Vincent Berg
email: vincent.berg@cea.fr

Jean-Baptiste Dor
email: jean-baptiste.dore@cea.fr

Valérian Mannoni
email: valerian.mannoni@cea.fr

Channel estimation strategy for LPWA transmission at low SNR: application to Turbo-FSK

Keywords: LPWA, Zadoff-Chu, Constant Envelope, NB-IoT

Turbo Frequency Shift Keying has been considered as a promising physical layer for low power wide-area network connectivity. Because of its constant envelope amplitude and the efficiency of its iterative receiver performance close to Shannon's limit can be achieved. However, results published so far in the literature for the waveform have assumed perfect channel estimation or Signal-to-noise (SNR) levels that are higher than the SNR levels considered for these applications. This paper analyzes a channel estimation strategy based on a specifically adapted pilot sequence. Simulations have been performed to evaluate the performance of the proposed approach. Performance loss induced by imperfect channel estimation algorithms is estimated.

I. INTRODUCTION

The installed base of the Internet of Things (IoT) devices is forecast to grow to almost 31 billion worldwide by 2020. Longrange, wide-area networks (LPWA) including narrowband IoT (NB-IoT) are expected to make a significant part of the almost 2.6 billion Machine-to-Machine (M2M) connections that will be in place by that time [START_REF] Columbus | Roundup of Internet of Things Forecasts[END_REF].

For LPWA systems, low energy consumption requirement directly results from the necessity to have a long battery life. Long range is achieved with performance at a very low level of sensitivity at the receiver and obtained by ensuring Quality of Service (QoS) for low levels of Signal-to-Noise Ratio (SNR). At the physical layer, this is achieved by selecting an energy efficient modulation combined with an efficient usage of the Power Amplifier (PA), the most power-consuming component of the transmission chain [START_REF] Raab | Power amplifiers and transmitters for rf and microwave[END_REF]. The efficiency of the PA is highly dependent on the peak-to-average power ratio (PAPR) of the signal [START_REF] Miller | Peak power and bandwidth efficient linear modulation[END_REF]. NB-IoT as defined in Release 13 of the 3GPP specification [START_REF] Gpp | Evolved Universal Terrestrial Radio Access (E-UTRA); Physical channels and modulation," 3rd Generation Partnership Project (3GPP)[END_REF] considers both Single Carrier Frequency Division Multiplexing (SC-FDM) and single-tone transmission to limit peak-to-average power ratio (PAPR). The use of single tone transmission is particularly interesting for LPWA communications. Beside, this is not without any drawback. As the bandwidth is small, the transmission time duration is significantly increased in comparison to SC-FDM transmission and hence the energy to transmit an information bit. However, PAPR levels of SC-FDM is significantly larger than 0dB. For these reasons, alternatives to SC-FDM should be considered.

Turbo Frequency-Shift Keying (Turbo-FSK) is a new waveform introduced in [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF] that combines a non-linear modulation with a convolutional code. Its receiver relies on turbo processing to meet a performance close to Shannon's limit for low spectral efficiency. Furthermore, the waveform exhibits a constant envelope (i.e.: its PAPR is equal to 0 dB). Turbo-FSK is therefore well adapted to future LPWA systems notably for the 5th generation of cellular systems.

The principles of Turbo-FSK have been first described in [START_REF] Roth | Turbo-FSK: A New Uplink Scheme for Low Power Wide Area Networks[END_REF]. The motivation for the scheme was to operate close to the Shannon capacity while using a constant envelop modulation. The structure of the transmitter (Fig. 1) is composed of λ repetition stages. For each stage, a differently interleaved version of the same input information bit sequence is encoded using a simple parity accumulator. The coded bits are then mapped onto a Turbo-FSK codeword: a combination of a linear N L -point phase shift keying (PSK) modulation and a nonlinear N ⊥ -carrier FSK modulation [START_REF] Roth | The physical layer for low power wide area networks : a study of combined modulation and coding associated with an iterative receiver[END_REF]. Efficient digital implementation can be realized by inverse Fast Fourier Transform (FFT) followed by a cyclic prefix insertion as for Orthogonal Frequency Division Multiplexing (Fig. 1). At the receiver, a soft FSK-detector estimates the probabilities of each possible Turbo-FSK codeword. These probabilities are then fed to the decoder, which uses them as channel observation, while output of the other decoders will be used as a priori information. A modified version of the algorithm proposed by Bahl, Cocke, Jelinek and Raviv (BCJR) [START_REF] Bahl | Optimal decoding of linear codes for minimizing symbol error rate (corresp.)[END_REF] is used to decode the trellis, and derive the a posteriori probabilities of the information bits. The association of the encoding with a non-linear modulation (FSK) allows to operate at very low levels of SNR. It has been demonstrated that the choice of the PSK is the optimal modulation minimizing channel capacity gap to Shannon's limit [START_REF] Roth | A Comparison of Physical Layers for Low Power Wide Area Networks[END_REF]. Practical implementations of the FSK-detector consist of a FFT combined with a frequency domain equalizer that relies on accurate channel estimation.

Expected SNR at the receiver should be well below 0dB and particularly since Turbo-FSK proposes to improve energy efficiency to within a few decibels of Shannon's limit, . A typical level of SNR at sensitivity limit is expected to be of around -10dB for LPWA applications. This imposes significant performance constraints on channel estimation at the receiver. Results published so far in the literature for the waveform have either assumed perfect synchronization and channel estimation or considered significantly higher levels of SNR [START_REF] Dehmas | Turbo-FSK, a physical layer for LPWA: Synchronization and channel estimation[END_REF]. This paper derives a channel estimation algorithm optimized for the very low levels of SNR required by low power IoT connectivity applications.

The paper is structured as follows: Section II presents a frame and pilot structure adapted to Turbo-FSK. In Section III, channel estimation algorithms are described and evaluated. Performance comparison against perfect channel estimation is given and an optimized structure is discussed. Section V concludes the paper.

Q = N q • q information bits π int 0 π int 1 π int λ-1
. . .

Parity-Accumulator Encoder

Parity-Accumulator Encoder Parity-Accumulator Encoder . . .

PSK Mapping

+ iFFT mapping iFFT + CP insertion . . . q bits q + 1 bits q + 1 bits

II. SYSTEM MODEL

A. Frame and Pilot Structure

A typical LTE uplink frame structure is considered for channel estimation. The resource grid structure where N ⊥ subcarriers are active is composed of data symbols interleaved with pilot symbols spaced every D symbols (Fig. 2). The structure is designed to give an estimate of the frequency response of the channel and to track its evolution during the duration of the transmission. In order to keep the lowpower advantage of the waveform, we imposed for the pilot symbol to have a low PAPR. As a consequence, constant amplitude, zero autocorrelation (CAZAC) sequences have been considered. Zadoff-Chu (ZC) sequences [START_REF] Chu | Polyphase codes with good periodic correlation properties (corresp.)[END_REF] are an example of such sequences and have already been widely used in cellular systems. Two approaches may be considered. For the first approach, the ZC sequence is generated in the time domain.

z[k] = αe -jπ k 2 N ⊥ N 2 e j2πν k N , ∀k ∈ [0, N ] (1)
where α is a scrambling factor that is constant for each pilot symbol, i.e. α ∈ {-1, 1}. The parameter N ⊥ is the number of active carriers and N the number of samples per symbol (the size of the FFT). The scalar ν is a normalized frequency offset used to shift the spectrum of the time sequence around the appropriate frequency resource block. A second approach considers the generation of the ZC sequences in the frequency domain as follows: The ZC sequence of ( 2) is then placed on the allocated resource carriers and the associated time sequence is generated using a N-point inverse FFT. Both alternatives have advantages and drawbacks. The ZC sequence based on (1), also called time ZC sequence, provides a constant envelop signal but localization in the frequency domain is not optimal: frequency orthogonality is not ensured between synchronized users at least a guard interval of one frequency tone should be considered. For instant, up to 17.3 dB of energy is leaked to the out-of-band adjacent carrier (carrier N ⊥ + 1). On the other hand, the ZC sequence based on (2), also called frequency ZC sequence, is by construction orthogonal between different users using different resource blocks, however when upsampled by the inverse FFT transform, the PAPR of the generated sequence is significantly larger than 0dB. Table I gives a performance comparison of both sequences in terms of PAPR and adjacent channel orthogonality for N ⊥ = 32 and N = 512 where adjacent channel power leaked on carriers N ⊥ + 1 and N ⊥ + 2 is given.

Z[k] = αe -jπ k 2 N ⊥ , ∀k ∈ [0, N ⊥ -1] (2)
To improve the PAPR properties of the frequency ZC sequence, we designed a modified frequency ZC sequence composed of the ZC sequence of (2) and an additive noise vector X selected in such a way that the resulting PAPR of the transmitted sequence is minimized and the signal-tointerference ratio (SIR) on each subcarrier is contained.

Z M od [k] = Z[k] + X[k], with 1 |X[k]| 2 > ρ, (3)
∀k ∈ [0, N ⊥ -1]. The resulting PAPR and leakage performance of the modified ZC frequency sequence is given in Table I.

Significant reduction of PAPR level is achieved: optimization gives a PAPR as low as 0.7dB when the allowed signal-tointerference level is set to 7dB.

In order to evaluate the impact on the autocorrelation properties of the ZC sequence performance, the ambiguity function of the modified frequency ZC sequence has been evaluated and . compared to the time ZC sequence. The ambiguity function is defined as follows:

A(f, t) = k z * [k -t]z[k]e j2πf t 2 (4)
Fig. 3 gives the ambiguity function for the modified ZC sequence when ρ is set to 7dB and compared to the time sequence. It demonstrates that the interference introduced to optimize PAPR has very little impact on the autocorrelation properties of the scheme and therefore has little impact on channel estimation performance.

B. Channel Estimation

The proposed pilot sequence is used to estimate the channel information state. A three step procedure is here considered. First, a least square (LS) estimation on the pilot subcarriers is performed. Then, a de-noising filter is applied on the N ⊥ LS estimates. Finally, channel estimation on pilot symbols is interpolated in the time domain to provide a channel estimate of each Turbo-FSK symbol over the duration of the transmission time interval. If Y (i.e.: Y [k], ∀k ∈ [0, N ⊥ -1]) is the multicarrier symbol at the receiver after FFT associated to the transmitted pilot symbol Z. Then in presence of a frequency domain complex channel H and frequency domain additive white Gaussian noise, N:

Y = ZH + N ( 5
)
where Z is the the ZC sequence and N is an additive white Gaussian noise. The least square estimate of the channel Ĥ is given by:

Ĥ = H + (Z H Z) -1 Z H N (6)
where operator H is the hermitian transpose. We will note Ñ the white Gaussian noise colored by the ZC sequence, Z:

Ñ = (Z H Z) -1 Z H N (7)
Improved estimation of the channel coefficients H[f ] may be derived from the observation of Ĥ[f ] using a filter W f designed to reject Z. To construct the filter W f , the following minimization problem should be solved:

arg min W f E W f Ĥ -H 2 (8)
Let Ω be defined as:

Ω = E W f Ĥ -H 2 (9)
Using ( 6) and ( 7), ( 9) may then be rewritten as:

Ω = E W f (H + Ñ) -H 2 = E tr W f HH H W f H -W f HH H -HH H W f H +HH H + W f Ñ ÑH W f H (10)
Then, if h is the N × 1 vector of the channel impulse response, H may be written as:

H = F(1 : N ⊥ , :)h = F N ⊥ h (11)
Finally ( 9) may be rewritten as:

Ω = tr W f F N ⊥ Φ h F N ⊥ H W f H -W f F N ⊥ Φ h F N ⊥ H -F N ⊥ Φ h F N ⊥ H W f H + F N ⊥ Φ h F N ⊥ H +W f R ÑW f H (12
) where Φ h is the time domain channel autocorrelation matrix of size N × N and R Ñ is the autocorrelation matrix of the noise colored by the pilot sequence. By taking the partial derivative of Ω with respect to W f and making it equal to zero, (12) becomes:

∂Ω ∂W f = 0 0 = (∆ + R Ñ)W f H -∆ W f H = (∆ + R Ñ) + ∆ (13) where ∆ = F N ⊥ Φ h F N ⊥
H . When pilots with constant amplitude are sent then the noise distribution after LS estimation is the same for each sample and therefore R Ñ = σ 2 n I with σ 2 n the variance of the noise. However, if a modified frequency ZC sequence (as defined in (3)) are transmitted, then the amplitude of each pilot differs from one to another. Consequently the statistic of the noise after LS estimation varies and in that case,

R Ñ = σ 2 n R NZ where R NZ (i, j) = 0 when i = j, ||Z(i) + X(i)|| -2 otherwise.
The question, then, is how the performance of the filter is impacted by the introduction of the interference added to limit PAPR in the case of the modified ZC sequence. Therefore performance of the proposed channel estimation scheme has been evaluated and compared to the performance of the time ZC sequence. Results of the comparison are given in the next section.

III. ESTIMATED PERFORMANCE

A. Channel estimation performance

The performance of the proposed filtered Least Square algorithm has been evaluated for both ZC sequences of ( 1) and ( 3). The size of the FFT, N , has been assumed to be equal to 512 points and N ⊥ is set to 32. The relative mean square error (RMSE) of the estimated channel coefficients is evaluated for various values of SNR assuming perfect knowledge of the channel statistics (i.e. Φ h ). Results show almost identical performance level between time ZC sequence and modified frequency ZC sequence. Fig. 4 gives the performance of the channel estimation method in presence of AWGN channel. The performance of the average filter is also given and is optimal in this case as the channel is a constant value. The gain of performance is significantly better than the LS approach without any filtering. In this case, an asymptotic gain of 10log(N ⊥ ) or 15dB is observed. The performance degradation introduced by the noise coloring of the modified frequence ZC sequence has been evaluated to be less than 0.2dB and is considered negligible.

Similarly, channel estimation performance is evaluated for the Extended Typical Urban (ETU) channel model of 3GPP. This channel emulates the impulse response of a signal received in a strong multipath environment and is typical of LPWA applications in a urban environment. In this case the coherence bandwidth is smaller than the bandwidth of the 32 active carriers (carrier spacing was 15kHz), hence the average filter is, as expected, inefficient as channel coefficients are not constant. Gain introduced by W f filter is approximately 10dB in comparison to the LS case.

B. Performance loss introduced by imperfect channel estimation

In order to evaluate the performance loss introduced by the proposed channel estimation scheme and the requirements put on the pilot period D as defined in Section II, performance of Turbo-FSK has been simulated under AWGN Channel (Fig. 6) for various values of N L from 4 to 32 with λ = 4 and different payload lengths (256bits and 1024bits). This configuration has been selected as typical for LPWA applications. As expected, as the constellation order N L is increased, spectral efficiency is also increased and the required level of SNR to reach a packet error rate (PER) less than 10 -2 is also increased. Furthermore, SNR requirement to attain performance is lower when longer packets are transmitted. SNR interval of interest is between -12 and -9dB and well below 0dB.

Then, for the same configurations, the performance loss introduced by a non-perfect channel estimation is evaluated. AWGN is added to the perfect channel state information of the received Turbo-FSK symbols and associated performance loss as a function of the added noise on the channel state information is measured by simulation. We defined as RM SE H , the amount of AWGN power that has been added. RM SE H is varied from -5dB to -20dB. Results can be found in Fig. 7 for a block length of 256-points and in Fig. 8 for a block length of 1024-points.

When RM SE H is lower than -10dB, performance loss may almost be approximated to an extra source of additive white noise. For instance, when RM SE H is equal to -10dB, performance loss introduced by an extra uncorrelated noise source should be of around 10 log(1 + 10 -10 10 ) = 0.4dB. This first order approximation proves relatively correct for lower FSK constellations (4-PSK ans 8-PSK). However, performance loss is significantly larger for higher values of N L (16-PSK and 32-PSK). Furthermore, when RM SE H is larger than -10dB, significant performance degradation is observed. For data frames with a shorter payload (256bits), performance loss ranges from 1.5dB for lower values of N L up to 3.3dB for

C. Recommended pilot structure

These results (from Fig. 4 to Fig. 8) help derive some recommendations for channel estimation as a function of the performance loss budget allocated to channel estimation. For Turbo-FSK, performance is obtained for values of SNR between -11.5dB and -9dB. For these levels of SNR, the channel estimation of the filtered LS algorithm gives a RMSE close to -5dB when AWGN channel is considered (Fig. 4). Further noise rejection is thus necessary if performance loss incurred by channel estimation need to be contained below 1.5dB (Fig. 7 and Fig. 8). This can be achieved by interpolating the resulting channel estimation across multiple pilot symbols (time domain interpolation). Hence, channel time interpolation should not only estimate channel variations between pilot symbols but also filter unwanted noise introduced by imperfect channel estimation. To obtain RM SE H levels of -10dB (resp. -15dB) interpolation over at least 3 pilot symbols (resp. 4) should be performed to avoid significant performance degradation.

IV. CONCLUSION

Turbo-FSK is a waveform that is well adapted to future LPWA systems for 5G systems: its performance is very close to Shannons limit and furthermore the waveform exhibits a constant envelope. However, the impact of channel estimation has so far not been thoroughly considered on the performance of the turbo-FSK receiver. This paper discussed and evaluated a channel estimation technique based on distributed pilot symbols. A pilot sequence based on a frequency domain ZC sequence has been studied and optimized for PAPR reduction. The PAPR of the pilot sequence has been reduced to below 1dB and the performance of channel estimation has been studied. Impact on overall performance of the receiver has been evaluated particularly for the very low levels of SNR that characterize the selected LPWA scenario.

V. ACKNOWLEDGMENT

The research leading to these results was supported by the French Agence Nationale de la Recherche (ANR), under grant agreement ANR-16-CE25-0002 (project EPHYL).

Fig. 1 .Fig. 2 .

 12 Fig. 1. Transmitter architecture of Turbo-FSK.

Fig. 3 .

 3 Fig. 3. Ambiguity function comparison. Left time ZC sequence as in (1) and right modified ZC frequency sequence as defined in (3)

Fig. 4 .Fig. 5 .

 45 Fig. 4. Performance of channel estimation under AWGN channel.

Fig. 6 .

 6 Fig. 6. AWGN performance of Turbo-FSK with N ⊥ = 32 sub-carriers and N L -PSK, N L = {4, 8, 16, 32}, λ = 4 to reach a PER of 10 -2 . Assuming perfect channel estimation. (red -block length equals to 1024bits -blue block length equals to 256bits)

Fig. 7 .

 7 Fig. 7. Performance loss introduced by imperfect channel estimation. Payload block length 256bits.

Fig. 8 .

 8 Fig. 8. Performance loss introduced by imperfect channel estimation. Payload block length 1024bits.

TABLE I .

 I ZC SEQUENCE COMPARISON WHEN N = 512 AND N ⊥ = 32

		PAPR [dB]	Adj. Leakage (N ⊥ + 1) [dB]	Adj. Leakage (N ⊥ + 2) [dB]
	ZC sequence (z)	0	17.3	40.8
	ZC sequence (Z)	2.6	<65.0	<65.0
	Modified
	ZC sequence (Z), ρ dB = 7	0.7	<65.0	<65.0


Keywords: harmony, stacks, superimposition, fourths, Mahler, Schönberg, Ravel, Milhaud, Bartók

The "stacking of fourths".

Introduction

At the end of the first movement of his Symphony No 7 (achieved in 1905), Gustav Mahler chose to try out a new melodic process. The melody of the brass section stacked up perfect fourths, in a globally ascending movement. The process was also more or less harmonic: the melodic stack of fourths sounded like a new type of arpeggio, no longer drawing a lot of thirds but fourths. Indeed, the traditional structure of all the chords, in tonal music, was always a superposing of thirds (at least concerning the root position of the chords), that's why a superimposition of fourths sounds like a new arpeggio, and that's probably what Mahler himself was intending to create, in the beginning of the 20th century. This new "harmonic creature" was assumed all the more by the Viennese composer since it was powerfully proclaimed by five horns in unison.

Schonberg and the fourths

It is not surprising that, only one year after Mahler, it was Arnold Schönberg who chose to renew the same harmonic idea, which could therefore be considered, at that time, to belong to the Viennese modernity. More than simply renewing the process, Schönberg magnified and even perhaps systematised it. Besides, this Chamber symphony No 1, op. 9 (composed in 1906), for 15 solo instruments, is traditionally called the Fourths Symphony. In this piece, Schönberg chose to break with traditional romantic symphony, notably with the so-called artificial succession of separated movements. Here, the movements are supposed to be played through. However, a division of the movements remains, and it is in the beginning of the third movement that a stack of fourths is systematically presented. Here, in an exemplary way (we might say a pedagogical way), the fourths build the whole movement. They begin it. The stack is clear and simple. It is melodic and harmonic, strictly ascending (not like in the Mahler's Symphony, where the melodic curve is only globally ascending and keeps some saw-tooth patterns). The notes come one by one, each time at the fourth above the previous one. It is clearly some kind of new "harmonic manifesto": from now on, chords are built with fourths, not thirds. However, this manifesto only concerns this particular piece by Schönberg. Superposing fourths never became a style in the Work of the Viennese composer, as it did in Bartók's (as we will see further on).

Ravel and Milhaud.

When Ravel chose to magnify the interval of fourth, he did it late in his career, and perhaps in a quite incidental waybecause he wanted to generate a very dark and raw atmosphere, in the beginning of the Piano concerto for the left hand (1929)(1930). The double basses just expose the four natural sounds of their strings (that are E A D G, each note separated by a fourth from the next one) in a fast sweeping up and down movement. Systematically stacking up the interval (very high), like Schönberg did in his Chamber symphony, was out of the question. However, such a "new harmonic systematic superposing" was attempted by the French, only six years after Schönberg, at the very beginning of the ballet Daphnis and Chloe (1912). But in Ravel's work, of course, it was a matter of superposing perfect fifths (and not perfect fourths)1 . But both stacks seem to find a common function of melodic "discovery", perhaps because both superpose perfect intervals and thus, make it possible to reach the twelve degrees (semi-tones) of the scale (or at least a new note at each degree, each occurrence of the superimposition). Like in the Schönberg's Chamber symphony, Ravel presented his new "chord" with slow successive sustained notes, from lower to higher parts, and in the beginning of the movement, as if the composer were softly opening something, introducing the different pitches or registers successively before writing anything. Some kind of mise en abyme seemed to occur in both cases: extremely progressive, the music seems to look for itself and then, to compose itself. Thus, it might "speak" about the beginning of every composition, slowly "finding its degrees on its own".

The French composer Darius Milhaud, in his ballet Le boeuf sur le toit (1919), also made use of a small stack of two fourths. In the refrain of this rondo, which displays the simple tonality of C major, the clarinet plays a fast three notes motif (B flat, E flat, A flat, ascending and descending). Thus, this motif clearly intends to exhibit two fourths stacked up upon each other. For Milhaud, the aim of the process was probably multiple. First of all, it made it possible to superpose, on the theme in C major, three notes quite far from this tonality and sounding as if in E flat major. In the end, the result sounds polytonal in a quite economical way (just adding three notes to the clarinet part). It certainly appeared interesting for a young composer already concerned by polytonality (as he was all his life long). Besides, a more famous user of polytonality, Stravinsky, had the idea of superposing fourths one year before (in his Four pieces for string quartet written in 1918), with the same aim.

Moreover, this process of stacking of fourths (and more generally "fourth exhibition"), was probably, already at this time, in the mood of the times (in the air du temps). Without speaking of a real "fashionable" technical idea, we could bet that several different avant-gardes had naturally thought about this simple way of building some modernity (during times when being modern appeared particularly crucial to almost all the artists), eventually without being influenced by each other. We shall come back to this later.

Bartók

Four years later, in the finale of the Dance Suite (1923), Bartók showed the same project as Schönberg (and as Ravel with his fifths): he superposed fourths in a very clearwe might again say "pedagogical"way. Perhaps "mise en abyme" again, the stack is simple, "natural", and high. It appears at the very beginning of the movement (like in the finale of the Chamber symphony by Schönberg). It begins with the low parts and ends with the highest ones (the double basses first present there note, then the cellos, etc.). The main novelty is that Bartók didn't write sustained notes but fast repeated ones: he might have preferred to generate a more rhythmical atmosphere, probably to achieve his usual folklorist project. Therefore, the process is subtly presented in a different way: in a more rhythmic, dancing, thus probably more popular way. As usual, the popular development comes after the first sophisticated experimentations (in this case, 17 years after Schönberg's Chamber symphony).

Above all, the use of the fourth in general, and sometimes of the more particular stack of fourths, developed itself enough, in the Work of Bartók, to become almost an element of style. The Concerto for orchestra (1942)(1943) begins with just such a progressive superimposition of fourths. The Second Concerto for violin and orchestra (1928) contains a powerful stack, claimed by the brass (and sounding like some kind of "fourth fanfare") and, for once, globally descending.

The fact that the fourth may have found such an importance in the Work of Bartók may be bonded to the frequent use of the pentatonic scale by the Hungarian. Indeed, Bartók, as ethnomusicologist, had discovered that pentatonic scales were part of the most authentic first Hungarian traditional music. Thus, the interval fourth seems "often found" by those scales and sounds, in a way, like a "pentatonic third" (the pentatonic interval that is often obtained when jumping a degree). Bartók probably considered that the fourths should have been "particularly Hungarian". We mentioned before that they seemed to be, in the beginning, more Austrian (Viennese). The purpose is not, therefore, to call them "Austro-Hungarian". It is better to notice that the process of stacking fourths seemed interesting enough to several composers from various countries, to make of them intimate parts of their most modern language, each time for a different reason and with a different aesthetic justification. Without mentioning in detail the case of Scriabin, it is well known that the fourth (perhaps not the stack of fourths in particular) was also an important harmonic (and melodic) element of his second style. Thus, this use might have found some totally different justification in the case of the Russian, and probably a mystical one.

After World War II

We have chronologically mentioned the cases of Mahler, Schönberg, Ravel, Stravinsky, Milhaud, Bartók and Scriabin. Many other composers could have been quoted, concerning the new harmonic stacks, during the first half of the 20 th century. Later, after the Second World War, superposing fourths had certainly become "classical enough" to lead Messiaen to mention the process in his Techniques de mon langage musical (1944). Thus, this process was not so obviously stylistic in his language.

That is precisely what is interesting in this explicit reference: probably the fourths had already become necessary ingredients of modernity, all the more necessary since they were used by different aesthetics. The French composer could have wanted to prove, in his so called Techniques (perhaps more like a manifesto, a justification), that he was using "all the elements of modernity in his composition laboratory". Besides, Messiaen's ambition (like that of Boulez and other composers, at this time) was of course to realise the difficult synthesis of the various movements of the early 20 th century. Thus, like real godsends, the fourths already seemed to build a small synthesis by themselves. However, Messiaen remained a harmonist, and superposing fourths could have appeared quite harmonically stiff to him. That's probably why the process stayed very occasional in his Work. It comes perhaps more often in what he himself called his "style oiseau" (his "bird style") insofar as this type of writing appeared, for once, more polyphonic than harmonistic. The style oiseau used to anable a particular part to build fourths in its corner, like a specific bird could build his nest while other birds would sing in their own various ways. The desired result was precisely rich and varied, anyway, like a mirror of abounding natural life. We can mention, for example, Le réveil des oiseaux, one bar before number 27. The divided second violins (5 to 8) play parallel stacks of fourths, while many other parts choose to draw more melodic birdsong-like motifs. At the same time, those fourths allow their post-impressionist way of moving themselves to be displayed (in a parallel way). We can also notice that not all the fourths are perfect. Some are augmented (tritones). Thus, the first stack is G, C, F sharp, B (and contains the tritone C-F sharp, the other fourths remaining perfect.)

In fact, after the war, in the works of various composers, the stacks of fourths seemed to "emancipate themselves" a little, admitting some augmented intervals in their constructions. Henry Dutilleux apparently liked to write such "irregular" stacks. For him like for Messiaen, they came probably in his Work, at first, because he also wanted to synthesize all the aesthetic movements of the passed. But he integrated them much more often than Messiaen, notably because he was probably less afraid than the latter of their dense supply of various degrees (we will return to this later on). Whereas Messiaen was often looking for a particularly clear style, Dutilleux perhaps allowed himself more involved, complex, mysterious atmospheres. It might be why those involved, "heavy" stacks currently integrated his pieces and became namely stylistic, perhaps like in the work of no other post war composer.

Jazz and movie musics [from here, English not corrected yet]

What may surprise, in the great development of this process during the 20 th century, is that the latter became classical in different aesthetics, when it seemed to be, in the beginning, a quite formal (we could say "intellectual") process: Schönberg seemed to look for an exit to the classical way of building chords (with thirds). But this exit, a priori, was only visual, theoretical, unfounded or at least without deep foundation. Indeed, the thirds, used in the chords, based themselves on real physical legitimacy: the fact that they "fast appear in the harmonic spectrum" (they are parts of the very first harmonics of each spectrum and, for example, separate already low harmonics like the 4 th , 5 th , 6 th and 7 th ). Thus, the traditional chords of tonal music imitate that spectrum and, in that way, imitate "nature". For the whole 18 th century (and notably Rameau), the Renaissance and the whole Antiquity since Pythagoras, the chords justified themselves from that spectrum. Therefore, breaking with that secular tradition, Schönberg showed his usual audacity and a formalist way of mind. But the remarkable thing is that he was followed, for once, by quite different aesthetics. Notably, his process, applied to his atonal music, was often renewed in tonal aesthetics. It is not surprising that Stravinsky also used the process, because the Russian was probably also often inclined to a certain formalism. But it might be more surprising that the French neoclassicism (and for example Milhaud), was tempted by such stacks. Finally, if it is often considered that the beginning of the 20 th century is represented by four great different avant-gardesthe French "impressionists" (notably Ravel), the Viennese school, Stravinsky on his own (for his invention of a great mechanical and/or primitivist ostinato) and perhaps also Varèse, for his attach to the pure timbrethree of these four avant-gardes and perhaps the most important of them (as a matter of fact all those composers, Varèse excepted) were concerned by this particular technical idea. The reason might be that the latter could fast become something else than a simple formal "alternative to the thirds". It contains several great technical advantages. Indeed, stacking up twelve perfect fourths, because of the intervallic exactitude, quickly presents all the semi-tones of the scale. In a way, it "justifies" their presence with a harmonic, (a vertical) argument, and not only a dodecaphonic one. This could have appeared practical for the Viennese school, but also for all the many composers interested by total-chromaticism. Also, stacking up only seven perfect fourths, on a certain root note (let's say C) makes, by chance, a "natural mode", and as a matter of fact, a very different mode from the one which's traditionally built on C (the C mode, called also major mode or Ionian mode): it builds the B mode (Locrian mode), thus, the strangestthen eventually the most modern?of the seven Greek modes. On this same C, the superimposition of fourths fast brings "strange notes", far away from the C major tonality: even without superposing seven fourths, two or three fourths will quickly bring B flat and E flat. That's what probably interested neoclassicism, often tempted by polytonality. Finally, we could afford that, during the beginning of the 20 th century, in any superimposition of fourths, the number of stacked fourths may be proportional to the radicalism of the composer: in the examples we shortly developed in this text, Milhaud, the neo-classicist, piled up only two fourths. The more radical Stravinsky and the impressionist Ravel stacked up four. And Schönberg, perhaps the symbol of formalism in itself, stacked up almost twelve.

From now on, we simply presented the development of a musical idea in various aesthetics, but always in written music (what our consumerist society calls "classical music" because that is the way music shops call it, by opposition to "music" which designs pop music or actual songs in general: something much more important for these shops). What may appear more interesting is that jazz also finally took over this process. During the 60's, the hard bop style, perhaps "invented" by John Coltrane, often used superimpositions of perfect fourths, probably because of its inclinations for modal music (let's not forget that a stack of seven fourth creates a Locrian mode on the root note). More than John Coltrane himself, it was probably McCoy Tyner, the pianist of his quartet, who thought about such stacks, because it remained a harmonic processs (and thus, concerning probably more a pianist than a saxophonist like Coltrane). The famous My favourite things, for example, currently stacks up perfect fourths.

Then, during the late 60's and the 70's, some Hollywood or TV composers, influenced by jazz (or jazzmen themselves in the beginning of their career), such as Lalo Schifrin or Henry Mancini, renewed this harmonic (vertical) idea. By the time, they composed the credit titles of famous serials. For example, Lalo Schifrin signed the score of Impossible mission and that of Bullet, Mancini that of The pink panther. That is probably how the stacks of fourths came finally to the (subconscious) knowledge of many.

John Williams himself, the musician of the first blockbuster of movie history2 , used to superpose fourths melodically or even harmonically in Star wars. During the famous beginning credit titles (where the text appears slanting), a short harmonic stack comes to the brass part, precisely when the credit titles end and when the "camera" goes down to show a great starship. The process seems to betray Williams's link to the great scores of the 20 th century, all the more since his own stack shows a real artistic ambition: an inventive way of difficult fast repeating notes to the brass section (perhaps already used by the French composer Florent Schmidt in the beginning of the 20 th century). Moreover, the composer used this process exactly during the first transition of the movie (the short moment between the credit titles and the beginning of action, when the camera comes down and literally "dives" into action). This might be because the stacks are heavy, heterogeneous to anything else and can not "live" very long, especially in thisstilltonal music.

Conclusion

As a matter of fact, the process might have appeared "heterogeneous" during the whole century, heterogeneous to any music. Schönberg, perhaps proud of his new technical idea, used it in the beginning of the finale of his symphony, like Bartók in his Dance Suite. Concerning that, we've employed the expression "mise en abyme of the beginning of any musical piece". But in a way, this mise en abyme betrays a relative failure. The process certainly appears too peculiar, rigid, systematic, heterogeneous and finally impossible to handle. It can only come at an important moment of a work, the beginning, or perhaps the end when, violently symbolic, it becomes a final signature. That is why it remains a sign, a symbol more than a real musical element. Symbol of the beginning for Schönberg, Ravel and Bartók, symbol of a certain (but particularly clear) transition for Williams, it strangely becomes, on the contrary, end signature in several movie or TV musics of the 70's and the 80's. Thus, for many of us, conditioned since our childhood by many of these musical slogans, the superimposition of fourths are finally the symbols of the end. The sounds seem to put themselves into stacks like into rows, in the manner of actors at the end of a show, before bowing to the audience. For example, the beginning credit titles of Staying alive (1983), staring John Travolta and directed by the actor Sylvester Stallone himself, end with such a characteristic stacking.

For Schönberg, in the beginning of his Chamber symphony, this technical new curiosity was opening something that the composer of Staying alive finally seemed to close. The Viennese composer may have thought that is process was "only a beginning" in the investigation of new musical processes. But the popular music may have answered: "no, on the contrary, this is the end, this is the most complicated process that I can tolerate, even if I'm proud of using it". But, for once, let's notice that Schönberg himself, who was often considered like the symbol of atonality and sometimes like the very image of "musical evil3 ", has finally influenced musicians who could symbolise the opposite (popular music). Thus, by the time, the famous concept defended by the avant-gardes since the Romanticism and before that, Kant's definition of the so-called "genius4 ", who was always a little fast supposed to foretell the future, founded here a small proof of its power of prediction.

Summay

The music of the early 20th Century namely developed a technical process that varies the traditional harmony. The latter used to build its chords stacking up thirds (imitating the harmonic spectrum). Abandoning this natural (acoustical) "determinism" of harmony, in a kind of abstract speculation, the new harmony chose to stack up also fourths. Several composers not only from different countries, but from different aesthetics, developed this ways of reaching a significant musical modernity: Scriabin, Milhaud, Bartók, but also Schönberg and before them, one of the privileged inspirers of the latter: Mahler. In this community of employment, and considering the variety of these aesthetics (starting from the neoclassicism of Milhaud to the so called "opposite": the atonal Viennese expressionism), does a phenomenon of absolute "technical need" have to be recognized? In this case, the latter curiously transcends all the aesthetic divergences until tonality/atonality antagonism. Nowadays, this very particular harmonic process is even employed in movie or TV music, sometimes like a new symbol of the end. This "new signal" may be one of the numerous "final popularisations" of musical processes which are born in very abstract contexts.

Jacques Amblard (b. 1970) is a musicologist, associate professor at the Aix Marseille University (France). His publications concern aesthetics, music of the XXth and the XXIst centuries (for instance in his recent books 20 regards on Messiaen, Aix-en-Provence, PUP, 2015 or in Micromusique et ludismes régressifs depuis 2000, co-authored with Emmanuelle Aymès, PUP, 2017); music and intonation, notably in a book about the French contemporary composer Pascal Dusapin (Dusapin. The Second Style or Intonation, Paris, mf, 2017); or musical pedagogy (Harmony Explained to Children, mf, 2006). He has also written several novels (like v comme Babel, Paris, Balland, 2001or Noé, mf, 2016). He conceived and presented a radio program, concerning musical pedagogy, on the cultural French public radio channel "France Culture", each week from September 1999 to june 2000. He gave two conferences at the Collège de France in 2007.

Later, in the very beginning of his Violin Concerto "to the memory of an angel", Berg stacked up fifths (G, D, A, E) to magnify the strings of violin, as Ravel did, in a way, to "celebrate" those of the double bass in the beginning of his Left hand concerto.

According to James Buhler , Williams generated a great part of the film success by giving back a new "mythic content" to the Wagnerian leitmotiv. See « Star wars, music, and myth », Music and cinema, University Press of NewEngland, 2000,

See Esteban Buch, « Figures politiques de la technique sérielle, Résistances et utopies sonores, Paris, CDMC, 2005, 213.

Emmanuel Kant. Critique of Judgement. § 49.


Jacques Amblard

Nationalism and intonation

Some composers of the late 19th century seemed to be interested in musical translation of the intonation of their own language, for example Mussorgsky in Russia and Hugo Wolff in Austria. We shall study the Hugo Wolff's Lied Wieviel Zeit verlor' ich dich zu lieben, and see if it's possible to find the trace of intonation of German in the singer's melody. If we find such a correlation, then it will be possible to conclude that Wolff has written, perhaps not typical Austrian music, but typical 'Germanic' music. Then, the shadow of intonation of speech, found in vocal music, can generate not a precise nationalism, but a nationalism created by a common language, as a matter of fact, a common intonation.

When children, confronted by a musicianship exercise, try to solfa the rhythm (and just the rhythm, without singing the pitch of the notes) of a piece, they usually mark the note coming on the beat (the first note of each beat, for example the first eighth-note or the first sixteenth-note of each beat) in two different ways at the same time. The first part of the beat is accentuated with a stronger intensity of the voice, of course, but also with a higher pitch (the other notes taking the same lower and monotonous pitch). This natural procedure comes from a characteristic of the human voice, the concordance of the stress (intensity) accent and the pitch accent. It's difficult for the voice to sing loudly in a low register, or softly in a high one. As a matter of fact, this concordance isn't obvious. It doesn't rule all the sonic phenomena: even if one finds an equivalent stress/pitch concordance in the technical characteristics of many wind instruments, like the flute or the clarinet, some other wind instruments reverse this concordance, for example the bassoon and the oboe. It's problematic for the bassoon to play piano in a low register or forte in a high one (as it is for the oboe). This correlation is first a phenomenon of the voice, any voice, not specially the singing voice, also the voice of the speech where the accent is generally higher and louder at the same time: the stress accent (as the linguists call it) corresponds to the pitch accent.

One has often noticed the shadow of intonation of speech in the German lieder of the 19 th century (Cohen-Lévinas, 1987, pp. 29-34). The Austrian composer Hugo Wolf seems to respect particularly the lexical accents of German when he sets poems to music. Mosco Carner insists on this fact, and moreover entitles the first chapter of his Wolf's biography: "In the beginning was the word" (1988, pp. 13-23). About the song Wer rief dich denn? (1890), Carner even employs the term recitativo accompagnato (ibid., p. 82). Henry Baraud, about the lieder of Wolf in general, doesn't hesitate to find a "musical notation where the succession of values and pitches reproduces the tiniest inflexion of the speaking voice". As a matter of fact, these "inflexions" of the speech are the characteristics of the lexical and/or prosodic German accents, which the composer respects and even emphasizes in the setting of the text to music. The concordance of pitch and stress must be correctly translated by musical notations. But in the case of German, the accent is not only louder and higher than the other syllables, it's also longer. Then Wolf has to respect a triple concordance of the accent (stress, pitch and duration). The beginning of the song Wie viele Zeit verlor' ich dich zu lieben (1896) shows such a triple correlation. When a German says the sentence of the title, he might mark a light accent on "Zeit", and a second accent, stronger, on the "ie" of "lieben". Thus, "Zeit" is placed on the melodic summit (the highest pitch) of the bar (admittedly also attempted in the three first syllables) and above all on the first beat (the strong beat of the bar). Consequently, Wolf respects the concordance stress/pitch. But the composer also gives to the syllable "Zeit" the longest rhythmical value of the bar (a pointed eighth-note, while the other notes of the bar are just eighth-notes or sixteenth-notes). The concordance is triple, as it must be in the setting of a German text to music, according to Wolf. The phenomenon is even clearer in the long "ie" of "lieben". Placed on the first beat of the next bar (consequently on the strong beat, once again) and on the melodic summit that this time it alone occupies, it lasts twice as long as the previous syllable ("zu"). Perhaps that's the way Wolf's songs become national music. It's not only the words (and thus the language that they carry) that represent cultural specificities, but also the intonations of German. Intonation of speech may be the deepest bond between people of the same culture, of the same country (and sometimes more precisely of the same region). This bond is affective.

Thus, Diderot calls intonation the language of "feelings" (1970, p. 171) and Rousseau calls it the language of "passions" (chapter II, in the begining). The linguist Ivan Fónagy specifies that "the emotive intonations reveal above all national […] secrets" (1971, p. 562).

However, the result that we have analysed may not be specific to German or Austrian music.

Italian or English composers, for example, might also respect a triple concordance of the speech accent a priori, because the accentuation also produces longer syllables in their respective languages. Therefore, we can only notice, in the example of Wolf, the composer's effort to make his music culturally specific. We can just conclude that Wolf distinguishes his music, for example, from French music (the accentuation is not a matter of duration in the intonation of French, as we'll see further), or from Russian music.

Before Wolf, Mussorgsky tried to translate the intonations of his language into music. Around the1870's, Mussorgsky writes: "Whoever speaks and above all whatever he says, my brain immediately works on the musical reproduction of what I hear" (Samagnac, 1992, p. 861). When Marcel Marnat reveals that in Pirushka, Mussorgsky uses an archaic Greek melody (1962, p. 85), it's because rudimentary tetrachords can be found in it, those that Odile Vivier (1973, p. 111) sees in Ecuatorial (1933-34) by Edgar Varèse. For us, however, these tetrachords might instead be those restricted scales (often with four degrees, "fuzzy" or not; for all these questions, see Amblard, 2002) that one can analyse in the intonation of speech. This interest of Mussorgsky in the musical transcription of Russian prosody begins "at the end of the 1868-1869 season" (Rimsky-Korsakov, 1938, p. 79) when he writes the Marriage. The Circle of Five is immediately impressed. Theorist and best technician of this group, Rimsky-Korsakov straight away notes in his diary: "All of them were touched by Mussorgsky's work, and enthusiastic about the originality of its recitatives" (ibid.). Myriam Samagnac confirms: "In The marriage, the recitative follows the incidences of speech so closely that one might think about a style exercise" (1992, p. 861). During the 1868-1869 season, according to Rimsky-Korsakov, Mussorgsky also outlines his Boris Godunov, in which he emphasizes his peculiar sense of recitative, and then composes the first songs of the cycle The nursery. In this cycle, the singer plays the role of a little boy; thus, a spirit of lightness arises which seems to authorise all liberties, including a modern transcription of speech intonation. In To the corner (1870), the singer sings in a traditional way only when his character, the little boy, is supposed to cry. Elsewhere, his singing is almost a recitation. In effect, Mussorgsky has noticed that the child's crying voice is high, melodic and, in fact, very close to singing. In The evening prayer, the child integrates numerous names in his prayer to Jesus in a monotonous tone, on two notes separated by a minor third, the interval that certain French phoneticians find very often in intonation (specifically, intonation of French, but apparently this interval is common in other languages as well; see Rossi, 1981, p. 117) (1866)(1867)(1868)(1869)(1870)(1871)(1872)(1873), that "Hans von Bülow liked", according to the composer, and that announces the other scherzo in 5 of Tchaïkovsky's Pathetic symphony (1893), and later the coda in sixth journal of Bartók's Mikrokosmos). Perhaps there is a correlation with the Slavish "subjective lyricism" that Adorno noticed (1976, pp. 115-116; this concept would bee specific to the "prebourgeois East" in which Adorno precisely integrates Hugo Wolf, Mahler, Mussorgsky and Janácek, in a way that is especially relevant to this work).

However, the irregular number of beats between the bars is a much more modern phenomenon; according to Marie-Claire Mussat, Bartók himself will wait until his violin Sonatas to make them widely used. Above all, this juxtaposition of bars with different numbers of beats comes, in the work of Mussorgsky, from the will of the composer to correlate the stress and the pitch accent. In the Ex. 1, one notices that the changes of the bar type mostly occur in order for the high pitches to be placed on the first beats (thus the strong beats). Those high pitches are obviously used for the lexical and/or prosodic accents. The irregularity of the bars, then, mirrors the fact that Russian words all have one lexical accent, but not the same number of syllables. Finally, we notice that Mussorgsky does not treat the accent with a longer duration. The syllables of Russian words, according to Mussorgsky, seem to be pronounced in a very regular rhythm (a quarter-note for each syllable). In this way, Russian and German music (as it appears in the mentioned Wolf's lied) find different melodic features which reflect the intonations of their respective cultures. But setting more specific intonations to music, like intonations of Moravian, could create more of a national identity. The case of certain pieces by Janácek, then, could de interesting to study.

From Jenufa (1894-1994) onwards, i.e. in almost all of Janácek's important pieces, one can notice a modern and repetitive aspect in the melodies. These melodies often seem built with the repetition of some very short cell of three or four notes. We can find an example of this in Jenufa itself. The second scene of the first act (Wien, Universal, 1917, p. 31) showcases, with the low strings (reinforced by the bassoon), the tireless repetition of four notes on three degrees (B flat, C, E flat, C), always in the same order. Such continuously repeated cells, built on three degrees, can be found in many works of Janácek. Those cells are often tri-polar, contained in a fourth (one successively hears a second and a third, or the contrary), the degrees always mechanically visited in the same order (which depends on each cell). For example, consider The Diary of one who vanished (1917-19, Praha, Státni Nakladatelství Krásné Literatury, 1953, p. 11), the second act of The cunning little vixen (1922-23, Wien, Universal, 1977, p. 63), Mládí (1924, Praha, Hudebné Matice Umelecké Besedy, 1947, pp. 27-28), or Capriccio (1926, Praha, Státni Nakladatelství Krásné Literatury, 1953, p. 23, bars 7-8.). The first movement of Sinfonietta (1926, Wien, London, Universal, 1954, p. 6, bars 1-3 of number 3) uses different types of dipolar or tri-polar cells, contained in fourths or fifths (and, as mentioned, the tri-polar cells contained in fourths).

Of course, we're talking about tonal music. Before analysing these cells as translations of intonation, we must consider the harmony, the mode and all the tonal concepts which also explain the presence of these cells. However, these cells seem to be of little tonal interest. They seem all the more monotonous since they come in the foreground, not in the background as they do when they're used as simple accompaniment (in the works of so many composers of the two previous centuries). Thus, the reason for their existence might not be purely musical. Perhaps they are more an imitation of very repetitive sonar phenomena, existing in nature or in the human, such as intonation of speech. As a matter of fact, the composer himself has explained his interest in intonation of speech (2000, pp. 54-57): "At the time I was composing Jenufa, I was also absorbing the melody of the speech […]. Discreetly, I was listening to words of the passerbys, reading the expressions on their faces, observing their environment, their companions, the time it was, whether it was clear or dark, cold or hot. I was feeling that all things were mirrored by the melody I was noting". About the repetitive cells we have been discussing, Guy Erisman claims that these "notes grouped in little packages", mechanically pronounced, are effectively the musical translation of a "jerky and economic" Moravian language (1992, p. 882).

Thus, what we hear as mechanical in Janácek's music might betray, more than a evocation of modern machine noises, a subtle reproduction of a specific and particularly jerky intonation, that of the Czechs.

In the first fifteen bars of the sixth number of The diary of one who vanished (see Ex. 2), a recurrent cell appears (sometimes truncated, see the piano part at the 4 th and the 15 th iteration of the cell, the singing part at the 4 th and the 8 th ), all the more interesting because it's exposed by the piano and then by the singing, coming thus in each moment in the foreground. Moreover, the piano plays the cell three times on the same degrees (C, G and B flat, then these degrees become E flat, C and D in the singing part, but the rhythm and the melodic shape stay the same), reinforced at the octave and non-harmonised (this phenomenon comes back in the piano part, no less than six times in a row between bars 9 and 14, in the same way for the piano and for the singing, then for the piano alone). The monotonous result of such repetitive procedures, then, might translate an intonation and more precisely, as we shall try to show, a Moravian intonation.

Bruno Nettl indicates that the Moravian intonation distinguishes itself with an obligatory accentuation on the first syllable of the word (and that this phenomenon, he continues, influences the Czech music, vocally as well as instrumentally, see Bright, 1971, p. 69). Thus, it's primarily the absence of anacrusis, in our example, that seems to translate this phenomenon.

Then, the singer strikes up the first syllable of the first word on the strong beat. In effect, the first syllable must be strongly marked because it carries the accent (as in all Moravian words, according to Nettl). Later, each strong beat (with the exception of bar 8) carries the first syllable of a word (see Ex. 2 again). For the same reasons, it's on the strong beat that the melodic summit is generally attempted; consequently, the concordance stress/pitch is respected. Even more specific to Moravian is the relative shortness of the accent. The accent makes neither a longer syllable as in German (or Italian and English), nor a syllable of equivalent duration as in Russian (and maybe also French as we shall see further), but a shorter sixteenth-note syllable, thus three times as short as the syllable carried by the pointed eighth-note. Does the "jerky" aspect of this language, which Erisman discusses, come from this sudden fall succeeding a short accent?

According to Fernand Carton (quoted by Fónagy, 1991, p. 110), the vowel would also get shorter on the accent in the intonation found in northern France. But intonation of French, compared to that of other European languages, may sound particularly flat. Finally, the syllables seem to be pronounced in a very monotonous and regular rhythm, and the accent (lexical or even prosodic) appears almost non-existent, at least according to Debussy. Consider Debussy's Pelleas et Mélisande (1901Mélisande ( -1902)). In the second scene of the first act (Paris, Frommont, 1902, p. 27), the composer employs his famous singing/recitative. Thus, the melody juxtaposes numerous repeated notes, which is remarkable in such a moderated tempo. There's almost no accentuation, no higher pitch or louder notes, and no longer values (all the syllables are almost isorhythmical). If the text had been in English, German, Russian or Moravian, that setting to music would have seemed strangely monotonous. But the fact that the language is French justifies the procedure and, moreover, creates a specific translation of the French intonations (particularly flat, "by default" intonations). That's one of the reasons why Pelleas et Mélisande is understood to be a real national French opera. It's a work that emphasizes the weakness of the French accentuations. In effect, as it is shown by the American linguist Ernest F. Haden (1972, p. 209), French does not even have an obligatory lexical accent. When Debussy completely suppresses it, he underlines this national particularity. As in Russian, the prosodic accent doesn't have a longer duration. Séguinot indicates that in French, the vowel almost never gets longer (in only 2% of the pronounced vowels, see Fónagy, 1991, p. 110). We may finish with a peculiar comparison. English sometimes marks its accents with a longer duration. We know that the jig is born in the United Kingdom (in Scotland for some, in Ireland for others). Thus, the jig, a dance in three, is based on the rhythm half-note/quarter-note, in which the half-note comes on the first and strong beat and lasts two beats, and the quarter-note (lasting one single beat) comes on the third and weak beat. The fundamental structure of the jig respects the stress/duration concordance, as in the language of people who have invented this dance. Perhaps this fact is not a coincidence. Conversely, a typical French dance in three is the minuet. Therefore, this dance is characterised by the relative and strange equivalence of its three beats. Thus, when Richard Strauss reproaches Debussy for his "irregularities" (see Fónagy, 1979, p. 171) in his settings to music, he might not be fully aware of the distinctions between German and French intonations. Perhaps he ignores this "heavy syllabic waist" that characterises French, according to the linguist Mario Rossi (1999, p. 183). But if Strauss believes he can judge the way Debussy treats the problem of prosody, that's because he himself is very concerned by the question of intonation, not only in vocal music, but also in instrumental music. The nationalism of the latter can not come from the use of German words (there are no words), but can notably proceed in the assignment of intonation-like melodies to instruments. In the symphonic poem A hero's life (1899), the hero is once calumniated by his enemies. The intonations of this slander are then imitated by the woodwinds, in unceasing melodic round-trips (Leipzig, Leuckart, 1899, p. 28, number 19). The melodic curve, in a saw-tooth pattern, seems to translate the fact that German strongly marks each lexical accent (one per word) with great pitch differences between accentuated and non-accentuated syllables. We also notice that the high pitches come on the beat (or on the second eighth-note of the beat, each syllable having the value of a sixteenth-note), so that the stress/pitch concordance is respected. We now realise that Strauss couldn't understand

  . As early as in the first song, Oh tell me a story, Nianiushka (1868), Mussorgsky imposes this shrewd distance, perhaps characteristic of every successful work, between the lightness of the subject matter and the modernity of the technique. We notice a technical progress since The marriage in the treatment of the stress/pitch concordance. Likely the first to try this procedure in the history of savant occidental music, Mussorgsky juxtaposes bars with different numbers of beats. Some of those bars are already "irregular" (e.g. 5 or 7 beats per bar, see Ex. 1). Even long before Stravinsky, each bar contains its own number of beats. Bars 4 and 6 are irregular, containing 5 beats. The irregularity exists, therefore, both between the bars and within certain bars. If the simple irregularity of the bar itself (bar with 5 or 7 beats) is quite modern in savant music in 1870, it has its precedents, especially in the music of composers coming from Slavish or Finno-Ugrian cultures. Traditional songs aside, one can quote the 17 th century canticle composed by the Hungarian duke Pál Esterházy, Az földi àllatokroll ("About the beasts of this earth") which contains bars of 11 beats, or Chopin's larghetto of the First sonata in C minor op. 4, containing bars of 5 beats (then the irregular bars will still be particularly attached to Eastern Europe; consider the scherzo in 5 of Rimsky-Korsakov's Third symphony

  Debussy's work. No other intonations seem as different as those of French and German, the French intonations caricatured as perfectly flat by Debussy, and German intonations as particularly mobile, almost instable, by Strauss. A certain "technical nationalism" seems to be reached, at least in some bars of those different works, as if France were accustomed to a sonar flatness, and Germany to the opposite. Jacques Amblard University of Provence (France) Bibliographic references ADORNO, T. W., Mahler, Paris, Minuit, 1976. AMBLARD, J., Pascal Dusapin. L'intonation ou le secret, Paris, Musica Falsa, 2002. BARRAUD, H., « Lied », in Encyclopaedia universalis, CDrom. BRIGHT, W., " Points de contacts entre langage et musique ", in Musique en Jeu, n° 5, Paris, Seuil, novembre 1971. CARNER, M., Les lieder de Wolf, Arles, Actes Sud, 1988. COHEN-LÉVINAS, D., "A propos du lied romantique allemand", in La voix au-delà du chant, Paris éd Michel de Maule, 1987. DIDEROT, D., Salon de 1767, Paris, Société encyclopédique française et Club du livre, 1970. ERISMAN, G., "Janácek", in Encyclopoedia Universalis, tome XII, 1992. FÓNAGY, I., « Bases pusionnelles de la phonation », in Revue française de psychanalyse, tome 35 (juillet 1971).

  FÓNAGY, I., « L'accent français : accent probabilitaire », in Studia phonetica, n° 15, Paris, Didier, 1979. FÓNAGY, I., La vive voix, Paris, Payot, 1991. HADEN, E. F., « The accentual system of French », in Papers in linguistics and phonetics in the memory of Pierre Delattre, The Hague, Paris, Mouton, 1972. JANÁCEK, L., "Autour de Jenufa", in Hudebni Revue, 1915/1916, reproduced in the concert program Vec Makropulos published by the Festival International d'Art Lyrique, Aix-en-Provence, juin/juillet 2000. MARNAT, M., Moussorgski, Paris, Seuil, 1962. RIMSKI-KORSAKOV, N., Journal de ma vie musicale, Paris, Gallimard, 1938. ROUSSEAU, J.-J., Essai sur l'origine des langues. ROSSI, M., Di CRISTO, A., HIRST, D., MARTION, P. et NISHINUMA, Y., L'intonation. De l'acoustique à la sémantique, Paris, Klincksieck, 1981. ROSSI, M., L'intonation, Paris, Ophrys, 1999. SAMAGNAC, M., « Moussorgski », In Encyclopaedia universalis, tome XV, 1992. VIVIER, O., Varèse, Paris, Seuil, 1973.


Margaux Régniez

Adrien Pelat

Charles Pézerat

François Gautier

Vibration damping of honeycomb sandwich panels induced by micro-perforations

Keywords: acoustic impedance, vibration damping, micro-perforations. 1

INTRODUCTION

Sandwich materials composed of two skins glued on both sides of honeycomb cells are widely used in several domains because they offer the possibility to make lightweight and stiff structures. In term of vibroacoustic characteristics, honeycomb sandwich materials have however high acoustic radiation efficiencies, or reciprocally, they are sensitive to acoustic excitations. The aim of the present study is to investigate the capability of micro-perforations to reduce vibrations of honeycomb sandwich materials.

Micro-perforations are well known devices used for the attenuation of acoustic levels. Indeed, micro-perforations provide an acoustic dissipation on the surface of the treated structure due to the viscous effects inside the micro-holes whose dimensions are less than the boundary layer of the fluid. Micro-perforations were particularly studied for rigid structures [START_REF] Maa | Theory and design of microperforated panel soundabsorbing constructions[END_REF][START_REF] Maa | Wide-band sound absorber based on microperforated panels[END_REF][START_REF] Maa | Potential of microperforated panel absorber[END_REF], [START_REF] Dupont | Transparence et absorption acoustiques des structures microperforées[END_REF][START_REF] Atalla | Modeling of perforated plates and screens using rigid frame porous models[END_REF] and more recently for vibrating structures [START_REF] Lee | Effects of the panel and Helmholtz resonances on a micro-perforated absorber[END_REF][START_REF] Bravo | Sound absorption and transmission through flexible micro-perforated panels backed by an air layer and a thin plate[END_REF], where the objectives are always the improvement of the acoustic absorption. Of course, when the structure is excited by an acoustic field, the decrease of the wall pressure due to the presence of micro-holes implies that the vibration level is also reduced. The reduction of the wall pressure can be quantified by the well known acoustic absorption coefficient quantifying the ratio between the absorbed energy and the incidence energy. This treatment is also particularly effective for high levels, where the dissipation phenomenon becomes non linear [START_REF] Boumda | On the variations of acoustic absorption peak with particle velocity in micro-perforated panels at high level of excitation[END_REF]. Otherwise, whatever the excitation (airborne or structure-borne excitations), the dynamic comportment of the structure can be changed due to fluid motion (generated by the vibration) inside micro-holes. Viscous effects can then introduce dissipation leading to a non-negligible damping when numerous micro-perforations are considered. This phenomenon was shown on micro-perforated membranes [START_REF] Cackley | The modeling of unconventional sound absorbing materials: microperforated films and closed cell foams[END_REF]. The goal of this paper is to see if this damping effect is also interesting for flexural motion of honeycomb sandwich panels.

The proposed modeling is divided in three parts, corresponding to models at three different scales: the microscale, the mesoscale and the macro-scale. The micro-scale corresponds to the scale of the honeycomb cell, where the physical phenomena are described and modeled to compute the dissipation delivered by one hole. The mesoscale contains several perforated cells, where the damping effect can be homogenized. The macro-scale is the scale of the structure, where the effects on modal properties can be obtained. , ω being the circular frequency.

At the micro-scale

The honeycomb sandwich plate is supposed to be modeled by the Kirchhoff's plate theory. As a consequence, at the scale of a single honeycomb cell, the motions of the wall cells (or plate cross-section) can be described by a superposition of two elementary vibratory motions: a translation and a rotation.

Both structural motions are coupled to the internal fluid motions, inducing the damping effect to be modeled.

Translation of the cell induces an air flow in the hole which is supposed to be be done on one side of the cell. Air

Vibration damping of honeycomb sandwich panels induced by micro-perforations

Margaux Regniez 1,2 , Adrien Pelat 2 , Charles Pézerat 2 , François Gautier 2 1 CNES, 18 Avenue Edouard Belin, 31400 Toulouse, France 2 Université du Maine, CNRS UMR 6613, LAUM, Avenue Olivier Messiaen, 72085 LE MANS CEDEX 9, France emails: margaux.regniez@univ-lemans.fr, adrien.pelat@univ-lemans.fr, nicolas.joly@univ-lemans.fr, charles.pezerat@univlemans.fr, francois.gautier@univ-lemans.fr flow is also induced by the cell rotation, but is supposed to be very weak. Rotation effects are such ignored in the modeling.

To calculate the equivalent viscous force ⃗ f v resulting from the microperforated honeycomb cell translation (see figure 1), the fluid motion is described by a mass-springdamping system as shown on figure 2, for which the equation of motion is:

j ω M f ẋ f = -K f jω ( ẋ f -ẇ) -C f ( ẋ f -ẇ) (1)
where

K f = P 0 γ S f 2 V cav corresponds to the effect of the cell cavity, M f =S f ℑ( Z Maa ) ω and C f =S f ℜ( Z Maa )
are respectively the mass and dissipation of the fluid inside the micro-perforation, computed using respectively the imaginary and real parts of the acoustical impedance given by Maa [START_REF] Maa | Theory and design of microperforated panel soundabsorbing constructions[END_REF][START_REF] Maa | Wide-band sound absorber based on microperforated panels[END_REF][START_REF] Maa | Potential of microperforated panel absorber[END_REF]:

Z Maa = 32 ν ρ 0 t d 2 ( √ 1+ x 2 32 +√2 x d 8 t ) + j ωρ 0 t ( 1+ 1 √ 3 2 + x 2 2 + 8 3 π d t ) (2)
with, √ ωρ 0 ν d 2 , d the hole's diameter, t the hole's thickness, ν the kinematic viscosity, ρ 0 the air density and ω the circular frequency.

The applied force on the honeycomb cell due to the fluid dissipation is written as:

⃗ f v = [ C f ( ẋ f -ẇ) + K f j ω ( ẋ f -ẇ) ] . ⃗ z .
(3) Expression ( 3) is valid only if we consider that the one degree of freedom model described in figure 2 is valid. For an air piston inside a perforation, this is not the case since the stiffness term in (3) is not applied to the plate. This point is not considered in the paper.

It is important to note that the observed reaction is local, which means that each elementary force ⃗ f v associated to one cell is independent from the forces associated to the neighboring cells, No interaction between adjacent cell is considered.

At the mesoscale

At the mesoscale the considered surface is noted dS , it is sized to be higher than the cell dimension but small enough compared to the acoustic wavelength.

The relationship between the viscous forces ⃗ f v (in N) and the resultant force distribution ⃗ f Rv (in N/m²) on dS is:

⃗ f Rv dS= N ⃗ f v = σ S f ⃗ f v dS ( 4
)
where N =n t/ c n c is the number of holes in the surface dS . The number n h / c is the number of holes per cell and n c the number of honeycomb cell in dS ,

σ= n t / h S f S c
is the perforation rate and S f the surface of a hole. In this study, the number of holes per cell n h /c is fixed to one. The motion of the elementary surface dS is supposed to be described by the classical bending equation:

-ω 2 ρ h w( x , y , ω) dS+ D Δ 2 w( x , y , ω)dS =Δ p dS+ f Rv dS (5) where w( x , y ,ω) corresponds to the transverse flexural displacement, ρ h is its surface density, D the flexural rigidity and Δ p the difference of pressures at each side of the panel. The resultant force distribution ⃗ f Rv given by (6) allows us to define the damping coefficient C v which will have sense, as expressed, if it is real and positive.

⃗ f Rv dS= σ S f ( C f + K f j ω ) ( ẋ f -ẇ ) . ⃗ z dS =-C v ẇ .⃗ z dS

At the macro-scale

The macro-scale corresponds to the scale of the complete structure. In the following, a rectangular honeycomb sandwich panel is considered . It is considered to be simply supported on its edges. The dimensions are expressed on the figure 4. Equation of motion ( 5) is considered on the whole the surface of the structure and established for the harmonic regime:

-ω 2 ρ h w( x , y ,ω)+ j ωC (x , y)w( x , y ,ω)+ D Δ 2 w( x , y ,ω)=Δ p [START_REF] Rayleigh | Theory of sound[END_REF] where C ( x , y)= I ( x , y)C v (8) is the damping operator, taking into account the microperforation positions, specified thanks to the indicator function I ( x , y) ( I ( x , y)=1 if there is a hole at point (x,y), and I ( x , y)=0 otherwise).

The damping properties of the micro-perforated plate are investigated by looking at the modes of the damped plate described by equation [START_REF] Rayleigh | Theory of sound[END_REF]. Since the damping for the microperforated plate is expected to be low, modes of the damped plate are supposed to be closed to the ones of the undamped plates and are computed using a perturbation technique.

The eigen values λ 0k and the mode shapes ϕ 0k of the undamped plate are the solutions of (ρh λ 0k 2 + D Δ

2 )ϕ 0k =0 , (9) associated to the imposed boundary conditions. The eigenvalues λ k and the mode shapes ϕ k of the damped plate are the solutions of

(λ k 2 ρ h+ λ k C ( x , y)+D Δ 2 ) ϕ k =0
(10) and are supposed to be written as

λ k =λ 0k +δ λ (11) ϕ k =ϕ 0k +δ ϕ ( 12
) where δ λ and δ ϕ are correction terms, supposed to be small.

Reporting [START_REF] Bravo | Sound absorption and transmission through flexible micro-perforated panels backed by an air layer and a thin plate[END_REF] and ( 12) into (10) leads to:

(DΔ 2 +λ 0k 2 ρ h) ϕ 0k +2 λ 0k ρ hδ λ ϕ 0k +λ 0k C( x , y)ϕ 0k +(δλ ) 2 ρ h ϕ 0k +( D Δ 2 +λ 0k 2 ρ h) δ ϕ+C (x , y)δλ ϕ 0k +2 λ 0k δλ ρ h δϕ+(δ λ) 2 ρ hδ ϕ +λ 0k C ( x , y) δ ϕ+δ λ C ( x , y) δ ϕ=0 . ( 13
)
The third line, the last three terms of the second line and the last term of the first line of equation ( 13) are second order terms since they correspond to products of small values. The first term of the first line of the same equation is equal to zero because it corresponds to the equation ( 9) of the conservative associated system. Finally, equation ( 13) leads to:

(D Δ 2 +λ 0k 2 ρ h)δ ϕ+(2 λ 0k ρ h )δ λ ϕ 0k +λ 0k C ( x , y) ϕ 0k =0 . ( 14
)
Projecting equation ( 14) on the mode ϕ 0k gives the correction terms δ λ as

δ λ= -∫ S ϕ 0k 2 C ( x , y) dS 2 ∫ S ρ h ϕ 0k 2 dS ( 15
)
where S is the panel surface. This leads to a modal damping coefficient for the plate equal to

ξ k = -δ λ ω 0k = ∫ S ϕ 0k 2 C ( x , y) dS 2 ω 0k∫ S ρ h ϕ 0k 2 dS . ( 16
)
If microperforations are uniformly spread over the structure, we get C ( x , y)=C v and thus,

ξ k = C v 2 ω 0k ρ h . ( 17
)
The special case where the hole distribution is uniform leads to the maximum value of the modal damping. In the general case, we have

ξ k = C v 2 ω 0k ρ h - ∫ S ϕ 0k 2 C v d S 2 ω 0k ∫ S ρ h ϕ 0k 2 d S ( 18
)
where S corresponds to the non-perforated surface.

Results are presented in the next paragraph for several perforation rates σ .

Parametric study

The analysis of the term C v (eq. ( 6)) is made using a parametric study. C v is written as (19) where

C v = σ S f ( C f + K f j ω ) ( j ω M f K f j ω +C f + j ω M f ) = R v + j ω I v
R v = σ (C f +K f M f C f ) S f ( C f 2 -( ω M f -K f j ω )
2

)
and

I v = σ ( M f C f 2 -K f M f 2 - K f 2 M f ω 2 ) S f ( C f 2 -( ω M f -K f j ω )
2 )

, and plotted on the figure 5 These curves show the classical effect of the resonance of the one-degree-of-freedom oscillator, whose natural frequency Using equation ( 17), the modal damping of a rectangular simply supported sandwich plate (dimensions 0,5m x 0,33m x 0,02m and bending stiffness 150 kg.m².s -2 ) is computed for the first five modes (between 200 and 1000 Hz). Figure 6 plots the results as a function of perforation rate σ .

is f 0 = 1 2 π √ K f M f .
Eigenfrequencies of the plates are found to be 210 Hz, 405 Hz, 648 Hz, 729 Hz and 842 Hz.

The order of magnitude of the modal damping is rather low and is found to be 1.10 -8 for first and second modes. For modes of higher frequency and for higher perforation rates, the modal damping is found to be 2.10 -6 . A simple computation allows us to estimate the value that such modal damping should have to obtain a mobility difference of 1 dB between the non-perforated and the perforated panels. This value is found to be 10 -3 . Hence, the current treatment as dimensioned here, leads to a too low damping to reach this mobility difference.

CONCLUSION

In this paper, we propose a methodology for computing the modal damping induced by micro-perforations performed on one side of a honeycomb sandwich panel. This method is based on the Maa model for the fluid motion inside each hole, which allows us to estimate an equivalent viscous force (micro-scale model). The resultant force is then computed considering a group of cells (meso-scale model) and allow us to express a term, whose real part expresses apparent damping for the global structure (macro-scale model).

A perturbation technique is used to estimate the modal damping coefficient. The model can take into account a nonuniform distribution of microperforations and the upper bound result for the modal damping has been found. Indeed, it corresponds to the case where the distribution of microperforations is homogeneous on the structure.

The order of magnitude observed for such modal damping is found to be rather low for the proposed geometry.

  equations of the paper are written at the harmonic regime. The convention chosen is e j ωt

Figure 1 :

 1 Figure 1: Single microperforated cell and resultant viscous force.

Figure 2 :

 2 Figure 2: One degree of freedom representation of the fluid movement inside the micro-perforated honeycomb cell.

Figure 3 :

 3 Figure 3: Assembly of microperforated honeycomb cells to form the surface element dS.

  for three different values of perforation rate, depending on the following holes and cavity dimensions:

Figure 4 :

 4 Figure 4: Plate's notations.

ACKNOWLEDGMENTS

The authors would like to thank Didier Gangloff from the Centre National des Etudes Spatiales (CNES) and Eric Labiole from Thales Alenia Space, for the financial support of the thesis work.