String-scaling design

            Recent work that I have done on the use of the local unit of measurement to design and build historical keyboard instruments suggests that the use of simple units of the local unit of measurement was as fundamental to the design of the string scalings as it was to the design and construction of the case[1].  For example, instruments made in different centres throughout Italy and designed for yellow brass stringing and tuned to normal reference pitch R would have treble scalings based on a length for c² given in the following table[2]:

Table 3 - Common c²-equivalent scalings of brass-strung harpsichords and virginals

 

Indeed it is likely that the local variation in pitch from one centre to another in Italy and elsewhere arose because the design and construction of instruments, and especially of organs, was based on the different local units of measurement.  Brass-strung harpsichords, virginals and clavichords designed using each of these local units of measurement would, however, all sound well at roughly the same pitch.  This suggests that instruments using other stringing materials like iron and red brass were also designed using scalings based on a simple number of the local units of measurement.

 

            Hence it would seem to be worthwhile to look at normal iron, yellow brass and red brass string scalings and to compare them with simple numbers of Venetian once when considering a clavichord like the one made and designed by Pisaurensis.  This is done in Table 4 below. 

 Table 4 – Typical French and Flemish scalings compared with some simple Venetian measurements

 

            From Table 4 it is clear that the normal seventeenth and eighteenth-century scalings found in North-European instruments are close to c² scalings based on 12, 10 and 8 Venetian once for iron, yellow brass and red brass stringing respectively.  For the note g² a fifth higher than c², the corresponding scalings would just be  those of c² and these are shown near the middle of Table 4  Thus for an instrument pitched a fifth higher than reference pitch at R + 5, iron, yellow brass and red brass stringing would correspond to c²–equivalent scalings of 8, 6 and 5 Venetian once respectively (shown on the right-hand side of Table 4).

 

            Let us look therefore at the scalings of the Pisaurensis clavichord to see whether or not any aspect of the stringing of this instrument fits into this pattern.  The string lengths for each note with the bridges in their present positions are given in Table 5 below.

 

Table 5 – String scalings as measured with the bridges in their present positions

Dominicus Pisaurensis clavichord, Venice, 1543

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

 

            These scalings are plotted in the graph shown in Figure 2 below.  This graph shows the pitches of the played notes at the bottom of the graph and, assuming that the instrument was designed to sound at a pitch of R + 5, with the actual sounded pitch given at the top of the graph.  In all of the discussions below unless otherwise stated the notes referred to are the played notes, not the sounded notes.

 

            The graph shown in Figure 2 is plotted on semi-logarithmic graph paper.  This is used because it makes Pythagorean scalings, which double and halve with each octave drop and rise in pitch, plot as straight lines.  Indeed Pythagorean scalings for c²-equivalent scalings given by 8, 6 and 5 Venetian once have been drawn on this graph, and all of these correspond to straight lines.

 

            Comparing the theoretical thin lines for Pythagorean c² scalings of 8, 6 and 5 Venetian once with the measured scaling in the treble it is clear that the line formed by the measured points is close to the theoretical line for iron scalings (c² = 8 once at a pitch of R + 5) but deviates gradually more and more for the notes at the extreme treble end of the compass.  The measured length of c² itself is 234½mm, about 2.7 mm longer than 8 Venetian once = 231.84mm.  The measured lengths in the extreme treble are also about 2.7mm too long when compared to the theoretical line, and this suggests that the whole treble bridge is now about 2.7 mm too far away from the tangents.

 

            On the other hand the measured length of tenor c is somewhat short of the theoretical line for yellow brass stringing based on c² = 6 Venetian once.  At a pitch of tenor c two octaves lower than c² the equivalent of c² = 6 once is  once = 26 once = 772.8mm.  The measured length of tenor c is 756mm which is about 16.8mm less than this value.  Therefore moving the middle bridge away from the tangents by this amount would bring it into agreement with the theoretical value for yellow brass stringing above this note.  In a parallel way the third bridge would also have to be moved away from the tangents by 8.4mm at the note F in order to position it at a distance of 4x8 once = 32 once = 927.36mm.  Instead of giving an F scaling of 919mm as it does in its present position it would then be in a position where it would correspond to a red brass f²-equivalent scaling of exactly 8 once (ie a c² scaling of 5 once transposed).[3]

            The lengths c² = 8 once, c = 26 once, and F = 32 once could be considered to be the critical design scalings for this clavichord corresponding to notes critically stressed just at their breaking points for the materials iron, yellow brass and red brass respectively.  It is also, in my view, these lengths that determine the correct positions of each of the three moveable bridges.  This has been shown diagramatically below in Figure 3 where a plan view of the instrument is shown along with the fret positions and the string pairs for each of the critical design notes.  The green lines show the new positions of the bridges which would have to be moved from their old positions in order to achieve these critical design scalings.

 

Figure 3 – Critical design scalings, the proposed bridge positions (in green), and the fret positions

Dominicus Pisaurensis clavichord, Venice, 1543

Leipzig Musikinstrumentenmuseum der Universität, Clavichorde Cat. No. 1

  

A numeric analysis of all of the string lengths is given below in Table 6