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CARL FRIEDRICH ABEL, A GAINSBOROUGH PAINTING, AND VIOL TEMPERAMENT – SOME EVIDENCE AND ENIGMAS
[1] The Consort (Journal of the Dolmetsch Foundation), Summer 2003, Vol. 59, 51 - 74. Gainsborough’s
portrait of Carl Friedrich Abel and his viol,[2]
now in the National Portrait Gallery (but currently not on display), is
an arresting work both in its light and colour, and in its composition.
Unlike Gainsborough’s portrait of Abel in the Huntington Library and In the painting, the brilliant golden yellow of the light on Abel’s attire draws the eye, and in contrast, the dull brown colouring of the viol’s frets - being almost identical to that of the fingerboard - cause this detail of the instrument to recede. Indeed, as the portrait was displayed in 1994, above eye level and with the canvass protected by glass, the 'higher' frets (further down the fingerboard towards the bridge) were barely visible. Close scrutiny however, reveals significant detail in the frets, including a ‘correct’ graduation in their thicknesses from nut to bridge. On the higher, thicker frets (those nearer the nut), the double strand of gut is clearly depicted, showing the detail of the two strands separating slightly at the knot end, as one would expect to see on an actual instrument. When viewed from a normal distance, there is one outstanding feature of the painting that any viol player will notice as strange. This is the 'unrealistic' spacing of the first three frets, and in particular the gap between the first and second frets. Viol players will be familiar with the subject of viol frets and their positions on the instrument – it is not something that violists can ignore for long. The gut frets are tied around the neck of the viol – they are moveable, and replaceable. Indeed, they have to be replaced from time to time, as they are subject to wear, and eventually will break if left unattended. The frets also have to be moved from their initial positions as the viol’s strings age, because as the strings become older and more worn, there will be changes to the precise pitches of the notes they produce at given stopping positions. Experienced players will also move the frets for many other reasons, including the accommodation of a tuning or temperament on other instruments, with which the viol is being played. For most viol players with some experience, the fret positioning is never merely an accepted feature of the instrument. On the contrary, the fret positions are usually a matter of conscious choice, depending on the tuning results for which the player is aiming, and the condition of the instrument and its strings. Could we regard Gainsborough’s painting as providing ‘iconographical evidence’ for Abel’s fret positions in practice? If so, then we might also have evidence for an unusual tuning system that Abel used, or perhaps preferred. We might in naturally expect an artist of Gainsborough’s calibre to reliably depict detail, but as it happens, in this particular case our expectations would be heightened more by what we know about Gainsborough, Abel and the viol. Gainsborough was a
very close friend and pupil of Abel, and after arriving in So what does Gainsborough’s portrait of Abel and his viol tell us, in this respect? As is often the case in examining ‘evidence’, the process of looking for ‘answers’ produces more questions. The actual measurements for the fret positions[14] are as follows (the measurements were taken along the surface of the glass covering the canvass, so any normal expected error is likely to have increased by error due to parallax): :
From these measurements alone it is clear that the viol in the painting is probably very close to ‘life size’. A maximum string length of around 73 or 74 cm is about what we might expect for a large, seven string viol. Once in possession of measurements, it is easy to empirically recreate the fret positioning on a viol of the same string length, or proportional positioning on a viol of similar string length. If we do this, the empirical result is interesting, not because it provides an immediate insight into certain tuning preferences, but precisely because it appears to be entirely impractical. The frets turn out to be literally nowhere near positions that would allow a ‘normal’ workable tuning of the instrument, even taking into account the variations that might occur due to ageing strings of different kinds and/or different conditions. They are not even where one might expect to find them if they had ‘migrated’ from their proper positions, perhaps on an instrument that it not very much used now. Would Gainsborough have ignored this detail of an instrument, or exercised ‘artistic license’ to such a degree on this one aspect of the portrait? The portrait was apparently in poor condition when it arrived at the National Portrait Gallery, and was subject to restoration work. Could the restoration have provided the painting with spurious fret positions? Surely we should not assume this kind of error, without further evidence. So what interpretation can we make? The frets could have been added to the instrument just for the purposes of the portrait, without the instrument having been played first, or Gainsborough could simply have attached no importance to the accurate depiction of fret positions. I would think the latter is also unlikely. It is probably more sensible to argue that the fret positions are depicted according Gainsborough’s sense of ‘visual proportion’ appropriate to the composition of the painting, thus representing real fret positions in a visually correct, but not quantitatively correct way. If this is the case, then there is still some interesting information to be gleaned from their depicted positioning. Temperament Even had there been unquestionable fret positions in the portrait, we would not expect to demonstrate this or that specific temperament to have been definitely in use, but rather, a more general ‘class’ or ‘type’. To see the reason for this, before we make an interpretation, and to see how the land lies with respect to viols and temperaments generally, we should first take a detour into the subject of temperament. The concept of musical temperament is mentioned as early as 1496 in Gafurio’s Practica musicae,[15] in which he refers to an already existing practice of tempering rather than a new idea. The general concern with the relationship of musical intervals and arithmetic ratios has far older documented origins. Even the well known, explicit (but enigmatic) references to these relationships that appear in Plato’s Timeaus, reiterate an existing lineage of ideas to which allusions also appear in pre-Socratic sources, and whose origin is attributed to Pythagoras (fl. BC 530).[16] The basic tenets of temperament theory then, concerning the way ‘untempered’ musical intervals are represented mathematically, are not disconnected from the ‘Pythagorean tradition’ in general. The ‘traditional’ mathematical expression of temperament theory is (like many scientific theories) an idealised, approximate theory – it is a further development from the ancient (and by ascription, Pythagorean) science of ‘harmonics’. It is built on certain ratios assumed to ‘define’ musical intervals arithmetically, that we now know can indeed be justified as approximately true, on the basis of the acoustical structure of some musical tones, especially those from strings and pipes. Hitherto, the ‘inventors’ of temperaments did not have the knowledge of the complex acoustical structure of musical tones in mathematical terms, that we have today. What was originally known, and attributed to Pythagoras, was the reliably accurate (but not mathematically perfect) relationship between tensioned string lengths and musical ‘pitch’, a relationship approximately reiterated by organ pipes. This relationship does yield simple, whole number ratios, like 3:2 for a perfect fifths and 2:1 for an octave. The 'history of science' would tend to consider Pythagoras for his 'contributions to science', regarding him in the role of an 'early scientist'. But a notably different Pythagoras is seen by other disciplines, and by a considerable body of sources from before the so-called 'scientific revolution', and also by the many later sources that were still part of the 'intellectual culture' of deference to the ancients. What we see here, is Pythagoras the Divine, the mystic, the numerologist, the healer, and the 'hearer' of the 'harmony of the spheres'. It is not from a sheer confidence in the discovered laws of natural phenomena, that the simple arithmetic ratios, and eventually even the mathematics of temperament theory, acquired their original 'authority'. Rather, this was inherited from their relationship with the venerable ‘Pythagorean tradition’ that gave the original endorsement to the ratios for the perfect intervals. This ‘tradition’ has not been without its opponents,[17] but nonetheless has enjoyed considerable influence as being based on the ‘authority’ of ancient wisdom. We find this influence repeatedly appearing, whether overtly or covertly, in many important sources including, for example, Boethius and Ptolemy. Boethius will already be familiar to musicologists. Claudius Ptolemy (c. AD 75), who may be less familiar to musicians and musicologists, was responsible for the model of the ‘Ptolemaic universe’ or celestial system,[18] with the Earth at the centre, that predominated for fifteen centuries until the ‘scientific revolution’ of the seventeenth century. At the beginning of his extensive three -book treatise entitled Harmonics , he states that the student of harmonics must aim ‘to preserve the hypotheses of the kanon ’ (a stringed instrument), and that the astronomer must aim ‘to preserve the hypotheses concerning the movements of the heavenly bodies...’[19] These ‘hypotheses’ derive from the Divinely ordained connection between certain whole number ratios, musical intervals, the motions of the planets, and the human soul.[20] The Church’s endorsement of geocentricity may have been connected with ‘metaphysical’ dogma, but the relevant sections of Ptolemy’s Harmonics are mathematically intense – so much so that he has been quite justly described as a ‘number cruncher’.[21] Being based on ‘ancient authority’ transmitted by Plato, and according to the attestations, derived from Pythagoras, the quantitative part of Ptolemy’s work was very influential, very complex, – and whilst mathematically correct, very mistaken. In a sense, temperament theory has in the past enjoyed the same umbrella of authority, even though in its own right, its ‘fit’ to the acoustics of musical instruments is arguably better than the ‘fit’ of Ptolemaic astronomy to actual planetary motions. Today, temperament theory remains a very useful way of sensibly modelling acoustical tuning results on musical instruments – best of all pipe organs and stringed keyboard instruments.[22] But in itself it is essentially an arithmetic theory of number relationships, really applicable only to instruments whose tones have acoustical structures that are reasonably close to harmonic.[23] It is, or should be, regarded like many other scientific theories - as a convenient approximation, and not as a Divinely ordained absolute. The musical perfect fifth is not in itself a string length ratio. It is an acoustical consonance. The consonance occurs because of the acoustical structure of the tones, whether in strings, pipes, or the human voice. On strings, it is this that causes the Pythagorean string length ratios to exist, not vice versa. When, for example, we apply harmonic temperament theory to the metallophones of the Gamelan, bells, gongs, or tuned drums, it becomes relatively vague, inappropriate, or even meaningless. Even applied to organs and harpsichords it should always be remembered that it is not an ‘absolute’ model or a true physical theory for the acoustics of these instruments, despite its precision in dealing with say, ‘1/4 comma’ as distinct from ‘1/5 comma’ or ‘2/7 comma’. Most temperaments, however vaguely described when they were first documented, are now ‘translated’ into a mathematically defined form. Without the pre-defined association of ratios such as 3:2 for a perfect fifth, or 2:1 for an octave, temperament theory as we know it and use it today, ceases to exist. Being based on the pure mathematical relationships of numbers, and in the first instance on simple, whole number ratios, the construction of different temperaments as various twelve note divisions of the octave, is very much a matter of mathematical precision.[24] In acoustical terms, the practical need for tempering arises from the fact that very small, microtonal intervals (for example the commas, dieses or schisma), appear between the different results of tuning a given set of notes through different tuning sequences. These microtonal intervals are themselves divided and distributed in various ways through the musical intervals that occur in the chromatic scale, according to the specific temperament in use. They include, for example, the syntonic comma whose typically small (close to unity) associated ratio is 81:80. This microtonal interval is then divided in various temperaments into even smaller intervals, up to around a twelfth of the size, and sometimes even smaller. What mathematically defines one temperament as distinct from another then, is a distribution of ratios defined to this order of accuracy. When we distinguish between say, quarter-comma meantone and fifth comma meantone, we are talking about differences between the two scales ranging from about a twelfth of a semitone to less than a ninetieth of a semitone.[25] Only if when in use, an instrument's tuning stability remains well inside these limits, can we truly of speak of such clearly defined different temperaments as existing on the instrument in question. Tuning in
practice As far as practical tuning is concerned there are three things to consider. The first is how well the ‘whole number ratio’ premisses of temperament theory ‘fit’ the actual acoustical nature of the musical instrument to which we are attempting to apply it. The second - if such a good ‘fit’ exists - is what means are employed to ensure that the tuning result in practice, represents with fidelity the theoretical prescription. The third, is whether this precision of tuning actually remains in place whilst the instrument is in use. The idea of this or that temperament being present on an instrument, as defined by temperament theory, is therefore only meaningful to the degree to which (a) the acoustics of the instrument comply with the premisses of temperament theory, and (b), a properly accurate means of applying the temperament is employed, and (c), the tuning stability of the instrument in use, matches the precision with which the temperament is defined. In the case of an organ or harpsichord, we could argue the use of an electronic meter would provide an accurate means of tuning. (However, we should certainly not trust the popular notion that a single meter dial, being electronic, must therefore provide the ‘last word’ in accuracy). [26] Fortunately, harpsichord strings do generally conform closely to the expectations of temperament theory, provided the instrument’s tuning stability is satisfactory. Viols on the other hand, behave quite differently. Compared to the tuning of temperaments on the harpsichord, significant changes to the note produced by the viol string arise not only from turning the peg (the equivalent of turning the wrest pin on a harpsichord), but also from the pressure of the bowing, and the pressure applied by the finger for a stopped noted. The note also varies with the exact finger stopping position. Variations caused by any of these factors can cause changes that in play, would violate the level of precision necessary for a genuine temperament definition, as accurately applied to the keyboard. In addition to the notes having tuning characteristics that are unstable or not fixed, viol strings themselves present further difficulties. A characteristic known in musical acoustics as inharmonicity, which causes the actual physical behaviour of strings to ‘disagree’ with the premisses of temperament theory, is inevitably present to a relatively high degree in the relatively short, low tension, gut based viol strings. Wear and ageing of the strings can make the effect of inharmonicity very large indeed.[27] The effect of this alone, and not least the fact that it will be present in markedly different degrees in the different strings across the viol, including strings of different ages, construction and condition, compromises the ‘fit’ between the acoustics of the strings and the expectations of temperament theory. One of the least recognised features of viol tone, but one disruptive to temperament definition, is falseness. Many viol players will have been frustrated by occasional acute string falseness at some time, when attempting to tune the viol, perhaps assuming this characteristic of string behaviour to be some kind of ‘wolf’ in the instrument. In musician's terminology, a false string has more than one pitch. Its pitch will typically cycle between a maximum and a minimum if it is plucked. The difference between the two pitches can in a bad case be in the order of a whole comma (about a quarter of a semitone) or more, when the string is plucked. It would not be an exaggeration to say that most viols strings are false, to some degree – but if the rate of alternation is slow enough compared with the decay rate, or the pitch variation is small enough, it will not be noticed. Bowing can steady the tone, but the same level of imprecision can remain in the effect of bowing. Thus we may tune and fret a viol according to a temperament prescription for a keyboard, or a mathematical definition, but in the event of playing we will not genuinely be playing in that temperament as mathematically defined. Furthermore the acoustical characteristics of our scale and intervals as played on the viol, will not be precisely the same as those on the harpsichord tuned to the same temperament. The original 'idea' or 'purpose' of the temperament as devised the keyboard, in terms of interval qualities, may be sometimes be compromised, and other times even lost on the viol. Remarkably, this will be the case even if we tune and match note for note unisons with the harpsichord.[28] What all this means is that we cannot measure precise fret positions to several decimal places and expect to ‘reverse engineer’ this data to indicate a precise temperament 'designed' for a keyboard, and defined in terms of a precise division of a comma. Nevertheless, there are meaningful, general statements we can make about viol temperament, provided we understand that there is a difference between the meanings of ‘temperament’ applied to a viol, a harpsichord, or a mathematical description. Even with the vaguest data we might be able to do this, for the simple reason that all temperaments can be classified in broad generic classes, and some classes produce recognisable patterns in fret spacing. Thus, we can distinguish between say, fretting that looks like 'a meantone', or fretting that looks like 'an extreme meantone'. We can also recognise an 'equal temperament' fretting, roughly speaking. We cannot really work with precise data to produce precise conclusions, but we can use precise data to produce general conclusions, probably best expressed simply through the visual matching of fret space patterns. Temperament
classes ‘Traditionally’,
temperaments are often mathematically defined in theory using the Fig 1:
In this particular example (Fig 1), rising fifths appear clockwise round the circle as far as G sharp, and falling fifths appear anticlockwise to E flat. Nominally, the interval G sharp rising to E flat, is a diminished minor sixth, not a fifth. The practical counterpart to this model is that starting on C, we could tune a whole chromatic scale by tuning first the clockwise sequence of notes C, G, D, A, etc. as far as G sharp, and then the anticlockwise sequence. In practice we can ‘invert’ any rising fifth and tune a falling fourth instead. Similarly we can invert any falling fifth and tune a rising fourth instead. For example, a practical sequence using perfect fifths or their inversions to construct a chromatic scale within one octave of the compass might be: Fig 2
If we complete the scale by tuning the C an octave above the starting C, as a ‘perfect’ octave, we would then find that the rising fourth and fifth formed from G and F in the scale to the upper C, would turn out to be perfect too. However, the untuned interval formed between G sharp and E flat - the notes at the ends of the sequences - will not be a perfect fifth. Nor will it sound like one. It will be the infamous ‘wolf’ interval that is musically unusable – a ‘fifth’ that is far too small or ‘narrow’. The amount by which this ‘wolf’ deviates from a perfect fifth is the Pythagorean comma, about 24 cents or a quarter of a semitone.[29] A comma is only a microtonal interval, but a fifth needs only to be ‘mistuned’ by a microtonal interval in order to become too grossly mistuned to be musically acceptable. In short, twelve perfect fifths will not ‘fit into’ an octave, whichever sequence of fifths and inversions we choose. The root physical cause of this lies in the ‘internal’ acoustical structure the musical tone produced by things like strings and pipes. Pythagoras, whose name the comma bears, did not know this, but apparently did know about the relationship of tensioned string lengths and musical pitch, which in fact has its roots in the same cause. The mathematical part of the model consists of representing the musical intervals, and all the relationships between them, as ratios round the Circle.[30] In Fig 1 or Fig 2 the
ratio for any interval can be derived from the product of the ratios of
all the intervals sequenced to make it. As long as we obey the necessary
rules for manipulating the arithmetic, we will get the same results
whether we use sequences round the We could of course draw straight lines connecting notes that ‘define’ semitones, rather than major thirds, Fig 3: Fig 3
Clearly then, the size of any semitone will also be ‘defined’ by the chosen sizes of the tempered fifths (or ‘non fifths’) in the sequence or arc of the Circle across its straight line. Temperaments do not necessarily consist of a set of eleven deliberately tuned or tempered fifths, always leaving the last ‘fifth’ or wolf interval untuned. Many temperaments employ the deliberate tuning or tempering of all twelve fifths. The former kind, require tuning or tempering two sequences round the Circle, first one way, and then the other, leaving one interval untuned. These are thus called non circulating temperaments. The latter kind could be tuned in a continuous sequence right round the Circle. These are circulating temperaments. If all the tuned fifths are tempered by the same amount, the temperament is said to be a regular temperament. If they are tempered by different amounts, the temperament is said to be irregular. All non circulating, regular temperaments that temper eleven of the twelve intervals round the Circle, are now referred to in contemporary theory as ‘meantone’ temperaments, because the size of the tone is half the size of the major third, across arcs that do not include the ‘wolf’ interval. This set of all such temperaments is complemented with the addition of Equal Temperament, in which all the fifths round the Circle are tempered by the same amount. Equal Temperament is the only circulating meantone temperament.[31] Pietro Aron appears to be have been the first to document a meantone temperament in 1523,[32] without any mathematical description, and what he described is now recognised mathematically as ‘1/4 comma meantone’. In this temperament eleven fifths are tempered narrow by ¼ syntonic comma (about 1/20 semitone), a reasonable maximum amount of tempering one would expect to find in fifths. Despite the fact that history shows a process of ‘inventing’ or ‘discovering’ numerous temperaments, sometimes empirically, sometimes mathematically, all the possibilities for temperaments of any kind, derived from the ‘Pythagorean’ or harmonic ratios, can be encapsulated in, and predicted from generalised laws.[33] Any temperament, whether circulating or non circulating, can in fact be simply written as an array of twelve quantities, including the ‘wolf’, if necessary. Today we have computer power to do any tedious mathematical ‘donkey work’ for us, so it is actually just as easy to express any temperament as a mathematical vector, and to manipulate data in matrices. The fret positions in Gainsborough’s painting, I first analysed in 1994 using computer spreadsheet,[34] but the following results were obtained by direct graphing from vector formulae.[35] Abel’s
frets
We can represent the fingerboard horizontally, for convenience,
with the nut on the left and the bridge off towards the right. The
strings in this case would be, from top to bottom, D(1st),
A(2nd), E(3rd), C(4th), G(5th),
D(6th), A(7th). The height of the frets in the
diagrams is arbitrary, and we will number the frets 1 to 7 from left to
right. Abel’s frets in the painting can then be represented as
follows, with the positioning of frets for Equal Temperament on the same
strings, compared above: Fig 4
It
is convenient to speak of musical interval sizes in cents (hundredths of
a semitone). We can then also speak of fret positions and spacings in
cents. Thus to say fret 1 is 100 cents from the nut would mean the fret
is positioned so that the note stopped on it is 100 cents above the open
string note. We are, of course, now deliberately neglecting variations
that would be due finger stopping positions or bowing pressures. Taken
literally, the fret positions do not correspond to any recognisable
positioning that one would expect to find if the viol was tuned to any
practical temperament. The second, third and fourth frets are all
impracticably low (near the bridge), and the sixth and seventh frets are
impracticably high (near the nut). It is unlikely that the seventh fret
would be positioned unfavourably with respect to the note it stops on
the first string, so the interval from the open first string (D) to its
seventh fret stopped note (A) would reasonably be expected to form a
musically usable interval. The
diagrams give some immediate idea of the situation, but the severity is
not immediately obvious. For a better appreciation, consider the
following: The diagram shows the seventh fret is quite ‘flat’
compared to its Equal Temperament position. Theoretically, if the
seventh fret were tuned 1/4 comma flat (1/4 comma being a reasonable
maximum tempering to expect in the interval D-A), it would be positioned
696 cents from the nut. It is in fact positioned 17 cents flat of the
'quarter comma' position, which makes the interval with the open string
definitely a 'wolf'. Inharmonicity due to string age and wear may cause
frets to be moved back towards the nut in some instances, but by this
distance would be most unusual, and this does not account for the very
'sharp' positions of the second, third, and fourth frets.
Viol
players will easily grasp the impracticality of the fret positioning
from the following: If the open first string (D), were tuned to make an
octave with the D stopped on the second fret of the fourth (C) string,
then the open second string (A), would have to be tuned 36 cents (more
than a third of a semitone) flatter than a pure fourth below the first
(D) string, in order for the C stopped on its third fret to make an
octave with the open C string. There
is one way that we could explain the overall positioning of the sixth
and seventh frets. If the action was very high, the distance from the
strings to the fingerboard in this region would be great. Stopping the
strings on such a viol in these high positions, considerably increases
the string tension, so the frets are correspondingly moved back towards
the nut. What this does not explain, is the positions of the first three
or four frets.
‘Scaled’ positioning Assuming Gainsborough used a sense of ‘artistic visual proportion’, rather than taking measurements, we can make adjustments to take into account Abel’s seventh fret position as depicted by the painting, so that our representation of Equal Temperament, given a suitable scale mapping from the data, agrees with the painting on the position of the seventh fret. By doing this, we can see if the painted frets appear to be a good ‘artistic visual proportion’ representation of an Equal Temperament spacing. The results are then: Fig 5
The results are not a very convincing portrayal of Equal Temperament spacing. Similarly, we can carry out the same procedure for meantone varieties:
Fig 6
Fig 6 shows one natural extreme, the quarter comma meantone (with the wolf nominally G sharp to E flat). Here, the first fret is positioned for a B flat on the A string rather than a C sharp on the C string, which is an option that might typically be chosen if one had to tune the viol to meantone. One characteristic effect of the wolf in the meantone Circle, is that any intervals between notes joined by an arc that includes the wolf, will be a different size to the same intervals between notes joined other arcs of the Circle. The consequence of this that there are two sizes of semitone in meantone temperament, which are clearly visible in the diagram. Fig 7 shows the same for the less extreme example of 1/8 comma meantone. Fig 7
In Equal Temperament all intervals of any one kind are the same size, resulting in spaces between the frets that diminish exponentially. The resultant visual pattern of smoothly decreasing spaces can be seen from Fig 4 or Fig 5. It is clear even visually, from the Abel frets, that we would be dealing with an unequal temperament, and from the above, not a meantone. Any temperament other than Equal Temperament will demand that some frets (notably the first and sixth) be in different positions for different strings. One practical answer to this is to position the fret correctly for one or more strings, at the expense of the others. We did this with the first fret in the meantone diagrams. Splitting the two strands of fret-gut apart to provide more than one position is another possibility, but the painting definitely shows no signs of this. It is of course also possible to place the fret at a compromise position between the ‘correct’ positions for different strings. These factors make the painted frets particularly difficult to interpret. In practice, frets may also be moved from recognisable positions based on a recognisable temperament, purely to accommodate particular intervals in a particular piece of music. This is done to improve sonority, and to overcome stopped note pitch anomalies caused by inharmonicity in the string behaviour. It is possible for a string to ‘stop flat’ on certain frets, requiring the fret to be moved towards the bridge, but it is more common for it to ‘stop sharp’, in which case the fret is moved back towards the nut. In the scaled depiction where the seventh fret is assumed to be in a ‘sensible’ position, the most striking feature of the Abel frets is that all the frets except the first and seventh, are positioned very ‘high’, i.e. shifted towards the bridge. Another way of ‘normalising’, ignoring the seventh fret position, is to make the second fret positions agree for both Abel and Equal Temperament mapped for the same strings. Why the second fret? On the seventh fret none of the stopped notes are ‘accidentals’ for which there may be enharmonic variations for its position in an unequal temperament. The second fret is similarly a good choice because there is only one accidental, the F sharp on the E string, but there are other good reasons. The open strings effectively make the viol an ‘instrument in D’. It would be arguably very unlikely to favour a tuning without good octaves between the open D strings and the stopped D on the second fret of the C string. Similar arguments apply also to the two open A strings on the seven string instrument, and the stopped A on the second fret of the G string. If we scale to align the second fret of a sequence for Equal temperament with the second of the painted frets on Abel’s viol, we get the following visual result:
Fig 8
Now we can see that except for the first fret, the Abel frets could arguably be a depiction of Equal Temperament, but with the ‘artistic license’ allowing the exponential diminution of fret spacing towards the bridge, to be shown rather overzealously. What of that first fret? The answer will be immediately apparent to viol players themselves. It is not unusual for the first fret to be moved back towards the nut, for several reasons, on a viol otherwise basically tuned in Equal Temperament. This may be to provide a ‘sweeter’ (less tempered wide, or more ‘pure’) third between F sharp/G flat on the second fret of the E string, and A sharp/B flat on the first fret of the A string. It also often provides a better F on the first fret of the E string, where this note stops sharp (as it frequently does, on the thicker E string). A flattened F is usually desirable to improve the fourth from the open C string to the F, which because it involves one open and one stopped string, can be particularly sonorous. Lastly, it may also assist the tuning of the F major chord from the F on the bottom D string. This interpretation in no way conclusive, but constitutes a reasonable guess to explain what is otherwise a puzzling lack of accuracy in the painting. The outstanding problem that this argument does not resolve, is that the gap between the first and second frets is still visually, simply too large to be convincing. This is in fact the one immediately disquieting feature of the fret spacing, when one sees the painting itself. The
tonality of Abel’s viol compositions Can we glean any clues
from the tonalities that Abel uses in his compositions? Abel’s
compositions, as might perhaps be expected, do not exploit remote
tonalities. His symphonic writing never explores keys requiring more
than four sharps or flats, and three flats are more common than three
sharps.[36]
The viol works are even more restricted. The manuscript in One could argue these
restrictions were to suit the technique of the pupil for whom the pieces
were written. Elizabeth Pembroke (1738 – The key restrictions are consistent with those that would be expected had a meantone temperament been in use, and would also be consistent with the use of an unequal temperament that in the usual way favoured the quality of the major thirds in the ‘home’ keys. However, these kinds of temperament would not normally prevent the use of the keys that remain unexploited in the manuscript. The natural resonance characteristics of the viol would explain the avoidance of the flat keys which can be very dull on the viol, but this does not apply to D minor, not least because the viol has two open D strings and at least one open A string. The consideration of resonance fails also to explain the avoidance of A minor and to a lesser extent G minor. My own impression is that the key restrictions are probably not connected with temperament choice, and only partly, if at all, with resonance or technique considerations. The key restrictions seem likely to remain an unanswered question. Lbl Add.
34007 Some more insight might be gained from a manuscript fragment in the British Library, inscribed The Pure Method of Tuning the Harpsichord, According to Abel, in Lbl Add. 34007. This manuscript consists mostly of compositions, but also contains the fragment purporting to be Abel’s method of harpsichord tuning. Of the baroque composers, it is probably JS Bach whom musicians and musicologists would first think of in relation to harpsichord tuning. The notion of JS Bach as an adept of tuning ‘well tempered’ harpsichords is as familiar as the title Das Wohltemperierte Klavier, of the forty eight preludes and fugues. However, it is of course not just Bach who would have needed to be skilled at harpsichord tuning, but harpsichordists in general, including Abel, who was himself listed in Mortimer’s London Universal Directory, 1763, as a harpsichord teacher.[40] Harpsichordists have a choice of temperaments to which the harpsichord can be tuned, and we would expect competent harpsichordists to have a knowledge of tempering principles, and perhaps temperament theory. We would expect, as we find today, that certain temperaments would be preferred. Where viols are played with a harpsichord, then the viol tuner is often subservient to the harpsichordist’s tuning – the harpsichord will be tuned as desired, and the viol must accommodate. But the reverse could be true, especially if the harpsichord tuner or ensemble director is a violist, as would have been the case with Abel. However, whatever the apparent choices of temperament, the ‘note for note’ matching of a harpsichord tuning with a viol’s notes - both the open strings and the stopped notes - is not actually possible, in theory, except in the case where both instruments are tuned to Equal Temperament. Equal Temperament, in which all semitones are acoustically the same ‘size’, might therefore seem to be the natural choice for a violist who has to play together with a harpsichord. The problem is, that equal tempering gives both the harpsichordist and the violist equally tempered major thirds, which are considerably wide (2/3 comma) and which many harpsichordists and violists find objectionable, at least in the ‘home’ keys. The instructions given in the manuscript are reproduced in Fig 9: Fig 9
The first sequence, to bar 24, is
circulating, and the second sequence, from bars 25 to 44 is
non-circulating, the break in the sequence being after bar 38, and the
position of any potential 'wolf' fifth being in the conventional
position of G sharp to E flat. Both sequences are irregular, so we would
not expect any interpretation to look like Equal Temperament in fret
spacing, but it might look something like a meantone. Since
there are no other tempering instructions, it is reasonable to suppose
that the instruction to 'bear up a little' at the vertical line mark,
signifies points of deviation from an otherwise regular temperament. The
unmarked fifths would have to be tempered narrow, otherwise the result
of both sequences taken literally, would consist of a temperament
comprising 8 untempered ‘pure’ fifths, 2 fifths tempered
wide, and a ‘wolf’. Deliberately tempering fifths wide
exacerbates the very need of temperament, and seems illogical, however,
such a thing is not unheard of - schemes including wide tempered fifths
do exist, for example, the Temperament ordinaire. Instructions
for harpsichord tuning in late eighteenth century The first sequence The
first sequence involves tuning all twelve fifths in the cycle, which
means there is no 'wolf'. The instruction to 'bear up a little' at the
mark, probably means to raise the pitch a little from its otherwise
tempered position, so the fifths A-E and C sharp – G sharp are to be
wider than they would otherwise be, and B flat -F will be narrower. (The
mark on the manuscript is no more than a vertical dash of about the same
height as the semibreves, and its identification with one particular
note of each pair does seem to be quite clear).
The
instruction at bar 17 is ambiguous, because it seems to require
re-tuning the G sharp (just tuned as an octave from the preceding A
flat) from a C sharp that has not yet been tuned. The widening of this
fifth could only sensibly be achieved by flattening the C sharp.
Although the mark is associated with the G sharp, one could argue it is
the C sharp that is to be raised, which would narrow the fifth more. The
same point in the second sequence, however, clearly indicates the G
sharp is to be raised. The second sequence
The
second sequence follows a pattern identifiable with that used in
meantone tuning. The fifths circulate 'clockwise' from C as far as G
sharp, and 'anticlockwise' from C as far as E flat, leaving the interval
G sharp – E flat untuned. G sharp- E flat is one of the usual
positions to have the 'wolf' in meantone tuning, and in the tuning under
examination here, would be the most tempered fifth, or a 'wolf',
depending on the amount of tempering in the regular fifths.
There
are more possible practical interpretations of this sequence because the
untuned interval G sharp – E flat can be tempered by any amount
required, in order to maintain a total tempering round the circle of
'fifths' of 24 cents. General interpretation The
general idea given by the instructions can be represented on the Circle,
Fig 10: Fig 10
Here,
the asterisks mark the intervals where we are told to ‘bear a little
upward’, and the W marks the
potential ‘wolf’ interval in the second sequence. This latter does
not have to be a ‘wolf’, but in the second sequence it is not
actually tuned, so it will absorb and contain any difference between a
Pythagorean comma and the total accumulative tempering in the other
eleven intervals. Given
the ambiguity in the manuscript, there are various interpretations one
could make considering tuning on the harpsichord alone. We do not know
the origin of the manuscript, or even if the ambiguities arise simply
from the fact that its author’s own understanding of the tempering
principles was questionable (assuming the author was not Abel himself).
But let us for a moment postulate that for the reasons stated above, the
intended temperament ‘had the viol in mind’, in order to make a
better match between keyboard and viol, and accommodate the viol’s own
tuning parameters. Then the possibilities are more limited. Rather than
converting the Circle to a data vector and making an impartial analysis,
some good old fashioned lateral thinking is called for, using the Circle
itself, and knowledge of the viol’s characteristics. Ou | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
