input. This can be used to approximate the delay-and-subtract operation of the harmonic cancellation filter, or the SDF. The delay can be applied indifferently to the inhibitory input (left) or to the excitatory input (center).

a mechanism has been hypothesized within the medial superior olive (MSO) in a circuit that estimates interaural time differences [START_REF] Lu | Activity-dependent synaptic integration and modulation of bilateral excitatory inputs in an auditory coincidence detection circuit: Synaptic integration and modulation in nucleus laminaris[END_REF][START_REF] Van Der Heijden | Directional Hearing by Linear Summation of Binaural Inputs at the Medial Superior Olive[END_REF]).

An analogous circuit can approximate the harmonic cancellation filter (de Cheveigné, 2021). A neuron is contacted by an excitatory and an inhibitory synapse, one fed from a direct pathway, the other via a delay (Fig. 4, bottom left). The EPSP and IPSP (inhibitory post synaptic potential) summate as in Fig. 4 (bottom right). The neuron's threshold is such that it fires at each excitatory spike, unless a spike arrives together at the inhibitory synapse within some short window. Instantaneous output spike probability then reflects the difference of input spike probabilities, as required to approximate Eq. 2. Such a gating

Why is the perceptual octave stretched? mechanism has been found within the lateral superior olive (LSO) in a circuit that estimates interaural differences in level or onset time [START_REF] Franken | Principal cells of the brainstem's interaural sound level detector are temporal differentiators rather than integrators[END_REF]. h. Asymmetry. We are now in position to suggest an explanation for the mistuning asymmetry. First, let us suppose that PSPs on direct and delayed pathways have a similar duration: the mistuning curve is roughly symmetrical around 1 kHz (Fig. 5 top right), with slopes that depend on PSP duration, being shallower for longer PSPs. If instead the PSPs differ in duration, the mistuning curve is asymmetrical relative to 1 kHz. In particular, if the PSP duration of the delayed pathway is longer than that of the direct pathway, there is a greater tolerance for a positive mistuning (stretched octave) than negative. A threshold value of filter output (green dots) is reached for a smaller value of negative than positive mistuning, qualitatively consistent with most behavioral results. Interestingly, the position of the minimum does not shift, just the slopes.

Why is the perceptual octave stretched? The origin of the asymmetry can be understood intuitively by looking at Fig. 5 (left, bottom). Increasing the higher-tone period by 100 µs (compressed octave) would shift the onset of the delayed IPSP beyond the offset of the direct EPSP, abolishing the inhibitory effect. In constrast, decreasing it by the same amount (stretched octave) would only partially reduce the overlap. If, instead, the delayed path had the shorter PSP, the asymmetry would be reversed (dotted line in Fig. 5 bottom right). The asymmetry is also contingent on the asymmetric shape of the PSPs (steeper onset than offset).

Why is the perceptual octave stretched?

The demonstration that asymmetry may arise within a plausible mechanism of octave mistuning detection is the main result of this paper. However, the predicted direction depends on time constants for delayed and undelayed pathways. To match experimental results, the delayed pathway must have a longer time constant than the direct pathway, if not, the model fails. The model is applicable directly to the harmonic octave (concurrent sounds); the degree to which it accounts also for melodic octave affinity is examined in the Discussion.

i. Simulation. The model was simulated and its predictions compared with behavioral data from [START_REF] Demany | On the perceptual limits of octave harmony and their origin[END_REF]. Stimuli in that study consisted of the sum of two octavespaced pure tones of duration 0.5 s with equal amplitudes and lower tone frequency 300 Hz, 500 Hz, 1000 Hz or 2000 Hz. Here, the stimulus is fed to a gammatone filter bank [START_REF] Slaney | An efficient implementation of the patterson-holdsworth auditory filter bank technical report 35[END_REF] with bandwidth parameters from [START_REF] Moore | Suggested formulae for calculating auditory-filter bandwidths and excitation patterns[END_REF]. A single channel was selected, with CF equal to the frequency of the higher tone, and the half-wave rectified channel output scaled to drive an inhomogenous Poisson process with a 1 ms dead time (refractory effect). That was implemented as a homogenous Poisson process followed by a thinning process to enforce a time-dependent instantaneous rate and an interval-dependent refractory function [START_REF] De Cheveigné | A nerve fiber discharge model for the study of pitch[END_REF][START_REF] Lewis | Simulation of nonhomogeneous Poisson processes by thinning[END_REF]. The scaling factor was adjusted to ensure a physiologically-realistic average discharge rate R. To simulate the group of N auditory fibers that innervate an inner hair cell (or a group of inner hair cells with similar CF), N such processes were run in parallel and their spike times merged. This assumes that subsequent neural stages have access to such a compound spike train (ignoring for simplicity the distinction between high and low spontaneous rate fibers). Optionally, a Gaussian-Why is the perceptual octave stretched?

distributed jitter with standard deviation σ can be added to each spike time to approximate synchrony roll-off or other sources of temporal imprecision.

Spike times (produced with arbitrary temporal precision) were converted to a pulse train (100 kHz sampling rate) that was split into a "direct pathwhay" convolved with a decaying exponential kernel modeling an EPSP, and a "delayed pathway" (delay T ) convolved by a second kernel modeling an IPSP, with time constants τ e and τ i respectively. The delay T was set to the period of the lower tone. Direct and delayed signals were added (with negative sign for the IPSP-convolved signal), and the sum was half-wave rectified and averaged over the duration of the stimulus, resulting in a single number representing the "activation" of the gating neuron. To simulate the decision made by a subject on every trial (160 trials per condition), the activation for the mistuned pair was compared with that for the in-tune pair, and the decision deemed correct if the former was greater than the latter. This simulation was repeated 1000 times for each condition.

The results presented here are for one channel (haircell), R=134 spikes/s, N =10 fibers per hair cell, σ=0 (no added jitter), τ e =100 µs, τ i =150 µs. Several points are worth noting. First, the model reproduces listeners' sensitivity to mis-217 tuning: detection is near chance for small mistunings and near-perfect for large mistunings,

218

with slopes roughly similar to the behavioral data, shallower at high than low frequencies.

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Second, sensitivity is greater for negative than positive mistunings, with a hint of bias at 220 small positive mistunings (scores reliably below 50 %). Third, asymmetry is seen at all 221 frequencies, as observed for some (albeit not all) subjects.

222

Why is the perceptual octave stretched?

Between-subject differences, seen also in other studies (see Discussion), cannot be predicted with a single set of parameters (error bars are too small), but can be reproduced if parameters are allowed to vary between subjects. The asymmetry can be abolished or reversed (as observed for some subjects/conditions) by making the delayed pathway time constant equal to, or shorter than, that of the direct pathway (not shown). Increasing both time constants reduces asymmetry at high frequencies (as observed for subjects LD and CS), decreasing them reduces asymmetry at low frequencies (not shown). A non-zero value of σ also abolishes mistuning sensitivity at higher frequencies (not shown). Increasing N (e.g. fibers from several inner hair cells pooled) increases mistuning sensitivity, reducing N (as might occur due to deafferentation) reduces mistuning sensitivity (not shown). The simulation is based on a single peripheral channel, drawing from a wider range (c.f. Fig. 1) would further increase mistuning sensitivity (not shown), and so-on.

This flexibility is fortunate in that it allows a wide range of observed patterns to be accounted for. It is unfortunate in that the model is difficultly falsifiable, which is objectionable from a Popperian perspective. Better knowledge about the physiology (see Discussion) might allow tighter predictions, for now the values plotted in Fig. 6 carry limited predictive weight.

III. DISCUSSION

Two things need explaining. The first is octave affinity, for both harmonic and melodic octaves. The second is asymmetric sensitivity to mistuning. This paper attempts to explain both, but the explanation raises some issues that are discussed here. Anticipating, evidence

Why is the perceptual octave stretched?

from physiology leads us to expect an asymmetry, but only weakly predicts its sign. The explanation is directly applicable to the harmonic octave, but accounting for the melodic octave requires additional assumptions.

a. Affinity or Equivalence?. The octave has been recognized as remarkable since antiquity, and it plays an important role in the music of most cultures [START_REF] Burns | Intervals, scales, and tuning In Deutsch D[END_REF][START_REF] Shepard | Geometrical approximations to the structure of musical pitch[END_REF]. Tones separated by an octave are treated as equivalent for the definition of a chroma class [START_REF] Bachem | Tone height and tone chroma as two different pitch qualities[END_REF][START_REF] Burns | Intervals, scales, and tuning In Deutsch D[END_REF][START_REF] Deutsch | Music recognition[END_REF][START_REF] Harrison | Simultaneous consonance in music perception and composition[END_REF]. For two complex tones separated by an octave, all the harmonics of the higher tone (including its fundamental) are also harmonics of the lower tone. Likewise, subharmonics of the lower tone are subharmonics of the higher tone. The lower tone thus "contains" the octave tone, but the converse is not true: octave similarity thus lacks the symmetry property that would be required for "equivalence" in a mathematical sense.

b. Harmonic vs Melodic Octave. Affinity has been observed for both concurrent sounds (harmonic octave) and sequential sounds (melodic octave). For pure tones, the melodic octave is somewhat elusive as discussed by [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF]. While some studies suggest perceptual similarity or even equivalence [START_REF] Demany | The perceptual reality of tone chroma in early infancy[END_REF], others found little evidence that melodic octave intervals are special [START_REF] Kallman | Octave equivalence as measured by similarity ratings[END_REF]). An interpretation is that two perceptual qualities are involved, one salient for all listeners (difference in tone height)

and the other more subtle and requiring musical literacy (chroma) [START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF].

Tasks that probe the latter require ignoring the former.

Parsimony lobbies for a common basis for the harmonic and melodic octave, but experimental support has been ambiguous. Of three careful studies by the same group, two

Why is the perceptual octave stretched?

concluded that there are distinct mechanisms [START_REF] Bonnard | Harmonic fusion and pitch affinity: Is there a direct link?[END_REF][START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF], and the third that there is a common mechanism [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF], see also Demany andSemal (1988, 1990); [START_REF] Demany | On the perceptual limits of octave harmony and their origin[END_REF]. For a harmonic octave, mistuning is harder to detect if the higher tone exceeds ∼2 kHz [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF], which is possibly related to the finding of [START_REF] Hartmann | Hearing a mistuned harmonic in an otherwise periodic complex tone[END_REF] that subjects cannot detect mistuning of a partial of a harmonic complex beyond ∼2 kHz. For a melodic octave, mistuning is detectable up to ∼4 kHz or higher [START_REF] Ward | Subjective Musical Pitch[END_REF], however, mistuning is easier to detect for harmonic than melodic octaves in the low frequency region [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF].

c. Detecting mistuning. For a harmonic octave, detection is straightforward. Any mechanism that can detect inharmonicity (in the frequency domain) or aperiodicity (in the time domain) can detect octave mistuning. The model presented in this paper uses a cancellation filter for that purpose. For the melodic octave, the situation is more subtle because memory is involved: a trace of the first tone must be retained for comparison with the second tone. The trace of the first tone (which might be the lower or the higher) can be compared to a similar trace derived from the second tone, or instead used as a parameter of a mistuning detector applied to the second tone (as in this paper). [START_REF] Terhardt | Pitch, consonance, and harmony[END_REF] assumed that every harmonic of a complex tone evokes a "spectral pitch"

which is transformed into a series of "virtual pitch" cues via a harmonic template. Cues from all harmonics are then tallied in a weighted histogram (see also [START_REF] Schroeder | Period histogram and product spectrum: new methods for fundamentalfrequency measurement[END_REF]) that shows a series of peaks, the largest of which signals the pitch. However, there are other peaks, and Terhardt noted that the largest among them tend to be separated by an octave interval. The trace, in this account, is the subharmonic histogram of the first tone, compared

Why is the perceptual octave stretched?

to the subharmonic histogram of the second tone to detect an octave relation (or mistuning thereof).

This explanation relies on the regular pattern of harmonics (overtones) of most musical sounds. Their perceptual reality was noted by Oresme in the XIVth century [START_REF] Bod | A new history of the humanities: the search for principles and patterns from antiquity to the present[END_REF],

and their physical reality worked out by a line of thinkers from Mersenne in the XVIIth up to Fourier in the XIXth century [START_REF] De Cheveigné | Pitch perception models In[END_REF]. Interestingly, the mathematician Fermat, writing to Mersenne in 1636, raised the possibility that the "true" octave might differ from a ratio 2:1 (https://www.musicologie.org/Biographies/m/mersenne.html).

The implication of harmonics in octave affinity is attributed to Sauveur and Rameau in the 18th century [START_REF] Christensen | Eighteenth-Century Science and the "Corps Sonore:" The Scientific Background to Rameau's "Principle of Harmony[END_REF], and learning of a template from lifelong exposure to harmonic sounds, notably speech [START_REF] Terhardt | Pitch, consonance, and harmony[END_REF], to Jean Jacques Dortous de Mairan, contemporary of Rameau [START_REF] Christensen | Eighteenth-Century Science and the "Corps Sonore:" The Scientific Background to Rameau's "Principle of Harmony[END_REF].

Instead of harmonics, one can invoke subharmonics (super-periods). [START_REF] Ohgushi | The origin of tonality and a possible explanation of the octave enlargement phenomenon[END_REF] invoked for this purpose the first-order interspike interval (ISI) histogram of auditory nerve fiber discharge patterns, that typically shows a series of modes near period multiples. The ISI histogram for the first tone is compared to a similar histogram from the second tone.

At the octave, some peaks of the histogram for the higher tone coincide with peaks for the lower tone, hence affinity. A similar idea has been proposed for other auditory nerve fiber interval statistics such as autocorrelation [START_REF] Cariani | Neural correlates of the pitch of complex tones. i. pitch and pitch salience[END_REF][START_REF] Tramo | Neurophysiology and neuroanatomy of pitch perception: Auditory cortex[END_REF]. The principle has much earlier roots in the "coincidence theory of consonance" elaborated in the 16th and 17th centuries [START_REF] Christensen | Rameau and musical thought in the enlightenment[END_REF][START_REF] Green | The harmonic series from Mersenne to Rameau: An historical study of circumstances leading to its recognition and application to music Ph[END_REF], and indeed in the association of consonant intervals with small integer ratios usually attributed to Pythagoras.

Why is the perceptual octave stretched?

The model proposed in this paper is of the "subharmonic" flavor. The trace, in this case, is the delay parameter T of the first filter, conserved and used to tune a new filter applied to the second tone (or the filter itself, "frozen" until the second tone). This sequential operation is straightforward in the case of an ascending octave, because the filter fit to the lower tone also cancels the higher, but for a descending octave the filter must be fit to twice the period of the first tone. This is less straightforward, but perhaps a skill that a musicallyinclined subject might have learned to master. Incidentally, data specific to the descending melodic octave are somewhat sparse. [START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF]; [START_REF] Cuddy | Octave Discrimination: Temporal and Contextual Effects[END_REF] presented notes only in ascending order, while [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF]; [START_REF] Demany | Harmonic and melodic octave templates[END_REF]; [START_REF] Ward | Subjective Musical Pitch[END_REF] presented notes in alternation, so ascending and descending intervals contributed to each judgement. However, [START_REF] Bonnard | Harmonic fusion and pitch affinity: Is there a direct link?[END_REF] presented half their subjects with ascending intervals and the other half with descending intervals, and found no reliable difference, consistent with reports of [START_REF] Hartmann | On the origin of the enlarged melodic octave[END_REF] and [START_REF] Rosner | Stretching and Compression in the Perception of Musical Intervals[END_REF]. [START_REF] Luo | Melodic interval perception by normal-hearing listeners and cochlear implant users[END_REF] observed that musical intervals are generally less accurately ranked when descending than ascending (but they did not test octaves), and [START_REF] Vos | Ascending and Descending Melodic Intervals: Statistical Findings and Their Perceptual Relevance[END_REF] analyzed two music databases and found that descending octaves were half as common as ascending.

d. Octave stretch and mistuning asymmetry.. Octave stretch (preference for a frequency ratio larger than 2) and mistuning asymmetry (greater tolerance for positive than negative mistuning) are distinct phenomena, but that could conceivably be related: in the presence of jitter or noise, stretching the octave might be a "safe" strategy to reduce the likelihood of a perceptually noticeable mistuning. If such is the case, a model of mistuning

Why is the perceptual octave stretched?

asymmetry could account for both phenomena, and data related to both would be relevant to constrain that model.

Mistunings are easier to detect for positive than negative deviations for both the harmonic octave [START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF][START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF][START_REF] Demany | Dichotic fusion of two tones one octave apart: Evidence for internal octave templates[END_REF][START_REF] Demany | On the perceptual limits of octave harmony and their origin[END_REF] and the melodic octave [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF][START_REF] Demany | Harmonic and melodic octave templates[END_REF][START_REF] Dobbins | Octave discrimination: An experimental confirmation of the "stretched" subjective octave[END_REF]. It is however unclear that this is specific to the octave, as a similar stretch has been reported for intervals of minor sixth or greater, whereas smaller intervals are compressed [START_REF] Burns | Intervals, scales, and tuning In Deutsch D[END_REF][START_REF] Rakowski | Intonation Variants of Musical Intervals in Isolation and in Musical Contexts[END_REF]. The melodic octave is stretched for pure tones [START_REF] Hartmann | On the origin of the enlarged melodic octave[END_REF][START_REF] Ohgushi | The origin of tonality and a possible explanation of the octave enlargement phenomenon[END_REF] as well as complex tones and sounds of musical instruments [START_REF] Jaatinen | Octave stretching phenomenon with complex tones of orchestral instruments[END_REF][START_REF] Sundberg | Musical octaves and pitch[END_REF].

The melodic stretch is typically a few percent, so a subject must be sensitive to pitch differences of that order, as is the case for most musicians but not all non-musicians [START_REF] Micheyl | Influence of musical and psychoacoustical training on pitch discrimination[END_REF]. For at least some subjects, the standard deviation of octave matches was indeed smaller than the magnitude of the stretch [START_REF] Demany | Harmonic and melodic octave templates[END_REF]. The magnitude of the stretch varies markedly across subjects [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF][START_REF] Demany | Harmonic and melodic octave templates[END_REF][START_REF] Hartmann | On the origin of the enlarged melodic octave[END_REF][START_REF] Ward | Subjective Musical Pitch[END_REF], and for some subjects, in some conditions, it is close to zero or even reversed [START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF][START_REF] Hartmann | On the origin of the enlarged melodic octave[END_REF][START_REF] Rosner | Stretching and Compression in the Perception of Musical Intervals[END_REF]. This is an extra burden for a model as it must explain why the phenomenon is observed in some subjects and not others. Large inter-subject differences were also observed for pitch discrimination [START_REF] Micheyl | Influence of musical and psychoacoustical training on pitch discrimination[END_REF], and sensitivity to consonance [START_REF] Mcdermott | Individual Differences Reveal the Basis of Consonance[END_REF].

The stretch is typically larger at higher frequencies than low, at least for the melodic octave [START_REF] Demany | Harmonic and melodic octave templates[END_REF][START_REF] Jaatinen | Octave stretching phenomenon with complex tones of orchestral instruments[END_REF][START_REF] Ohgushi | The origin of tonality and a possible explanation of the octave enlargement phenomenon[END_REF]. For a harmonic

Why is the perceptual octave stretched?

octave, as mentioned earlier, the ability to notice mistuning itself is reduced in the upper range. [START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF] found that sensitivity to mistuning of either direction improved with level, which they attributed to the increased salience of beats, however, the asymmetry vanished at higher levels. For the melodic octave, [START_REF] Sundberg | Musical octaves and pitch[END_REF] found a complex dependency of asymmetry on subject, frequency, tone nature (pure or complex), and intensity of both tones. [START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF] measured discrimination between various frequency ratios in the vicinity of 2:1 for concurrent pure tone pairs. Discrimination from a reference ratio of 1.96:1 was unexpectedly better for a test ratio of 2:1 than for a larger test ratio. From this, they concluded that the true octave ratio (2:1) is "special". Apparently, the subjective versus objective mistuning function is "V"-shaped with a minimum at 2:1, but with a steeper slope on the lower (<1 octave) than higher (>1 octave) side, consistent with Fig. 5 (bottom right). [START_REF] Bonnard | Harmonic fusion and pitch affinity: Is there a direct link?[END_REF] also found that perceptual fusion was maximal for a ratio close to 2:1, decreasing more rapidly for downward than upward mistunings, which again suggests an asymmetric tolerance for mistuning from the optimal ratio 2:1, rather an optimal ratio larger than 2:1. The asymmetry was not observed for mildly hearing-impaired subjects [START_REF] Bonnard | The Effect of Cochlear Damage on the Sensitivity to Harmonicity Ear & Hearing[END_REF].

It has also been suggested that the asymmetry might reflect an aesthetic bias (e.g. [START_REF] Demany | The perception of octave pitch affinity and harmonic fusion have a common origin[END_REF], which begs the question of what drives the aesthetics. A possibility is that mistuning is unpleasant, and the bias results from upwards mistunings being less noticeable, but the question then is why mistuning is unpleasant. A speculative explanation is that harmonicity-based segregation mechanisms help maintain the clarity of a complex musical Why is the perceptual octave stretched?

scene, pleasurable according to the concept of perceptual fluency [START_REF] Reber | Effects of Perceptual Fluency on Affective Judgments[END_REF]. If so,the aesthetics themselves would be an emergent property.

e. Physiological Basis. Excitatory-inhibitory (EI) interaction is found throughout the nervous system. High-bandwidth auditory representations are supported by temporallyspecialized neural circuits, some excitatory and others inhibitory [START_REF] Joris | Early Binaural Hearing: The Comparison of Temporal Differences at the Two Ears[END_REF][START_REF] Stange-Marten | Input timing for spatial processing is precisely tuned via constant synaptic delays and myelination patterns in the auditory brainstem[END_REF]. Fast EI interaction has been identified as early as the cochlear nucleus [START_REF] Campagnola | A Map of Functional Synaptic Connectivity in the Mouse Anteroventral Cochlear Nucleus[END_REF][START_REF] Joris | Temporal and Binaural Properties in Dorsal Cochlear Nucleus and Its Output Tract[END_REF][START_REF] Kuenzel | Modulatory influences on time-coding neurons in the ventral cochlear nucleus[END_REF][START_REF] Ngodup | Identification of an inhibitory neuron subtype, the l-stellate cell of the cochlear nucleus[END_REF][START_REF] Paolini | Fast Inhibition Alters First Spike Timing in Auditory Brainstem Neurons[END_REF][START_REF] Xie | Target-Specific IPSC Kinetics Promote Temporal Processing in Auditory Parallel Pathways[END_REF] and as late as the the dendritic fields of the inferior colliculus [START_REF] Caspari | Heterogeneity of Intrinsic and Synaptic Properties of Neurons in the Ventral and Dorsal Parts of the Ventral Nucleus of the Lateral Lemniscus[END_REF][START_REF] Chen | A temporal integration mechanism enhances frequency selectivity of broadband inputs to inferior colliculus[END_REF], in addition to intermediate stages within the superior olivary complex [START_REF] Franken | Principal cells of the brainstem's interaural sound level detector are temporal differentiators rather than integrators[END_REF][START_REF] Grothe | New roles for synaptic inhibition in sound localization[END_REF][START_REF] Myoga | Glycinergic inhibition tunes coincidence detection in the auditory brainstem[END_REF][START_REF] Roberts | A Mechanistic Understanding of the Role of Feedforward Inhibition in the Mammalian Sound Localization Circuitry[END_REF] and nuclei of lateral lemniscus [START_REF] Caspari | Heterogeneity of Intrinsic and Synaptic Properties of Neurons in the Ventral and Dorsal Parts of the Ventral Nucleus of the Lateral Lemniscus[END_REF].

In some instances, EI interaction has been attributed to binaural processing, for example estimation of interaural level differences (ILD) [START_REF] Tollin | Interaural Phase and Level Difference Sensitivity in Low-Frequency Neurons in the Lateral Superior Olive[END_REF], detection of onset timing differences [START_REF] Franken | Principal cells of the brainstem's interaural sound level detector are temporal differentiators rather than integrators[END_REF], or sharpening of sensitivity to interaural time difference (ITD) [START_REF] Myoga | Glycinergic inhibition tunes coincidence detection in the auditory brainstem[END_REF]. In others, the processing seems to be monaural.

For example, neurons in the ventral nucleus of the lateral lemniscus (VNLL) receive, via calyx-like synapses, excitatory input from contralateral cochlear nucleus (CN) and ipsilateral medial nucleus of the trapezoid body (MNTB) [START_REF] Caspari | Heterogeneity of Intrinsic and Synaptic Properties of Neurons in the Ventral and Dorsal Parts of the Ventral Nucleus of the Lateral Lemniscus[END_REF], neurons in superior periolivary nucleus (SPON) and VNLL receive input from contralateral octopus cells and ipsilateral MNTB [START_REF] Felix | Octopus Cells in the Posteroventral Cochlear Nucleus Provide the Main Excitatory Input to the Superior Paraolivary Nucleus[END_REF]. Cells in MSO also receive EI input and a certain number of them are monaural [START_REF] Grothe | The function of the medial superior olive in small mammals: temporal receptive fields in auditory analysis[END_REF].

Why is the perceptual octave stretched?

Inhibition requires an extra synapse and thus might be expected to lag excitation. This has indeed been observed [START_REF] Xie | Target-Specific IPSC Kinetics Promote Temporal Processing in Auditory Parallel Pathways[END_REF], but the opposite has also been found [START_REF] Nayagam | Powerful, Onset Inhibition in the Ventral Nucleus of the Lateral Lemniscus[END_REF][START_REF] Paolini | Fast Inhibition Alters First Spike Timing in Auditory Brainstem Neurons[END_REF][START_REF] Roberts | A Mechanistic Understanding of the Role of Feedforward Inhibition in the Mammalian Sound Localization Circuitry[END_REF], presumably reflecting properties of the inhibitory pathways that favor transmission speed, such as thick myelinated axons or secure synapses [START_REF] Grothe | New roles for synaptic inhibition in sound localization[END_REF][START_REF] Stange-Marten | Input timing for spatial processing is precisely tuned via constant synaptic delays and myelination patterns in the auditory brainstem[END_REF].

The dynamics of interaction are usually thought to be governed by the time constants of excitatory and inhibitory post synaptic events, for example via a subtractive mechanism analogous to Fig. 4 (right) [START_REF] Roberts | A Mechanistic Understanding of the Role of Feedforward Inhibition in the Mammalian Sound Localization Circuitry[END_REF][START_REF] Van Der Heijden | Directional Hearing by Linear Summation of Binaural Inputs at the Medial Superior Olive[END_REF]. Inhibitory events (IPSPs or IPSCs, inhibitory post-synaptic currents) are often reported to be more sluggish than excitatory (EPSPs or EPSCs, excitatory post-synaptic currents) [START_REF] Campagnola | A Map of Functional Synaptic Connectivity in the Mouse Anteroventral Cochlear Nucleus[END_REF][START_REF] Caspari | Heterogeneity of Intrinsic and Synaptic Properties of Neurons in the Ventral and Dorsal Parts of the Ventral Nucleus of the Lateral Lemniscus[END_REF]. Examples of decay time constants reported for inhibition and excitation, respectively, are ∼10 ms vs ∼ 0.5 ms in mouse AVCN bushy cells [START_REF] Campagnola | A Map of Functional Synaptic Connectivity in the Mouse Anteroventral Cochlear Nucleus[END_REF], ∼2.5 ms vs ∼0.3 ms in mouse ventrolateral VNLL [START_REF] Caspari | Heterogeneity of Intrinsic and Synaptic Properties of Neurons in the Ventral and Dorsal Parts of the Ventral Nucleus of the Lateral Lemniscus[END_REF], or ∼1.5 ms vs 0.3 ms in mouse MSO [START_REF] Myoga | Glycinergic inhibition tunes coincidence detection in the auditory brainstem[END_REF], etc. In contrast, [START_REF] Pilati | Acoustic trauma slows AMPA receptor-mediated EPSCs in the auditory brainstem, reducing GluA4 subunit expression as a mechanism to rescue binaural function: Acoustic trauma slows EPSCs in the LSO[END_REF] reported closely matched time constants of EPSCs and IPSCs in mouse LSO.

However, the dynamics of EI interaction might not be directly governed by the duration of post-synaptic current or potential events: [START_REF] Franken | Glycinergic axonal inhibition subserves acute spatial sensitivity to sudden increases in sound intensity[END_REF] observed in LSO a window of binaural interaction much shorter (∼0.5 ms or less) than the IPSP decay time constant (∼5 ms), and with remarkably steep slopes (transition from maximum to suppressed firing on the order of 0.1 ms). Thus, the relatively short time constants (100 and 150 µs) chosen for the simulation (section Methods and Results) are not entirely implausible.

Why is the perceptual octave stretched?

In summary, there is ample evidence that dynamics of excitation and inhibition might differ between neural pathways, some delayed relative to others, as required by the account proposed in this paper. However, the evidence is sufficiently complex and diverse to support either direction of octave shift. Slower dynamics for the delayed pathway should lead to a positive shift (consistent with behaviour), whereas faster dynamics would lead to the opposite shift. Better knowledge about the locus of interaction and its properties is required for tighter predictions.

f. Other Explanations.. A simple explanation is a response or classification bias due to the structure of the response or category set. If responses are limited to chroma within the octave, the listener might maintain a sharp boundary between the major seventh and the octave, but not necessarily between the octave and the minor ninth. A similar hypothesis could explain why the perceptual boundary between minor and major second is low [START_REF] Rakowski | Intonation Variants of Musical Intervals in Isolation and in Musical Contexts[END_REF][START_REF] Vurma | Production and Perception of Musical Intervals[END_REF], and more generally the tendency for large intervals (not just the octave) to stretch and small intervals to shrink. This explanation seems less likely in a non-musical psychoacoustic setting, such as discriminating mistuned from in-tune intervals [START_REF] Bonnard | Auditory discrimination of frequency ratios: The octave singularity[END_REF].

Another proposal, also rooted in music, relates octave enlargement to the "Pythagorean comma", which is the frequency ratio between twelve fifths (1.5 12 ) and seven octaves, (2 7 ), which amounts to roughly 3.4 cents per octave [START_REF] Hubbard | The Pythagorean comma and preference for a stretched octave[END_REF]. This is in the ballpark of some perceptual shifts, but it is not clear why they would not then be equally distributed along the seven octaves, rather than the stretch being small (even negative) at low frequencies and large (up to 50 cents) at high frequencies. Nor is it clear how shifts could be subject-Why is the perceptual octave stretched?

dependent, or level-dependent. A different, physics-based explanation is that the scale of notes on the piano is stretched to reduce the perceptual effects of beats that result from the inharmonic spectrum of strings [START_REF] Giordano | Explaining the Railsback stretch in terms of the inharmonicity of piano tones and sensory dissonance[END_REF]. The piano is often used as a reference to tune other instruments, and it is conceivable that this led musicians to expect octaves to be mistuned. The so-called "Railback" tuning curve is steep at the low-and high-frequency ends, consistent with a larger octave stretch in the high register but not with the small or negative octave stretch observed in the low register.

As mentioned earlier, [START_REF] Terhardt | Pitch, consonance, and harmony[END_REF] proposed that mutual masking between partials shifts their "spectral pitch", resulting in a shift of the "virtual pitch" cues that they evoke via a harmonic template, and a mismatch when cues for the lower and higher tone are lined up. The same effects could also affect learning of that template. The explanation is weakened by the failure of [START_REF] Peters | Pitch of components of complex tones[END_REF] to replicate the shifts upon which this model is predicated.

Also mentioned earlier, [START_REF] Ohgushi | The origin of tonality and a possible explanation of the octave enlargement phenomenon[END_REF] explained octave shifts on the basis of the firstorder interspike interval (ISI) histogram of auditory nerve fiber discharges (e.g. [START_REF] Rose | Phase-locked response to low-frequency tones in single auditory nerve fibers of the squirrel monkey[END_REF]. His explanation exploits two quirks of first-order ISIs: (a) Spikes cannot occur at intervals shorter than the absolute refractory period of the auditory nerve (∼1 ms), and their probability remains depressed over the next ∼5 ms (relative refractory period) so the histogram is truncated near its origin and its first mode or modes may be missing or shifted rightward. (b) At low frequency and/or high discharge rate, multiple spikes may occur within a period, in which case intervals that span periods are shortened, leading to a leftward shift of the histogram modes. Leftward shifts are more common for low frequencies, Why is the perceptual octave stretched? rightward shifts more common for higher frequencies, hence a mismatch when histograms for octave-spaced tones are lined up, reduced by increasing the frequency of the higher tone.

This explanation is not entirely satisfactory. The first-order ISI histogram was used in early studies for technical reasons, but it does not well represent the information carried by the spike train. More recent studies prefer the all-order ISI histogram (autocorrelation) (e.g. [START_REF] Cariani | Neural correlates of the pitch of complex tones. i. pitch and pitch salience[END_REF] and shuffled versions thereof [START_REF] Joris | Correlation Index: A new metric to quantify temporal coding[END_REF][START_REF] Joris | Interaural time sensitivity dominated by cochlea-induced envelope patterns[END_REF]. A first issue is that the first-order ISI histogram is relatively noisy because a N -spike train offers only N -1 intervals, in contrast to the ∼ N 2 /2 all-order intervals exploitable by the all-order ISI histogram, or the ∼ (KN ) 2 /2 all-order intervals within a group of K fibers as reflected by a shuffled all-order histogram [START_REF] Joris | Correlation Index: A new metric to quantify temporal coding[END_REF][START_REF] Joris | Interaural time sensitivity dominated by cochlea-induced envelope patterns[END_REF]). A second issue is that the shape of the first-order histogram changes drastically with discharge rate (and thus level): a high rate induces multiple spikes within the same period, leading to an exagerated zero-order mode and a leftward shift of subsequent modes (this is the quirk (b) mentioned above). These reasons make the first-order ISI histogram an implausible basis for a model. [START_REF] Hartmann | On the origin of the enlarged melodic octave[END_REF] and [START_REF] Mckinney | A possible neurophysiological basis of the octave enlargement effect[END_REF]. [START_REF] Mckinney | A possible neurophysiological basis of the octave enlargement effect[END_REF] adapted Ohgushi's model to the more plausible all-order ISI histogram. [START_REF] Mckinney | A possible neurophysiological basis of the octave enlargement effect[END_REF] recorded from a large number of auditory nerve fiber recordings in the cat and developed a quantitative model of melodic octave shifts. The model is highly constrained by the physiological data, with only one free parameter, an advantage in terms of falsifiability. However, this feature also makes it harder to account for inter-individual differences.

Furthermore, the model predicts only small asymmetries at low frequencies (unless the implausible first-order ISI histogram is used), and requires relatively long delays (on the order Why is the perceptual octave stretched? of 5 periods). Perhaps the most serious weakness is that the refractoriness-based property that it relies on is lost when spikes from multiple fibers (e.g. from the same haircell) are pooled. Forbidding such pooling means assuming that each neuron contributing to calculate the histogram receives input from at most one fiber, wasting much of the information. This is a strong assumption.

To summarize, multiple explanations of octave stretching can be envisaged, in addition to that proposed in this paper. Each is deficient in certain aspects, so we cannot say, at this point, that the problem is solved. Hopefully, the present model is a step in the right direction.

g. Limitations.. A first weakness of the model is that, while it accounts for the possibility of a systematic mistuning of the octave, it only weakly predicts its magnitude or sign. To do so would require identifying the locus of its implementation within the auditory system, and verifying that the time constants agree with the assumptions of the model.

A second weakness is that the model is not immediately applicable to the melodic octave.

To account for mistuning asymmetry of both ascending and descending octaves, effects must apply equally to the first tone (when the trace is established) and at the second tone (where it is applied or compared with a trace derived from the second tone), but with a greater impact on high than low frequency tones. The details of how this would happen have not been worked out.

h. Significance.. Keeping limitations of the model in mind, we can revisit the claim, made earlier, that the careful observation of a relatively subtle effect may yield wider insights into hearing. The logic is that, since the effect is reliably established, and cannot easily Why is the perceptual octave stretched?

and the asymmetry resulted from the asymmetric shape of PSPs rather than different time constants along different pathways.

To the extent that octave mistuning asymmetry is indeed explanable with these assumptions, it may offer wider insights as to how hearing works.

CONCLUSION

The asymmetry of mistuning of the harmonic octave can be explained by a model of excitatory-inhibitory interaction within the auditory brain. A difference in time constants of excitation and inhibition causes the neural circuit to be more sensitive to a negative than a positive mistuning of the higher tone. This paper reviews the phenomenon, describes the model, reviews relevant data from auditory physiology, and discusses evidence for, and against, the model.

  FIG. 1. (a) Power as a function of filter center frequency (CF) for a pair of pure tones separated by

  FIG. 2. Period estimation based on autocorrelation (left) or cancellation (right). For a periodic

FIG. 3 .

 3 FIG.3. Harmonic cancellation filter. Left: comb filter with impulse response δ 0 (t) -δ T (t). Right:

FIG. 4 .

 4 FIG. 4. Neural approximations of autocorrelation (top) or cancellation (bottom). Top: a neuron

  g. A neural octave mistuning detector. The neural cancellation mechanism can be used to detect octave mistuning. To be concrete, let us suppose that the lower tone has a frequency of 500 Hz. A harmonic cancellation filter is implemented as in Fig.4(left or center) with EPSP and IPSP shaped as in Fig.5(top left), with the inhibitory pathway delayed by T = 2 ms (inverse of 500 Hz). The spike train is split in two pathways, one convolved with the EPSP, and the other delayed and convolved with the IPSP (Fig.4bottom left). The firing probability, modelled as one minus the normalized cross-product between direct and delayed PSPs, is plotted in Fig.5(top right) as a function of frequency circa 1 kHz. It is minimal at 1 kHz (in tune) and increases on either side (mistuned). The increase in firing probability with mistuning would offer a cue to mistuning from the octave relationship.

FIG. 5 .

 5 FIG. 5. Detection of mistuning with a neural cancellation filter. Left: schematic PSPs on direct

  FIG. 6. Model simulated data (left column) and behavioral data replotted from Demany et al.

ACKNOWLEDGMENTS

This work was supported by grants ANR-10-LABX-0087 IEC, ANR-10-IDEX-0001-02 PSL, and ANR-17-EURE-0017. Laurent Demany contributed many ideas and useful criticism to earlier versions of this paper, which also benefitted from useful comments from the Editor (Philip Joris) and two anonymous reviewers.

Why is the perceptual octave stretched? be explained otherwise, each of the model's assumptions receives some degree of support.

Reviewing:

• Pitch perception and and harmonic fusion involve processing of time domain patterns in the auditory brain, in contrast to a purely spectral account (e.g. [START_REF] Terhardt | Pitch, consonance, and harmony[END_REF] • Processing involves a spike coincidence or anticoincidence counting process akin to autocorrelation or cancellation (e.g [START_REF] Cariani | Neural correlates of the pitch of complex tones. i. pitch and pitch salience[END_REF][START_REF] De Cheveigné | Cancellation model of pitch perception[END_REF][START_REF] Meddis | Virtual pitch and phase sensitivity of a computer model of the auditory periphery. I: Pitch identification[END_REF]) in contrast, for example, to first-order interspike interval statistics [START_REF] Ohgushi | The origin of tonality and a possible explanation of the octave enlargement phenomenon[END_REF]).

• Cancellation is favored over correlation as it can help ensure invariance to competing periodic sounds (de Cheveigné, 2021), suggesting that asymmetry is an emergent property of a mechanism that addresses wider needs. Cancellation can be implemented on the basis of correlation (in this sense the two are equivalent) but this entails an additional implementation complexity (de Cheveigné, 2021).

• Cancellation (and correlation) may be affected by details of their neural implementation, such as the shape of PSPs involved in coincidence detection.

Neural cancellation was previously effective in a model of pitch shifts of mistuned partials [START_REF] De Cheveigné | Pitch shifts of mistuned partials: A time-domain model[END_REF][START_REF] Hartmann | On the pitches of the components of a complex tone[END_REF][START_REF] Hartmann | Hearing a mistuned harmonic in an otherwise periodic complex tone[END_REF]. The asymmetric shape of PSPs was also invoked in a model to explain why the pitch of a peak in frequency modulation is more salient than that of a trough [START_REF] De Cheveigné | A model of the perceptual asymmetry between peaks and troughs of frequency modulation[END_REF][START_REF] Demany | The perception of frequency peaks and troughs in wide frequency modulations[END_REF], although that model involved autocorrelation rather than cancellation,


Doctorante Chaymaa Lotfy

La diversité en genre dans les comités directeurs des Fédérations Royales Marocaines Sportives Gender diversity in the steering committees of the Royal Moroccan Sports Federations

Keywords: Diversité des sexes, représentation féminine, Fédération Royale Marocaine, poste de direction Classification JEL : L83, Z20 Gender diversity, female representation, Royal Moroccan Federation, leadership positions JEL Classification: L83, Z20 Paper type: Empirical research

Déclaration de divulgation :

Les auteurs n'ont pas connaissance de quelconque financement qui pourrait affecter l'objectivité de cette étude.

Conflit d'intérêts :

Les auteurs ne signalent aucun conflit d'intérêts.

Introduction

Le secteur sportif au Maroc, et dans plusieurs pays, porte ces dernières années une attention particulière aux différentes formes de diversité, notamment à la diversité du genre, considérant que les femmes sous représentée dans les divers organes d'une organisation sportive (conseil d'administration, assemblée générale, bureau), etc. Plusieurs sessions de formation, références bibliographiques, voire offres d'ateliers et de conférences se sont mises en oeuvre guidée par des politiques de gestion qui peuvent soutenir la place de tous pour assurer une meilleure inclusion et plus d'égalité. Malgré toutes ces initiatives, nous remarquions toujours une représentation de minorité des femmes dans les institutions sportives. L'importance du sport pour l'équité des genres et le développent économique à très grande échelle a été fortement démontrée dans les recherches académiques (Cunningham, 2019 ; Step Up Equality, 2020b). Les auteurs affirment que la présence des femmes contribue pleinement à la performance de l'organisation sportive (Step Up Equality, 2020a ; 2020b). Selon la thèse réalisée par Abdelaziz, B. (2018), il est clair pour que les femmes remplissent efficacement leur fonction dans les institutions sportives, il est impératif que les Fédérations Nationales Sportives (FSN) jouent un rôle primordial pour lutter contre le déséquilibre du genre et promouvoir la place des femmes, et par conséquent, c'est au sein de ces organisations que l'égalité homme-femmes doit débuter Les recherches universitaires affirment ainsi que les femmes restent sous représentées et sont confrontées à des freins qui entravent leur participation dans la gestion, l'administration, les postes d'entraineurs et d'officiels, ce qui accentue davantage les inégalités (Morgenroth & Ryan 2018, Janet al. 2019, [START_REF] Mazerolle | Barriers to the role of the head athletic trainer for women in National Collegiate Athletic Association Division II and III settings[END_REF], Foppiano-Vilo al. 2022). Cela demande l'adoption de politiques et de programmes de gouvernance solides en faveur de l'avancement de la situation pour les femmes. Le Maroc n'échappe pas à cette réalité malgré les efforts qui ont été faits dans ce sens depuis une dizaine d'années. En outre, les données nécessaires pour faciliter l'élaboration des actions les plus adéquates sont très rares et quand elles existent, elles sont difficilement exploitées et diffusées (PWC, 2021). Dans cette recherche, nous cherchons à savoir jusqu'à quel point les Fédérations Royales Marocaines Sportives (FRMS) ont pu mener une meilleure diversité en genre, l'objectif est donc d'examiner la représentation de l'ensemble des acteurs (Sportifs et sportives ; dirigeants et dirigeantes) des FNS au Maroc. Le plan de notre article sera structuré en deux parties, la première offre un aperçu général sur les théories en relation avec la gouvernance des conseils d'administration et le sport. Alors que, la deuxième consacre une présentation sur le genre dans le contexte marocain, en étudiant la participation des femmes dans les domaines de prise de décision au sein des institutions sportives au Maroc. Pour ce faire les données résultantes de l'étude méthodologique, nous dévoilons la problématique qui a été remontée dans cette recherche.

La diversité des genres dans le milieu sportif

L'approche des organisations sportives par genre

À la base d'un aperçu des travaux scientifiques menés dans le domaine sportif, l'approche la plus importante dans le domaine du sport établissant à notre avis une véritable réponse qui cause inaccessibilité des femmes à intégrer le système sportif, est l'approche « genre -organisationsystème » (GOS). Nous pouvons distinguer deux autres approches adoptées par cette approche, www.ijafame.org la première nommée « l'approche par le genre » qui considère que la faible personnalité, le manque de la motivation et la mauvaise attitude sont les principaux facteurs selon lesquelles les femmes n'arrivent pas avancer au sommet de la hiérarchie. Cette carence pousse les femmes à se comporter nuisiblement face à leurs succès et à leurs progressions au sein de l'organisation, ce qui pourrait expliciter leurs faibles progrès en matière d'égalités des sexes dans le sommet de la hiérarchie. Et la deuxième appelée « l'approche par l'organisation », soutienne que les caractéristiques organisationnelles affectent le comportement managérial des femmes. La façon dont les femmes dirigent est la raison pour laquelle elles sont relativement sous-représentées dans les postes de direction [START_REF] Kanter | Men and women of the corporation[END_REF]. Tandis que notre approche genre -organisation -système, suggère que la place des femmes au sein des organisations est influencée à la fois par leur « genre » et par la structure organisationnelle (Fagenson, 1990). Ce cadre théorique permet ainsi d'envisager les statuts professionnels des deux sexes dans le cadre organisationnel et sociétal qui contribue à la différence et à l'égalité. D'après les travaux de Parker et Fagenson (1994), l'évolution des individus, des organisations, des rôles et des sociétés se fait simultanément en rapport avec les variations de l'environnement et du rythme. Ce postulat pourra montrer les raisons pour lesquelles l'avancement des femmes en termes d'égalité managériale est varié mondialement.

La diversité du genre dans les organisations sportives

Une conscience de la contribution du sport au bien-être des femmes a vu le jour dès les années 70, la raison pour laquelle il y avait la naissance des organisations sportives féminines à travers le monde entier, y compris les pays en voie de développement. Le mouvement sportif (le Comité International Olympique (CIO), les Comités Nationaux Olympiques (CNO), les Fédérations Internationales (FI) et Nationales (FN), et les divers Organisations Sportives (OS)), a progressivement fait part pour que les femmes trouvent leur place et grave les échelons en tant qu'entraîneuses, arbitres et cadres. En effet, aux Jeux Olympiques (JO), le nombre des femmes participant augmentait avec le changement des mentalités, passant de 0 à 34% entre la période de 1900 et 1996, à une égalité presque parfaite aux jeux de Tokyo en 2020, soit 48% du total des pratiquantes. Ceci était grâce à « Alic Milliat », la première femme qui a pu accéder aux lieux de déroulement des JO et a pu donner ce pouvoir, d'assister aux grandes compétitions et d'y participer, aux autres femmes, elle a créé la Fédération Sportive Féminine Internationale (FSFI) en 1992, et contribuait à élaboration des initiatives destinées particulièrement aux femmes. En fait, même si la participation des femmes à la pratique des activités sportives et aux jeux Olympiques est en nette progression, il se trouve que le pourcentage des femmes dirigeantes et responsables au sein des instances de décisions reste loin d'être sur le même niveau d'égalité avec les hommes. L'objectif pour lequel le Comité International Olympique (CIO), a recommandé aux membres du mouvement olympique de combler cet écart en adoptant des mesures d'actions qui contribuent à atteindre l'égalité entre les genres1

Le comité directeur diversifié

À travers la revue de la littérature existante, la majorité des auteurs jugent le travail des femmes se diffère à celui des hommes (Moreno-Gómez et al., 2018), y compris pour le secteur sportif et de l'activité physique, bien que la composition par sexe soit une variété non fonctionnelle. Le Comité Directeur est le noyau d'une institution sportive. En ce qui concerne la diversité de ses membres, elle peut se caractériser à travers plusieurs aspects démographiques, humaines et sociales [START_REF] Ahern | The Changing of the Boards: The Impact on Firm Valuation of Mandated Female Board Representation[END_REF]Boubaker, et al., 2014 ;[START_REF] Tian | Difference of general-graphs and its corresponding matrices[END_REF] ; [START_REF] Westphal | How Experience and Network Ties Affect the Influence of Demographic Minorities on Corporate Boards[END_REF]Johnson, et al., 2013) où le caractère sexe de la dimension démographique, dont www.ijafame.org Les thématiques les plus traitées relatives au genre (et les femmes en particulier) visent à analyser la sous-représentation des femmes dans les organes décisionnels du milieu sportif. C'est pour cela que plusieurs chercheurs se sont plus concentrés pour examiner les facteurs empêchant la progression des femmes dans les postes de direction dans le milieu sportif [START_REF]Factors influencing career paths and progress of Kenyan women in sport leadership[END_REF]. Ces études révèlent qu'il existe plusieurs obstacles liés aux situations familiales, à l'existence de préjugés, à la sous-estimation des opinions des femmes elles-mêmes et au développement de positions dans des environnements masculinisés, comme l'affirme Foppiano-Vilo et al., (2022). Pour arriver à la parité hommes-femmes dans le monde du sport, une bonne gouvernance des FRMS doit être adoptée. Cela a permis à des chercheurs de se focaliser sur le Conseil de direction et les postes de

'association dans tous les actes de la vie civile et à l'égard des pouvoirs publics ; -Assure l'exécution des décisions de l'association de l'assemblée générale et du comité directeur ; -veille au fonctionnement régulier de l'association ; -Élabore l'organigramme de l'administration de l'association et le soumet à l'approbation du comité directeur ; -Assure le bon déroulement des assemblées fédérales et des réunions du comité directeur ; -Signe toute décision, correspondance ou tout document engageant l'association ; -Ordonnance les dépenses, et ce dans la limite du budget approuvé par l'assemblée générale ; -Négocie des appuis financiers à court terme auprès des établissements bancaires ; -Conclut, sur autorisation de l'assemblée générale, des emprunts bancaires à moyen et long terme ; -Gère le patrimoine de l'association, sur autorisation de l'assemblée générale ; -Désigne le président délégué de chaque section relevant de l'association (pour le cas des associations sportives multidisciplinaires) ; -Recrute et révoque le personnel de l'association ; Il peut déléguer une partie de ses attributions à l'un des vice-présidents qui le seconde dans l'exercice de ces fonctions et le remplace en cas d'absence ou d'empêchement ». Article 26 : Le secrétaire général « Le secrétaire général est chargé de : -La coordination des activités des organes de l'association, et du suivi des relations avec les organisations sportifs ; -La préparation des élections et des réunions des assemblées générales et du comité directeur ; -La préparation du rapport moral de l'association à soumettre à l'approbation de l'assemblée générale et sa publication sur le site web de l'association ou dans un journal d'annonces légales après approbation ; -L'établissement des procès-verbaux des délibérations des assemblées générales et

Données et méthodes

Dans le cadre de cette étude, nous avons adopté l'approche transversale descriptive qui nous a permis d'analyser l'état des questions étudiées telles que définies dans le contexte réel. Cette méthode consiste à décrire, comprendre et expliquer avec précision un phénomène dans un temps et un lieu particuliers. (Williams, 2007, P.66). Puisque notre étude est limitée par des contraintes de temps et par la rareté des ressources, la recherche secondaire a considéré la plus adéquate pour la présente étude, car elle est plus rentable que la recherche primaire et permet d'offrir des résultats larges et rapides (Heydenrych et Case, 2018). Pour les besoins de cette étude, nous avons privilégié l'étude de toutes les FRMS (Tableau 1) pour doubles raisons : Premièrement, nous souhaitons enrichir la base de données et la rendre accessible au public puisque les statistiques ne sont pas facilement disponibles et les informations ne sont bien archivées qu'à partir de l'année 2019. Deuxièmement, nous voulons analyser en profondeur la structure organisationnelle de toutes les fédérations en termes de diversité des sexes pour contribuer à améliorer la réalité de leur participation dans les instances de décision au sein ces organisations sportives. Les variables étudiées sont au nombre de deux : l'effectif des acteurs et la diversité des sexes. En ce qui concerne la collecte des données secondaires, nous avons fait appel à des statistiques descriptives obtenues auprès du personnel du CNOM et MENPS et en consultant les pages web de toutes les FRMS. Le but était de recueillir les informations liées aux licenciés sportifs et aux personnels salariés et bénévoles en fonction des critères suivants : le pourcentage d'hommes et de femmes adhérant à des clubs sportifs, la proportion d'hommes et de femmes dans les commissions administratives et techniques ainsi que le taux des membres hommes et femmes dans les bureaux directeurs. Il convient de souligner que les données offertes visent à examiner et analyser la représentation des sportives et la représentation des femmes qui détiennent le titre de responsable, directeur ou président, afin de comparer leur participation avec celles des hommes.

4.

Résultats

Pour les trois catégories étudiées, nous avons : -Pratiquant-e-s adhérant-e-s aux clubs sportifs : À partir d'un échantillon adhérant aux différents clubs sportifs existants au Maroc, qui sont affiliés à une moyenne de 45 fédérations sportives marocaines entre 2009-2012 et 57 en 2019, répartis en cinq grandes catégories (Sport collectif, sport individuel, sport nautique, sport de salle et le sport motorisé). Nous notons en total des pratiquants dans les clubs et les ligues (avec le ratio des femmes), d'après l'étude réalisée par Abdelaziz (2018), une moyenne de 293039 licenciés pendant la période de (2009, www.ijafame.org 2010, 2011 et 2012), tandis que les données que nous avons retiré à partir des rapports moraux des FRMS pour l'année 2019 est de 208553 licenciés. Les résultats du tableau 1 illustrent que les pratiquants hommes représentaient 240 678 licenciés, soit 67%, ce qui est largement supérieur à celui des pratiquantes femmes dont le pourcentage est de 33% (67 875 licenciées). Nous pensons que, probablement, la pratique du sport dans le milieu scolaire à travers l'Education Physique et Sportive (EPS) a été moins pratiquée par les filles, cela ne leur permet pas d'adhérer aux clubs sportifs. En outre, en nous basant sur la même étude précitée, nous constatons une grande différence entre l'effective des sportives estimé en période de (2009, 2010, 2011 et 2012) qui est de 46 563 licenciées soit 15,9% et celui de la période de 2019 qui est de 67875 licenciées. Et donc, la double augmentation du nombre de licenciées (+17,1) laisse à penser qu'il y a une politique positive de la culture sportive et des changements de mentalités pour l'amélioration de la représentativité des femmes sur le plan pratique.

. Représentation des dirigeant-e-s dans les instances de responsabilités des FRMS par sexe Sources : Les données relatives aux membres des bureaux directeurs sont issues des rapports de projet de loi de Finances, ou des données issues du ministère de l'Éducation nationale, du Préscolaire et des Sports (MENPS) Figure 2. Représentation féminine dans les instances de responsabilités des FRMS par organe Sources : Les données relatives aux membres des bureaux directeurs sont issues du MENPS

Composition des bureaux exécutifs des FRMS

Comme indiqué dans la figure 3

Figure 3. La diversité des sexes au niveau des bureaux directeurs des FRMS Sources : Les données relatives aux membres des bureaux directeurs sont issues des rapports moraux des FRMS, ou de leur site internet. Figure 4. La diversité des sexes des bureaux directeurs des FRMS par postes

Sources : Les données relatives aux membres des bureaux directeurs sont issues des rapports moraux

des FRMS, ou de leur site internet. Cependant, certains efforts sont déployés pour promouvoir la participation féminine, en se basant sur la réalité qui a placé les femmes devant un avenir plein d'obstacles et de défis, afin qu'elles puissent se positionner dans le domaine sportif, notamment pour occuper des postes à responsabilités dans les organes de direction. Les femmes demeurent faiblement représentées dans la société marocaine en termes d'égalité entre les femmes et les hommes, surtout dans les sports de haut niveau qui se caractérisent par le monopole du côté masculin, tandis que leur présence se limite le plus souvent à leur accès aux institutions sportives privées. En ce qui concerne cette tendance, il est nécessaire de doubler les efforts pour surmonter les obstacles qui empêchent les femmes d'accéder à des postes haut placés (dans le domaine du sport). Le but est de les accueillir et les garde dans des postes de direction dans les organisations sportives et améliorer la réalité de leur participation dans les instances de décision afin qu'elles puissent jouer leur rôle à côté de l'homme en tant que partenaire, et non en tant qu'assistante.

5.

Discussion

6.

Conclusion

L'analyse de la composition liée à la diversité du genre dans les conseils d'administration au sein des Fédérations Royales Marocaines Sportives nous a permis de nous renseigner sur le taux de la visibilité des femmes relatives à la pratique, gestion et encadrement. Les résultats de notre étude expérimentale prouvent que les FRMS se sont marquées par l'inégalité entre les sexes, où les femmes subirent une discrimination en matière d'accès aux postes de direction qui leur rendent presque invisibles dans l'ensemble des organes des FRMS. L'introduction des femmes à la composition des fonctions de responsabilité à un effet positif. D'une part, il vise à perfectionner le résultat des instances sportives de décisions. D'autre part, il permet à favoriser la qualité de différents services au sein des organisations sportives. Dans ce contexte, l'évolution effective des femmes dans les conseils d'administration des fédérations royales marocaines sportives nécessite du mouvement sportif fédéral, des décideurs nationaux et des femmes elles-mêmes, qu'ils obéissent aux règles de gestion et la gouvernance appropriée, à la transparence et lucidité dans la réalisation des objectifs guidés par les conventions d'objectifs annuels, pour mieux réussir les buts tracés de la diversité du genre et réduire l'écart qui existe entre les deux sexes dans le milieu sportif. Bien que les résultats de cette recherche soient satisfaisants, il est primordial d'entamer les limites qui seront traitées ultérieurement dans d'autres articles tels que le choix méthodologique adopté qui sera l'objet de notre futur article.

  les femmes et les hommes sont représentés, apportent des avantages énormes au sein d'un comité directeur sur tous les plans. Composer un comité directeur d'une organisation sportive qui compte entre 5 et 15 membres, plus le président(e), il est important qu'une assemblée générale annuelle ait lieu pour réaliser une élection de ses membres qui seront élus pour une durée de 4ans renouvelables une seule fois selon les conditions régies par le Dahir n° 1-58-376 du 3 joumada I 1378 (15 novembre 1958) règlements le droit d'association, tel que modifié et complété et aux dispositions particulières de la présente loi. Le bureau du comité directeur est composé du président (e), 1er vice-président(e), 2ème viceprésident(e), directeur(rice) général(e), secrétaire général(e), secrétaire-adjoint(e), trésorier, trésorier-adjoint(e) et … (entre 3 et 9) assesseurs(es). Chacun d'entre eux a des rôles et des responsabilités pour la protection des intérêts généraux de l'organisation sportive et la bonne manière de gérer sa gouvernance conformément aux dispositions prévues par le Dahir susvisé. Dans les articles 25, 26 et 27, la Loi 30_09 relative à l'éducation physique et aux sport (promulguée par le Dahir N°1-10-150 du 13 Ramadan 1431-24 Août 2010 publiée au Bulletin Officiel N°5888 du 4 Novembre 2010), évoque les fonctions des responsables clé du comité directeur : Article 25 : Le président « Le président du comité directeur est, de droit, le président de l'association. A ce titre, il : -Représente l

  responsabilité en termes de leadership et d'encadrement technique (Cosentino et al. 2021, Megheirkouni et al. 2017). En particulier, les études entamées ont été menées sur l'exclusion des femmes des postes de dirigeants sportifs (Terret, T. 2004), l'expérience de leadership au sommet hiérarchique de leurs organisations (Belghiti-Mahut et al. 2014). Ainsi sur la compréhension de la vie et des expériences des femmes dans le leadership sportif (M'mbaha et al. 2016), sur les meilleures pratiques pour remédier au manque de femmes dans les rôles d'entraîneurs et de dirigeants sportifs (Culver 2019) et sur les freins influençant les cheminements de carrière des femmes occupant des postes de direction dans le sport (Janet al. 2019). En ce qui concerne le contexte marocain, il y a peu de recherches empiriques qui argumentent les causes de la faible présence des femmes dans les conseils d'administration, la gestion et l'encadrement technique vu le manque des données statistiques en termes des représentations féminines dans l'ensemble des commissions des Fédérations Royales Marocaines Sportives d'où la nécessité de cette étude pour combler cette lacune. Cela suppose que le mouvement sportif fédéral marocain subit toujours d'un écart entre les femmes et leur taux de leurs www.ijafame.org représentations, ce qui ne facilite pas la diversité du genre dans l'ensemble des Fédérations Royales Marocaines Sportives (FRMS). Dans ce contexte, la présente recherche vise à répondre à cet appel en effectuant une analyse sur la représentation des femmes dans les dits organisations. De manière particulière, cette étude s'occupera d'analyser les trois populations suivantes : -Les licenciés masculins et féminins affiliés aux organisations sportives -Le personnel élu en position de leadership dans les organes administratifs et techniques au sein des FRMS -La proportion de femmes sur les conseils directeurs des FRMS 3.

Figure 1

 1 Figure 1. Représentation des dirigeant-e-s dans les instances de responsabilités des FRMS par sexe

  , les femmes représentent 12% des membres occupant des postes dits clés dans les conseils directeurs des FRMS. Ce constat met en valeur l'importance du processus de recrutement pour un équilibre dans la répartition des postes dans le but d'atteindre l'égalité de genre. La figure 4 présente un diagramme à barre. Les données collectées pour l'ensemble des 57 fédérations qui existent au Maroc : quatre (04) femmes sont présentes (Sports Aérobics, Fitness, Hip Hop et Disciplines Assimilées, Volleyball, sport pour tous et Bridge). Une seule d'entre elles occupe le poste de Présidente d'une fédération Olympique. Sur quatre-vingt-dix (90) viceprésidents de fédérations, onze (11) sont des femmes. En ce qui concerne les secrétaires généraux qui sont au nombre de quarante-huit (48), trois (3) sont des femmes. Au niveau des quarante (40) secrétaires généraux adjoints, il y a cinq (5) femmes. S'agissant des quarante-sept des ligues et associations sportives www.ijafame.org (47) postes de trésoriers, les femmes n'occupent aucun poste et sur quarante-et-un (41) trésoriers-général adjoints, il y a cinq (5) femmes.

  participation féminine dans les commissions du comité nationale olympique marocaine (CNOM) n'est pas meilleure. Elle ne compte qu'une seule femme dirigeante : Nawal EL MOUTAWAKEL Présidente de la Commission Femme et sport et représentante du CIO dans Comité exécutif au sein du Comité Nationale Olympique Marocain, parmi les dix (10) commissions du CNOM. Alors qu'aucune femme marocaine n'assurait la présidence du comité exécutif olympique marocain. En outre, d'un point de vue statistique des structures dirigeantes au niveau des fédérations sportives (FS), il y a trois (3) femmes présidentes dans les 57 Fédérations Sportives du Maroc, alors que sur les 30 fédérations sportives olympiques (FSO), il n'y a qu'une seule femme dirigeante (Taleb, Y. 2020). Même si les femmes au Maroc sont engagées de plus en plus au niveau du processus décisionnel du monde du sport et qu'elles démontrent leur existence dans le monde entier. Et bien qu'elles prouvent qu'elles sont des femmes à compétences certaines et capables de performances comme les hommes, il reste qu'elles sont d'un nombre insignifiant par rapport à la représentation masculine. Au niveau international, pour lutter contre les inégalités, le Comité International Olympique (CIO) a mis en place un engagement en matière d'égalité des sexes d'ici 2024. Il y a également appel à la promotion de la représentation des femmes au poste de direction dans l'ensemble du mouvement olympique, en vue d'atteindre au moins 30 % de la présence des femmes dans les organes de décision des organisations sportives, en cherchant à encourager les Parties Prenantes (P.P) -plus précisément les Comités Nationaux Olympiques (CNOs) et les Fédérations Internationales (FIs) -à promouvoir la participation significative des femmes dans le sport (CIO, 2021) Dans ce contexte, le CIO a pris plusieurs recommandations lié à la gouvernance : article 18, 19, 20, 21 et 24 du projet d'analyse de l'égalité des sexes du CIO (IOC, I. 2018). La communication s'est faite à travers une série de séminaires sur l'égalité des genres, organisés en coopération avec l'Association des Fédérations Internationales des Sports Olympiques d'Eté (ASOIF) et l'Association des Fédérations Internationales des Sports Olympiques d'Hiver (AIOWF), tenus en ligne en raison de la pandémie de COVID-19, du 30 juin au 2 juillet 2021, dans le but d'augmenter le nombre de femmes dans les organes de direction et d'améliorer leur représentation aux postes de décision. A cet égard, la présidente de la Commission des Femmes dans le Sport du CIO a souligné que : « Le CIO reconnaît la nécessité de compter sur davantage de femmes dirigeantes et ne cesse de montrer l'exemple en faisant la promotion d'une forte présence féminine à ces fonctions, au sein de tous les organes de direction et d'administration du Mouvement olympique. » (IOC, I. 2018). Il ne s'agit pas seulement de parler d'augmentation de la représentation féminine au sein des organes de décision ; il s'agit de le faire par des mécanismes et des obligations et d'exiger également des comptes aux dirigeants du monde sportif, pour veiller à ce que ces quotas et objectifs soient mis en oeuvre à tous les niveaux du sport. Au Maroc, pour combattre la réalité du faible taux de femmes dirigeantes en raison d'une forte « tradition sexiste » et des fonctions de pouvoir qui demeurent très masculinisées. La présidente de l'Agence Marocaine Anti Dopage, Dr Fatima Abou-Ali, a souligné, lors de ses interventions dans le séminaire numérique (qui a eu lieu le 8 Mars 2021) mené sur la question de l'accès des femmes dans le sport « l'importance de soutenir une approche genre qui donnerait aux femmes des opportunités pour gérer et diriger des institutions sportives grâce à leur travail assidu et responsable et à leur diligence continue ». De plus, elle a ajouté que la convergence et la coopération de toutes les parties prenantes créeraient une atmosphère et des conditions favorables à l'accès des femmes aux postes de décision dans le domaine sportif en collaboration avec les hommes. www.ijafame.org

Le genre dans le contexte sportif marocain

  La gouvernance du sport et la diversité (des sexes) sont des sujets à attention particulière dans les études, vu qu'ils contribuent à la recherche d'une performance plus solide dans le secteur sportif (Spaaij, R. al. 2018). Les chercheurs sont largement focalisés sur les questions Genre et Sport dans certains pays en Europe et en Amérique du Nord, bien que des études récentes commencent à se pencher sur le terrain du contexte africain, à savoir l'Afrique du sud, l'Egypte, le Kenya, le Maroc et le Sénégal (PWC, 2021). Les Organisations sportives les plus examinées dans la revue de littérature selon la classification de BAYLES, E (2005) sont de niveau 1 : les Organisations du mouvement sportif (Fédérations Sportives et Associations affiliées aux Fédérations Ligues et Clubs Sportifs Professionnels)

	Article 27 : Le trésorier
	« Le trésorier est chargé de :
	-Gérer les ressources de l'association. À ce titre, il procède au recouvrement des recettes, à
	la liquidation des dépenses ordonnancées par le président, et à la tenue de la comptabilité
	de l'association qui doit être certifiée par un commissaire aux comptes ;
	-Co-signer avec le président les chèques et titres de paiement émis au nom de l'association ;
	-Préparer le projet de budget de l'exercice suivant et le soumettre à la délibération du comité
	directeur ;
	-Préparer le rapport financier de l'association à présenter à l'assemblée générale.
	Le trésorier est assisté dans ses fonctions par le trésorier adjoint qui le remplace en cas
	d'absence ou d'empêchement.
	2.2.2.

du comité directeur. www.ijafame.org Le secrétaire général est assisté dans ses fonctions par le trésorier adjoint qui le remplace en cas d'absence ou d'empêchement ».

Tableau 2 : Nombres et ratios des pratiquant-e-s par sexes

		Nombres		Pourcentage (%)
	Licenciées	46563*	67875**	15,9*	33**
	Licenciés	246476*	240678**	84,1*	67**
	Total	293039*	208553**

**

Les données sont issues des rapports moraux des FRMS pour la période 2019. Source : *D'après la thèse réalisée par Abdelaziz, B. en 2018.

-Cadre en gestion et encadrement technique

La figure

1

illustre le ratio de dirigeantes qui occupent les postes les plus élevés hiérarchiquement dans les FRMS. Les résultats présentent une faible proportion des femmes cadres (15%), soit 1677 dirigeantes, contre un pourcentage de 85%, soit (9503 dirigeants). Plus précisément la répartition des femmes dans les rôles de responsabilité est éclairée à travers la figure

2

, comme suit : 50 femmes au comité directeur. 58 femmes à l'organe disciplinaire. 82 femmes dans la structure administrative. 16 femmes dans la structure technique. 28 femmes dans la structure médicale. 35 autres femmes collaboratrices. 753 femmes officielles. 646 femmes cadres techniques des ligues et associations sportives. 5 femmes dans l'organe de formation et 4 femmes dans l'organe de communication. Soit un total de 1677 dirigeantes. En outre, l'exploitation de données des rapports liés aux projets de lois de Finances, nous a permis de confirmer la domination masculine dans les hautes sphères des FRMS (une moyenne de 10% des taux de représentativité des femmes dans les fédérations royales marocaines sportives en 2016/2017 et 15% en 2018/2019), même si l'évolution féminine continue d'augmenter. Divers obstacles peuvent peut-être contribuer à cette inégalité, comme des problèmes d'ordres familiaux et sociaux. www.ijafame.org

  Au Maroc, nous constatons que la participation des femmes à des postes de direction est faible, d'après les données précitées. Ce constat est confirmé par Abderrazak Elakari, directeur du Bureau des Sports du Ministère de la Jeunesse et des Sports ; « Peu de femmes pratiquent le sport et peu de femmes sont en responsabilité … » (C. Guibert & C. Arab, 2016) Dans le cadre de la Stratégie Nationale pour le Développement du Sport (2008-2020), le Ministère de la Jeunesse et du Sport (MJS) a pris un certain nombre de mesures, relatives à la politique suivie dans le domaine sportif. Cette politique a été adoptée conformément aux instructions de Sa Majesté le Roi Mohammed VI suite à sa Lettre royale lors des assises de Skhirat en 2008, pour renforcer la position des sportifs (es) marocaines, augmenter leur participation : « Promouvoir le sport et la pratique sportive, moderniser la gouvernance » (Pomé[START_REF] Eloi | La filière sport au Maroc Etude Filière Sport Pour l'Agence Française de Développement (AFD)[END_REF], et rétablir des politiques basées sur l'intégration du genre. Cela menait selon les données issues du rapport de MJS en 2020 à doubler le nombre de femmes dans son secteur pour inclure 1 476 femmes, soit 44%. Cependant, ces mesures ne suffisent pas à réaliser les aspirations du Maroc en faveur de l'égalité hommes-femmes, parce que les femmes marocaines sont toujours marginalisées dans les postes de responsabilité au sein des sphères sportives. Il se trouve aussi, malgré la présence significative des femmes marocaines diplômées et expérimentées, issues des institutions spécialisées en management et gouvernance du sport, la

	ISSN: 2658-8455
	Volume 4, Issue 1-2 (2023), pp. 167-180.
	© Authors: CC BY-NC-ND
		Proportion des
		femmes
		12%
				Proportion des
				hommes
				88%
	Membres
	Trésorier adjoint
	Trésorier général
	Secrétaire adjoint
	Secrétaire général
	4ème Vice President
	3ème Vice President
	2ème Vice President
	1er Vice President
	President
	0	20	40	60	80	100	120
	Proportion des femmes	Proportion des hommes
							177
		www.ijafame.org

https://olympics.com/cio/egalite-des-genres/aux-jeux-olympiques ISSN:

2658-8455 Volume 4, Issue 1-2 (2023), pp. 167-180. © Authors: CC BY-NC-ND