Nicola Montecchio
email: nicola.montecchio@dei.unipd.it

Arshia Cont
email: arshia.cont@ircam.fr

ACCELERATING THE MIXING PHASE IN STUDIO RECORDING PRODUCTIONS BY AUTOMATIC AUDIO ALIGNMENT

We propose a system for accelerating the mixing phase in a recording production, by making use of audio alignment techniques to automatically align multiple takes of excerpts of a music piece against a performance of the whole work. We extend the approach of our previous work, based on sequential Montecarlo inference techniques, that was targeted at real-time alignment for score/audio following. The proposed approach is capable of producing partial alignments as well as identifying relevant regions in the partial results with regards to the reference, for better integration within a studio mix workflow. The approach is evaluated using data obtained from two recording sessions of classical music pieces, and we discuss its effectiveness for reducing manual work in a production chain.

INTRODUCTION

The common practice in productions of studio recordings consists of several phases. At first the raw audio material is captured and stored on a support. This material is subsequently combined and edited in order to produce a mix, which is finalized in the mastering phase for commercial release. Nowadays, the whole process revolves around a computer Digital Audio Workstation (DAW).

In the case of instrumental recording, the initial task involves capturing a complete reference run-through of the entire piece, after which additional takes of specific sections are recorded to allow the mixing engineer to mask performance mistakes or reduce eventual environmental noises. The role of a mixing engineer is to integrate these takes within the global reference in order to achieve a seamless final mix [START_REF] Bartlett | Practical Recording Techniques: The step-by-step approach to professional audio recording[END_REF]. The first step in preparing a mix session consists in arranging the takes with regards to the global ref-Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. c 2011 International Society for Music Information Retrieval. erence. Figure 1 shows a typical DAW session prepared out of a reference run-through (the top track) and additional takes aligned appropriately. Those takes usually require further cleanup as they commonly include noise or conversation that are not useful for the final mix. This means that, in addition to alignment, the mixing engineer identifies cutpoints for each take that correspond to relevant regions in the reference. The additional takes are finally blended with the reference by crossfading short overlapping audio regions to avoid perceptual discontinuities. The purpose of this work is to facilitate the process of mixing by integrating automatic (audio to audio) alignment techniques into the production chain. Special care is taken to consider existing practices within the workflow, such as automatic identification of interest points. In contrast to most literature on audio alignment, we are concerned with two essential aspects: the ability to identify a partial alignment with an unknown starting position and the detection of regions of interest inside the alignment. Moreover our approach permits to achieve different degrees of accuracy depending on efficiency requirements.

Using audio material collected from two real-life recording sessions, we show that it is possible to optimize the operations of sound engineers by automating time-consuming tasks. We further discuss how such framework can be integrated pragmatically within common DAW software.

RELATED WORK

At the application level, alignment techniques were already introduced in the literature in [START_REF] Dannenberg | Polyphonic Audio Matching for Score Following and Intelligent Audio Editors[END_REF]. Alignment of audio to the symbolic representation of a piece was integrated into the workflow, permitting the automation of the editing process through operations such as pitch and timing corrections. The application of these approaches is precluded in the present context by the requirement of accessing a symbolic representation of the music. Nonetheless, despite this limitation, the work provides important insights in the integration within a DAW setup.

At the technological level, audio alignment has often been the subject of extensive research; an overview of classical approaches in literature can be found in [START_REF] Montecchio | A Unified Approach to Real Time Audio-to-Score and Audio-to-Audio Alignment Using Sequential Montecarlo Inference Techniques[END_REF]. In contrast to traditional methods, an important aspect of this work is the consideration of partial results and detection of interest regions. An audio alignment method with similar aims was introduced in [START_REF] Müller | Path-Constrained Partial Music Synchronization[END_REF], that explicitly deals with the synchronization of recordings that have different structural forms.

GENERAL ARCHITECTURE

The proposed methodology was devised assuming that a generic algorithm is available that is capable of aligning audio sequences without a known starting position. Even though methods such as HMM or DTW [START_REF] Dixon | Match: a Music Alignment Tool Chest[END_REF] could have been used for this aim, we chose to exploit our previous work [START_REF] Montecchio | A Unified Approach to Real Time Audio-to-Score and Audio-to-Audio Alignment Using Sequential Montecarlo Inference Techniques[END_REF] on sequential Montecarlo inference because of its straightforward applicability to the present context, its flexibility regarding the degree of accuracy given by the availability of smoothing algorithms and the possibility to trade accuracy for computational efficiency in an direct way.

In the first phase a rough alignment is produced as in Figure 2(a); the initial uncertainty in the alignment is due to the fact that the initial position is not known a priori. In a second phase we identify a sufficiently long region of the alignment that can be reasonably approximated by a straight line, as in Figure 2(b); this region intuitively corresponds to the "correct" section of the alignment. These two phases solve the task of placing the takes along the reference (Figure 1).

The remaining steps address the tasks in which a more accurate alignment is required. In the third phase, the initial portion of the alignment is corrected, starting from a position inside the region found in the previous phase and using a reversed variant of the alignment algorithm (Figure 2(c)). Finally, a refined alignment is produced by exploiting a smoothing algorithm for sequential Montecarlo inference, as shown in Figure 2(d).

METHODOLOGY

The four phases described in the previous section are highlighted in Figure 2 and described below in detail.

Initial Alignment

The alignment problem is formulated as the tracking of an input data stream along a reference, using motion equations.

System State Representation

The system state is modeled as a two-dimensional random variable x = (s, t), representing the current position in the reference audio and tempo respectively; s is measured in seconds and t is the speed ratio of the performances. The incoming signal processing frontend is based on spectral features extracted from the FFT analysis of an overlapping, windowed signal representation, with hop size ∆T . In order to use sequential Montecarlo methods to estimate the hidden variable x k = (s k , t k ) using observation z k at time frame k, we assume that the state evolution is Markovian.

Observation Modeling

Let p(z k |x k ) denote the likelihood of observing an audio frame z k of the take given the current position along the reference performance s k . We consider a simple spectral similarity measure, defined as the Kullback-Leibler divergence between the power spectra at frame k of the take and at time s k in the reference.

System State Transition Modeling

Let p(x k |x k-1 ) denote the pdf for the state transition; we make use of tempo estimation in the previous frame, assuming that it does not change too quickly:

p(x k |x k-1 ) = N ( s k t k | µ k , Σ) µ k = s k-1 + ∆T t k-1 t k-1 Σ = σ 2 s ∆T 0 0 σ 2 t ∆T
Intuitively, this corresponds to a performance where tempo is rather steady but can fluctuate; the parameters σ 2 t and σ 2 s control respectively the variability of tempo and the possibility of local mismatches that do not affect the overall tempo estimate.

Inference Algorithm

Sequential Montecarlo inference methods work by recursively approximating the current distribution of the system state using the technique of Sequential Importance Sampling: a random measure {x i k , w i k } Ns i=1 is used to characterize the posterior pdf with a set of N s particles over the state domain and associated weights, and is updated at each time step as in Algorithm 1. In particular, q(x k |x k-1 , z k ) is the particle sampling function. In our implementation this corresponds to the transition probability density function; in this case the algorithm is known as condensation algorithm.

An optional resampling step is used to address the degeneracy problem, common to particle filtering approaches; this is discussed in detail in [START_REF] Arulampalam | A Tutorial on Particle Filters for Online Nonlinear/Non-Gaussian Bayesian Tracking[END_REF][START_REF] Douc | Comparison of Resampling Schemes for Particle Filtering[END_REF] and in the next paragraph.

The decoding of position and tempo is carried out by computing the expected value of the resulting random measure (which is efficiently computed as

E[x k ] = Ns i=1 x i k w i k ).
Algorithm 1: SIS Particle Filter -Update step

for i = 1 . . . N s do sample x i k according to q(x i k |x i k-1 , z k ) ŵi k ← w i k-1 p(z k |x i k )p(x i k |x i k-1 ) q(x i k |x i k-1 ,z k ) w i k ← ŵi k j ŵj k ∀i = 1 . . . N s N ef f ← ( Ns i=1 (w i k ) 2 ) -1 if N ef f < resampling threshold then resample x 1 k . . . x Ns k according to ddf w 1 k . . . w Ns k w i k ← N -1 s ∀i = 1 . . . N s

Initialization

Initialization plays a central role in the performance of the algorithm; in a probabilistic context this corresponds to an appropriate choice of the prior distribution p(x 0 ). In a real-time setup the player is expected to start the performance at a well known point of the reference; this fact is exploited in the design of the algorithm by setting an appropriately shaped prior distribution, typically a low-variance one around the beginning.

In the proposed situation however the initial point is not known (it represents indeed the aim of our interest). To cope with this, the prior distribution p(x 0 ) is set to be uniform over the whole duration L of the reference performance; the algorithm is expected to "converge" to the correct position after a few iterations. Figure 3 shows the evolution of the probability distribution for the position of the input at different moments of the alignment.

Degeneracy Issues w.r.t. Realtime Alignment

A relevant parameter of Algorithm 1 is the resampling threshold. The variable N ef f , commonly known as effective sample size, is used to estimate the degree of degeneracy which affects the random measure; degeneracy is related to the variance of the weights {w i k } Ns 1 , and it is proven to be always increasing in absence of resampling. In a degenerate situation most particles have close-to-zero weight, resulting in most of the computation being spent in updating particles which are subject to numerical approximation errors. Resampling is introduced to obviate this issue. Intuitively, resampling replaces a random measure of the true distribution with an equivalent one (in the limit of N s → ∞) that is better suited for the inference algorithm. Since resampling introduces other problems (in particular, sample impoverishment, i.e., a small number of particles is selected multiple times) its usage should be limited, thus producing the necessity for a threshold on the effective sample size.

In the real-time score following case [START_REF] Montecchio | A Unified Approach to Real Time Audio-to-Score and Audio-to-Audio Alignment Using Sequential Montecarlo Inference Techniques[END_REF] the mass of the distribution is always concentrated around a small region of the domain thus allowing the resampling threshold to be relatively low. In contrast, in a situation such as the one depicted in Figure 3, the sparsity of the distribution in the initial phases of the alignment imposes a much higher resampling threshold, otherwise many relevant hypotheses are soon lost in the resampling phase and cannot be recovered.

Identification of the Interest Region

This phase aims at identifying a region of the alignment obtained previously where it is certain that the alignment is indeed "correct". As depicted in Figure 2(b), a typical alignment can be subdivided into two regions, the first one being characterized by irregular oscillations (because not enough data has been observed yet in order to select the most probable hypothesis with enough confidence) and the second one resembling a straight line; we will refer to the former as convergence region and to the latter as interest region.

As can be inferred by observing the plot in Figure 2(b), the most important characteristic of the interest region is its slope. From a technical point of view, the slope should be as constant as possible for the alignment region to be significant. From a musical perspective it should be roughly unitary, implying that the performance tempos of the single take and the reference are approximately the same. In addition to that, the duration of the interest region should be long enough to discard noisy sections of the alignment.

The interest region is identified in the following man- ner: each of many initial candidate regions w 1 . . . w W is iteratively expanded as long as it meets the criteria exposed above; the longest of the resulting intervals is elected as the interest region, unless none of them matches the requirements, in which case the alignment is identified as incorrect. The process described above is depicted in Figure 4 (dashed horizontal lines represent the regions progressively examined by the algorithm) and formalized in Algorithm 2.

Algorithm 2:

Identification of interest region w 1 , . . . , w W ← regularly spaced intervals in [0, L] candidates ← ∅ for i = 1 . . . W do while |w i | < L do w i ← max(0, w start i -∆T ), min(L, w end i +∆T ) a i ← slope of LS-fit line for points in w i e i ← mean difference with LS-fit line in w i if a i ∈ [1 -∆A, 1 + ∆A] ∩ e i < ∆E then candidates ← candidates ∪ i else break if |candidates| > 0 then interest region ← max i∈candidates w i else alignment is incorrect

Correction of the Convergence Region

In order to fix the convergence region of the alignment, we exploit again the sequential Montecarlo inference methodology of 4.1, with some adaptations. The general idea is to run the algorithm "backwards", i.e., to align the timereversed audio streams, starting from a point in the previous alignment that is known to be correct. The starting point B is chosen inside the region of interest. The prior distribution for the backward alignment is equal to that of the forward alignment at B, however with the value of the velocity for each particle inverted: p(x

(b) B ) = diag(1, -1)p(x B |z 0 . . . z B ).
The audio stream of the take is then reversed and processed by Algorithm 1, as in Figure 2(c). Experimentation shows that a narrow uniform or gaussian prior centered in (B, -1) T are for practical purposes equivalent to the form of p(x (b) B ) mentioned above.

Smoothing Inference

Sequential Montecarlo inference algorithms are typically for online estimation; this implies that at each instant only the information about the past is exploited, instead of the whole observation sequence. In the context of an offline application however these real-time constraints can be dropped. Both the Forward/Backward and Viterbi inference algorithms can be deduced, respectively estimating the probability distribution at each instant given the full observation sequence and the Maximum A Posteriori alignment. The running time of both algorithms is quadratic in the number of particles, however this issue can be mitigated by an appropriate choice of the prior distribution p(x

(f b) 0
) such as a resampling of diag(1, -1) p(x (b) 0 ) with a smaller numer of samples.

EVALUATION

An ideal evaluation of the efficacy of the proposed methodology in the context discussed in Section 1 should aim at measuring the amount of work saved in production with respect to the current workflow. A discussion of our current work in this area is presented in Section 6.

Below we evaluate the efficacy of the proposed approach regarding the initial phase of laying out the takes as in Figure 1. The accuracy of the alignment in terms of latency and average error was evaluated in our previous work [START_REF] Montecchio | A Unified Approach to Real Time Audio-to-Score and Audio-to-Audio Alignment Using Sequential Montecarlo Inference Techniques[END_REF]; a similar analysis could not be performed in this case, due to the lack of a (manually annotated) reference linking the timings of each musical event for all takes to the reference recording. Moreover, in this situation the aim is rather to position correctly the highest number of takes against the reference, rather than to align them with the highest possible precision.

Dataset description

We collected the recordings produced in two real-life sessions by different groups of sound engineers, consisting of approximately 3 hours of audio data. The first one is a recording session of the second movement of J. Brahms' sextet op. 18; the second one was produced shortly after the premiere of P. Manoury's "Tensio", for string quartet and live electronics, in December 2010. Table 1 summarizes their characteristics.

Experimental Results

We performed the alignment of each take in the two databases according to the procedure introduced in Section 4. We select the center point of the interest region identified in the second phase as the alignment reference for the whole take (we do not performs the optional two last steps).

In all the test we executed, we set the number of particles N s to be proportional to the duration of the reference (60 particles per second). Our implementation aligns a minute of audio in 2.29s for N s = 10 5 on a laptop computer with a 2.4 Ghz Intel i5 processor (a single core is used).

Brahms Dataset

For this dataset, a manual placement of all the takes with respect to the reference recording was performed using a musical score, in order to evaluate the correctness of the automatic procedure. Aural inspection of the data showed that none of the recordings but one presented undesired noises.

All the takes but one were correctly aligned. In the unsuccessful case, the length of the recording itself was one second shorter than the minimum length for an interest region (15s); using last alignment point as a reference, the placement of this take also results to be correct.

Manoury Dataset

The dataset contains a complete run-through and 49 separate takes. The particularity of this dataset is the presence of undesired material for the final mix in many of the individual takes (such as speech, practice sessions, volume and calibration tests). Out of 49 takes, 14 contain exclusively noise and 21 partially. In the former case we consider the alignment correct if the file is discarded, in the latter we aim at aligning correctly the interesting portion of the take. This is in sharp contrast with the "cleanness" of the Brahms set and presents difficulties that were not foreseen when formulating the alignment procedure.

Contrarily to the Brahms dataset, the evaluation of the alignment precision was done a posteriori: instead of performing a manual alignment in advance, the results of the automatic alignment were checked. The reason for this lies behind the length (approximately 40 minutes) and complexity of the music: even with the score at our disposal, it was immediately evident that a manual alignment would have taken a very long time. It is precisely this difficulty that sound engineers had to face.

Our first experiment aligning this dataset yielded rather poor results on the 21 files containing noise regions of significant length (in some cases up to more than one minute); since in almost all cases the noisy portion was at the beginning, we decided to directly align the reversed audio streams in the first phase. With this simple adaptation the results are as follows: of the 35 files containing interesting regions, 26 were correctly aligned; all of the 14 takes that contained exclusively noises were correctly discarded by the algorithm.

The absence of false positives (no noise-only takes were mistakenly aligned) and the correct positioning of all the aligned files suggest that the simple algorithm for identification of the interest region is robust enough to be applied to rather short audio segments, yielding the possibility of repeating the alignment algorithm multiple times on different subregions of the audio in order to avoid noisy sections.

WORKFLOW ADAPTATION

The audio industry has established over the years common standards for mixing that are adopted in most professional studio records worldwide. Integration of new technologies within existing workflow therefore requires special attention to existing practices within the community. To this end, we conducted several interviews with sound engineers.

From an R&D standpoint, an ideal integration would be a direct implementation of this technology into the graphical user interface of common DAW softwares to maximize usability. Such integration would allow novel possibilities, such as linking two tracks by means of their alignment and defining the placement of transition points between them for crossfading, avoiding any destructive editing regarding the discarded audio regions. Such integration requires direct contact with software houses which are mostly close to public domain development.

An alternative solution is represented by standalone alignment tools, whose outputs should be directly importable into a commercial DAW. Virtually all the major DAWs and video post production systems support the Open Media Framework (OMF) and the Advanced Authoring Format (AAF), respectively owned by Avid Technology, Inc. and by the Advanced Media Workflow Association (AMWA) 1 . These are employed as interchange formats to allow interoperability between different software. An alignment software, that we are currently developing, could automatically construct an initial session using an interchange format that audio engineers can use in their DAW to start the mixing process.

CONCLUSIONS AND FUTURE WORK

In this paper, we attempted to address two issues: Introducing novels tools generalizing audio matching algorithms to partial alignment with relevant region detection, and their integration within realistic studio mixing procedure to accelerate mixing session preparation for audio engineers. The first task involves adapting audio alignment techniques to situations where there is no specific prior knowledge on the 1 http://www.avid.com, http://www.amwa.tv starting point of the alignment. Such considerations would allow audio engineers to automatically obtain a global view of many different individual takes with regards to a reference run-through recording in a typical recording session, as well as providing access to relevant parts within each take; this is a time-consuming task if done manually. We further discussed how this procedure can realistically be integrated into common mixing workflows.

Applications of the proposed technology are not limited to the preparation of the initial mixing session: mid-level information obtained during the alignment task can in fact be further integrated in a studio mixing workflow. For example, our audio alignment provides useful information about the tempo of a performance with regards to the reference that can be employed as an important factor for the mixing engineer. Such integration requires further collaboration with audio engineers to determine an optimal exploitation of these informations in the context of existing practices.

Figure 1 .

 1 Figure 1. A typical DAW mixing session.

B

  Identification of the interest region of the alignment. ) = diag(1, -1) p(xB|z0 . . . zB) (c) Correction of the beginning of the alignment.

  . . . z0) (d) Final alignment obtained using smoothed inference.

Figure 2 .

 2 Figure 2. Alignment methodology.

Figure 3 .

 3 Figure 3. Evolution of p(s k |z 1 . . . z k ).

Figure 4 .

 4 Figure 4. Identification of the interest region.

Table 1 .

 1 Datasets used for evaluation.

	dataset	n. of rec.		duration [s]
			ref. takes (avg,std)	total
	Brahms	20 + ref.	588.8	112.8, 92.0 2844.0
	Manoury 49 + ref. 2339.4	113.5, 94.0 7900.4


Jean Michel Chapuis
email: jchapuis@free.fr

Price Fairness versus Pricing Fairness

Keywords: consumer perceptions, price fairness, pricing fairness, tourism study

This research note discusses the distinction between these two concepts of perceptions of fairness, based on the theory of distributive justice and procedural justice, in order to helps understand consumer behavior. With a sample of 250 tourists in French Polynesia and a structural equation model, tourists do not confuse price fairness and pricing fairness. The theoretical implications are that future research should use two distinct scales. For managers, the study suggests that the attention devoted to explaining the fairness of the pricing has more impact on consumer satisfaction than some attempts to explain the price.

Introduction

Long haul trips are usually book well in advance of departure, as much as 6 months on average in the case of tourism to French Polynesia. Moreover, the purchasing behavior for such services is also usually long, taking up to several days. In the while, firms such as airlines set prices according to the available capacity to sell. Their latest technologies of data collection and computing allow real time pricing based on level of demand and individual level tracking of sites visited and actions taken for example.

Price variations resulting from such pricing scheme are particularly salient at the stage of choice in the buying process. This research aims at exploring whether perceived fairness of such practices relate to price and pricing dimensions.

Both price offered and the rationale for offering a certain price may lead to perceptions of fairness [START_REF] Xia | The price is unfair! A conceptual framework of price fairness perceptions[END_REF] and to negative consequences for the consumer, such as dissatisfaction, distrust [START_REF] Chapuis | Perceived fairness and trust in consumer"s reactions to revenue management[END_REF], and lower intentions to repurchase (e.g., [START_REF] Oliver | Consumer perceptions of interpersonal equity and satisfaction in transactions: A field survey approach[END_REF] and by extension for the seller (e.g., [START_REF] Campbell | Perceptions of price unfairness: Antecedents and consequences[END_REF]. Price fairness is a buyer's judgment of a seller"s price [START_REF] Haws | Dynamic pricing and consumer fairness perceptions[END_REF].

Although consumers are able to quickly identify unfair situations, it is conversely more difficult for consumers to assess whether a policy is fair [START_REF] Bolton | Consumer perceptions of price (un)fairness[END_REF][START_REF] Taylor | How hotel guests perceive the fairness of differential room pricing[END_REF].

However, the relationships between consumers" perceptions of pricing fairness and their price fairness perceptions are a two-way causality. Pricing fairness can be inferred from certain types of outcomes and be a consequence of perceived price fairness [START_REF] Garbarino | Consumer response to norm-breaking pricing events in e-commerce[END_REF]. The process that leads to the final price may play a significant role in price fairness perceptions. The effect of price perceptions of price fairness can be mediated by perceived pricing fairness [START_REF] Xia | Effects of Consumers' Efforts on Price and Promotion Fairness Perceptions[END_REF]. The purpose of this research is to understand the interactions between consumers" perceptions of price fairness and pricing fairness. The research has implications for future researches and for managers.

The structure of the paper is the following. The first section reviews the literature. The second section presents an exploration study. The third section discusses and draws the implications of the results.

3/12

02/03/2012

A theoretical background of perceived fairness

People judge price as an outcome as well as a process, referring to equity theory and procedural justice respectively.

Price fairness

Equity judgment considers the comparison the ratios of inputs and outputs of each partner within a relationship or with another person as a reference. [START_REF] Adams | Inequity in social exchange[END_REF] defines a state of inequity when one perceives that the ratio of ones inputs and outputs is not equal to that of the other. Equity theory attempts to explain transactional satisfaction in terms of perceptions of fair or unfair distributions of resources within interpersonal relationships. Then, the breakdown of the proportionality results in the feeling of being treated unfairly. Individuals who perceive themselves in inequitable relationships will experience distress because they value fair treatment. They are motivated to reduce the inequity within the relationship either by distorting contributions (e.g. trust) and/or rewards (e.g. satisfaction) in their own minds and altering costs and/or benefits (e.g. complaining) or by leaving the relationship. Price fairness is a consumer"s assessment and associated emotions of whether the difference (or lack of difference) between a seller"s price and the price of a comparative other party is reasonable, acceptable, or justifiable [START_REF] Xia | The price is unfair! A conceptual framework of price fairness perceptions[END_REF]. Price fairness judgments may be based on previous prices, competitor prices, and profits [START_REF] Bolton | Consumer perceptions of price (un)fairness[END_REF].

Pricing fairness

Attribution theory explores the cognitive perceptions of individuals when they "attribute" causes to events [START_REF] Weiner | An attributional theory of achievement motivation and emotion[END_REF]. Inferences are important determinants of consumers" affective and behavioral responses to price discriminations. Procedural justice refers to the perceived fairness of policies, procedures, and criteria used by decision makers by which ends are accomplished [START_REF] Thibaut | Procedural justice: A psychological analysis[END_REF]. Pricing fairness considers the comparison of the pricing process with social norms [START_REF] Cox | Can differential prices be fair?[END_REF].

In this case, the social norms are the rules that the community agrees sellers should follow when setting prices [START_REF] Garbarino | Consumer response to norm-breaking pricing events in e-commerce[END_REF]. For ex., Rohlfs and Kimes 4/12 02/03/2012 (2007) explored the consumers" perceptions of the fairness of blended and non-blended best-available price policies in the hospitality industry. As a result of a day-to-day seasonal price discrimination, the room price changes from night to night in the nonblended policy. The offer is economically equivalent to quoting a single average rate for a multiple-night stay. The authors found that respondents rated average price policy less fair than the non-blended prices. Whether or not a pricing scheme improves the firm"s profit, the attribution of a negative motive to it will cause the perception of pricing (un)fairness [START_REF] Campbell | Perceptions of price unfairness: Antecedents and consequences[END_REF]. From the above framework, this note posits that Hypothesis 1: Perceptions of fairness is a multidimensional concept, dealing with price and pricing fairness.

If so, how do they interact? Pricing fairness can be inferred from perceived price fairness. A perceived fair price may increases the perception of a fair pricing scheme [START_REF] Kimes | Has revenue management become acceptable? Findings from an international study on the perceived fairness of rate fences[END_REF]. Buyers, having perceived that the rules pertaining to the price have not been followed, will infer that the rules of pricing have also not been followed [START_REF] Maxwell | Rule-based price fairness and its effect on willingness to purchase[END_REF].

On the other way, perceptions of price fairness may be affected not only by the price paid but also by the rules that were used to set the prices. Kimes and Wirtz (2003, p. 128) assumes that a fair price is one that results from a fair pricing rule. However, a price can break the equity ratio while the pricing policy used to set it does not break a norm, or conversely. For example, advance purchase discount is quit rational and accepted but yields lower outcomes (higher prices to people who book latecompensated by more flexibility and/or less uncertainty about the value of the product) to people who book at the last moment. Promotion fairness appears as a mediating factor between perceived input and price fairness perceptions [START_REF] Xia | Effects of Consumers' Efforts on Price and Promotion Fairness Perceptions[END_REF]. Finally, pricing fairness can constitute a unique source of fairness alongside price fairness considerations. Therefore: Hypothesis 2a: There is a positive relationship between consumers" perceptions of pricing fairness and their perceptions of price fairness.

Experimental design

A common research method in studies of fairness is to ask participants of experiments to imagine themselves in hypothetical situations. Five percent of the November 2009 flow of passengers (n=250) were conveniently chosen at the queues of outbound flights from Tahiti International Airport (PPT)1 . A scenario concerning an airline which makes a pricing decision that questions the fairness of the transaction is offered to each participant. Price discriminations are manipulated within scenarios in a between-subject experiment2 . The manipulation of a competitor"s price along with a price increase decided by the airline leads to two situations expected to trigger fairness perceptions. First, the participant confronts a price increase during the buying process (manipulation to USD 2,000). Second, it provides a relative position, as the participant encounters a price increase higher or lower than what he would afford on the market (manipulation to USD 1,500 or 2,500). Having read a randomly assigned scenario, participants answered several questions about the fairness of their situation (Table 2).

A confirmatory factor analysis (CFA) investigates the validity of the constructs.

The procedure checks whether the measurements reflected the latent variables by assessing the convergence validity according to the goodness-of-fit and factor loadings statistics [START_REF] Gerbing | An updated paradigm for scale development incorporating unidimensionality and its assessment[END_REF].

The price fairness and pricing fairness as endogenous constructs are measured using 5-point Likert-type scales, all anchored at endpoint from "strongly disagree"=1 to 6/12 02/03/2012 "strongly agree"=5. The scales are adapted from [START_REF] Campbell | Perceptions of price unfairness: Antecedents and consequences[END_REF] and [START_REF] Xia | Effects of Consumers' Efforts on Price and Promotion Fairness Perceptions[END_REF] modifying them to fit the study settings. The reliability for the price fairness is high (Jöreskog"s Rho=0.88). The scale for the pricing fairness is reliable (Rho=0.75). The overall satisfaction with the reservation process is measured by two items based on the studies of Oliver and Swan (1989) (anchored "strongly disagree"=1 to "strongly agree"=5). The scale is reliable (Rho=0.75). The CFA model is estimated using a bootstrap procedure available in AMOS ( 16). The related goodness-of-fit statistics are large (χ 2 (6) =12.63, p=0.05; AGFI=0.94 and RMSEA=0.066). All standardized residual covariances were lower than 2. Table 3 presents the standardized estimates of the factor loadings and other statistics. An evidence of internal consistency has stemmed from Jöreskog"s Rhos between 0.75 and 0.88. As evidence of convergent validity, all measurement factor loadings were significant (all t-values >2). All items loaded only on their respective factor.

Results

The discriminant validity of the constructs

The study explicitly addresses the dimensionality of fairness constructs taking into account the random error. The hypothesis 1 assumes that fairness consists of distinct, although possibly interrelated, price fairness and pricing fairness judgments.

The convergence validity of a two-dimensional construct of fairness is not rejected as the items loaded only on their respective construct. The discriminant validity presumes that a construct should share more variance with its measures than it shares with other constructs of the same causality level within the model. The perception that a price is unfair is expected to trigger the judgment of the pricing process as unfair, suggesting a positive coefficient between them and vice versa (H2: >0). The significant positive correlation (=.74; p<.01) between the price fairness and pricing fairness is lower than the root of the average variance extracted of each factor (respectively .79 and .61 -Fornell and Lacker, 1981). A formal test of the discriminant validity assumes as null hypothesis that the two constructs are exactly similar (=1.00). Relaxing this constraint 8/12 02/03/2012 leads to a significant decrease in the χ 2 (χ 2 (1) =42.54; p<.01) meaning a better goodnessof-fit of the model rejecting the null hypothesis. So, there is evidence to reject the absence of discrimination and support H1 and H2a that price fairness and pricing fairness are two distinct, positively interrelated constructs.

Perceptions of price fairness and pricing fairness influence satisfaction

Model 1 in Table 4 shows that 32% of the variance of satisfaction is explained by price fairness and pricing fairness. However, only the influence of pricing fairness is significant (=0.74, t=4.25) while price fairness is not (=-0.22, t=-1.64).

Discussion and managerial implications

Overall, the analysis is consistent with the hypotheses 1 and 2 that fairness is a two-dimension construct and an antecedent of satisfaction. The respondents, who had experienced a price increase during the buying process, were concerned with fairness in the price as well as in the pricing process as predicted. The mains contributions of the research are distinguishing perceptions of price fairness and of pricing fairness, triggered during the buying process.

First and consistent with the current trend in consumer studies [START_REF] Xia | Effects of Consumers' Efforts on Price and Promotion Fairness Perceptions[END_REF], the theoretical conceptualization of perceptions of price fairness and pricing fairness appears to be statistically significant. This is consistent with [START_REF] Maxwell | Rule-based price fairness and its effect on willingness to purchase[END_REF] results that price fairness and pricing fairness are two distinct concepts with a positive two-way relationship. The findings mean that fairness perceptions are not only induced solely by the fact that a consumer has to pay a price higher than previously promised but also by the pricing procedure used to set that price. The results are also consistent with the Oliver and Swan"s (1989) conclusion that fairness emerges as the dominant determinant 10/12 02/03/2012 of consumer satisfaction. The study shows that using two constructs increase the explanatory power of the model.

Second as a main result, the study shows that perceptions of fairness appear early, during the buying process, not only at the time of consumption. This is consistent with both [START_REF] Bolton | Consumer perceptions of price (un)fairness[END_REF] and [START_REF] Xia | The price is unfair! A conceptual framework of price fairness perceptions[END_REF] that fairness judgments consider self-comparison and other-comparison. Overall, the study has implications for researchers and practitioners alike.

The findings suggest that research working with only one construct of fairness about price as an outcome as well as a process is limiting the empirical relationships.

For example, [START_REF] Choi | Hotel revenue management and its impact on customers' perceptions of fairness[END_REF] and [START_REF] Lindenmeier | The effects of inventory control and denied boarding on customer satisfaction: The case of capacity-based airline revenue management[END_REF] found that the lowest satisfaction is associated with unfavorable price comparison. The findings here extend our knowledge and are consistent with their results by showing how price and pricing fairness perceptions affects satisfaction after a price discrimination.

Moreover, the proposed framework may help explain the inconsistent results across studies.

Implications for managers are twofold. The managers should act very early in the buying process and do not wait hoping the customer will not meet (during the consumption) another who pays a lower price. Second, managers should adapt the information they provide about any price difference in any situation as perceptions of fairness may occur at the price or pricing level. The effect of perceived pricing fairness on consumers" price fairness perceptions suggests that managers should be concerned as the whether consumers are actually paying attention to and understand the price restrictions [START_REF] Xia | Effects of Consumers' Efforts on Price and Promotion Fairness Perceptions[END_REF]. Managers should frame the price tag to lessen the adverse impact at the transactional level. They should explain the rational of the pricing scheme at the relationship (social norms) level. In the sample, this is the more efficient solution as, when pricing fairness goes up by one standard deviation, satisfaction increase on average by .72 standard deviation. While increasing price fairness by one standard deviation does not significantly increase satisfaction.

11/12 02/03/2012

Conclusion

This research contributes to introduce a distinction between perceptions of price and pricing fairness. Moreover, the results show that those perceptions occurring during the buying process have an impact on satisfaction. The results from a sample of 250 passengers at an international luxury destination supported that experiencing price discrimination while booking a flight leads to perceptions of fairness having an influence on consumer satisfaction.

The caveats of the limitations of this research are warranted. While scenarios may not be perfect, they are useful for exploratory purpose. An implication of the study is that researchers should carefully distinguish the source of perceived fairness.

However, the current study is based on scales already available in the literature without formally developing a dedicated scale according to the dimensions of the fairness concept. This calls for a future research to assess the validity of a more relevant instrument.

Hypothesis 2b :

 2b Perceptions of price and pricing fairness influence consumer satisfaction. 5/12 02/03/2012 3 Empirical Analysis in the tourism industry This section aims to empirically test the interactions of price fairness and pricing fairness in a survey-based study with a Structural Equation Modeling (SEM). Its first part designs the experiment and presents the latent variables. The second provides the empirical results of various models.

Table 2 :

 2 Constructs and descriptive statistics of items(N= 250)

	Construct	Item description
		Mean	Std.Dev.	Skewness	Kurtosis -3
	Price	x 1 : I paid the right price for my ticket
	Fairness	2.53	0.84	0.09	-0.44
		x 2 : The price that I was charged for my ticket was acceptable
		2.62	1.03	0.20	-0.43
	Pricing	x 3 : I was treated fairly by the airline"s pricing process
	Fairness	2.58	0.97	0.15	-0.59
		x 4 : The price that I was charged for my ticket was reasonable
		2.43	0.96	0.38	-0.50
	Satisfaction	x 5 : Overall, I am satisfied with the airline"s reservation service
		2.80	0.95	-0.05	-0.67
		x 6 : My experience with the reservation process of the airline is better
		than expected
		2.92	1.01	-0.31	-1.00

Table 3 :

 3 CFA

	Construct (AVE)	Item	 t-value	R²
	price fairness		x 1	0.89	16.12 ***	0.79
	(0.79)		x 2	0.89	16.06 ***	0.78
	pricing fairness	x 3	0.77	13.01 ***	0.60
	(0.61)		x 4	0.78	13.15 ***	0.61
	Satisfaction		x 5	0.82	10.68 ***	0.68
	(0.60)		x 6	0.73	9.80 ***	0.53
	χ 2	(6) =12.63; χ 2	/d.f. =2.10; p=.05; AGFI=0.94; RMSEA=0.066

Legend:  are the standardized factor loadings. *** indicates p-values lower than .01, AVE = average variance extracted, AGFI = adjusted goodness-of-fit index, RMSEA = root mean square error of approximation.

Table 4 :

 4 Standardized parameter estimates of modelsUp to this point, the analysis indicates a strong support for the H2b. Two competing models are also assessed for completeness especially if only one factor is considered. The Model 2a considers the influence of price fairness only. Then price fairness is a significant determinant of consumer satisfaction (=0.35, t=4.56) but the model has a low explanatory power (R²=.12). The Model 2b considers the influence of pricing fairness separately. The total effect of pricing fairness on satisfaction is positive and significant ( 31 =0.52, t=5.62) with R²=.27. Moreover, the χ 2 test shows that not taking into account price fairness does not reduce significantly the goodness of fit, as the model 2b is nested with the model 1 (χ 2(1) =2.8; p=.09). Lastly, the Model 3 provides an indirect test of H1. The model assumes that a single variable fairness from the 4 items of price fairness and pricing fairness. The results show that fairness is a significant determinant of consumer satisfaction (=0.38, t=4.82) but the model has a low explanatory power (R²=.14), similar to the model 2a. This competing model was less fitted to the data than the proposed model 1 (χ 2 (38) =77.68, p<.01). Overall, the

	Models	goodness-of-fit		Path	Sd Est	t-value	R²
	Model 1		χ 2	(6) =12.63; p=.05			0.32
				Price fairness	<-> Pricing Fairness	0.74	7.42 ***
				Price fairness	->	Satisfaction	-0.22	-1.64
			Pricing Fairness ->	Satisfaction	0.72	4.25 ***
	Model 2a	χ 2	(7) =38.1; p<.00	χ 2	(1) =25.5; p<.00	0.12
				Price fairness	<-> Pricing Fairness	0.75	7.58 ***
				Price fairness	->	Satisfaction	0.35	4.56 ***
	Model 2b	χ 2	(7) =15.5; p=.03	χ 2	(1) =2.8; p=.09	0.27
				Price fairness	<-> Pricing Fairness	0.72	7.35 ***
			Pricing Fairness ->	Satisfaction	0.52	5.62 ***
	Model 3		χ 2	(13) =77.68; p<.00			0.14
				Fairness	->	Satisfaction	0.38	4.82 ***
	Legend: Sd Est are the standardized estimates of covariance (<->) and path (->). ***
	indicates p-values lower than .01.

results support that both perceptions of fairness influence consumer satisfaction.

The descriptive statistics are available from the author. The tests address the bias of nonresponse, law of distribution, representativeness of the sample, relationship between variables and characteristics of participants.

 2 The method to ensure the realism of the data offered in the scenario, its test of realism as well as the manipulations checks, are available from the author.