Proceedings paper


Intonation and Interpretation in String Quartet Performance: the case of the flat leading note

Peter Johnson, Birmingham Conservatoire (UK)



In a famous paper from the 1930s, Carl Seashore observed a tendency among violinists to adopt the very wide major thirds and narrow semitones of Pythagorean tuning (Seashore 1938: 218-224). His data, however, reveal extreme deviation in the size of these intervals, and this is inconsistent with any formal temperament. The minor seconds vary from 38 cents smaller than equal temperament to 18 cents larger, a difference of more than half a semitone (p.221). Either he was using poor performances, or the variety of intonational practice must be regarded as musically significant. Studies of unaccompanied cello playing, including recent and admired recordings by Mischa Maisky and Anner Bylsma, show a similar predilection for wide major thirds and narrow semitones together with a striking range of actual tunings (Johnson 1999b).

Casals has argued that good playing demands what he calls 'expressive intonation', where tones are sharpened or flattened in accordance with the direction of melodic movement (Blum 1987). This principle is consistently applied in his own recording of Bach's C minor Sarabande (Johnson 1999b). Intonation thus serves as an indexical sign of voice-leading. But is this its only expressive function? It is widely assumed in the profession that intonation has a more general expressive function, and it is well-known that we tend to interpret small deviations of tuning qualitatively rather than quantitatively (Makeig 1982). And what happens when a performer reverses the tendency, by playing a falling leading-note sharper than a rising one in the same phrase? In the Notes that append this paper, I discuss an example from Beethoven's last slow movement, the Lento assai from the Op.135 quartet (Example 1). In bars 3 and 5, there are conjunct falling and rising leading-notes in the first violin. Out of 25 recordings, sixteen violinists take the falling leading-note in bar 3 sharper than the rising, five tune the notes almost identically, and only four conform to 'expressive tuning' (Johnson 1999b). Explanations can no doubt to be found for this reversal of normal practice in the special qualities of Beethoven's music, but these would confirm an expressive function for intonation transcending the strictly syntactical.

In fact, whatever principle we devise to justify normal practice, the anomalies remain problematic. From the study of leading-notes at the start of the Lento assai, one recording in particular stands out as highly idiosyncratic. This is No.17, the Lindsays' 1987 recording, in which the leader consistently tunes all the leading-notes in bars 3 and 5 as pure major thirds against the dominant in the second violin. Against the normative tunings of Equal Temperament, this gives very flat leading-notes and correspondingly wide semitones to and from the adjacent tonics.

What are we to make of such divergent practices? In particular, how do we 'read' and experience the idiosyncratic tunings in the Lindsays' recording? In this paper, I propose to address these questions by comparing this recording with two others. One will be an example of very sharp tuning in the same bars of the Lento assai, the other an application of Just Intonation by the Lindsays in the very different context of the Heiliger Dankgesang from Op.132. First, however, we need critically to review our methods.

Example 1

1. Method

An obvious problem in dealing with an idiosyncratic performance is how we assess its competence. How do we know that our analyses are not revealing simple errors or miscalculations in the execution of the performance? By using commercially released recordings, we have a strong, although not foolproof, assurance of an error-free performance that has met the approval of its performers, the producer and engineer, and eventually the wider community of experts and the listening public. On the other hand, recordings made in the laboratory carry no such assurances, neither are they normally available for external scrutiny or further analysis.

How, then, do we accurately and reliably analyse intonation from a recording? Necessarily, we have to work from small samples, and these need to be representative or symptomatic of the larger musical context. My examples have been supported by note-by-note analyses of adjacent events, which confirm the stylistic integrity of the performances. A preparatory survey exposes the moments of special interest, which, perhaps not surprisingly, tend to involve the leading-notes.

We can define the frequency spectrum of any complex musical event quickly and efficiently using Spectrum Analysis. In ensemble performance, the chief question raised by this method is whether the fundamental frequencies are accurate indicators of our perceptions of pitch and interval. In a paper on intonation in Barbershop singing, Hagerman and Sundberg support the contention that they do (1980), and empirical tests in which single-beat extracts are compared with synthesised tones of know frequency, also confirm this in most cases (Johnson 1999b). Nevertheless, there are anomalies, and the Lindsays' recording of the Lento assai provides an interesting example.

Figure 1 shows the spectra of two chords. In the upper plot, the chord is from the opening chorale of the Heiliger Dankgesang from Op.132, and represents the frequency-content of the C major chord in bar 4. The leading-note is E4 in first violin, and C3, G3 and C4 complete the dominant harmony in the lower strings. The lower plot is from the Lento assai, and represents the last beat of bar 3, a more complex second inversion dominant seventh, but also with the leading note, C4, in melodic prominence in first violin (see Example 1).

Figure 1

The accuracy of the frequency-data shown in Figure 1 is determined by the value of k, shown to the right of the title-line. The analysis resolves the source signal into discrete bands each of width k Hz, and shows the strength of each band. By adjusting this value, more precise readings are available without overtaxing a standard personal computer, but there is a trade-off between accuracy and efficiency. Amplitude levels in the figure are shown as decibel-difference calculated from the strongest peak in the signal. For a more technical description see the Notes appended to this paper.

In both cases, the analysis has generated a spectrum from k-11025Hz, but it is evident that in this quiet music, most of the relevant acoustical information is contained within the 50-3000Hz range. We can see a marked difference between the two plots, the lower showing a tailing off of peaks in the spectrum above about 900Hz, whereas the upper plot suggest that we shall need to rescale the plot to gain access to all the significant harmonics. To this extent, the analyses give a good visual analogue of the differences of tone-colour of the two chords. Although both are played quietly and in the same tessitura, the lower has a noticeably darker quality of tone.

2. Just Intonation

The two extracts analysed in Figure 1 are chosen because they illustrate Just Intonation in the tuning between first and second violins. The relevant calculations are, from the upper plot:

C4 at 263.45 x 5/4 gives E4 at 329.3Hz

and from the lower plot:

Aflat3 at 209.44 x 5/4 gives C4 at 261.8Hz.

Both results are within 0.4 Hz of the actual peaks for the leading-notes, a barely perceptible difference of 2.2 cents. The margin of error is < (k x 5/4) = < 0.21 Hz or about 1 cent. For a brief explanation of the relevant acoustic theory see the Notes at the end of this paper.

Examination of the lower-order harmonics in the upper plot reveals close conformity to the exact integer multiples of acoustical theory. In the lower plot, that is the case for all the harmonics of the first violin (< 2 cents) but not for those of the second violin's Aflat3. The second harmonic at 416.03Hz is flatter than the theoretical second harmonic (209.44 x 2) by almost 12 cents. In other words, the interval between the second harmonics of first and second violins is an equal tempered major third. In fact, the peaks at Aflat4 and C5 are close integer multiples of the cello's Aflat2. The fundamental of this tone is unclear in Figure 2, but by replotting the figure with a higher sample-size and hence greater precision (see Notes), Aflat2 is shown to be 104.2Hz. The multiples of this frequency are within 4 cents of the peaks at Aflat4 and C5, < 4 cents being the level of precision of these calculations. The tuning of the other cello note, the Eflat, is very close to Just Intonation in relation to the two violin tones. One other detail to emerge from this spectrum is that the viola's Gflat3 is tuned as an almost true 7th harmonic against the second violin's A flat (183.54 x 8/7 = 209.76). This Just dominant seventh is some 25 cents flatter than the Just major second from A flat.

3. Comparison and Interpretation

We have, then, two examples of applied Just Intonation, from the same ensemble and in a similar repertoire. But many other factors are different. In the case of the Heiliger Dankgesang, the intention to find pure tuning is explicit throughout the opening statement of the Chorale, which is to say that we hear the passage in terms of the tones of Just Intonation. However, the execution does not always match this ideal, and in the other chords there is a liveliness in the sonorities arising from intonation that is very nearly but not quite pure. And we have seen that in the Lento assai, there is a direct conflict between the second violin and cello A flats, creating on the one hand a Just major third and on the other, an equal-tempered tenth. Both performances are therefore less than ideal. Should we therefore conclude that our sources are invalid as exemplary or at least expert performances? The expert community has clearly judged otherwise, for these recordings have been in the CD catalogues already for some 13 years. Let us make up our own minds by hearing the extracts. Here, first, is the start of the Heiliger Dankgesang.


OP.132iii b.1-6 (Lindsays)]

What I find interesting here is the way we differentiate between intention and execution. We can handle the differences either evaluatively or interpretatively, but we elect to listen evaluatively, we must bear in mind that the apparent imperfections have not been edited away in the recording process, as they could have been. If instead we listen interpretatively, we assume that what we hear is intentional or acceptable to the performers and, in some musical sense, construct meaning from what we hear. In the Lindsays' Chorale, for instance, there is no singing persona from the first violin, as there is in some other recordings. With no hint of vibrato, there is a clear intention of creating a blended, single sonority for each chord. Yet the first violin is still the top line and the step-wise movement confirms that this is a chorale melody. There is therefore a deictic tension between what we know the music could be, and what we hear. This tension, when appropriately used, is, I suggest, one of the most positive aspects of good performance. It explains why we need more than one good performance of the same work, and why there is no single definitive interpretation.

If the peculiar tension in the Lindsays' playing of the chorale arises from the gap between intention and realisation, it is reinforced by the withdrawal of persona from the first violin, which plays the melody almost as if it were not the principal line. This is signified by the lack of vibrato and by using Just Intonation, thus denying the normative sharpening towards the tonic. But Just Intonation, I suggest, has its own surcharge of meaning, and this may even transcend the semiotic in its strictly natural origins. It is perhaps its very neutrality that has made Just Intonation unusual in mainstream classical performance over the last seventy years. On the other hand, period instrument performance, in which Mean-tone tuning with its pure thirds is widely used, serves to remind us that any interpretation of performance practice is style-specific, and that sharp tuning and vibrato are not the only way of signifying salience or a soloistic persona.

This example also highlights the inevitable interaction between work and performance. Beethoven's hymn is addressed, as it says in the score, 'an die Gottheit', and can thus be read as aspirational, a glimpse of an ideal that remains inattainable. This invites an interpretation of the Lindsays' performance in terms of embodiment or enactment of that aspiration. In this context, Just Intonation together with the senza vibrato may symbolise an aspirational loss of individuality, the transcendence of self. Yet in the Lindsays' performance, Just Intonation is noit perfectly achieved, and the performance itself demonstrates the extreme difficulties of sustaining pure tuning on modern stringed instrument over the duration of these slow-moving minims. It is the very impossibility of achieving a clearly defined ideal that emerges, metaphorically, in this performance. Its warm, deeply human properties derive from its quite specific vulnerability. To this extent, the performance ironizes Beethoven's Dankgesang, although without a trace of the cynicism we have come to associate with postmodernist and deconstructive performances in the theatre. Here are four human beings, highly skilled and imaginative in their art, utterly dedicated to performing Beethoven's music as an engagement with the transcendental, and fully aware that their aspiration is idealistic. Were it to achieve the perfection of pure, natural tuning, we would read it very differently.

This, at least, is my interpretation, offered as a metaphor of a strictly musical experience of the embodied sonorous properties of the recorded performance. But how, then, are we to interpret the more complex signifiers in the Lindsays' performance of the opening bars of the Lento assai?

[HERE PLAY Op.135iii b.1-6 (Lindsays)]

At first glance, intentions again seem explicit. Here is a song, a 'Süsser Ruhegesang' as Beethoven wrote in a sketch-book. And in some respects, the Lindsays' leader, Peter Cropper, adopts the contemporary conventions of a singing line, with a steady, continuous tone and normal vibrato. Here, deictically, we do have our vocal persona, standing before the accompanying tonic and dominant chords of the lower strings. The quiet, undemonstrative playing is not incompatible with this reading, but significantly qualifies it. But there are three factors in the performance that undermine this idealised image. Firstly, the tempo is extraordinarily slow, by far the slowest of any post-war recording of this movement. It can be seen from Figure 1 that the single note has a duration of 1.5s., and beat-timings indicate that there is a long tenuto over this upbeat, as if the bar-line were problematic to cross. Thus the intonational anomalies discussed above are emphasized, even dwelt upon, and the step from leading note to tonic is easily heard as a wide interval in the recording. And yet, the sustained 6th beat is not heard unambiguously in terms of the resonances of Just Intonation, as are the chords in the Chorale from the Heiliger Dankgesang, and instead tend to sound a little flat, particularly in relation to the next tonic. What was presumably a minor aberration in tuning between second violin and cello here becomes meaningful. The serendipedous, we know, is not necessarily inartistic.

Beethoven's 'Ruhegesang', therefore, is performed as a song enveloped in difficulties: the 6/8 metre doesn't flow, the alternating tonic and dominant chords don't sway; there is no trace of the gentle lullaby that is implicit in the score. Intonation is here one of a complex set of factors. Again, therefore, Beethoven's score is ironized. But here, the tension is between the simplicity of the score-content and the complex and contradictory signifiers of the performance.

And yet, if the Lindsays' interpretation is a manifestly inauthentic reading of the opening bars of the movement, it is arguably implicit in the way Beethoven develops the theme in the first two variations. Already at bar 22 we come to a deeply tragic, C sharp minor variation, complete with the slow tread of a funeral march and strange, stalking chromatic harmony. In most performances, this variation arrives as an abrupt fall, or perhaps a reminiscence. The Lindsays' is the only recording that performs this moment as if it were the inevitable consequence of the Theme itself. It is, they seem to say, what the movement is all about. I simplify, of course, but I hope my words give some intimation of the richness of this interpretation.

And so, what of very sharp tunings of the leading notes in the Theme of the Lento assai? In the Leipziger Quartet's 1999 recording, the rising leading note is tuned 37 cents sharper than Just (No.25 in Figure 2 below). Checks on the higher harmonics show that there are no peaks flatter than the integer multiples, and there is no doubt that the reading is accurate. And, surprisingly, the tuning does not seem to offend. Perhaps this is because it is consistent with other signifiers: Beethoven's 'cantante tranquillo' is presented very much cantante, but less tranquillo; the playing is markedly soloistic, as if inviting us to admire the singing tone of the first violin, even to participate mentally in its production. But we are not taken into the deeper recesses of musical experience as we are by the Lindsays, and this may be because of the lack of ambivalence in the performance and its relation to the score.

Theodor Adorno argued that the task of the performer is to problematise the composer's work (1997: 106). The Lindsays' performance shows some of the ways in which this can be achieved to artistic purpose. The possibility of a radical rethinking of the score through performance may be one measure of the value both of the performance and of the work itself. Beethoven's music invites a multiplicity of readings, whether in the spirit of authenticity or of reinterpretation and re-evaluation. The Lindsays demonstrate that performance can be both creative and critical. Our task, as listeners, is to carry this process to the music as we hear it, as the product of both compositional and performance artistry and of interactive listening on our part. I have tried to show that a useful preliminary step is to isolate salient properties of the performance, such as intonation and vibrato, but I have also argued that interpretation necessarily follows as a process of reintegration or synthesis between score, performance and our own interactive response as listeners. We cannot adequately describe our musical experiences, but words are there to help us, and as Nicholas Cook has argued, without them, our experience would be impoverished (Cook 1999).





Additional Notes

1. Pitch-names, cents and intervals

I use the American Standard pitch-names, where the cello C-string is C2, middle c is C4 and soprano top-c is C6. A440 is therefore A4. I refer to performed notes as 'tones'.

The interval, I, between two frequencies f1 and f2 is defined by I = f1/f2. This is converted to cents, c, by:

c = 1200 x log2(I).

An equal tempered semitone is 100 cents. At C4, a difference of 2 cents is about 0.3 Hz, and the equivalent in other registers can quickly be calculated by multiplied or divided by a factor of 2 per octave. So at C2, 2 cents represents a difference of 0.075 cents. Note that in Seashore (1938), intervals are shown in fractions of a whole-tone, where 0.01 = 2 cents.

Just Intonation is defined by the convergence of the lowest common harmonics between any two tones. For two frequencies, f1 and f2, the Just major third is given as f1x 5 = f2 x 4, so that f2 = f1 x 5/4 (386 cents). The Just perfect fifth is 3/2, and the major seventh is 15/16.

2. Spectrum Analysis

The accuracy of a spectrum analysis depends upon the ratio between the sample-rate of the source recording and the size of sample submitted to analysis. The resultant analysis is a plot of frequency against amplitude, the readings along the frequency axis proceeding in steps of k Herz, where

k = sample-rate/sample-size.

It is this ratio that determines precision in the reading of frequency. In Figure 1, k is set at 0.1682Hz, this being the ratio between a sample-rate of 22.05kHz and a sample-size of 217. The latter in fact represents a duration of almost six seconds, but shorter extracts may be used with the same sample-size by padded the source-file with zeros. This generates mathematically predictable anomalies in the form of smooth curves connecting the peak readings, but does not affect the peaks. A Hanning Window is applied to reduce this effect. In Figure 1, the original sound-source has a duration of 1.5s, 1/4th of sample-size.

The software I am using is SPAN, which is a purpose-specific implementation of the signal processing routines in Matlab (Johnson 1999c). For a more detailed implementation and discussion of spectrographic analysis see Johnson 1999a. The somewhat complex mathematics is explained in Poularikas & Seely 1991: 259-260, or in any standard text on signal processing.

3. Comparison of Leading Notes in Op.135iii, bars 3 and 5

Figure 2 shows how 25 string quartets handle the tuning of the melodic leading-notes in bars 3 and 5 of Op.135iii. The ensembles are arranged in chronological order along the x-axis, from the Flonzaley Quartet's recording of 1927 to the Leipziger Quartet's of 1999, and the sample includes a quite remarkable performance of an arrangement for the strings of the Vienna Philharmonic under Bernstein (No.22). This example can be included because a well-trained string section plays with sufficient precision of intonation to generate very clear peaks in a spectrum analyses.

The upper plot relates to bar 3, and the lower to the equivalent tones in bar 5. The signs Ú and Ù indicate the tuning of the falling and rising leading notes respectively, calculated as intervals in cents from the sustained Aflat3 in second violin. The zero line represents a Just major third (386 cents) between first and second violin. The equal tempered major third would be about 14 cents sharp, the Pythagorean, 22 cents sharp.

Contrary to the principle of expressive tuning, the falling tone is on average the sharper, by a factor of about 5 cents in the upper plot and 6 cents in the lower. Contrary to 'expressive tuning', 16 (13) out of the twenty-five recordings tune the rising leading-note flatter than the rising one, whereas only 4 (8) show the ascending leading higher than the lower; 5 (4) are identical, within a safe margin of error of <3 cents (figures for bar 5 in brackets). And we can note that all the entries below the zero line, i.e. flatter than Just, relate to the rising leading note in both plots. We can similarly compare the melodic semitone between these leading notes and the preceding and following tonics. As we would expect, the sharp major thirds are reflected by narrow semitonal steps in the melody (Johnson 1999b).

Figure 2

























Alban Berg(1989)


Fine Arts(c.1960)


New Budapest(1991)














La Salle(1976)












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