E Dupoux

C Pallier

N Sebastian

J Mehler

A Destressing "Deafness" in French?

Spanish but not French uses accent to contrast between words (e.g., tópo vs topó). Two populations of subjects were tested on the same materials to determine whether this difference has an impact on the perceptual capacities of listeners. In Experiment 1, using an ABX paradigm, we found that French subjects had significantly more difficulties than Spanish subjects to perform an ABX classification task based on accent. In Experiment 2, we found that Spanish subjects were unable to ignore irrelevant differences in accent in a phoneme-based ABX task, whereas French subjects had no difficulty at all. In Experiment 3, we replicated the basic French finding, and found that Spanish subjects benefited from redundant accent information even when phonemic information alone was sufficient to perform the task. In our final experiment, we showed that French subjects can be made to respond to the acoustic correlates of accent; therefore their difficulty in Experiment 1 seem to be located at the level of short-term memory. The implications of these findings for language-specific processing and acquisition are discussed.

. Much less is known about the way in which rhythmical and structural properties of the native language affect encoding of foreign words. This study reports four experiments to assess the claim that nonsegmental properties of the native language greatly affect the way in which unfamiliar items are processed and represented.

 for an early formulation of this view and Miller and Jusczyk (1990) for a later one. There is no question that segments are an essential ingredient of the sound pattern of language though many other properties of speech are also important. Foreign accents reflect segmental difficulties but also supra-segmental ones. For instance, informal observations suggest that French speakers who ac-

quire English after puberty produce segments that are not prototypical of English, and prosodic output is usually deficient as well. It is not uncommon to hear speakers of French who fail to reduce vowels or who make the wrong vowel prominent. Many mistakes are, in fact, due to the inability of native speakers of French to place stress in the correct place. How is one to explain these consistent mistakes? Why is it that even proficient French speakers of English produce anomalous prosodic structure?

In this paper we try to show that such informal observations as the ones alluded to above reflect the essential processing routines and representation structures elaborated specifically for one's native language. [START_REF] Cutler | The periodicity bias[END_REF] and also Christophe and Dupoux (in press) argue that prosody is essential during first language acquisition. When the child first learns a language, he or she extracts its rhythmicalperiodical properties. Those properties facilitate language acquisition and trigger adjustments on perception and production routines that allow efficient language-specific processing. Moreover, in adults, these adjusted perception and production routines are no longer very flexible and disrupt in certain ways the processing of foreign languages. This helps us to explain how natural languages are processed differently by native and by proficient foreign speakers of those languages.

These considerations lead us to conjecture that speakers of different languages become sensitized, not only to different phonetic contrasts, but also to different prosodic properties. If this hypothesis is correct, determining which rhythmical and prosodic structures create problems for foreign speakers should help us to understand how the native language is acquired and how foreign languages are subsequently learned.

Consider the accent differences between Spanish and French. In Spanish, the default position for accent1 in multisyllabic words is on the penultimate syllable. However, many words have accent in other positions. Indeed, there are minimal pairs that have exactly the same segments but differ in meaning because of the accent location. For example, bébe ("drink" present tense) vs bebé ("baby"), or t ópo ("mole") vs top ó ("met") drawn from Spanish, exemplify this property (see Navarro Tomas, 1946). In French, however, accent does not carry lexical information; there is no such contrast as bébe versus bebé. In this language, accent is either described as being not specified lexically, or as falling on the last full vowel of content words [START_REF] Dell | Les développements récents en phonologie : quelques idées centrales[END_REF].

Thus, speakers of Spanish have to process and represent accent to identify the lexical item(s) intended by the speaker. Speakers of French, however, do not need to process accent, at least not in the same way. As far as lexical identification is concerned, only segmental content matters, and accent information could be completely left unspecified in the French lexicon.

The following experiments investigate this hypothesis by testing the ability of French and Spanish speakers either to detect stress differences (Exp. 1 and Exp. 3) or to ignore them (Exp. 2). The final experiment addresses the issue of the level at which French/Spanish differences in performance may arise.

Experiment 1

Our first study tested native speakers of French and of Spanish with an ABX discrimination task involving an accent contrast. Subjects were presented with three items that varied only in accent location and had to press a button to indicate whether the last item corresponded to the first or to the second item. In this experiment, we tested two accent contrasts on trisyllabic CVCVCV items: bópelo vs bopélo (1 st vs 2 nd syllable with accent), and bopélo vs bopel ó (2 nd vs 3 rd syllable with accent).

Note that these accent patterns are all possible in Spanish, although the penultimalte pattern (bopélo) may be said to be the unmarked case. Since our two experimental contrasts involve items with accent in the penultimate syllable, there is no reason to expect that, for Spanish subjects, 1 st vs 2 nd syllable accent would be easier or harder than 2 nd vs 3 rd . In contrast, French is a language with accent on the last full vowel of content words. One might expect accent-final items to be perceived as natural, whereas items with nonfinal accents should be perceived as foreign-sounding. In this case, 2 nd vs 3 rd syllable accent involves a difference in legality, and should be easier for the French than 1 st vs 2 nd (which are both illegal). This is not, of course, the only possibility. French subjects might assimilate all accent patterns to one, or simply, may not represent accent at all. In this case, both contrasts would be equally difficult.

French and Spanish differ in more than placement of accent. First, French has been described as a syllable-timed language. Spanish is also a syllable-timed language, but it also involves an alternating pattern of prominence peaks that reflects a higher order prosodic unit: the foot. Nonetheless, previous psycholinguistic investigations found that the two languages processed syllabic structures in very similar ways [START_REF] Bradley | The status of the syllable in the perception of spanish and english[END_REF][START_REF] Mehler | The syllable's role in speech segmentation[END_REF][START_REF] Pallier | Attentional allocation within syllabic structure of spoken words[END_REF][START_REF] Sebastian-Galles | Contrasting syllabic effects in catalan and spanish[END_REF]. Second, French has 14 vowels, and Spanish only five. Similarily, Spanish and French differ in the consonantal inventory and in allowed consonant clusters. In order to cope with these differences, our materials only used a subset of segments and syllable structure legal in both languages.

The following experiments use the exact same stimuli for French and Spanish hearers. However, since recording either French or Spanish speakers might have too obviously advantaged one population over the other because of the fine difference in the phonetic realization of phonemes in the two languages, we decided to record the stimuli using speakers of a third language: Dutch. Dutch is a stress-timed language, which allows accent to appear in difference places in words. The phoneme subset that we selected for French and Spanish contains phonemes that are all pronouncable in Dutch. In an informal pretest, Dutch realization of these phonemes were judged to be slightly foreign-sounding but highly acceptable both to French and Spanish listeners.

Method

Materials. Twelve CVCVCV triplets of the form (b ópelo, bopélo, bopel ó) were constructed (see Appendix A). Each member of a triplet had the same phonemic content, and the main accent falling on the first, second or third syllable. None of the vowels were reduced, and they belonged to the set [e],[i],[o],[a], [u]. All items were nonwords both in French and Spanish.

The 12 triplets were recorded by three speakers of Dutch (two females, one male) and checked by a phonetician. They were instructed to mark word accent clearly on the first, second or third syllable, while keeping the other vowels unreduced. Each item was digitized at 16KHz at 16bits (on an OROS AU22 A/D Board), digitally edited, and stored on a computer disk.

From the recorded experimental triplets, 96 trials were constructed. Each trial consisted of three stimuli: A, B and X, the first two said by the two female speakers and the third by the male speaker. The A and B items always had the same segmental content, but differed in accent. Two contrasts were tested: 1 st vs 2 nd and 2 nd vs 3 rd . For a given contrast, A and B received the two accents in the two possible orders: (e.g., bópelo bopélo, or bopélo bópelo). This resulted in 4 different A-B combinations for each experimental triplet. The X item always had the same segmental content as A or B, but on half of the trials, X had the same accent as A and on the other half, it had the same accent as B. The overall design was 2 x 2 x 2: Accent-Contrast x Accent-Order x X-identity.

The 96 experimental trials were split into two blocks (with each condition and stimuli represented equally often in each block). An extra practice block of 10 trials was constructed that contained the same conditions but with different items from the experimental blocks.

Procedure. Each experimental trial presented the three stimuli (A, B and X), separated by an interval of 500 ms. Subjects were instructed to listen to words in a foreign language. They were told that the first two stimuli were different, and that the third one was identical to the first or to the second. They were required to press a button on their left or on their right to indicate whether X was identical to A or to B respectively. Subjects were given a 4000 ms deadline to respond. The next trial started 1000 ms after each response, or after the deadline.

In the ten practice trials, subjects received feedback as to whether their response was correct or not. Feedback consisted in the word "Correct", "Incorrect", or the string "The response is A/B" when subjects failed to respond before the deadline. It was displayed for 1000 ms and then erased from the screen. For incorrect responses, the same trial was presented again, until the response was correct. In the two experimental blocks of 48 trials, no feedback was presented. The blocks were separately randomized for each subject. A short pause separated the two experimental blocks. Responses were recorded and reaction time was measured from the onset of the X stimuli by the EXPE software package [START_REF] Pallier | Expe5: an expandable programming language for on-line psychological experiments[END_REF].

Subjects. Sixteen French and 16 Spanish subjects participated in the experiment. The French subjects were students recruited in Paris. There were 10 men and 6 women, 14 right handers and 2 left handers in the group. Their median age was 25. The Spanish subjects were students at the University of Barcelona who received course credit for the experiment. There were 4 men and 12 women, 13 right handers and 3 left handers. The median age was 20. None of the subjects understood the other language nor did they understand Dutch.

Results

One item (vasuma) generated many errors in this experiment and in the next, and was therefore removed from subsequent analyses.

Mean reaction times and error rates broken down by Language (French or Spanish), Accent Contrast (1 st vs 2 nd or 2 nd vs 3 rd ) and Response Type (A-Response or B-Response) are displayed in Table 1. These two dependent variables were subjected respectively to two ANOVAs, one with subjects and one with items as random variables. Language was a between-subject factor, Response Type and Accent Contrast were within-subject factors, and all three factors were withinitem.

In the errors analyses, French subjects made significantly more errors than Spanish subjects (19% vs 4%, F1(1,30)=17, p .001; F2(1,10)=158, p .001).

No effect of Accent Contrast was found (Fs 1), and this factor did not interact with Language. A-Responses yielded more errors than B-Responses (13% vs 10%, F1(1,30)=4.0, p .05; F2(1,10)=20, p .001), but this effect was mostly due to French subjects, and mostly for the accent 2 nd versus 3 rd contrast. The three-way interaction between Response Type, Language, and Accent-Type was only significant in the subjects analysis (F1(1,30)=4.9, p .03; F2(1,10)=4.3, p .1), and the twoway interactions between Response Type and Language, and Response Type and Accent Type were only significant in the items analysis (p .05). No other interaction was significant.

The reaction time analyses revealed that French subjects were slower than Spanish subjects, although the effect was only significant in the items analysis (94ms, F1(1,30)=1.8, p¡ .1; F2(1,10)=77, p .001). There were no effects of Accent Type. Finally, A-Responses were slower than B-Responses (37ms, F1(1,30)=7.5, p .01; F2(1,10)=9.6, p .01) and there was no significant interaction with the other factors.

Discussion

French and Spanish subjects were asked to perform an ABX task based on two accent contrasts. Spanish Subjects made very few mistakes but French subjects had difficulty performing the task. This difference can be seen mostly in the error rate, although there was also a trend in reaction times.

Interestingly, there was no noticeable difference between the accent 1 st vs 2 nd contrast and the 2 nd vs 3 rd contrast, either for the French or for the Spanish subjects. For the Spanish, both comparisons involve accent-second items (the unmarked case) and two other accent patterns that are possible in this language. It is, therefore, not very surprising that the two contrasts were equally easy for the Spanish population. However, one might expect the French listeners to find the accent-final items more prototypical than the other two accent patterns. Yet there was no difference between 1 st vs 2 nd and 2 nd vs 3 rd contrasts, suggesting that the difficulty that French subjects have with accent is general.

Experiment 1 demonstrated that Spanish subjects are overall more efficient than French subjects in discriminating accent contrasts. However, it might be that the Dutch materials contained vowels or consonants that were more prototypical for the Spanish than for the French subjects. In order to control for these potential effects, it would be useful to reverse the previous pattern of results, that is, to find a situation that should be easy for the French to perform and difficult for the Spanish.

Experiment 2

In this experiment, we asked subjects to ignore accent, and to respond on the basis of segmental information only. Accent was varied orthogonally with phonemic structure so that relying on accent information would lead to chance performance. The reasoning is that French subjects who have difficulty hearing accent in the first place, would have no difficulty in ignoring it. In contrast, for the Spanish speakers, accent plays an important role in word identification. Hence, re- quiring Spanish subjects to ignore accent should be difficult. This experiment used the same materials as Experiment 1, and the same subjects population.

Method

Materials. Twelve CVCVCV quadruplets of the form (b ópelo, bopélo, sópelo, sopélo) were constructed (see Appendix B). They resulted from the crossing of two contrasts: an accent contrast (accent in the first or accent in the second syllable) and a phoneme contrast (a single phoneme change in one of the syllables). The segmental content of the first two members of the quadruplets was the same as in Experiment 1. The phoneme change occurred equally often in the first, second or third syllable. All items were nonwords both in French and in Spanish.

The 12 quadruplets were recorded by the same three speakers of Dutch as in Experiment 1 and were digitized in the same way. From the recorded experimental quadruplets, 192 experimental trials were constructed. Each experimental trial consisted of three stimuli: A, B and X, the first two spoken by the two females and the third by the male. The A and B items always differed in one phoneme and in placement of accent. In half of the trials, A had accent first, and in the other half it had accent second. In half of the trials, A was a stimulus from the Experiment 1 set, and in the other half it was from the 'phoneme-change' set. This resulted in 4 different A-B combinations for each experimental quadruplet (e.g. bópelosopélo, bopélo-sópelo, sópelo-bopélo, and sopélo-b ópelo). X always had the same segmental content as A or B, and had the same accent as A or B. This resulted in 4 different cases that we will consider as the crossing of two factors: whether X had the segments of A or B (Response Type), and whether X had an accent congruent with its segments (Congruency). As far as the last factor is concerned, there are two cases: in one condition, X was identical to A or B both in terms of accent and phoneme. This was called the Accent Congruent Condition. In the other, X had the segments of one stimulus, and the accent of the other. This was the Accent Incongruent Condi-tion. The overall design was thus 4 x 2 x 2: A-B combinations x Response Type x Congruency.

The 192 experimental trials were split into four blocks of 48 trials each (A, B, C and D). Each condition and stimulus was represented equally often in each block. An extra practice block of 10 trials was constructed that contained the same conditions but with different items from the experimental blocks.

Procedure. Stimulus presentation and feedback were as in Experiment 1. Subjects were instructed to listen to words in a foreign language in which accent contrasts were irrelevant. They were told that the first two stimuli were words that differed by one speech sound, and that the third one was identical to the first or to the second in terms of their speech sounds. They were required to press a button on their left or on their right to indicate whether X was identical to A or to B respectively.

After the instructions, subjects performed the ten practice trials, receiving feedback as to whether their response was correct or not. As in Experiment 1, subjects were not allowed to pass on to the experimental session unless they had correctly responded to all the items in the practice session. After the practice trials had been completed, subjects received two experimental blocks of 48 trials (half of the subjects received Blocks A and B, and half received Blocks C and D), with no feedback. The blocks were separately randomized for each subject. A short pause separated the two blocks. Response deadline, intertrial interval and reaction time measurement were as in Experiment 1.

Subjects. Sixteen French and sixteen Spanish subjects participated in the experiment. They were drawn from the same population as in Experiment 1. The French subjects included 11 men and 5 women, 13 right handers and 3 left handers. Their median age was 22. The Spanish subjects were 2 men and 14 women, 15 right handers and 1 left hander. The median age was 21. None of the subjects understood the other language.

Results

As in Experiment 1, the item vasuma generated many errors and was removed from subsequent analyses.

Reaction times and error rates were subjected to subjects and items ANOVAs (see means in Table 2). Language (French or Spanish) was a between-subject factor, Response Type (A-Response or B-Response), and Congruency (Congruent vs Incongruent) were within-subject factors. All three factors were between items.

The error analyses revealed an effect of Language, with Spanish subjects making more errors than French subjects (11.6% vs 6.6%, F1(1,30)=4.0, p¢ .05; F2(1,10)=17, p¢ .002). The Incongruent Condition generated more errors than the Congruent Condition (10.9% vs 7.3%, F1(1,30)=7.4, p¢ .01; F2(1,10)=8.7, p¢ .01), but this was mostly due to the Spanish subjects. However, the interaction between Language and Congruency was only significant in the items analysis (F1(1,30)=2.4, p£ .1; F2(1,10)=6.8 p¢ .03). A-responses generated more errors than B-responses (11.7% vs 6.5%, F1(1,30)=11.6, p¢ .002; F2(1,10)=11.5, p¢ .007), although this was mostly due to Spanish subjects. The interaction between Response Type and Language was significant in both analyses (F1(1,30)=4.8,p¢ .04; F2(1,10)=10.0, p¢ .01). No other interaction reached significance.

The reaction time analyses showed that Spanish subjects had significantly longer reaction times than French subjects (178ms, F1(1,30)=4.1, p¢ .05; F2(1,10)=59, p¢ .001). Congruent responses did not differ from Incongruent responses (p£ .1). A-responses yielded slower reaction times than B-responses (41ms, F1(1,30)=4.2, p¢ .05; F2(1,10)=6.2, p¢ .03). There was an interaction between Congruency and Response type which was only significant in the items analysis (F1(1,30)=3.8, .05¢ p¢ .1; F2(1,10)=5.7, p¢ .04), and no other interaction was significant.

Discussion

We found that French subjects were both faster and more accurate than Spanish subjects, reversing the pattern ob-served in Experiment 1. This suggests that the French can concentrate on the segments and ignore irrelevant variations in accent. In contrast, Spanish subjects were slower and made significantly more errors, suggesting that they cannot ignore irrelevant accent variations. Although the Spanish listeners made more errors in the Incongruent than in the Congruent condition, the effect was not reflected in reaction times, suggesting that ignoring accent was difficult for the subjects and globally increased their reaction times.

Figure 1 shows the overall results of Experiments 1 and 2. An overall ANOVA reveals a significant interaction between tasks and Language for both Reaction Times ( p¢ .02 in both items and subjects analyses) and errors ( p¢ .001 in both items and subjects analyses), confirming that subjects of the two languages process phoneme and accent information in a very different way.

Experiment 3

In the previous two experiments, we compared French and Spanish subjects on ABX tasks that focused either on the accent or on the phoneme. We found that French listeners had difficulties responding on the basis of accent whereas the Spanish listeners had trouble ignoring accent. However, these differences all rest on comparisons between groups of subjects drawn from similar but different populations (students in Paris vs students in Barcelona). Another shortcoming of the previous experiments is that the differences found might be due to the way in which subjects understood the task, rather than to the way in which they perceived the stimuli. This is especially true in Experiment 2 where subjects were asked to ignore accent. French and Spanish subjects may differ in their understanding of the notion of accent.

In order to evaluate these possibilities, we carried out a further experiment comparing directly for each individual subject responses based on accent and responses based on phonemes, while keeping the other dimension constant. We predicted that unlike Spanish Subjects, French subjects would have greater difficulty in discriminating accent as com- pared to phonemes. We also included a redundant condition in which both accent and phoneme information were associated with the same reponse. Because Spanish subjects represent both phonemic and accent information, they should be faster in the redundant condition than in either the phoneme condition or the accent condition. French subjects, in contrast, should respond in the redundant condition as if accent was unchanged. Because of the higher number of conditions in this experiment, and because we found no differences between 1 st vs 2 nd accent locations in the previous experiments, we decided to limit ourselves to just these two positions.

Method

Materials. Twelve new CVCVCV quadruplets of the form (fídape, fidápe, lídape, lidápe) were constructed (see Appendix C). They resulted from the crossing of two contrasts: an accent contrast (accent in the first or accent in the second syllable) and a phoneme contrast (a single phoneme change in one of the syllables). The phoneme change occurred equally often in the first, second, or third syllable. The vowel set was

[a], [e], [i], [o], [u] and the consonant set [p], [t], [k], [b], [d], [g], [f], [s], [l], [m],
[n] which are all common segments in Spanish, French (and Dutch). All items were nonwords, both in French and in Spanish. The 12 quadruplets were recorded by two speakers of Dutch (one male, one female), and were digitized as in Experiments 1 and 2.

From the recorded experimental quadruplets, 288 experimental trials were constructed. Each experimental trial consisted of three stimuli: A, B and X. A and B were spoken by the same female speaker and the X stimulus by the male speaker. A and B either differed by one phoneme only (fídape, lídape: Phoneme Condition) by accent only (fídape, fidápe: Accent Condition) or by both accent and phoneme (fídape, lidápe Redundant Condition). The trials were counterbalanced such that all members of a quadruplet (fídape, fidápe, lídape, and lidápe) appeared in both position A and B, for each of the above defined matching conditions. This resulted in 12 different A-B combinations for each experimental quadruplet. X was identical (both in accent and in phoneme) to either A or B. There were two response types (A-responses trials, and B-responses trials). The overall design was thus 3 x 4 x 2: Matching Condition x Quadruplet Counterbalancing x Response Type.

The 288 experimental trials were split into four blocks of 72 trials each (A, B, C and D). Each condition and stimulus were represented equally often in each block. A practice block of 10 trials was constructed that contained the same conditions but with different items as the experimental blocks.

Procedure. The scheduling of practice and experimental trials was as in Experiments 1 and 2. Subjects were instructed to listen to words in a foreign language. They were told that the first two stimuli were different words, and that the third was identical to the first or to the second. They were required to press a button on their left or on their right to indicate whether X was identical to A or to B, respectively.

In the ten practice trials, subjects received feedback as to whether their response was correct. For incorrect responses, the same trial was presented immediately again until the response was correct. After practice, subjects received two experimental blocks of 72 trials (half of the subjects received Block A and B, and half received Block C and D), with no feedback. The blocks were separately randomized for each subject. A short pause separated the two blocks. Responses were recorded and reaction times measured from the onset of the X stimuli by the EXPE software package.

Subjects. Twenty French and twenty Spanish subjects participated in the experiment. The French subjects were students from the Ecole Polytechnique, Palaiseau. There were 15 men and 1 woman, 11 right handers and 5 left handers. Their median age was 22. The Spanish subjects were psychology students at the University of Barcelona who received course credit for their participation. There were 3 men and 13 women, 15 right handers and 1 left hander, with a median age of 20. About half of the French subjects had learned Spanish as a third language (the second one being English) at the age of 14. None of the Spanish subjects had learned French. None of the subjects knew Dutch.

Results

Reaction times and error rates were subjected to subjects and items ANOVAs (see means in Table 3). Language (French or Spanish) was a between-subject factor, Response Type (A-Response or B-Response) and Matching Condition (Phoneme, Accent and Redundant) were within-subject factors. All three factors were within-item.

The two populations did not differ in their mean error rate (all ps¤ .1). However, there was a main effect of Matching Condition (F1(2,76)=21, p¥ .001; F2(2,22)=17, p¥ .001) as well as an interaction with Language (F1(2,76)=5.3, p¦ .007; F2(2,22)=7.0, p¦ .004).

The Accent Condition yielded more errors than the Phoneme Condition, especially for the French subjects (10.8% vs 2.7%, both p¦ .001) and to a lesser extent for the Spanish subjects (8.3% vs 5.6%, p¦ .05). The Phoneme and Redundant Conditions did not differ in terms of error rates. A-responses yielded more errors than B-responses, (13.8% vs 5.3%, F1(1,38)=11.6, p¦ .002; F2(1,11)=16.2, p¦ .002) and Response Type interacted both with Language (F1(1,38)=4.9, p¦ .03; F2(1,11)=20, p¦ .001) and with Matching Condition (F1(2,76)=10.5, p¦ .001; F2(2,22)=4.9, p¦ .01). No other interaction was significant.

Overall, Spanish subjects had significantly faster reaction times than French subjects (191ms, F1(1,38)=4.8, p¦ .03; F2(1,11)=169, p¦ .001). The Matching Condition factor was significant (F1(2,76)=27, p¦ .001; F2(2,22)=15, p¦ .001), as was an interaction with Language (F1(2,76)=10, p¦ .001; F2(2,22)=25, p¦ .001). To understand the interaction, we ran a series of post-hoc F-tests. The Accent Condition was significantly slower than the Phoneme Condition for French listeners (146ms, p¦ .001 both by items and by subjects), but not for Spanish listeners (both ps § .1). In contrast, the Redundant condition was significantly faster than the Phoneme Condition for Spanish (49ms, p¦ .05 by subjects, and p¦ .02 by item), but not for French listeners (ps § .1). A-responses were significantly slower than B-responses (108ms, F1(1,38)=25, p¦ .001; F2(1,11)=34, p¦ .001). The Response Type factor interacted with the Matching Condition (F1(2,76)=12, p¦ .001; F2(2,22)=6.0, p¦ .01), and there was a three-way interaction between Response Type, Matching Condition and Language (F1(2,76)=4.6, p¦ .01; F2(2,22)=3.5, p¦ .05). These interactions can be accounted for by the fact that the difference between accent and phoneme is much larger in A-Responses than in B-Responses (and mostly due to the French listeners). Also, the difference between the Redundant and Phoneme Conditions is larger in B-responses than in A-Responses (and mostly due to the Spanish listeners).

Discussion

In this experiment, French subjects were slow in the Accent Condition and relatively fast in the two other conditions. The error data corroborate the pattern observed in the response latencies. Spanish subjects were fastest in the Redundant Condition and equally slow in the Accent and in the Phoneme Conditions.

This pattern of data supports our hypothesis. French subjects have difficulty in perceiving accent and do not use accent in the Redundant Condition. Consequently, their performances are identical in the Phoneme and in the Redundant Condition. Spanish subjects, in contrast, perceive both accentual information and phonemic information with equal ease. These two conditions therefore show similar reaction times and errors. When accentual and phonemic information are redundant, Spanish subjects are faster2 .

Notice that in the present experiment, the accent condition produces errors in French listeners (10%), but relatively less so than in Experiment 1 (20%). One possible reason for this is that only one accent contrast was used in the present experiment, whereas Experiment 1 used two contrasts: 1 st vs 2 nd and 2 nd vs 3 rd . Having only to deal with one accent contrast may allow French subjects to focus more successfully way on the acoustic information necessary to perform the task. The other possibility is that we only used two voices in the present experiment, thereby allowing subjects to compare A and B stimuli in terms of rather low level characteristics (since they were spoken in the same voice). In Experiment 1, in contrast, all three stimuli were spoken in a different voice, thereby inducing subjects to use a more abstract level of representation. This predicts that if we used only one speaker, French sub-jects may use low level representations and have even less trouble with accent contrasts. We examine this possibility in the next experiment.

In the first three experiments, we consistently found differences according to Response Type. Overall, A-Responses were more difficult than B-responses. This may reflect the fact that judging immediate identity (B-Responses) may be performed on a shallow memory store, whereas judging identity when there is some intervening material (A-Responses) requires holding several stimuli in memory, and keeping track of the order of each stimulus. Overall, A-Response situations may just be more confusing and open to different strategies than B-Response situations.

Apart from this general effect, response type interacts with the matching conditions in interesting ways. In particular the difficulty that French subjects have with accent (relative to some baseline) is always larger in A-responses than in Bresponses (Exp 1 and 3). One possible explanation for this is that, as the task gets harder, existing differences in difficulty are amplified. Even in B-responses, the French subjects have difficulty with accent. One other possible explanation is that, for the French subjects, accent information is mostly represented in some acoustic store, which decays rather quickly and is poorly represented or not represented at all in a more abstract and longer term storage. In B-response situations, French subjects can judge accent identity based on this low level store. In A-response situations, the strategy would be less probable.

This raises the issue of the level at which the French deficit should be located. Is it the case that French subjects are impaired in hearing the acoustic correlates of accent, or do they not code accent in some short term memory buffer? The next experiment was designed to examine this question.

Experiment 4

The results obtained so far suggest that French subjects have difficulty with accent. However, we do not know at which level this difficulty arises. Is it the case that French subjects have lost perceptual sensitivity to accent contrast or rather that they have trouble with representing and storing in working memory accent patterns that are otherwise accurately perceived? In order to explore this issue, we placed French subjects in a simplified situation in which they only had to discriminate between two stimuli that were spoken by the same speaker. Subjects heard two stimuli (separated by a tone of varying duration) and had to judge whether the stimuli were the same tokens or varied in their pronunciation. There were two Different conditions, one in which the stimuli differed in accent (fídape, fidápe), and one in which they differed in one phoneme (fídape, lídape). In the Same condition, subjects heard two different recordings of the same stimuli.

In order to assess the potential effect of memory load, a pure 2KHz tone of either 200 ms or 2200 ms was inserted between each member of a stimulus pair. The idea is that for long intervening tones, echoic/acoustic memory traces of the first stimulus in a pair should be less accessible than for short tones, forcing subjects to rely on a more central representa-tion. If French subjects are impaired in the early processing of the acoustic correlates of accent, they should have problems with the Accent Condition (compared to the Phoneme Condition), irrespective of tone duration. However, if the problem is linked to later processing/encoding stages, one would expect to see no problem with accent in this simplified task, or only in the long tone condition.

Method

Materials. The 12 experimental quadruplets of Experiment 3 were used and 288 new experimental trials were constructed. Each experimental trial consisted of two stimuli, A and X, which were spoken by the same Dutch female as in Experiment 3. A and X were either identical in accent and phoneme (although they were different tokens, fídape, fídape), differed by one phoneme only (fídape, lídape: Phoneme Condition) or by accent only (fídape, fidápe: Accent Condition). The trials were counterbalanced so that all members of a quadruplet (fídape, fidápe, l ídape, and lidápe) appeared in position A, for each of the above defined matching conditions. This resulted in 12 different A-X combinations for each experimental quadruplet. Each A-X combination could appear in two ISI conditions, separated by a 200 ms 2kHz intervening pure tone, or a 2200 ms 2kHz pure tone. The overall design was thus 3 x 4 x 2: Matching Condition x Quadruplet Counterbalancing x ISI.

The 288 experimental trials were split into four blocks of 72 trials each (A, B, C, and D). Each condition and stimulus were represented equally often in each block, except that blocks A and C had only short ISI conditions, and blocks B and D had only long ISI conditions. A practice block of 10 trials was constructed that contained the same conditions as the experimental blocks but with different items.

Procedure. Each experimental trial consisted of the following sequence of three events: stimulus A, tone, stimulus B, each separated by an interval of 300ms. Subjects were instructed to listen to words in a foreign language separated by a tone. They were told that the two words were either identical or pronounced in a different way. They were required to press a button on their right or on their left to indicate whether X was identical to or different from A, respectively. The deadline and intertrial intervals were the same as in the previous experiments.

In the ten practice trials, subjects received feedback as to whether their response was correct or not. For incorrect responses, the same trial was presented immediately again, until the response was correct. After practice, subjects received two experimental blocks of 72 trials with no feedback presented (half received Blocks A and B, and half received Blocks C and D). The blocks were separately randomized for each individual subject. Subjects were distributed into two groups according to whether they received the short ISI or the long ISI first (orders A-B, and C-D, for short ISI first, and B-A and D-C for long ISI first). A short pause separated the two blocks. Responses were recorded and reaction times measured from the onset of the X stimuli by the EXPE software package.

Subjects. Twenty French students recruited in Paris participated in the experiment. There were 10 women, and 10 men in the group. Their median age was 30. None of the French subjects had learned Spanish as a foreign language or knew Dutch.

Results

Table 4 shows reaction times and error rates on Same and Different responses across the different conditions. We analysed the Different responses, with ISI and Matching Condition as within subject factors and Group as a between subject factor. All three factors were within-item.

In the error analysis, there was no main effect but there was an interaction between condition and ISI in the items analysis (F1(1,19)=2.5, p¨.1; F2(1,11)=10.3, p© .008). This interaction was due to the fact that, in the long ISI condition, the Accent condition tended to generate more errors than the Phoneme condition (4.3% vs 1.7%, F1(1,19)=1.4, p¨.1; F2(1,11)=17.4, p© .002), whereas no such trend was found in the short ISI condition (2.1% vs 2.7%, ns).

Reaction times in the Accent Condition were not significantly different from those in the Phoneme Condition (F1(1,19)=1, p¨.1; F2(1,11)© 1). Long ISIs resulted in longer reaction times than short ISIs but this difference was only significant in the items analysis (41ms, F1(1,19)=2.6, p¨.1; F2(1,11)=25.2, p© .001). The two factors did not interact (p¨.1).

Discussion

In this experiment, we had subjects perform an AX discrimination task on either accent or phoneme contrasts. The results show that French subjects can discriminate stimuli that differ only in accent, and do so with few errors (3.2%) compared to what we found in Experiments 1 and 3 (19% and 11%). The performance was similar for accent or phoneme contrasts and remained accurate when the two stimuli were separated by a 2200 ms tone, although there was a trend towards more errors for accent contrasts in the long tone condition. This suggests that French subjects can detect the acoustic correlates of accent and maintain a working memory of these correlates for more than 2 seconds. In this simplified paradigm, French subjects have very accurate access to the acoustic correlates of accent contrasts.

We can offer two reasons why the accent distinction may have been easier in the present paradigm than in Experiments 1, 2 and 3. One would be due to the fact that the present AX paradigm only involves two stimuli (the tone being ignored), which decreases the memory load compared to the ABX paradigm. Subjects may only have used some kind of immediate buffer3 rather than having to store the identity and sequential order of three trisyllabic items (which is almost a supra-span situation in terms of number of syllables). The other reason may be that the present situation involved no change in talker, allowing the task to be performed at a less abstract level than in the previous two experiments. Hence, subjects might now use an acoustic representation to discriminate accent, a strategy less available when there was a speaker change.

Both accounts converge on the same idea. We suggest that acoustic information is processed in basically the same way in French and Spanish subjects. However, in order to retain such information in a short term memory store, this information has to be recoded into a more abstract level. We further suggest that language-specificity comes into play at this level. After acoustic information is processed, it is recoded in a different linguistic format by speakers of different languages. Word accent plays no lexical role in French, and is hence not represented at this level. Why this should be so will be discussed below.

General discussion

In this study, we have documented the poor ability of French speakers to deal with contrasts in accent. French subjects, contrary to Spanish subjects, have difficulty in making discriminations based on accent as indicated by slow reaction times and numerous errors (Experiment 1). This deficiency is coupled with the absence of any beneficiary or detrimental effect of concomitant accent variations when they perform judgments based on phonemes (Experiments 2 and 4). In contrast, Spanish subjects readily discriminate among accent patterns (Experiment 1). Moreover, they can be shown to extract and represent such information automatically, even when they are asked to focus on segmental information. Orthogonal variations of accent slow them down when they have to perform a task based on phonemes only (Experiment 2). Mutatis mutandis, Spanish subjects respond faster when accent and phonemes are redundantly varied compared to cases when only one dimension is available for decision (Experiment 3). For Spanish subjects, accent appears to be a nondetachable aspect of phonological information, whereas for the French, this information is not represented, at least not at the level that is supposedly tapped by this task. Stress is probably used at a different level by French speakers, e.g., for finding word or phonological phrase boundaries. This would be compatible with claims made by Frasier (1987) and [START_REF] Church | Phonological parsing and lexical retrieval[END_REF] that noncontrastive/allophonic information is used to parse speech into phonological constituents. Our final study shows, however, that French subjects are not altogether insensitive to differences in lexical stress, in that, under appropriate circumstances, they can detect the acoustic correlates of accent (Experiment 4).

This research sheds new light on past studies of languagespecific processing. Languages are often characterized by the set of phonemes that they use to distinguish words. For example, English uses the contrast between [r] and [l] (race versus lace), whereas Japanese does not. These differences affect the ability of adults and young infants to distinguish speech sounds [START_REF] Best | Examination of perceptual reorganization for nonnative speech contrasts: Zulu click discrimination by english-speaking adults and infants[END_REF][START_REF] Goto | Auditory perception by normal japanese adults of the sounds 'r' and 'l[END_REF][START_REF] Mann | Distinguishing universal and languagedependent levels of speech perception: Evidence from japanese listeners' perception of english [l] and [r][END_REF][START_REF] Miyawaki | An effect of linguistic experience: the discrimination of /r/ and /l/ by native speakers of japanese and english[END_REF]Werker & Tees, 1984a). Similarily, languages are characterized by higher order units (such as syllables, or morae) which allow the specification of restrictions on the co-occurrence of individual segments. Evidence for cross-linguistic variation in the way these structures are used in perception have been found [START_REF] Bradley | The status of the syllable in the perception of spanish and english[END_REF][START_REF] Cutler | A language specific comprehension strategy[END_REF][START_REF] Cutler | The role of strong syllables in segmentation for lexical access[END_REF]Mehler, Segui, & Frauenfelder, 1981;[START_REF] Otake | Mora or syllable? speech segmentation in japanese[END_REF][START_REF] Pallier | Attentional allocation within syllabic structure of spoken words[END_REF][START_REF] Sebastian-Galles | Contrasting syllabic effects in catalan and spanish[END_REF][START_REF] Zwitserlood | The role of the syllable in the perception of spoken dutch[END_REF].

Our study shows that language-specific effects are not restricted to differences in the segmental inventory (or in the inventory of higher order units). Suprasegmental information, such as accent is also treated in different ways by listeners of different languages4 . We believe it is likely that our results will generalize to other accent distributions. For instance, we predict that listeners who speak languages with fixed initial accent will behave like French listeners. We also predict that our results will extend to other dimensions (tone, pitch accent, etc).

Current models have viewed speech perception as consisting essentially in the discovery of phonemes arranged in their sequential order [START_REF] Mcclelland | The TRACE model of speech perception[END_REF][START_REF] Norris | Shortlist: a connectionist model of continuous speech recognition[END_REF]. Although suprasegmental information was never explicitly excluded from these models, studies dealing with segments are much more common. On the basis of the above reported studies we are inclined to favor a view according to which segmental information should be complemented with a variety of abstract representations of prosodic dimensions. Moreover, the way in which these dimensions are represented is not universal. Depending on the language's phonology, dimensions that have a contrastive value will be specified in full detail, and others that have a fixed or predictable distribution will not be represented at all. Such a view is compatible with studies showing that knowledge about the use of suprasegmental structure in the native language begins to be acquired very early by the young infant, perhaps earlier than knowledge about permissible segmental units [START_REF] Jusczyk | Preference for predominant stress patterns of english words[END_REF][START_REF] Mehler | A precursor of language acquisition in young infants[END_REF][START_REF] Mehler | The role of suprasegmentals in speech perception and acquisition[END_REF].

Eimas and his colleagues published a number of experiments that show that shortly after birth infants can discriminate contrasts that are used in a natural language [START_REF] Eimas | On infant speech perception and the acquisition of language[END_REF]. Werker and her colleages, pursued this line of work and demonstrated that before the end of their first year infants begin to neglect some contrasts absent from the language they are mastering (for instance, Werker & Tees, 1984a[START_REF] Polka | Developmental changes in perception of nonnative vowel contrasts[END_REF], but some contrasts remain, see Best, McRoberts & Nomathemba, 1988). [START_REF] Jusczyk | Infant's recognition of foreign versus native language words[END_REF] found that infants show some sensitivity to their language's phonotactic properties around the age of 9 months. Notice, however, that Kuhl and her colleagues [START_REF] Kuhl | Human adults and human infants show a "perceptual magnet effect" for the prototypes of speech categories, monkeys do not[END_REF] have shown that the vowels are learned around 6 months, that is, a few months prior to the consonants.

Mehler and colleagues have argued that infants learn about the prosody much before they learn about the segments of their own language. Indeed, [START_REF] Mehler | A precursor of language acquisition in young infants[END_REF] have shown that very young infants react differentially to novel sentences from their native language very shortly after birth. Moreover, infants still display such a differential reaction when the utterances are filtered to ensure that most of the segmental information is removed and only the prosodic information remains available. [START_REF] Moon | Two-day-olds prefer their native language[END_REF] report similar results. Thus, infants individuate prosodic properties of their native language before they begin to specify its segmental aspects. Incidentally, it can be argued that the early individuation of prosodic properties protects the infant who is confronted from birth by multilingual input. [START_REF] Jusczyk | Perception of acoustic correlates of major phrasal units by young infants[END_REF] have shown that infants around 6 months are sensitive to sentence level prosodic boundaries. Jusczyk et al. (1993) have shown that by 6 months, English infants start to be sensitive to the typical stress pattern of their language (that is, strongweak). We conjecture that around the same age, French infants should start losing sensitivity to accent contrasts.

Although much remains to be done, we believe that prosodic information will be added to our current models of adult speech perception and improve our understanding of how speech is acquired and used by human infants.

Experiment 2: bopelo-sopelo, povami-lovami, tamido-pamido, vasuma-fasuma, detoma-deboma, lumisa-ludisa, picadu-piradu, rimato-ripato, mepado-mepato, metilo-metibo, someta-somefa, baveta-baveka. Experiments 3 and 4: bedapi-nedapi, kidosa-kimosa, dabitu-dalitu, nelufo-nepufo, dufoga-dufola, nibako-tibako, fidape-lidape, nolaku-nosaku, fubeno-fubeto, poleda-poleka, kebuli-pebuli, tamido-tamipo.

Figure 1 .

 1 Figure1. Reaction times (in grey) and error rates (in black) to ABX judgments in French and Spanish Subjects. Fig1a: ABX on accent only; phonemes fixed (Experiment 1). Fig1b: ABX on phonemes only; accent varied orthogonally (Experiment 2).

Table 1

 1 Mean reaction time, standard error, and error rate to ABX judgments based on two accent contrasts (Experiment 1): accent 1 st versus 2 nd , and accent 2 nd vs 3 rd .

	Language	Response Type	Mean	(se)	Err	Mean	(se)	Err	Mean	(se)	Err
			accent 1 st vs 2 nd		accent 2 nd vs 3 rd		Mean
			(bópelo bopélo)		(bopélo bopeló)
	FRENCH
		X=A	1230	(57)	21%	1261	(54)	23%	1245	(39)	22%
		X=B	1226	(61)	19%	1228	(49)	14%	1227	(39)	16%
		Mean	1228	(41)	20%	1244	(36)	18%	1236	(27)	19%
	SPANISH
		X=A	1163	(52)	5.1%	1178	(49)	3.9%	1171	(35)	4.5%
		X=B	1120	(41)	3.1%	1107	(39)	3.9%	1114	(28)	3.5%
		Mean	1142	(33)	4.1%	1143	(31)	3.9%	1142	(23)	4.0%

Table 2

 2 Mean reaction time, standard error, and error rate to ABX judgments based on a phonemic difference as a function of whether accent was congruent or incongruent with phonemic information (Experiment 2).

	Language	Response Type	Mean	(se)	Err	Mean	(se)	Err	Mean	(se)	Err
				Congruent			Incongruent			Mean
			(bópelo sopélo bópelo)	(bópelo sopélo bopélo)
	FRENCH
		X=A	1072	(69)	6.5%	1062	(67)	8.5%	1067	(47)	7.5%
		X=B	1016	(58)	5.1%	1056	(57)	6.2%	1036	(40)	5.7%
		Mean	1044	(44)	5.8%	1059	(43)	7.4%	1051	(31)	6.6%
	SPANISH
		X=A	1258	(82)	13.9%	1252	(64)	17.9%	1255	(51)	15.9%
		X=B	1173	(57)	3.7%	1233	(61)	11.1%	1203	(42)	7.4%
		Mean	1216	(50)	8.8%	1243	(44)	14.5%	1229	(33)	11.6%

Table 3

 3 Mean reaction time, standard error, and error rate to ABX judgments in a redundant, phoneme only and accent only condition (Experiment 3).

	Lang	Response Type	Mean	(se)	Err	Mean	(se)	Err	Mean	(se)	Err
				Redundant			Phoneme			Accent
			(fídape lidápe fídape)	(fídape lídape fídape)	(fídape fidápe fídape)
	FRENCH
		X=A	1005	(41)	2.7%	1019	(44)	2.1%	1234	(54)	13.7%
		X=B	1004	(47)	2.9%	1023	(39)	3.3%	1099	(51)	7.9%
		mean	1005	(31)	2.8%	1021	(29)	2.7%	1167	(38)	10.8%
	SPANISH
		X=A	958	(36)	7.9%	965	(31)	8.1%	1017	(36)	13.9%
		X=B	874	(30)	3.5%	965	(30)	3.1%	935	(32)	2.7%
		mean	916	(24)	5.7%	965	(21)	5.6%	976	(24)	8.3%

Table 4

 4 Mean reaction time, standard error, and error rate to acoustic same-different AX judgments, in an identity, phoneme change and accent change condition (Experiment 4).

	Condition	Mean	(se)	Err	Mean	(se)	Err
			Short ISI			Long ISI
				"Same" Responses
	Identity Condition	943	(29)	5.4%	972	(32)	5.2%
	(fídape fídape)
				"Different" Responses
	Phoneme Change	919	(27)	2.7%	942	(32)	1.7%
	(fídape lídape)
	Accent Change	918	(30)	2.1%	978	(42)	4.3%
	(fídape fidápe)

In this paper we use the term "accent" to refer to phonetic/perceptual prominence in a string of syllables. We stay neutral as to whether such prominence is correlated or not with the notion of metrical stress as used in phonology.

Note that the redundancy gain is only apparent in the reaction times, not in the errors. This could be due to the fact that there is some trial-to-trial variation as to whether the phonemic information is available earlier or later than the accent information. If in congruent trials, subjects use the first available information, they will then be faster on average than when they can only use one type of information. However accuracy will not be affected.

Note however, that any such immediate buffer seems to be not affected at all by an intervening 2200 ms tone.

[START_REF] Cutler | The role of strong syllables in segmentation for lexical access[END_REF] found that the distribution of stress plays a role in word segmentation in English. However, they claim that this effect is not really suprasegmental since it rests on a contrast between reduced and nonreduced vowels, a segmental distinction. In English, it is uncertain whether purely suprasegmental properties affect word recognition (see[START_REF] Cutler | The use of prosodic information in word recognition[END_REF].

This research was supported by grants from the Human Frontier Science Program, the Human Capital program, the Franco-Spanish Exchange Program for 1993-1995, the Fyssen Foundation and DRET. We thank Jennifer Glos, Xavier Hardy and Olivier Crouzet for help in stimulus construction and data gathering. We thank Stanislas Dehaene and Brit van Ooijen for very useful com-


Jean-Théo Jiolat
email: jean-theo.jiolat@sorbonne-universite.fr

Jean-Loïc Le Carrou
email: jean-loic.le_carrou@sorbonne-universite.fr

Christophe D'alessandro
email: christophe.dalessandro@sorbonne-universite.fr

A parametric study of finger motions when playing the clavichord : towards characterization of expressive control

Keywords: Clavichord, gesture, robotic finger

The clavichord is considered to be the most demanding keyboard instrument in terms of finger control. This is because of its direct mechanisms: the key works as a lever. When the finger presses the key, the tangent (metal blade) on the key's extremity goes up and strike the string. And as long as the finger remains pressed on the key, the tangent remains in contact with the string, leading to string's tone variation. The loudness of the sound is proportional to the velocity of the key's displacement. Then there is a duality between loudness and pitch accuracy. This is the paradox of the clavichord. The objective of the study is to analyze experimentally the vibro-acoustical consequences of the instrument with respect to the gestural strategies of the finger. To proceed, an experimented player performs in different configurations two main gestures: the pushed and pulled gesture. A robotic finger is used to simulate different trajectories in terms of downward displacement and velocity. The study shows that the pushed and pulled gestures have opposed influences on the fundamental frequency and on the sound level. The robotic finger demonstrates that a rise in sound level without a rise in fundamental frequency is possible.

INTRODUCTION 1.The clavichord's paradox

The clavichord is the earliest stringed keyboard instruments, dating back to the XVth century [START_REF] Brauchli | The clavichord[END_REF]. Its sound level is low compared to other stringed instruments, and it is the only keyboard instrument allowing for some pitch control. When a key is pressed, the corresponding pair of strings is impacted by a small metal blade (the tangent) placed at the end of the key. As long as the key is pressed, the tangent remains in contact with the strings. The tangent is at the same time the nut (i.e. one extremity) of the string and the string exciter (the string is then excited at a vibration node). It has been showed experimentally that the sound level of the clavichord is proportional to the tangent velocity at impact [START_REF]On the dynamics of the clavichord: From tangent motion to sound a[END_REF]. So the faster the key is pressed, the louder the sound becomes. However, when a key is pressed with a high velocity, the key's displacement tends to be higher. The tangent raises the string, then increases its tension, and thereby increases the vibration fundamental frequency. As a result, playing louder ends up in raising the pitch, if the key is pressed in a simple vertical motion. To control independently loudness and intonation would require a paradoxical gesture: at the tangentstring contact, the tangent should have enough velocity, but should not raise the string. In other words, the key should transfer all the tangent momentum to the string, but without raising it, or losing contact. This dependence between loudness and pitch accuracy is coined "the clavichord's paradox" [START_REF]Le paradoxe du clavicorde[END_REF][START_REF]Le paradoxe du clavicorde et la technique de Bach au clavier[END_REF]. It is difficult, at least for human players, to achieve exactly such a motion. However compromises between tangent impact velocity and string displacement are possible.

Historical clavichord techniques

The clavichord is considered as the most demanding among keyboards instruments in terms of finger control. This is because every nuance of pressure of the finger on the key is likely to change loudness and intonation. In addition both finger velocity and displacement must be controlled because of the clavichord's paradox. To deal with these constraints, specific performance practice have been elaborated. Because of these constraints, the clavichord has always been highly praised as a pedagogical instrument. Several texts describing clavichord performance around Johann Sebastian Bach's circle, "Every Players first Grammatica" to quote J.G. Walther (1732) (see [START_REF] Walther | Musicalisches Lexicon oder Musicalische Bibliothec[END_REF], page 169), mention a specific technique called "Schnellen" [START_REF] Van Delft | Schnellen. a quintessential articulation technique in eighteenth-century keyboard playing. The Keyboard in Baroque Europe[END_REF], which can be translated in French by "tire" [START_REF]Le paradoxe du clavicorde et la technique de Bach au clavier[END_REF] and in English by "pulled" (see for instance, J.J. Quantz, 1752, C.P.E. Bach 1752, Forkel 1802). In this technique, the finger tip is drawn back quickly after contact with the key, in a sliding motion.

Measurement and simulation of fingers motions

In a preceding work [START_REF]Le paradoxe du clavicorde[END_REF], the effect of vertical finger motion ("pushing motion") and sliding finger motion ("pulling") on loudness and pitch of clavichord tones have been studied. It has been shown that the pulling gesture is a better compromise for dealing with the clavichord's paradox: loudness and pitch are controlled more independently with pulled than with pushed motions. The aim of the present work is to study the clavichord's paradox with the help of new measurement techniques and robotic simulation: 1/ to measure accurately finger trajectories and their consequences on vibration and sound patterns (Section 2); 2/ to reproduce these trajectories using a robotic finger, in order to study the limits of the clavichord's paradox, and then the "optimal" trajectories, decoupling key velocity at impact and string displacement (Section 3). The objective is to measure the vibration of the excited string resulting from the motion of the musician's finger. In preceding works, measurements were performed with the help of an accelerometer near the tangent, a string-tangent contact signal and a measurement microphone. It appeared necessary to measure directly the finger motion and the string motion, using non-invasive measurement devices. Finger motions are filmed by a high-speed camera (Phantom Miro M 120) with a 2000 frame per second rate. Several marks are placed on the finger. Trajectories of these marks are estimated thanks to image processing. String vibrations are measured with the help of calibrated opto-switch sensors [START_REF] Le Carrou | A low-cost high-precision measurement method of string motion[END_REF]. These sensors are optical forks, positioned around the string. The string motion in one direction is measured with accuracy and without contact. Only the vertical displacement of the string is considered here (although the horizontal displacement can also be significant). The string chosen for our measurements is the G2 string (length is 70 cm, fundamental frequency 185 Hz). The sensor is placed at 2 cm from the extremity of the string, near the bridge in order to be within its measurement range. Sound pressure is measured with the help of an omnidirectional DPA 4006-TL microphone placed at 30 cm above the soundboard. A set of 8 trajectories are recorded, using index and middle fingers, pulled and pushed motions for long and short notes.

MEASUREMENTS OF FINGER AND VIBRATORY MOTIONS

Results

In figure 2, we used the videos to extract the trajectories representative of the two distinct motions : the pushed and pulled gestures. The pushed motion refers to a vertical trajectory, the finger going mostly downward. The pulled motion corresponds to a vertical and horizontal trajectory, the finger sliding on the key and going downward at the same time. The string height is also small at this position, about 0.2-0.3 mm. It is much larger at the tangent position.

The string is much more elevated in the case of the pushed gesture than the pulled one (see figure 3 and4). Because of this difference in string height, the string tension and then the sound fundamental frequency is higher for the pulled motion. Note that the vibration amplitude is also larger in the case of the pushed gesture, resulting in a louder tone. Fundamental frequency is measured on the sound and vibration signals using the Yin algorithm [START_REF] De Cheveigné | Yin, a fundamental frequency estimator for speech and music[END_REF] implemented in Matlab. Fundamental frequency with respect to time (G2 string) is displayed in figure 5. As predicted, the fundamental frequency is higher for the pushed gesture compared to the pulled gesture. The difference between the pushed and the pulled gesture is more than 4 cents for some conditions. Such a difference is perceptually noticeable. Fundamental frequency gives information about the way the musician deals with the contact between the tangent and the string with respect to time. In figure 5, one can observe that the fundamental frequency for the pushed gesture decreases with respect to time, whereas that of the pulled gesture remains around the same fundamental frequency although with some little hills. This shows that the key control differs for both gestures. These variations of finger depth after the string-tangent contact are certainly perceived in terms of quality of touch. The sound level in dB for the different microphone signals are displayed in figure 5 (integration time 250 ms). Pushed gestures produce higher sound levels than pulled gestures. This has already been observed on the signal amplitude in figure 3 and4. In summary, different gestures, corresponding to different finger trajectories, are producing different vibratory patterns of the string, and then different sounds. In the small set of recordings available, the pushed motions always produce a larger string displacement : the string is always raised higher, and the amplitude of vibration is larger. A larger amplitude of vibration results in a louder sound. A higher string height results in a higher fundamental frequency. For the same reasons, the finger motion in the case of pulled gestures gives lower fundamental frequencies and also weaker sounds. Note that in previous studies it has been shown that pulled motions, to some extent, allows for independent loudness and pitch control, a result that cannot be observed here, because no sample have similar loudness. These measurements are the first direct measurements of string height, and are in good agreement with the theory developed in [START_REF]On the dynamics of the clavichord: From tangent motion to sound a[END_REF].

ROBOTIC SIMULATION OF FINGERS MOTIONS

3.1 Experimental setup: the robotic finger Measurements of finger motion show the dependence between string height, sound radiated and fundamental frequency. As predicted by the clavichord's paradox, it seems difficult to control simultaneously the key (then tangent, then string) velocity and displacement. The pulled motion provides a better control and a better management of the clavichord's paradox, because the finger trajectory is more complex: pressure on the string can be released after the tangent-string impact.

It is interesting to study the clavichord's paradox with the help of controlled and reproducible key trajectories.

For this purpous, a robotic finger is used. The DROPIC robot [START_REF] Le Carrou | Dropic: A tool for the study of string instruments in playing conditions[END_REF] has been initially developed for simulation of finger trajectories in plucking gesture of harps [START_REF] Chadefaux | Harp plucking robotic finger[END_REF][START_REF] Chadefaux | A model of harp plucking[END_REF]. It has been applied to keyboard instruments in studies of the plectra effects for the harpsichord. [START_REF] Paté | Influence of plectrum shape and jack velocity on the sound of the harpsichord: an experimental study[END_REF]. The robotic finger has two degrees of freedom. It can reproduce any trajectory in a plane parallel to the axis of the key. Note that the key itself has only one degree of freedom.

The effort for depressing a key is relatively weak, less than 2 N. For a given starting trajectory, two parameters are considered and modified: downward displacement (resulting in string height) and its maximal velocity (corresponding to loudness). The A2 string (length is 59.1 cm, fundamental frequency 205Hz) is studied. The initial position of the robotic finger above the key is set before modifying either the velocity or the displacement. A joint measure of the string vibration by means of calibrated opto-switch sensors is performed. Three different velocities with the same displacement, and three different downward displacements with the same velocity have been programmed. The displacement of the key corresponding to a referent trajectory performed by the robotic finger is displayed in figure 8. The trajectory has a typical shape with a notch followed by a plateau. It is possible to adjust independently the depth of the notch and the height of the plateau, that correspond roughly to the key velocity at contact and to the string height. Systematic variations of displacement and velocity are performed. Note that in this second experiment, the note studied is A2 instead of G2 studied in Section 2. These two notes are close enough to be compared. In figure 8, the key velocity is varying but the key depth is constant. The key depth is about 5-6 mm in this case. The resulting average string elevation is 0.1 mm. Figure 7 displays the fundamental frequency of the different signals measured when the key is played with the robotic finger. The resulting fundamental frequency does not change, while the amplitude of the signal increases. This shows that the trajectory of the finger is well repeated by the robot no matter the change in velocity. Moreover, it demonstrates that the clavichord's paradox can be managed with appropriate trajectories. These results also confirm that the displacement of the key is directly linked to the string's fundamental frequency. Conversely, changing the displacement of the key but maintaining the same velocity produces changes in fundamental frequency. However, fundamental frequency is very stable in the case of the robotic finger compared to a musician's finger (compare figures 7 and 5).

These results demonstrate that a robotic control is able to manage the clavichord's paradox. Whether human and robotic control are comparable is questionable. In the present experiment, the robotic finger has no haptic or sound feedback: the trajectories are optimized directly, without any perceptual loop. On the contrary, human control relies much on audio and haptic feedback. The musicians tend to control the contact between the tangent and the string after the excitation by modifying the key position according to the perceived effect of their initial motion. This variation in time of the key position is probably an essential feature of the specific style of a musical performance. Another difference between the robotic finger and human finger is their mechanical and dynamical properties. Human fingers have a much limited range of velocity and acceleration than the robot. Oscillations of the key-string-finger system that are observed in human control [START_REF]On the dynamics of the clavichord: From tangent motion to sound a[END_REF] seem negligible in the case of the robotic finger (see the displacement of the key in figure 8).

CONCLUSIONS

This work presents two experiments addressing the clavichord's paradox, i.e. simultaneous control of velocity and displacement of the string and tangent when playing the instrument. In a first part, new measurements using a new methodology is used. Two types of finger trajectories have been used for performing two different motions : the pushed and the pulled gesture. This experiment confirms the dependence of displacement and velocity, and the possibility to modulate this dependence with appropriate gestures. In the second experiment, a robotic finger is used to further optimize the key trajectory, by modifying in terms of velocity and downward displacement a referent trajectory. In this case it seems possible to manage the clavichord's paradox, and to control independently velocity and displacement, i.e. intonation and loudness of the instrument. Whether a musician would be able to effectively perform this type of movement remains an open question.

Figure 1 .

 1 Figure 1. Photo from above of the Hubert clavichord.

2. 1

 1 Experimental setup The instrument under study has been built by C. d'Alessandro and C. Besnainou, and completed in 2007 (at The Paris Workshop, led by M. Ducornet, in Montreuil). It is based on a kit designed by E. Dancet and M. Ducornet after XVIIIth century unfretted clavichord models by G. Hubert. The instrument is not an exact copy of an historical model. It has been built especially for acoustic investigations, but it has occasionally been played in concert. The instrument has 51 keys, from C to d3, with double strings in brass. Its dimensions are 1267 mm x 358 mm x 112 mm. It is tuned at A=415 Hz, in a Kirnberger II temperament. Vibrating string lengths C = 1097mm, c = 926mm, c1 = 509mm, c2 = 262mm, c3 = 122mm.

Figure 2 .

 2 Figure 2. (Top) Images of the pushed (left-hand side) and the pulled gesture (right-hand side) performed by the index finger. (Bottom) Trajectories of the pushed (left-hand side) and the pulled gesture (right-hand side) performed by the index finger (with t b the beginning time and t e the ending time).

Figure 3 .

 3 Figure 3. Vibratory signal of the G2 string excited by means of the two different trajectories done by the index finger with a short length (left-hand side), with a zoom at the beginning of the signals (right-hand side).

Figure 4 .

 4 Figure 4. Vibratory signal of the G2 string excited by means of the two different trajectories done by the index finger with a long length (left-hand side), with a zoom at the beginning of the signals (right-hand side).

Figure 5 .

 5 Figure 5. Fundamental frequency of the signals in the case of the pushed and pulled gesture done by the middle finger and the index finger (left-hand side). Sound level of the different exciting configurations (right-hand side) (I : Index finger, M : Middle finger, S : Short, L : Long).

Figure 6 .

 6 Figure 6. Measurement devices: (left-hand side) optical forks for string displacement. (right-hand side) DRoPi-Crobotic finger for key trajectory control.

Figure 7 .

 7 Figure 7. Tracking of the fundamental frequencies of the signals in the case where we modify the displacement of the key (left-hand side) and in the case where we modify its velocity (right-hand side).

Figure 8 .

 8 Figure 8. Temporal signals of the A2 string produced by the different velocities of the key (left-hand side). Displacement of the key in the case of the referent trajectory performed by the robotic finger (right-hand side).

ACKNOWLEDGEMENTS

The authors thank Sebastien Guy and Laurent Quartier for their help during the measurements.


A behavioral approach of decision making under risk and uncertainty

Thomas Garcia

To cite this version:

Thomas Garcia. A behavioral approach of decision making under risk and uncertainty. Economics and Finance. Université de Lyon; Queensland University of Technology. Brisbane, Australie, 2019. English. NNT : 2019LYSE2042. tel-02341761

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Michael A Sherbon
email: michael.sherbon@case.edu

Golden Ratio Geometry and the Fine-Structure Constant

Keywords: fine-structure constant, fundamental constants, history of physics, history of mathematics, golden ratio, quartic equation

The golden ratio is found to be related to the fine-structure constant, which determines the strength of the electromagnetic interaction. The golden ratio and classical harmonic proportions with quartic equations give an approximate value for the inverse fine-structure constant the same as that discovered previously in the geometry of the hydrogen atom. With the former golden ratio results, relationships are also shown between the four fundamental forces of nature: electromagnetism, the weak force, the strong force and the force of gravitation.

Introduction

From John Rigden, "The fine-structure constant derives its name from its origin. It first appeared in Sommerfeld's work to explain the fine details of the hydrogen spectrum. ... Since Sommerfeld expressed the energy states of the hydrogen atom in terms of the constant [alpha], it came to be called the fine-structure constant." [START_REF] Rigden | Hydrogen: The Essential Element[END_REF]. Arnold Sommerfeld states, "In our theory of the fine structure there is a confluence of the three main currents of modern research in theoretical physics, namely, the theory of electrons, the theory of quanta, and the theory of relativity" and the fine-structure constant "... could be interpreted more physically as the ratio of an electron's velocity in the first Bohr orbit to the speed of light." [START_REF] Sommerfeld | Atomic Structure and Spectral Lines[END_REF]. From previous work, "The electromagnetic coupling constant determining the strength of its interaction is the fine-structure constant α = e 2 /hc in cgs units with the elementary charge e, the reduced Planck's constant h = h/2π and the speed of light c." [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF].

The fine-structure constant has a variety of physical interpretations from which measurements and determinations have been made. Even though the fine-structure constant is often defined as a low energy limit, in high energy physics it has become commonplace to regard it of less importance because the gauge coupling runs with energy. As Toichiro Kinoshita explains, "... the detailed high energy structure of the ultimate theory is irrelevant to the analysis of low energy phenomena except insofar as it determines these parameters." [START_REF] Kinoshita | Quantum Electrodynamics[END_REF]. What actually determines the fine-structure constant remains questionable for high energy theorists.

A reminder from Jesse DuMond's work that is still of significance today:

These results illustrate clearly the highly interrelated nature of the fundamental constants. Dependent as our knowledge of them is upon many different fields of physics, we have here a good example of the importance of making occasional analyses of the consistency situations of sufficiently inclusive scope to serve as valuable guides to further research. The present example emphasizes especially that a better knowledge of the Sommerfeld constant, α, would be of great value to physics at the present time [START_REF] Dumond | The Present Key Importance of the Fine Structure Constant, α, to a Better Knowledge of All the Fundamental Physical Constants[END_REF].

From the writings of Wolfgang Pauli, "The theoretical determination of the fine structure constant is certainly the most important of the unsolved problems of modern physics." [START_REF] Pauli | Writings on Physics and Philosophy[END_REF][START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. Richard Feynman said it was the greatest mystery of physics, perhaps the most often quoted physicist on the fine-structure constant: "It has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. ... You might say the 'hand of God' wrote that number, and 'we don't know how He pushed his pencil'" [START_REF] Feynman | The Strange Theory of Light and Matter[END_REF].

In the Timaeus, Plato "... considered the golden section to be the most binding of all mathematical relationships and the key to the physics of the cosmos," as quoted by Robert Schoch and Robert McNally in Pyramid Quest [START_REF] Schoch | Pyramid Quest: Secrets of the Great Pyramid and the Dawn of Civilization[END_REF]. Peter Champoux and William Buehler state that, "The capstone of the Pyramid generates the golden spirals which creates the whole form." [START_REF] Champoux | Gaia Matrix: Arkhom and the Geometries of Destiny in the North American Landscape[END_REF].

The golden ratio φ = (1 + √ 5)/2 and is a root of its minimal polynomial x 2 -x -1. Both the golden ratio and its conjugate are roots of the quartic equation x 4 -2x 3 -x 2 + 2x + 1 = 0. For "... a particular quartic plane curve, different combinations of the coefficients of the general curve give rise to the lemniscate of Bernoulli. ... Gauss's and Euler's study of the arc length of Bernoulli's lemniscate, a polar curve having the general form of a toric section, led to later work on elliptic functions." [START_REF] Sherbon | Fundamental Nature of the Fine-Structure Constant[END_REF]. The inflection points of the quartic are also related to the golden ratio [START_REF] Totland | Quartic Polynomials and the Golden Ratio[END_REF].

Calculation of the fine-structure constant

An approximation from previous work [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF], the inverse fine-structure constant α -1 is a root of:

x 4 -137x 3 -10x 2 + 697x -365 = 0. ( 1
)
This equation gives a value of for x as α -1 ≃ 137.035 999 168. The latest determination from the Gabrielse group [START_REF] Gabrielse | Towards an Improved Test of the Standard Model's Most Precise Prediction[END_REF] is α -1 ≃ 137.035 999 150 [START_REF] Heyrovska | The Golden Ratio, Ionic and Atomic Radii and Bond Lengths[END_REF], from experimental measurement and quantum electrodynamics [START_REF] Aoyama | Theory of the Anomalous Magnetic Moment of the Electron[END_REF]. This calculation of the inverse fine-structure constant gives the same approximate value as ancient geometry combined with the extension of Raji Heyrovska's work on the golden ratio structure of the hydrogen atom [START_REF] Sherbon | Fine-Structure Constant from Golden Ratio Geometry[END_REF]. 697 -137 = 280 + 280. 365 + 365 -10 = 280 + 440 = 720. Also stated previously, "The scaling factor for the second pyramid on the Giza Plateau is 137, having a height of 2 × 137 = 274 Royal Cubits and a base of 3 × 137 = 411 Royal Cubits." [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF]. "The Great Pyramid has a combined height and base of 280 + 440 = 720 Royal Cubits, 280/220 = 14/11 ≃ √ φ and 220/136 ≃ φ ." [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF]. Colonel R. S. Beard states that, "Sir William Petrie himself was thoroughly convinced that the Egyptians constructed the pyramid with a height-to-width-of-base ratio of seven to eleven." [START_REF] Beard | The Fibonacci Drawing Board Design of the Great Pyramid of Gizeh[END_REF].

Ernest Pecci quotes Thoth: "The Pyramid is a representation of all created things. All matter consists of cosmic frequencies held in place by the angulations contained within specific harmonic triangles that are found within the Platonic solids. ... the principle of proportion determines and maintains all forms." [START_REF] Pecci | The Great Pyramid: A Temple of Created Life[END_REF].

Another approximation from our previous work [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF] also gives the same approximate value for the inverse fine-structure constant. α -1 ≃ φ x, where x is a root of:

3x 4 -250x 3 -346x 2 + 48x -36 = 0.
(2) π 7 /7π is from the Hebrew inverted heart Tetractys or Tetragrammaton from 'Tengri 137' [START_REF] Anon | [END_REF]. The π 7 is the perimeter of the Great Pyramid base and 7π is found in the Golden Apex of the Great Pyramid, see Eq. ( 9) discussion. Jerry Gin states, It is also interesting to note that the name of God in Hebrew is YHWH (or Yahweh) which is translated as 'Tetragrammaton' from Greek. Tetra refers to four and may refer to the four faces of a tetrahedron and grammaton may refer to weight or gravity [START_REF] Gin | Fundamental Pattern and Consciousness[END_REF].

138x 4 + 83x 3 -70x 2 + 7x -161 = 0. (3
Ernest Pecci explains, "One importance of the Tetractys lies in its expansion to 3-dimensional and multi-dimensional levels. The Star of David, The Tree of Life, and the Tetragrammaton of the heart are all based upon the simple equilateral triangle divided into the ten dots of the Tetractys." [START_REF] Pecci | The Sacred Geometry of the Great Pyramid[END_REF]. Gordon Plummer says, "Perhaps of paramount importance is that of the Tetraktys. This was the symbol probably most revered by the Pythagoreans." [START_REF] Plummer | The Mathematics of the Cosmic Mind: A Study in Mathematical Symbolism[END_REF][START_REF] Plummer | By the Holy Tetraktys!: Symbol and Reality in Man and Nature[END_REF]. For Manly P. Hall: "The tetrad -4-was esteemed by the Pythagoreans as the primogenial number, the root of all things, the fountain of nature and the most perfect number. ... Wherein the Pythagoreans expressed God as a tetrad is explained in a sacred discourse ascribed to Pythagoras, wherein God is called the Number of Numbers." [START_REF] Hall | The Secret Teachings of All Ages[END_REF].

Leon Marvell writing about Robert Fludd: "Interestingly, Fludd's conviction that the Tetragrammaton or deity was the infinite dimension within and between all things is an early precursor of late nineteenth-century conceptions (such as those of William Clifford) that saw the fourth dimension as precipitating phenomena such as the appearance of matter." [START_REF] Marvell | The Physics of Transfigured Light: The Imaginal Realm and the Hermetic Foundations of Science[END_REF].

Another quartic polynomial also gives the same approximate value for α -1 . The inverse fine- structure constant α -1 ≃ 4R sinh(x), with R as the Reality Number from the Great Pyramid [25] and x is a root of:

38x 4 + 313x 3 -273x 2 -44x + 50 = 0. (4)
This quartic has a root x ≃ φ /e, with the golden ratio and Euler's number. 4 = 44 + 273 -313, 280 = 4(38 + 38 + 44 -50), 160 = 4(313 -273), and 440 = 160 + 280. Also, α -1 ≃ R cosh -1 (2π) ≃ R √ 2π, see the reference to Alfred Landé in our previous work [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF]. The Reality Number, R, "... is an angle by which the least amount of energy is needed to accomplish the most amount of good when rotating from one dimension into the next." [START_REF] Miller | Pyramid Truth Gateway Universe[END_REF]. R = 54.277 degrees. This value is encoded into the structure in this way: since the Pyramid has the apothem angle of 51.853... degrees, and since the Ascending and Descending Passages needed a specified angle between them ... the value exactly midway between the apothem angle and the R number was chosen. Step one is from 51.853... degrees to 53.065 degrees. Step two is from 53.065 degrees to 54.277 degrees. ... and the three angles honor the ascension principle [START_REF] Miller | Pyramid Truth Gateway Universe[END_REF]. R ≃ 7AB, where A is the Golden Apex of the Great Pyramid and B = tan -1 (4/π) is the base angle of the Great Pyramid [START_REF] Sherbon | Quintessential Nature of the Fine-Structure Constant[END_REF]. θ g ≃ 26.57 • ≃ tan -1 (1/2) is the ancient Golden Angle of Resurrection noted by Robert Temple [START_REF] Temple | Egyptian Dawn: Exposing the Real Truth Behind Ancient Egypt[END_REF]. θ g ≃ (R+B)/4 and 7 = φ 4 +φ -4 ≃ sinh √ 7. Another approximation for the inverse fine-structure constant α -1 ≃ exp(4x), where x ≈ 2/φ is a root of:

7x 4 -28x 3 + 9x 2 -11x + 36 = 0. ( 5
)
This equation also gives the same approximate value for the inverse fine-structure constant as Eq. ( 1). 280 = 28(11+36-9-28). 440 = (9+11)(11+11). The inverse fine-structure constant α -1 ≃ (4 × 4)x, where x ≃ 5φ 4 /4 is a root of:

2x 4 + 44x 3 -550x 2 + 233x -56 = 0. ( 6
)
This equation also gives the same approximate value for the inverse fine-structure constant as Eq. ( 1). 280 = 550 -56 -56 + 2 and 440 = 44(56 -44 -2) = 2(44 -56 + 233) -2. The inverse fine-structure constant α -1 ≃ 372/x,where x ≃ e ≃ φ 2 √ 2 :

16x 4 + 25x 3 -37x 2 -97x -833 = 0. ( 7
)
4 × 4 = 16 is the main harmonic number of the Great Pyramid according to Karim [START_REF] Karim | Back to a Future for Mankind: BioGeometry[END_REF] and Witteveen [START_REF] Witteveen | The Great Pyramid of Giza: A Modern View on Ancient Knowledge[END_REF]. Karim states that,

The proportion derived from that number (1:1.6) is the famous Golden Proportion used in Ancient architecture and now the basis of 'Sacred Geometry'. Sixteen brings out the sacred quality hidden within the One to manifest in our time-space dimension [START_REF] Karim | Back to a Future for Mankind: BioGeometry[END_REF].

The ln 372 ≃ 1 + ln 137 and sinh -1 (372) ≃ A -1 , inverse Golden Apex of the Great Pyramid [START_REF] Sherbon | Fundamental Physics and the Fine-Structure Constant[END_REF].

The harmonic sum of 372 is part of a spectrum analysis related to crystallographic groups [START_REF] Sherbon | Fundamental Physics and the Fine-Structure Constant[END_REF]. From Robert Lawlor describing harmonic division with the square root of two, "Here is a means of drawing a beautifully proportioned cup or grail-shaped vessel, using only the harmonic division to establish its curves and measures. We may speculate that this is the geometric essence of the Holy Grail." [START_REF] Lawlor | Sacred Geometry: Philosophy and Practice[END_REF].

This grail geometry includes

√ 2 -1 ≃ φ 2 /2π and 2 - √ 2 ≃ 4/φ 4 ≃ 4A. Also, in radians cot( √ 2 -1) ≃ 1 + √ φ
, linked to Plato's beautiful scalene triangle found in the Cosmological Circle extension known as Plato's Wheel [START_REF] Pecci | The Sacred Geometry of the Great Pyramid[END_REF]. Ernest Pecci says:

It is of significance that √ 2-1, 2- √ 2,
1 creates a 3-term harmonic proportion that places the baseline of the Pyramid at the harmonic mean between the floor of the Subterranean Chamber and the floor of the King's Chamber! [START_REF] Pecci | The Sacred Geometry of the Great Pyramid[END_REF].

From the Pythagorean perspective the Foundation Stone number of 729 is a harmonic of the fine-structure constant value. Another equation from previous work [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF] that also gives the same approximate value for the inverse fine-structure constant as Eq. ( 1) is α -1 ≃ 729/x, where x is a root of:

10x 4 -37x 3 -113x 2 + 73x + 371 = 0. ( 8
)
The root x ≃ φ φ f , where φ f is the reciprocal Fibonacci constant. 280 = 10 + 10 -37 -73 + (10 × 37) and 440 = 73 + 371 -((113 -73)/10). Czech physicist Raji Heyrovska says, "On noticing the closeness of the fine structure constant ... to the ratio of the angles, 360 • /φ 2 ... the author suggested that the small difference ... could be due to the Sommerfeld's relativity correction factor." [START_REF] Heyrovska | The Golden Ratio, Ionic and Atomic Radii and Bond Lengths[END_REF]. "It was also pointed out that the ratio 360 • /φ 2 ... which is a Golden section of 360 • , differs from the inverse fine structure constant by ... 2/φ 3 ... probably due to the difference in the g-factors for the electron and proton ...," with the result of α -1 ≃ (360/φ 2 ) -(2/φ 3 ) [START_REF] Heyrovska | Golden Ratio Based Fine Structure Constant and Rydberg Constant for Hydrogen Spectra[END_REF].

α -1 ≃ 360 φ 2 - 2 φ 3 + A 2 Kφ 4 - A 3 K 2 φ 5 + A 4 K 3 φ 7 . (9)
This extension of Heyrovska's equation found previously also gives α -1 ≃ 137.035 999 168 [START_REF] Aoyama | Theory of the Anomalous Magnetic Moment of the Electron[END_REF]. Raji Heyrovska found it a ... surprise to find for the first time that the Bohr radius is divided into two unique sections at the point of electrical neutrality, which is the Golden point. The Golden ratio, which manifests itself in many spontaneous creations of Nature, was thus found to originate right in the core of atoms [START_REF] Heyrovska | Golden Ratio Based Fine Structure Constant and Rydberg Constant for Hydrogen Spectra[END_REF].

William Eisen describes the geometry of what he called the 'Golden Apex of the Great Pyramid' where dividing the sides of his mathematical model for the Great Pyramid by π lengths along with four curves of the exponential function results in a small square in the center called the Golden Apex, the geometry and symmetry thought to be associated with the generation of the four fundamental forces of nature [START_REF] Sherbon | Fundamental Physics and the Fine-Structure Constant[END_REF]. The Golden Apex A is the side length of the resulting square

. A = e π -7π -1 ≃ √ πα ≃ π/ √ 440 ≃ 3/20.
The inverse Kepler-Bouwkamp constant is the polygon circumscribing constant K ≃ φ 2 /2A ≃ 8.7 and sin α -1 ≃ 7!/(713 + 137)K ≃ (π -1)/π [START_REF] Sherbon | Fundamental Nature of the Fine-Structure Constant[END_REF]. Also of interest, ln(φ /α) ≃ φ /2A ≃ K/φ and 11/37 ≃ cos √ φ ≃ 2A [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF]. The polygon circumscribing constant "has interesting geometric connections with the torus topology relating the relativity of Einstein with the geometry of classical and quantum mechanics." [START_REF] Sherbon | Wolfgang Pauli and the Fine-Structure Constant[END_REF]. The Reality Number R ≃ 7AB ≃ 2πK ≃ 360A, see Eq. ( 4) discussion.

Approximations from previous work [START_REF] Sherbon | Fundamental Physics and the Fine-Structure Constant[END_REF] involving the four fundamental forces:

A ≃ 4πφ α ≃ α w / sin 2 θ w ≃ πφ α s /4 ≃ K √ π/ ln α -1 G .

With sin 2 θ w as the Weinberg angle or the weak mixing angle, α w is the weak nuclear force, α s is the strong nuclear force and α G is the gravitational coupling constant [35]. The inclusion of the four fundamental forces in the Golden Apex design is part of the symbolic interpretation presented by William Eisen [START_REF] Eisen | The English Cabalah Volume II: The Mysteries of Phi[END_REF]. The geometry of the Tetraktys has been associated with the four elements and the four fundamental forces.

Conclusion

The inverse fine-structure constant calculated with quartic polynomials and the main harmonic parameters of the Great Pyramid have suggested the golden ratio as the mathematical basis for the fine-structure constant. From previous work "... several more formulations for the finestructure constant with the same approximate value have connections with prime number theory, the real fixed point of the hyperbolic cotangent, anomalous magnetic moment of the electron, Laplace limit of Kepler's equation and harmonic proportions of the Cosmological Circle" [START_REF] Sherbon | Fine-structure Constant from Sommerfeld to Feynman[END_REF][START_REF] Sherbon | Fundamental Physics and the Fine-Structure Constant[END_REF][START_REF] Sherbon | Physical Mathematics and The Fine-Structure Constant[END_REF].

) 137 = 7 +

 7 70 + 83 + 138 -161, 280 = 83 + 83 + 137 + 138 -161 and 440 = 7 + 70 + 83 + 280.

37/7π ≃ 2 √ 2 ≃ 1 +

 221 sin α -1 and φ / ln φ ≃ 2 2 √ 2. 280 = 16(37 -25) + 97 -25 + 16 and 440 = 16 + 97 + 97 -25 -25 + 280. 440 + 440 = 97 -25 -25 + 833.

Acknowledgments

Special thanks to Case Western Reserve University, MathWorld and WolframAlpha.