Tuning Meister - Observations on cello Sarabande


    Below are some few thoughts on the intonational information revealed from TuningMeister's analysis of Bach's c minor Sarabande from the cello suites. Perhaps the score will be useful for our discussion:


Sarabande Complete Score
 
    I have set the pitch for this cut from the bottom note C, which is an open string. However, if you observe the bottom Cs you will notice that the pitch varies a bit - bow pressure seems to change the pitch. So even an open string is never quite in tune all the time!

   First let me discuss a few anomalies in the trace, they are unavoidable but if you understand the reason for the imperfections in the tuning trace you can skip over some seemingly false information. They all result from having more than one note sounding simultaneously. The orange boxes locate where the trace would lie if the results were perfect.


 Measure 3  






At the end of measure 3 you will notice the upper g does not register right away, in fact hardly at all; mostly it appears to be the lower G [outlined in blue]. This happens because the lower G is an open string so it continues to ring even while the bow has moved to the g. The pitch in the lower graph is probably a good approximation of the upper g, as it is quite close to the lower G. If they were different, the difference would be greater than shown, as the calculation produces a weighted average

 






measure 6




  In measure 6 the f doesn't appear right away as the preceding d continues to sound  - 
I believe that is an open-string d, or perhaps it is just the resonance of the room. Since the difference tone of d to f is a Bb, that appears for a short time before the f is graphed [outlined in brown, lower graph. The actual Bb lies below the bottom of the upper graph]. In fact, many notes may show a slight delay before being drawn on the graph, mostly because of the resonant room acoustic.







measure 11
  





 The first note in measure 11 in the repeat comes out F# [blue box] instead of db' - again because the previous open-string G is still sounding. And the final note in that measure, c', shows an octave lower [deep red box] because the open-string C continues to sound. In fact, when the c' is released, the graph briefly shows C still sounding [green box].








penultimate measure

 



  In the penultimate bar the low G on the last beat begins while the previous c is still sounding (either the string itself, or room resonance) and the resultant difference tone is a low CC [octave below the graph]; so the beginning of the G shows instead a blue trace from that CC [outlined in grey]. Then the following upper b makes a major tenth with the still-sounding open-string G; TuningMeister hears the difference tone which is two octaves below G, so the trace remains brown for a G
[outlined in blue, lower graph]. The strong b influences the pitch measurement. If the b were a pure major tenth it would not change the pitch of the fundamental at all. But since tenths are quite high in Equal Temperament (henceforth ET) and even higher if raised as a leading tone, the pitch of the G seems to go up. In general it doesn't give much information about the b at all. 


 







   Thus most of the strange results come about because of interference from notes still sounding, even if it is just from room resonance, with bass notes having the greatest staying power. So close microphone placement during practice sessions would remove many of these problems, but of course that is not what is done for the best sound in a recorded performance, which we are analyzing here. 


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    Now let us look at some of the details of the playing itself.

Sarabande F octave




    The most basic particular criticism that I can offer is that the octave descent at the end of measure 12 is woefully out of tune, even more so in the repeat, where the lower F is about 20 cents, a fifth of a semitone, sharp [blue box, lower graph - compare to previous red trace]. If played simultaneously they would sound horribly out of tune. Did you notice this when listening? [Typically a piano's octaves need to be less than 1/2 cent off from true. In fact that 20 cent difference is the difference between say an Eb and a D# in 18th century tuning.]


       It is fairly easy to hear in this recording the slender semitones of  the raised leading tones
of modern intonation. And these are fairly modest compared to some. They are quite consistent, and clearly the second note of the pair is tuned by relationship to the leading tone. That is, even if the overall placement of the notes wanders, the relative pitch of the leading tone and its resolution is pretty much the same throughout the piece. This relationship is also reflected in the descending semitone, appoggiatura-like movements of the melody. Almost all of these successive semitonal movements, both rising and falling. are about 10 to 15 cents narrower than ET. Even when the same interval isn't placed precisely in the overall pitch scheme, the interval itself is quite accurate. It is easy to understand that the intonation is guided more by a note-by-note process than by any precise long-term feel for a constant tonic.

    It looks like the open G string is tuned way too sharp (end of measure 3). If the string were tuned to a perfect fifth above the C, it would be 2 cents sharper than ET. Here it seems about 6 or 7 cents above ET, and although as mentioned, it is hard to get a precise measure of the open C string (end of measure 4), that C is very close to center.

    Here's something I don't really understand. Typically, modern intonation lowers the minor third and raises a bit the major third. The way the final (arpeggiated) chord is tuned 
is consistent with this idea, with the eb slightly lower than ET. But the modulation to Eb major (the relative major) in the first section ends up with the Eb quite a bit higher than ET. The repeat is remarkably consistent. Actually, quite a few of the flat notes in this section are higher than I would expect (note especially some Abs). The best explanation for the arrival at a higher Eb that I can give is that its leading tone D (end of measure 7) has become quite sharp; even with the drop of about 10 cents via the slender leading tone relationship to the Eb, the Eb ends up sharp. But all the Ebs in the first section are higher than ET. It is quite curious.

    Overall there is not a great tendency to raise the pitch in the upper octaves that we find with many string players, although there is some. The final high c' is very well in tune.

    This performance is by János Starker, his RCA recording from the 1990s.




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