The case, stringing and fretting design of the 1543 Venetian clavichord by Dominicus Pisaurensis

an article published in  De Clavicordio, Proceedings of the V International Clavichord Symposium/Atti del V congresso internazionale sul clavicordo.  Magnano, 5-5 September 2001, edited by Bernard Brauchli, Alberto Galazzo and Ivan Moody, (Musica antica a Magnano, Magnano, 2002), 91-107, by Grant O'Brien

The fretting design and the temperament

            Table 6 above also reveals that Pisaurensis used the Venetian oncia to set out the position of the tangents not only for the lowest notes of each of the fretted groups on the top bridge, but also for all of the lowest notes on each of the bottom bridges.  If one compares the corrected lengths with my proposed design lengths in once for the strings on the two lowest bridges, then it is clear that each of these string lengths must have been designed by Pisaurensis to be a simple number and fraction of Venetian once.  This result had not been expected when moving the bridges to give the critical lengths, but provides convincing evidence about how Pisaurensis designed the fretting of this instrument: 

1.      he used the Venetian oncia in the design of the string lengths of his instrument,

2.      the bridges were positioned in those locations given by the critical lengths for each bridge,

3.      the strings lengths of each of the lowest notes of the fretted groups on the top bridge were also designed in terms of a simple number and fraction of Venetian once and

4.      the string lengths and tangent positions for all of the notes on the lowest two bridges were also designed to be a simple number and fraction of Venetian once. 

            As is well known the position of the tangents on the ends of the keylevers in any fretted clavichord gives an indication of the size of the semitone for each of the fretted pairs producing the semitones[1].  The sizes of these semitones in cents have been calculated and are given in the last two columns of Table 6.  Here it is clear that a number of different sizes of semitones is produced by the fretting of this instrument, and that there is a clear division into chromatic and diatonic semitones in the usual places with the enharmonic equivalent note g^T/a^I being tuned to g^T giving a small chromatic semitone g-g^T.  The sizes of these semitones have been transferred to Table 7 where the sizes of each of the semitones can be compared in each of the three top octaves (the strings of the notes from c to d of the lowest octave are placed on the middle bridge and are unfretted).

Table 7 – The sizes in cents of each of the semitones produced by the top bridge fretting patterns

Dominicus Pisaurensis clavichord, Venice, 1543

Leipzig Musikinstrumentenmuseum der Universität, Clavichorde, Cat. No. 1

There is a number of unusual and inconsistent features among these results.  It is possible to imagine a tuning system in which there are different sizes of either or both the chromatic and diatonic semitones[2].  However, in order for the octaves to be in tune, any fretting system where there are combinations of treble and quadruple fretting across different notes in separated octaves should have the same size of semitone in each octave.  If the corresponding semitones are not of the same size then it is impossible for the octaves played on the top and bottom notes of the semitone pair to be in tune.  Generally the diatonic semitones are fretted only once in each of the top 3 octaves and so there is not a problem of tuning the octaves for any of these notes.  Where the same diatonic semitones are fretted more than once these semitones (g^T-a and b-c) occur in separated octaves so no problem of octave tuning would occur.  However for the chromatic semitones there are serious octave tuning problems for the intervals e^I-e , f-f^T and b^I-b where the varying sizes of the semitones themselves would give rise to octaves that are badly out of tune by 22 cents, 21 cents and 26 cents, respectively.  Other smaller problems would also occur for some of the other chromatic semitones.  Some of the apparent error may be the result of simple measurement inaccuracy, especially for the treble part of the compass where an error in the measured position of only one millimetre would make a difference of about 10 cents in the size of the calculated interval.

A related inconsistency in the tuning and temperament produced by the fretting pattern of this instrument involves the sizes of the major seconds formed by the triple and quadruple fretting groups.  The last column in Table 6 gives the size of each of the major seconds formed by these fretting groups.  These can be calculated either directly from the corrected lengths of the strings or simply by adding together the sizes of the chromatic and diatonic semitones which comprise the major second (here and there a slight rounding-off error occurs in the final decimal place of the two).  As can be seen the major seconds vary in size from 181 cents to 212 cents (an equally-tempered major second is 200 cents wide and a ¼-comma meantone major second is 193.2 cents).  In the top octave the major second b^I2-c³ has a size of 206 cents and in the bottom octave of fretted notes b^I-c¹ has a size of 191 cents.  This is a difference of 15 cents, an amount not accounted for by the error in measurement, and an amount that would be clearly audible.  This makes a nonsense of using the fretting patterns to tune the instrument in octaves using the diatonic notes as described above.

Nonetheless there seems to be a deliberate attempt on the part of Pisaurensis to use his fretting system to produce different sizes of chromatic semitones, and seemingly also different sizes of major seconds.  It seems equally clear, however, that the fretting pattern designed by Pisaurensis produces a temperament that is musically unviable:  namely it produces octaves that are not in tune.  No solution has so far been found to this predicament. 

From Table 7 it can be seen that the average sizes of the individual chromatic semitones vary from 74 cents to 88 cents and have an average size of 80.5 cents (r.m.s. deviation from ¼-comma meantone = 4.8 cents).  The diatonic semitones vary from 101 cents to 127 cents and have an average size of 114.2 cents (r.m.s. deviation from ¼-comma meantone = 7.9 cents).  Quarter-comma meantone tuning is a kind of equal temperament in those keys that are playable since each of the intervals are of exactly the same size regardless of the tonality.  Hence in ¼-comma meantone all chromatic semitones are of the same size and all diatonic semitones are different from this but also all of the same size as one another.  Although apparently not designed with equal sizes of either kind of semitone in mind, the correct position of the tangents for ¼-comma meantone tuning have been calculated for this instrument for the fretted notes on the top bridge and these are the lengths given in the design-length column of Table 6 for each fretting group.  Comparison of these with the actual lengths given in the columns of the corrected lengths expressed either in millimetres or in once shows that, although some groups are very similar, there is a small but significant difference between them for other groups.

            At the moment the Leipzig Musikinstrumentenmuseum is temporarily closed and the Pisaurensis clavichord is not accessible.  Without first-hand access to the instrument to enable the tangent positions to be re-measured it seems that the question of the exact temperament produced by the various fretting patterns of this instrument cannot be resolved[3].

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