The Temperament of the Donat Clavichord
I was asked by John Raymond to consider what might have been the original temperament of the Donat clavichord, no. 12 in the Leipzig University Museum of Musical Instruments. The clavichord is dated c. 1700 and is described in Hubert Henkel’s 1981 catalogue of the collection, Clavichorde (Verlag Das Musikinstrument, Frankfurt/Main). However, the string-lengths given there were found, in a subsequent restoration, to be slightly incorrect. The measurements in the table below were given in a letter dated 29 March 1987 from Henkel to the late John Barnes. As I understand the letter, measurements of the sounding length of each note were taken from the tangent top to the bridge (the tangents being as far as practicable vertical); these were supplemented for each pair of fretted notes by measurements of the fretting distance, taken from the foot of the tangents. As is only to be expected, some discrepancies result. I have followed John Barnes in taking the length of the longer string of the upper note of each pair, and then adding the fretting distance to find the sounding length of the lower note. These measurements are then used as the basis of the analysis of the temperament.
The fretting of the instrument is basically diatonic, but it is not the same in each octave. As far as I can tell, this is original.
Course Notes Lengths Ratio Cents 1-12 C-B (unfretted) 13 c/c# 724/701 1.0328 56 14 d (unfretted) 15 eb/e 650/630 1.0317 54 16 f/f# 597/572 1.0437 74 17 g/g# 553/530 1.0434 73 18 a (unfretted) 19 bb/b 503/483 1.0414 70 20 c1/c#1 443/428 1.0350 60 21 d1/eb1 411/392 1.0485 82 22 e1 (unfretted) 23 f1/f#1 357/342 1.0439 74 24 g1/g#1 326/312 1.0449 76 25 a1/bb1 296/281 1.0534 90 26 b1 (unfretted) 27 c2/c#2 253/241 1.0498 84 28 d2/eb2 226/213 1.0610 103 29 e2 (unfretted) 30 f2/f#2 189/178 1.0618 104 31 g2/g#2 167/156 1.0705 118 32 a2/bb2 144/135 1.0667 112 33 b2/c3 124/115 1.0783 130
I think it is best, in this kind of analysis, to disregard notes above c2: in the high treble, practical considerations tend to prevent the tangents being put in the theoretically correct positions (because this would require very, very thin keylevers); also very small errors of workmanship or of the measurer produce large movements in pitch which can be misleading. My hunch is that these high notes were always adjusted by bending the tangents after the instrument was made. The temperament was set, as it were, in the middle octaves.
That leaves the notes below c2. I notice that four of the semitones (f-f#, g-g#, f1-f#1 and g1-g#1) are close to 74¢ in size. Now 74¢ is the interval produced when you follow Claas Douwes instructions in Grondig Ondersoeck van de Toonen der Musijk, first published in 1690 (thus quite close in date to the Donat) and reprinted several times (there is a facsimile published 1970, but see also the article by John Barnes in De Clavicordio I, p. 75)
For the chromatic semitones, Douwes prescribes string-lengths in the ratio 24:23. This seems a practical, easy, rule-of-thumb method for a craftsman to use, and I suspect it was a widely known rule ‘in the trade’ at the time. It produces a semitone very close to the chromatic semitone of quarter-comma mean-tone (which is actually 76¢). I think it is likely that Donat used the 24:23 rule to set out his rack-ruler for these semitones and also for bb-b which is quite close. The semitone eb-e however is far too narrow to have been set out in this way, and is in fact too small for any known temperament. I am curious about it since in the other octaves Eb is linked to D. Could the eb lever be a replacement?
Another anomaly is a1-bb1, whereas in the lower octave bb is fretted to b and a stands alone. Donat does not seem to be able to make up his mind whether he is using a D/A, E/A or E/B system of fretting. (Incidentally, I notice that in the catalogue bb1 is indeed linked to b1.) Anyway, this semitone (90¢) and d1-eb1 (82¢) are obviously meant to be larger than the other semitones, but neither of them is large enough to be a true quarter-comma diatonic semitone (117¢). Perhaps they are compromises, intended to enable eb1 to serve, at a pinch, also as d#1 and bb1 as a#1. That just leaves the two semitones c-c# and c1-c#1 which are very markedly small, apparently deliberately so. They are close to the very narrow chromatic semitones you get in one-third-comma mean-tone (produced by string-length ratio 28:27 approximately) but it is hard to see why.
Conclusions: very tentative ones. Either:
1. The original temperament was a kind of modified quarter-comma meantone, with a very low C# (for reasons unknown) and with Eb and perhaps Bb given a kind of compromise tuning enabling them to serve for D# and A# (ridiculous on a harpsichord because they would be out of tune in both positions, but perhaps feasible on a clavichord because the out-of-tune concords, when they occur, can be improved slightly by adjusting the pressure on the keys). The circle-of-fifths diagram below gives some idea of the kind of temperament I mean.
C -1/4 -1/4 F G +5/12 -1/4 Bb D +5/12 -1/4 Eb A +5/12 -1/4 G# E 0 -1/4 C# B -1/2 -1/4 F#
(The approximate amount of narrowing or widening of each fifth is shown as a fraction of a comma; '0' indicates an approximately pure fifth.)
2. It was meant to be quarter-comma mean-tone all along, and the anomalies in the placing of the tangents were adjusted by bending them. One consideration makes this slightly more plausible: the gaps between the keylevers of the lowest two or three fretted semitones may have been made narrower than they should theoretically be to avoid weakening the levers by excessive cranking.
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Peter Bavington, 20 July 2002