The First Fifth of
The Development of Western European Stringed Instruments by Ephraim Segerman
Introduction
This book offers no new discoveries of historical evidence, but many new interpretations of the available evidence. It differs from previous histories in the questions I ask and how I try to answer them. The main questions I ask of the evidence are how the instruments (and their names) developed, and how they were tuned, played, made and sounded. Much historical information is not relevant to these quests, such as outstanding composers, players, makers, surviving instruments and repertoire, and so these issues are not discussed. Previous histories have answered most of the interesting questions for which the evidence clearly allows only single answers. The questions I ask go further, including those which may have more than one possible valid answer.
There is no objective way of choosing what questions to ask, but in evaluating answers, some ways are more objective than others. The usual way in the music world is to judge whether it makes sense in terms of what one believes one knows, and how much evidence it can explain. My approach, which is more objective, is to judge how well the answer (i.e. theory) can explain every bit of the relevant evidence. The difference is discussed in Chapter 1a.
Many of my suggested answers will be considered by some to be wild speculations, since the evidence does not unambiguously support them. I consider them to be valid theories (because they can reasonably explain how all of the evidence came to be what it is), and I accept that other valid theories could possibly answer the questions just as well. A collection of valid theories answering each question would be a much greater contribution to knowledge than concluding that the question is unanswerable (i.e. it remains a mystery) with the evidence available. In most cases, I strongly doubt whether there are many other theories with as much explanatory power as the ones I offer.
I have been trained as a scientist rather than as an historian or musician, so my indoctrination does not include the traditions of music scholarship, making my approach to how scholarship is performed different from most music historians in some crucial ways. One way is that I am more determined than most to avoid letting my enjoyment of performances of the music that I hear (or can imagine) to influence my judgements concerning the interpretation of evidence. It could be considered that this makes me rather old-fashioned. Westrop wrote that
“A friend of mine was once shown, by Johannes Wolf, a piece of old music. He remarked, innocently enough, that it ought to sound well, only to receive the austere reply: ‘I’m not interested in how it sounds’”.1
Wolf was a music historian, and apparently was aware that any attempt to judge how well it sounds involves modern criteria which would obscure the only relevant historical question about sound, which is how it was expected to sound when written.
Most music historians since Wolf’s time have been too interested in how it sounds to modern ears, considering that the major purpose of their work is to contribute to the enjoyment of music by the listening public (with the cooperation of early music performers). Making music enjoyable is the job of musicians, not historians. It is right and proper for musicians to exploit what the historians do when it suits their own purposes. But the music historian’s job is to contribute to building the most probable picture of the sounds of what was performed and enjoyed then (as well as the cultural environment around such performances), based on as full and objective an interpretation of all of the evidence as can be mustered. It is not the historian’s job to support the fantasy that many musicians and instrument makers have, that they are closely imitating the original musicians and makers, nor to support the fantasy that many listeners have, that they are hearing just what their ancestors heard. The historians have been giving this support by indulging in their own fantasy of expecting that the more historically accurate a performance of old music is, the more enjoyable it should be to modern ears. To protect this fantasy, they have avoided the pursuit of any scholarship that might reduce the attractiveness of the music, and by resisting acceptance of any research leading to such results.
Since development is my major theme, I pay rather more attention to chronology than most - what changes appear to have happened when. Thus, while most histories concentrate on characterising main-stream instruments, I am just as concerned with transitional instruments. Ordering the evidence in chronological sequence has allowed my suggested answers to development questions to include some detailed stages.
From my scientific training, I am more numerate than most in the field, and so am more likely to use quantitative methods whenever the questions are quantitative. These methods have mainly been applied to questions about the sizes of instruments, for which as a measure of size, I have focussed on the vibrating open-string length.2
There is an attempt here to be somewhat comprehensive, so an amount of rehashing of familiar material cannot be avoided. But I don’t adhere to the convention that the space given to each topic should be related to a judgement of its importance, either then or now. The space I give to each instrument is mostly related to how much I have to say about its main characteristics and its development.
In the current culture of music historians, it is commonly felt that the great scholars of the past have established a fairly complete broad picture of music history. This is no-doubt true. But it is very often also considered that what is not known are 'mysteries', not knowable with the evidence available, and that all that modern scholars can expect to do is to collect new evidence to fill in the details. Those with this view may not welcome my reinterpreting the evidence on some historical questions, considering that these issues were either settled ages ago, or if not, they should remain as mysteries. I am sure that the revered early scholars would have been much more generous in considering and debating alternatives. And they would have been more willing to correct their mistakes when pointed out to them.3
I never intended to write a book since my writing style is not as readable as most. My work involves trying to answer historical questions that haven't been adequately answered before. Since I could not have confidence that I had adequately considered all alternatives to my answers, initial publication in FoMRHI Quarterly, which always welcomed incomplete and controversial studies, has been appropriate. With the publishing of Quarterlies halted by the current leadership of FoMRHI, this no more provides an outlet for my work. Indeed, some of my interpretations published as Communications (Comms) in FoMRHI Q have had to be revised, but a surprising number of others have stood up to further consideration, and it would be appropriate to combine these into longer articles in a journal of record that has a wider distribution in academic circles. This view is apparently not shared by such journals. A recent such article offered to one of these journals was rejected on the basis that the ideas had previously been published in FoMRHI Q. It was suggested that publication in a book would be more appropriate. With both of these outlets effectively closed, I really have had no choice but to write a book.
Many people consider that a book has more authority than other scholarly publications. That is only true to the extent that the time it takes to produce a book is great enough for second thoughts to modify ideas and how they are expressed, and to correct errors. Every book I have seen that was supposed to be the final word on its topic has turned out not to be so. This one will most certainly not be the final word on any topic. This could be especially true here since many of the ideas presented have not had the luxury of previously appearing in FoMRHI Comms, and so they have not had the benefit of such further reconsideration.
If there is interest, I hope soon to produce an improved illustrated edition of this book. Readers are requested to help this improvement in any way they can, especially by informing me of whatever errors I've made (such as where there is evidence that contradicts any of my theories) and of omissions (such as alternative valid answers to the questions I have addressed).
I have greatly enjoyed putting this book together, and I hope that the reader will find the ideas presented stimulating, even if there is some reluctance to incorporate them into one’s views of instrument history. I expect few immediate converts to my methodology, but expect that there will be many more when fashions of thinking about scholarly questions in this field swing towards more logical and quantitative analysis, objectivity, and more respect for the evidence and for valid theories attempting to explain it.
May 2004, revised May 2006
The Development of Western European Stringed Instruments
Contents
Chapter 1a: Methodology - Approaches in scholarship
The scholarly method I follow . . . . . . . 9
The method usually followed in music scholarship . . . . . 10
Chapter 1b: Methodology - Determining the original sizes and pitches of instruments
Measuring pictures . . . . . . . . . 14
Size estimation from finger stretch . . . . . . . 16
Evidence from surviving instruments . . . . . . 16
Praetorius's pitch - organ evidence . . . . . . . 18
Praetorius's pitch - wind-instrument evidence . . . . . 22
Pitch and string length limits from string properties and Praetorius’s evidence . 24
The highest-pitch longest-length limit . . . . . 24
The lowest-pitch shortest-length limit - pitch instability and pitch distortion 26
The lowest-pitch shortest-length limit - inharmonicity . . . 27
TABLE: Gut string limits of pitch and string-length from Praetorius . . 30
Chapter 2: Performance practices: early compared to modern
Embellishment . . . . . . . . . 31
Fastest notes . . . . . . . . . . 33
Tempo . . . . . . . . . . . 34
Time alteration . . . . . . . . . 35
Note production on viols and voices . . . . . . . 36
Phrasing and style . . . . . . . . . 37
Standards of precision . . . . . . . . 38
Concluding comments . . . . . . . . 40
Chapter 3: Medieval stringed instruments before the 15th century
Ancient stringed instruments . . . . . . . . 41
Medieval instrument names and development. . . . . . 42
Cruit and crowd . . . . . . . . . 45
Harp . . . . . . . . . . . 46
Psaltery . . . . . . . . . . 47
Rotta . . . . . . . . . . . 47
Fiddle and gigue . . . . . . . . . 47
Jerome of Moravia . . . . . . . . 48
Other considerations . . . . . . . . 51
Rebec . . . . . . . . . . . 52
Symphony or organistrum . . . . . . . . 53
Citole . . . . . . . . . . . 56
Lute and gittern . . . . . . . . . 57
Monochord and keyboard instruments inspired by it . . . . . 59
Instrument construction . . . . . . . . 59
Some general points about instrumental music . . . . . 61
Chapter 4: Developments in the 15th century
General points, including developments on the harp, lute and gittern . . 62
New types of fiddles including those influenced by the lute . . . . 63
The demise of the lute in Spain . . . . . . . 66
Trumpet marine (and harp brays) . . . . . . . 66
Cetra . . . . . . . . . . . 67
Psaltery and dulcimer . . . . . . . . . 69
Chapter 5: The development of liras and sets of viols
Lira da braccio and lira da gamba . . . . . . . 70
From bowed vihuelas to viols and lironi in Italy . . . . . 72
Tunings of Italian viols in sets . . . . . . . 76
The development of viols in Germany . . . . . . 80
Early viols in Spain . . . . . . . . . 85
Sets of viols in France . . . . . . . . 86
Sets of viols in England . . . . . . . . 88
The construction and design of viols . . . . . . . 90
Chapter 6: Independent viols
Viols for vocal accompaniment . . . . . . . 93
Viola bastarda and lyra viol . . . . . . . . 93
Barytone viol . . . . . . . . . . 94
Viola d'amore . . . . . . . . . . 95
Division viol and violoncino . . . . . . . . 96
Miniature soloistic viols . . . . . . . . 97
Baroque and later double-bass viols . . . . . . . 98
Chapter 7: Renaissance and baroque fiddles
Ensemble fiddles in 16th century Italy . . . . . . 100
Soloistic fiddles in 16th century Italy . . . . . . . 103
Italian fiddles in the 17th century . . . . . . . 105
Large bass fiddles . . . . . . . . . 108
Fiddles in France and England . . . . . . . 108
Fiddles in Germany . . . . . . . . . 112
Chapter 8: Renaissance and baroque plucked fingerboard instruments usually strung with gut
Lutes with a single neck . . . . . . . . 116
Lutes with two necks (or with an extended neck) . . . . . 121
Vihuela and viola . . . . . . . . . 124
Four-course guitar (gittern) . . . . . . . . 126
Five-course baroque or Spanish guitar . . . . . . 127
Angel lute or angelique . . . . . . . . 129
Colascione, colachon and gallichon . . . . . . 131
Mandora and mandolin . . . . . . . . 132
Wire-strung mandoras . . . . . . . . 135
Chapter 9: Renaissance and baroque plucked wire-strung fingerboard instruments
Citterns in the first half of the 16th century . . . . . . 137
Citterns from the second half of the 16th century and from the 17th . . . 139
Sizes . . . . . . . . . . 139
Fretting . . . . . . . . . 140
Construction and design . . . . . . . 141
French citterns . . . . . . . . 142
German citterns . . . . . . . . 143
'French'-tuned English cittern . . . . . . 144
The Meuler steel revolution . . . . . . . 144
‘Italian'-tuned English cittern . . . . . . . 145
Guittern and late cittern . . . . . . . 147
5-course guittern and cithrinchen . . . . . . . 148
English guitar or cistre . . . . . . . . 148
Archcittern . . . . . . . . . . 151
Bandora and orpharion . . . . . . . . 152
Polyphont . . . . . . . . . . 156
Appendix 1: Development influence charts
Descendants of the earliest stringed instruments . . . . . 158
Development of the more modern bowed instruments . . . . 159
Development of the more modern plucked instruments . . . . 160
Appendix 2: Some topics involving string and wood technology
Polnische geigen, fingering past the fingerboard and violin fingerboard length . 161
Twisted and roped gut strings, and catlins . . . . . . 162
Modern misconceptions . . . . . . . 165
Bowed strings, bridges, soundposts and bass bars . . . . . 166
Sound absorption by creep in strings and instruments . . . . 167
Moisture content and swelling in gut strings and wood . . . . 168
The maturing and ageing of wood . . . . . . . 169
Peg fitting . . . . . . . . . . 171
Index . . . . . . . . . . . . 173
The Author . . . . . . . . . . . 178
The Development of Western European Stringed Instruments
Chapter 1a: Methodology - Approaches in scholarship
The scholarly method I follow
Everyone agrees that the purpose of scholarship in any field is to create knowledge, which is composed of evidence and theories. Evidence is the raw material worked with, and theories are generalisations that provide explanations of the evidence and apply beyond it. The basic criterion for acceptance of a theory is that it is consistent with the evidence. Beyond such basics, methodology can vary. My approach follows that most commonly followed in the sciences, where a theory is falsified (proved untrue) when it cannot explain any piece of evidence in a way that has a reasonably acceptable probability of being true. A falsified theory then has to be either abandoned or modified to remove the falsification. Scientists design experiments to obtain the critical evidence that will falsify one or more of the competing theories. No theory can be proven true because it is always possible that a new piece of evidence will appear that falsifies it. Knowledge is not to be believed as true, but is to be trusted as the closest to truth that scholarship can offer with the evidence available and the valid (i.e. non-falsified) theories that have been offered.
The job of a scholar, while collecting all of the relevant evidence, is to imagine the various possible theories that might explain the evidence, and to eliminate those that are falsified by evidence that cannot be adequately explained by them. The remaining theories are then evaluated according to the probability of each one's least probable explanation of any piece of evidence. This is the criterion for how well a theory explains all of the evidence. If that probability is clearly much higher for one theory than for the others, it is chosen and added to current knowledge. If anyone suspects that this particular theory should not be the chosen one, he or she tries either to collect new evidence that could falsify it, or to create a new theory (or to modify a preexisting one) that explains all of the evidence at least as well as the chosen one does.
If two theories equally well explain all of the evidence, the simpler one is preferred. This principle goes by the name 'Occam's razor'. 'Simpler' is defined by having fewer assumptions unsupported by evidence. A common misapplication of this principle is to pre-judge how simple the theory should be and reject one that works just because it is more complicated than expected. Another common mistake is to interpret Occam's razor so that a simpler theory is to be preferred regardless of how comprehensively the competing theories can explain all of the relevant evidence.
The probability of a theory's explanation of a piece of evidence is a matter of judgement. Since scholarship is supposed to be as objective as possible, the effect of bias in judgement should be kept to a minimum. Bias is difficult to avoid in judgement, and to be able to come to conclusions in the process for scholarship, judgement at some point is unavoidable. The process of judging the historical probability of a theory's explanation of a piece of evidence resonates much less strongly with the unavoidable biases of previous expectations and almost-unconscious vested interests than the usual alternative - judging the probability of a theory being true. The former allows one to be more rational and fair, thus maintaining a higher level of objectivity.
Maximum respect for the evidence is basic in this approach to scholarship, so the evidence should have maximum control over the choice of theory. When one piece of evidence is apparently in contradiction with another, the theory must be able to explain both reasonably well. A theory that explains both without assuming error is preferred to one that assumes one is in error. But mistakes in evidence do occur, and conflict with other evidence is the only way to detect it. The mistake could be the result of the source's incompetence, bias or misunderstanding, or it could not be what it seems to be, or it could be an error in methodology or recording, or be deliberately misleading. A theory's explanation for it would present a scenario of how it could have become what it is, citing support from other evidence for similar problems in that source or similar ones. The probability of the explanation reflects how readily such a problem could occur, which makes falsification of theories by mistaken evidence remarkably rare. It is also very rare for sources to deliberately mislead.
Trust is usually given when we perceive that the probable consequences of not trusting are more undesirable than living with the perceived probability that the trust would be betrayed. For example, we trust doctors properly to diagnose and treat our illnesses, though we know that there is some probability that errors will be made. They have our trust because it is considerably more probable that our health will be preserved by trusting them than by following any alternative course. It is in this spirit that we trust the evidence (unless there is other evidence that suggests otherwise), and we trust that the chosen theory is as close to truth as is currently possible (unless we have some good ideas that could lead to replacing the chosen theory with another one).
Accepting that knowledge is no more than the best that scholars can do with the evidence available and the theories they had been able to dream up, with no objective way to determine how close to truth the theories might be, there is no inhibition to forming theories when there is very sparse evidence indeed. This allows a large majority of realistic questions historians want to ask to have answers in theories. With little evidence, the decisive evidence that falsifies some theories might be lacking, but having a set of possible theories is a better position to be in than keeping these questions open as mysteries to which there are no possible answers.
The methods usually followed in music scholarship
In music scholarship, theories are usually treated quite differently. Training does not include the falsification of theories by contradictory evidence, so there is no way to clearly disprove any theory with decisive evidence that invalidates it. Since there is no recognised objective criterion for settling disputes, controversy is rather futile and so is frowned upon, with gentlemanly behaviour the rule. The approach is to try to 'prove' that a theory is true by arguing rhetorically that the theory must be true because of the impressive amount of evidence that is consistent with it. Some evidence can be mistrusted, requiring 'confirmation' by other evidence to be taken seriously, so evidence that contradicts one's theory can easily be rejected as probably wrong, without taking responsibility for showing how it could have become wrong. A theory is incorporated into knowledge when the leading authorities in the field are convinced of its truth and include it in their books. When they are not convinced of the truth of a theory, they usually just ignore it, but when pressed for a response, the reaction is 'it is not proven'.
The main model for this process appears to be the law. The law aims to make quick clear decisions punishing (and deterring potential) wrongdoers or settling disputes within currently acceptable criteria of fairness. Since there is lots to gain or lose, much testimonial evidence is expected to be false or misleading. The weight of evidence and its trustworthiness are paramount. Evidence is readily rejected if one can raise any doubt about its reliability, and the decision is made on the basis of judgement by a jury or judges. That judgement is considered to be the proof. Since the outcome of the proceeding is very often dependent on judgements concerning the truth of evidence, it is strongly influenced by the persuasiveness of the performances of witnesses and advocates.
Because the knowledge that music scholars produce is strongly based on judgement and consensus, it is subject to change when a new generation wants to make its mark, and thus knowledge is a creature of fashion. Recent fashions have been deconstruction (which attempts to 'debunk' accepted judgements), and the politically-correct promotion of the contributions of women and members of minority groups. Whether truth could or should be a matter of fashion is a matter of debate. Post modernism is rather popular in our modern culture, and it postulates that there is no truth other than what is believed to be true. No distinction is made between subjective truth, which is what is believed, and an objective truth, which is a reality out there that is independent of what anyone thinks about it. Having unreserved conviction that one's insights are true is an advantage for success in many fields, including being a musician. Many music scholars have it too, and their scholarship is what they do to convince others of this. A statement attributed to Howard Mayer Brown4 (though I know of no evidence that he followed it) illustrates this: 'Musicology is what you indulge in when you know something is true, and have to go out and prove it'. To members of this school of music historians, the only function of evidence is to bolster their claims.
Other schools accept that there is a truth independent of what is believed to be true. One is a school that is at the opposite pole, being skeptical about most theories. The members of this school will only give acceptance to a theory if it is an unambiguous consequence of the evidence. Theories that cannot qualify (including those that are needed to answer most of the interesting unanswered historical questions) are to be avoided. Scholars in this school (and H. M. Brown was an outstanding practitioner) are renowned for collecting evidence and presenting it in useful ways.
The majority of music scholars I've met belong to a third school. They believe that scholarship produces truth that they can believe, and they seek answers to the important historical questions. The only way to achieve both of these objectives is to rely on consensus amongst their peers as the criterion for worthiness of belief. They may be somewhat independent thinkers in their own narrow fields of study, but otherwise, they follow the crowd. To them, the evidence and theories that are agreed to be true by the consensus are considered 'facts', and theories that don't have such agreement are considered 'speculations'. Speculations are opinions that everyone is entitled to have and promote, no matter how well or poorly they can explain the evidence. This seems quite appropriately liberal-democratic, but liberal democracy only works if there is controversy and free debate. But in this field, these are discouraged as unseemly. Speculations offered by respected scholars that are not challenged for some time slide into being considered facts, and are thus added to knowledge that can be believed in. Once a theory is so accepted, any new competing theory faces an enormous struggle just to be considered seriously.
Professional success in any field depends on communication skills, charisma and a reputation for competence. In a field like this one, that finds controversy embarrassing, anyone not superbly endowed in such ways who disputes issues that are considered settled, or engages in disputes of any sort, will be mistrusted and considered a loose cannon that might lower public respect for the field.
Some questions in music history present difficulties for music scholars when the surviving evidence conflicts with their aesthetic understanding of the music (i.e. when the objective truth indicated by the evidence conflicts with subjective truth, which is strongly related to aesthetic expectation), so these areas are consigned to the category of 'mysteries'. One example of this is the level of improvisatory deviation from the written music in early performances. Another, which I will discuss now, is the history of tempo standards:
It is a tribute to the objectivity that musicologists can muster is the general acceptance that till well into the baroque, contrary to modern practise, tempo markings referred to tempo standards. When some of the evidence concerning what those standards were was discussed by eminent musicologists in the 20th century, some of it was disbelieved and some misinterpreted. The problem was that when the musicologists performed the music at the tempi indicated by the evidence, it moved much more slowly than they expected, and they could no more enjoy it or understand it in the way they were used to. They couldn't imagine how it could possibly ever have been appreciated that slowly. Musicologists seem to have convinced themselves that their understanding of the surviving music, which includes aural acceptability, is objective truth, and so evidence indicating otherwise cannot be trusted and must be wrong. That is probably why the topic was not seriously studied till my two papers published in 1996, where I analysed all of the evidence I could find up to 1700,5 and I linked all of the evidence in a theory explaining the evolution of tempi from the beginning of mensural notation. I consider this to be my most important contribution to music history.
I thought that the papers would be of interest to early music performers as well as musicologists, so I submitted them to a journal they both read. There was a question as to whether the papers would be accepted for publication, since the journal's policy has always been advocacy for the early-music movement, and my results contradicted the tempo assumptions of the movement, which were at least twice as fast. There was only one response to my study, a highly critical one, but the editor didn't publish it because she wanted to avoid an extended debate. To accommodate her concern for brevity, I suggested that my critic and I presented single position statements from each of us in full knowledge of what the other was writing, and she accepted the suggestion. We submitted the position papers, but they were never published. This was probably because he could not fault my analysis according to historical criteria. Tempo history will continue to be considered a mystery by the field until either someone can find an interpretation of the evidence that is felt to be believable (hoping that new evidence emerges which contradicts that which is known), or a new generation of music scholars demands more objectivity in their field. The study of music history is often primarily considered to be a service to our current music culture, with much less interest in it as an application of general principles of scholarship to the evidence on the history of music.
In my papers since, I have been promoting (with some resistance from editors) my more disciplined approach (based on how scientific scholarship is performed) to bringing more objectivity into the way theories relate to evidence in music history. I have had no indication of success in convincing others, (and have noticed a deterioration in my acceptance in the scholarly community). This is to be expected, especially in a culture in which the need for re-examination of one's ideas is rare. If one is satisfied with how one does one's job, one's colleagues agree, and then someone from a different tradition comes along with a different set of rules for doing it, claiming that those rules would make the job better, it is much simpler to dismiss him as a crank rather than to seriously consider the issues raised.
Whenever a new conclusion presented here differs from what people in the field prefer to be true, it is likely to be widely rejected as 'unproven', or just ignored.
The Development of Western European Stringed Instruments
Chapter 1b: Methodology - Determining the original sizes and pitches of instruments
Size clearly affects what an instrument sounds like. I focus on the vibrating string length as a good measure of instrument size since it can be related to string properties and pitches. It can be estimated from interpreting measurements of pictures, surviving early written measurements, the evidence on surviving instruments, finger stretches in surviving tablature music and the reported pitches that the strings were tuned to (converted to pitch frequencies by knowledge of the history of pitch standards).
Measuring pictures
Estimating an instrument's string length by measuring it in an undistorted picture is straightforward if the strings are close to being parallel to the plane of the picture, and for comparison, there is something else of known dimension at the same apparent distance from the viewer. Let us call Sp the string length measured in the picture, Rp the apparent length of the reference object measured in the picture, Rf the known full-size real length of the object and Sf the full-size string length we want to find. Then by proportionality, Sf = Rf(Sp/Rp), where / means divide, no symbol means multiply and parentheses () enclose values that are calculated before being multiplied or otherwise operated on by what is outside. The most obvious reference object in a picture of an instruments being played is some dimension on the player. One possibility is total height, which for a fully grown male could, I would suggest, be about 160 cm, with perhaps an uncertainty of maybe 20%. Another possibility would be a dimension of the head, which could be better because we have reason to expect that variation in head size would be less than variation in total height. But hair styles and head clothing usually obscure direct measurement of head size in the pictures, so I mostly use visible components of head size, namely the distance between the eyes or the distance between the mouth and the centre between the eyes, whichever distance line is closest to being parallel to the plane of the picture. From averages in a small study performed on my acquaintances, I use 6.2 cm for both of these reference dimensions, and expect that the uncertainty would be about 15%.
The above assumes that the artist accurately depicted what would be realistic, as if the picture was a photograph. Even those pictures that look photographic could be distorted for various reasons. One is that the artists often worked from pattern books rather than copying from life, and they could have altered pattern-book designs with other design components from memory, forming unrealistic hybrids. Another is that the artist could contract, expand or distort items in a picture to give a more desired visual balance, to fit into a space or to provide emphasis for symbolic or other purposes. Some instrument depictions are unrealistic since they would not work as musical instruments as shown (but we should be careful about assuming this because some instruments could work in ways that we are not familiar with). And, of course, some pictures have been changed by over painting at a later time to update the subject matter or to attempt to 'restore' it.
Clearly unrealistic depictions can sometimes be due to amateur incompetence of the artist, or it was intended to be an instrument of fantasy. The latter can become evident from what the picture appears to represent. In early times, artists were not respected for their creativity, and their objectives were to meet the expectations of the people for whom they made the pictures, which were to depict reality when there was no good reason to do otherwise. In many cases, the artist was trying to be more realistic than a photo-like image would be, by twisting design components around to show their most interesting and informative aspects. Picasso said that art is the lie that reveals the truth. How the lie reveals the truth involves using a visual language that needs to be understood by the intended viewers, who in earlier times would usually have been members of the affluent classes, and with modern art they are the artistic cultural elite.
When trying to use pictures for measurements, it is necessary to be aware of the methods, the culture and the probable objectives of the artists when making the pictures. When choosing an instrument for measurement, one should first survey other pictures of people playing what seems to be the same instrument, and pick those that appear competent, undistorted and typical.
I have encountered some historians that argue that there is no technical information of value to be had from early pictures of instruments. There are makers who research and service the instrument needs of early-music performers, and they routinely scale the dimensions of surviving instruments in their 'copies' to meet customer requirements. They emphasise uncertainties in scholarship (as a basis for rejection) when the evidence indicates that a typical historical instrument characteristic, such as size or pitch level, differs from what is considered normal in the modern culture they share with their customers. It certainly is difficult to distance oneself from the culture of music and history we are immersed in, and to try to be fair about scholarship that challenges any of it. Training in scholarship should develop such objectivity (learning to distinguish between subjective and objective truth), but it rarely goes beyond encouraging trainees generally to be skeptical. Skepticism and cynicism, generally popular nowadays, are towards any claim of authority, but rarely towards one's own judgements. In this spirit, many of these historians can only accept studies that use evidence and techniques that they have been trained to handle (based on surviving instruments), and reject other evidence and techniques that are at least as relevant to their conclusions.
The other group is scholars with the responsibility for cataloguing collections of instruments or pictures6, who agonise about how sure they can be about the information they put in their entries. I sympathise with their predicament. They are expected to produce catalogues that are authoritatively correct, and they feel that any needed subsequent modifications could raise doubts about their competence. The modern history of historical scholarship shows that almost all of the studies of topics that attempt to be complete and definitive are not, often needing modification after publication. One can't write anything that is 'fireproof', and it is best to accept the disappointment with equanimity when one has missed something. I am not sure which is sadder, being able to convince oneself (with the hope of convincing others) that one has achieved the wanted perfection, or realising that one cannot meet the standard of perfection that one thinks is expected. History is an ongoing research project approaching truths, not a collection of truths.
Size estimation from fingering stretch
There has been some controversy about the size of the English cittern used for playing the solo repertoire published by Holborne and Robinson and in surviving manuscripts from that period (c.1600). The contenders are the small English cittern depicted by Praetorius with a string length of 35 cm and the smallest size of surviving Italian citterns with a string length of about 45 cm. A way to estimate the maximum string length is to find the biggest stretch indicated by the tablature in the repertoire, and compare it with an assumed maximum stretch for a average hand. I have assumed that my hand is of average size, and my maximum stretch between the first finger on a barré and the stretched-out little finger is about 11.5 cm.7 If we assume equal-temperament fretting for simplicity, and n is the fret number of the index finger and m is the fret number of the little finger, then the stretch = string length times (2-n/12 - 2-m/12), where the higher symbols are powers to which 2 is raised (the calculation can be done on any school scientific calculator). In the case of this repertoire, the biggest stretch occurs between a barré on the 2nd fret and the little finger on the 9th fret. It occurs in a printed book of cittern lessons, so it is unlikely to be intended for a player with a particularly large hand.8 Then the maximum string length = 11.5/(2-2/12 - 2-9/12), or 39 cm. This is enough less than the 45 cm string length of of the majority of Italian citterns to strongly favour the small English cittern.
This approach is useful in estimating the string length of the lyra viol used to play Corkine's (1610) tablature. The greatest stretch is between the 1st and 5th fret. The maximum string length then calculates to 59 cm. That was the size of a tenor viol. This should be compared to the lyra viol played later in the 17th century. Mace's (1676) lyra viol music had the greatest stretch from the 3rd and 7th fret. The maximum string length calculates to be 66 cm. By the end of the century, the Talbot ms (c.1694) indicated that the string length of a lyra viol was 71 cm. This indicates that the most valued sizes of lyra viols increased during the 17th century.
Evidence from surviving instruments
Surviving early instruments are the most dramatic evidence of instrument history. They provide invaluable information on materials, making methods and details of design and construction. As with any other type of evidence, care needs to be taken in its interpretation. During an instrument's centuries of coexistence with people interested in music, it is most likely that there had been very many attempts to find out what it sounded like. If the sound was found interesting, it is also very likely that it was used for performances of some kind. For such performances, any deterioration in its integrity would probably have been repaired, during which it could have been modified to better suit the playing technique of the player. These modifications could easily be detected now if the repairs and alterations were either incompetent or in a very different style or it used different materials than the rest of the instrument, but some other modifications could be undetectable.
The question of which aspects of a surviving instrument are original is important for instrument historians. One often finds components of instruments that clearly had previously been parts of other instruments. There was a thriving 19th century instruments antique industry, mostly in Italy (the most famous firm that did this was that of Franciolini), in which parts of surviving instruments were used to create instruments that differed from the originals but were most in demand by the collectors. One remarkably influential overreaction to the uncertainties resulting from such fakes, has suggested that some well-known 16th century museum instruments that used worm-eaten wood or were composites were also later fakes.9 The problem with this suggestion is that good well-seasoned wood has always been highly prized by makers because of its greater stability. So wood with a few non-active worm holes that present no threat to structural integrity would gladly be used in making a new instrument (even by many modern makers), and parts of irreparable or redundant instruments were gladly recycled.
Instruments have been most likely to be discarded when they lost respect when fashions changed, and they had no function to perform in the new fashion. Many more instruments that could be used without modification survived than those that needed modification for use, while very few (other than those of high decorative value) survived if they had no musical use. The rate of loss with time decreased when they became uncommon and gained value as curiosities and antiques. When instruments came in different sizes, we can expect that the numbers of each size that survived were usually very unrepresentative of what they originally were.
Lutes, citterns and bandoras were very popular in Renaissance and baroque England, yet not a single example of any of these instruments made in England survives. Nevertheless, many dozens of English viols from then have survived. The vast majority of these viols were small bass soloistic ones that survived because they could be used later as cellos. Some original tenor viols have survived because they could be used as small cellos. Treble viol bodies have survived because they had been in demand from the late 17th to the 20th centuries for conversion to violas. Only one viol that approaches consort bass size (converted to a small double bass) has survived. When the playing of viol music in sets was revived late in the 19th century, the written evidence on original sizes was either unknown or disbelieved. The disbelief was because they thought they knew what the sizes of bass viols were from the predominant number of surviving ones of cello size. They then invented tenor and treble viol sizes by scaling down from the basses they knew. This new set of viol sizes, 20% smaller than the originals, became standard then, and they still remain standard in the current early-music culture.
The current viol culture does not deny the clear written evidence on original larger viol sizes, but modern sizes 'work' well at the modern early-music pitch standard of a' = 415 Hz, and original sizes would not because of excessive breakage of gut top strings. This issue is avoided as much as possible by scholars as well as musicians. The listening public would be annoyed (at least) if it was made aware of aspects of the performances it enjoys that are knowingly historically inaccurate, and it would not thank anyone who informed it of this. Very occasionally, musicians attempt to emulate the rich sonorous sound of viols of original sizes by playing the music on sets composed of modern tenors as trebles, modern basses as tenors and double bass viols as basses.
Many instrument historians (who are often makers) are so enamoured by the sound of music played on restored surviving instruments (or accurate copies) that they are much more willing to trust interpretations of measurements on such instruments than any other type of evidence. Surviving instruments are real, able to be appreciated by sight and touch as well as by sound, while pictorial and written evidence is, by comparison, very remote and lifeless. Subjective truth, associated with the perception of attractiveness, is confused with objective truth. When there is apparent conflict between a piece of evidence of each of two different types, these historians are biassed towards trusting the type of evidence that they are most familiar with, and tend to reject or ignore the other. I will illustrate this below with the interpretations of evidence on Praetorius's Cammerthon pitch by organ and wind-instrument specialists. This issue is important for my estimation of sizes of various stringed instruments.
If we apply the more objective approach I have outlined above, one has the obligation to present a scenario for how every piece of relevant evidence that is apparently inconsistent with what one's theory expects could have possibly became what it is. One cannot reject evidence because one does not trust it without presenting a good case based on other evidence for how it became 'wrong'.
Praetorius's pitch - organ evidence
Important examples of poor standards in current scholarship are concerned with the question of Praetorius's Cammerthon pitch standard10. Its frequency is essential for my calculations of limits on string lengths from nominal pitches outlined below. At the end of the book about instruments written by Praetorius11, after the Index, and before the list of errata, is a 2-page addition entitled only 'NB'12. It includes a diagram giving dimensions for making a chromatic octave of square wooden and round metal pitch pipes. The stated intention was to define his primary pitch standard, which he called rechten Chormass or rechten Thon, for organ makers and singers to tune to.13 This appears to have been a more precise version for organ tuning of the pitch standard he generally called Cammerthon (or the usual or rechte Chorthon).
In the 19th and 20th centuries, various scholars have used the dimensions specified to find the frequency of that standard either by measuring it from pipes made, or by calculating it directly from the physics of the air vibration in an open cylindrical organ pipe with a mouth opening. The earliest determination of Praetorius's pitch from the pitch-pipe diagram was a' = 423 Hz (0.7 semitones below modern) by A. J. Ellis14 in 1880. Early in the 20th century, A. J. Hipkins15 mistakenly assumed that the pitch standard represented by the pitch pipes was the Chorthon of Catholic churches that Praetorius preferred to his own, and so Hipkins assumed that Praetorius's Cammerthon was a tone higher than Ellis's determination, i.e. a' = 475 Hz.
The apparent origin of the modern early-music pitch standard of a' = 415 Hz is in Bessaraboff's famous 1941 book16. His suggestion was that, for practical purposes, we should approximate the original pitches with the closest pitches to whole semitone steps from modern a' = 440 Hz. Thus, accepting Hipkins's erroneous conclusions, Bessaraboff assigned Ellis's 423 Hz for Praetorius's Chorthon to a' = 415 Hz and his Cammerthon to a' = 466 Hz. He claimed that the Chorthon pitch 'is the tonality of the musical system of the classical period, which lasted from about 1600 until 1810-20'. We now know that this is a gross distortion and oversimplification17, but the grain of truth here is that Praetorius pitch of the pitch pipes continually remained as the usual standard for string ensembles in north and much of south Germany though the period stated.
One problem with Bessaraboff's proposal is that it was based on Ellis's determination of the pitch-pipe pitch. If he made the proposal later, when better determinations of the pitch (see below) indicated that it was up to 10 Hz higher, his pitches would have been 440 Hz for Chorthon and 494 Hz for Cammerthon. This highlights the other problem with his proposal, which is that the important pitch standards of the time fall near the middle of his semitone ranges, so a small shift such as this one is grossly amplified.
The great attraction of Bessaraboff's proposal to early musicians is that it blurs the picture enough that they can justify the use the same instruments for all baroque and classical music, including copies of the superior later-baroque French woodwinds which played at about a semitone lower than Praetorius's pitch.
Bunjes18 built a set of reproduction pipes with the resultant pitch being a' = 430 Hz, and Bormann19 did the same with the resultant pitch being a' = 427 Hz. Thomas & Rhodes20 calculated the pitch using the method of Ingerslev & Frobenius21, with the resultant pitch being a' = 426 Hz. D. Gwynn22 surveyed previous determinations and added his own corrections to that of Bunjes, which he considered most reliable, with the resulting pitch being a' = 433 Hz.
Organ historians try to follow an organ's pitch history by studying its records of repairs and alterations, and on each of the pipes, studying the nominal pitch names written on them, the styles of that writing, the signs of pitch alteration and the final pitches. When the pitch of an organ is changed, pipes can be shifted to be activated by different keyboard keys and their lengths can be shortened by trimming (or cutting scoops) or lengthened by adding an extension. Smaller changes can be made by widening or narrowing the tops of pipes. When a pipe was shifted to a new key, the new nominal pitch was sometimes marked. The trimming of pipe lengths can rarely be detected, nominal pitches on pipes are often missing and records of an organ's repairs and alterations are notoriously incomplete. Occasionally, original decoration on some pipes or the space inside an original organ case can put limits on some original pipe lengths. The original pitch of an old organ is usually estimated from the pitches of pipes with the earliest pitch-name markings that show the least evidence of alteration.
Some experts on German organs made in the 17th and 18th centuries make the generalisation that their original pitches tended to be at about a semitone above modern throughout that period. There is no question that this was the case late in the 17th century, but we are concerned with the situation on Praetorius's time, early in that century. One very highly regarded organ, in mostly original condition, is the 1616 Compenius organ in Frederiksborg. Its very unusual all-wooden piping resists the tinkering with pitch that metal pipes have always been subjected to, and it fits neatly into an original case so original pipe lengths couldn't have been longer. It appears to have been made originally at a pitch of about a semitone above modern, and Praetorius was consulted on its design. These experts are very impressed by the sound of this organ and its association with Praetorius, and so they are very skeptical about Praetorius's pitch-pipe evidence, which implies that his pitch standard was about a semitone lower than the pitch of this organ.
Praetorius wrote that most of the organs in his time were tuned to his pitch (Cammerthon or proper Chorthon), but that there also were many at a tone higher and lower, and 'not a few' a semitone higher.23 He mounted a spirited argument against the tendency in his time to raise the currently fashionable pitch to a semitone higher24. A likely scenario is that he lost the battle against the higher pitch for the Compenius organ, but hoped (vainly, it turned out) to win the war with the arguments in his book. That organ is the only one amongst the about three dozen organs he esteemed (listing their stop dispositions) that have survived well enough for modern researchers to be able to estimate what their original pitches were. The vast majority of his esteemed organs could easily have been at the pitch he specified. There are three other German organs that Praetorius could have known when writing the book that have had their original pitches estimated. We have no idea about what he thought of them. Two had the pitch of a semitone above modern, and one was approximately at modern pitch. Since the pitch of the first two remained in fashion later in the century, the probability of their survival would be greater than others. In conclusion, there can be no statistical case made from the pitches of the few early 17th century German organs estimated that the most prevalent pitch was different from what Praetorius claimed.
There is also written evidence indicating that the most popular organ pitch level early in the 18th century was a tone higher than Praetorius's pitch, and that it dropped by a semitone late in that century. This change in pitch recognition is not reflected in the general conclusions of the organ specialists. In my analysis (that accepts all of the written evidence), the fashion of German organ pitch changed as follows: Early in the 17th century (Praetorius's time) it was a semitone lower than the constant level assumed by the organ experts, it was at that level (a semitone higher than in Praetorius's time) later in that century (when Schnitger was the major maker), it went up another semitone around 1700 (to follow the pitch of the ancient organs), and it dropped a semitone about two-thirds into the 18th century. We would expect these organs to be at the organ experts' pitch levels by late in the 18th century. I would be very surprised if the organ experts can tell the difference between the pipes remaining where they were during all of the 18th century (which they claim) and their being shifted a semitone at the beginning of the century (with the longest pipes unused) and back again later in the century.
The two competing theories are that Praetorius's pitch was as deduced from his pitch-pipe diagram, and that his pitch was about a semitone higher, as usually found in the surviving German baroque organs. The subjective choice that is usually taken is to decide which evidence one trusts more. A more objective choice between them should depend on the relative probabilities of how well the pitch-pipe evidence can be explained assuming the higher pitch theory, and of how well the surviving organ evidence can be explained by the lower pitch-pipe theory. It was shown above that there is no statistical case for inconsistency between the surviving organ evidence and Praetorius's lower pitch-pipe pitch.
The organ specialists have not attempted to explain how the pitch-pipe evidence could be consistent with their higher-pitch theory, but a harpsichord specialist who supports that theory has attempted this25. He noted that Praetorius had neither specified the wind pressure nor the mouth dimensions of his pitch pipes, and he proposed that these could have been high enough to get a pitch a semitone higher. As a model, he picked a late 16th century Innsbruck organ with pipes having extraordinarily large mouth dimensions, which has been restored with an extraordinarily high wind pressure of 90 mm water column. Assuming room temperature, these parameters and Praetorius's dimensions, he got a good part of the way towards pushing the pitch up a semitone on a test pipe he made.
To support his theory that the mouth dimensions were larger than expected, he also presented the mouth dimensions and diameters of 19 pipes (marked with the same nominal pitch as one of the pitch pipes) from surviving German organs roughly contemporary with Praetorius (their lengths have most probably been altered, so that is not relevant evidence). I calculated the averages of the pipe diameters and the mouth dimensions. Assuming Praetorius's pipe length, a wind pressure of 75 mm water column (considered to be the maximum expected by a specialist on early German organs, who happens to advocate the higher-pitch theory for Praetorius's pitch) and the annual average temperature of 10 degrees Celsius in Praetorius's region in Germany (churches were not heated), I calculated the pitch of a pipe with the average mouth dimensions and Praetorius's diameter. The method of Ingerslev & Frobenius was used, with a slight correction for the average mismatch between their test pipes and their theoretical calculation.26 The result was a' = 437, 436, 435 and 434 Hz for the temperament being equal, sixth comma, fifth comma and fourth comma meantone respectively. If I use the average diameter of the pipes instead of Praetorius's diameter the results are 2 Hz higher. If I assume a wind pressure of 55 mm water column (like on the Compenius organ) instead of 75 mm, the results are 3 to 4 Hz lower.27 The uncertainty in the calculation method is about ± 6 Hz.
Thus the pipe information not given by Praetorius cannot provide an explanation of how the pitch-pipe evidence is what it is in a way that has a reasonable probability of being true. Koster seems to believe that just showing that a theory's explanation is a possibility is enough to give it validity.
Praetorius's pitch - wind-instrument evidence
The semitone-higher theory for Praetorius's pitch has been an article of faith amongst wind-instrument specialists since Anthony Baines suggested it in his famous book on woodwind instruments28. He wrote that "Recorders at Verona identical in shape and in size with those in Praetorius's scale drawings at 'chamber pitch', sound a good semitone above modern pitch; say about a' = 470". His criteria for being 'identical' must have been rather fuzzy since I (and others) have found that there is a systematic error in the sounding lengths of the recorders in Praetorius's drawing, so that as depicted, the pitch standard varies, with the smallest ones at a standard about a semitone lower than the largest ones.
We have reason to expect that a large fraction of the surviving wind instruments would sound about a semitone above modern because they were made in Venice, where they were played with organs, and that was the pitch standard of Venetian organs. Woodwind instruments made there were used extensively throughout Europe, and the woodwind specialists interpret this as suggesting that this pitch standard was largely universal (including the German regions Praetorius knew). This could well have been true for most bands of Venetian woodwinds, but the expectation of these specialists that this carried over to the pitch standards of string bands does not have any supporting evidence, and is unlikely because wind bands and string bands rarely played together (the difference in pitch standards probably was a factor). Praetorius's insistence that both types of instruments played at the same standard was very unusual for his time. A minority of surviving instruments (mostly transverse flutes and mute cornetts) were made at lower pitch standards, apparently for playing with stringed or keyboard instruments at lower standards.
The pitches of woodwind instruments other than recorders cannot be determined from Praetorius's drawings with enough accuracy to distinguish between the two theories a semitone apart. There is uncertainty concerning pitch-affecting factors that can't be seen, such as the plug positions on transverse flutes and the reed characteristics in reed-blown instruments, but in addition, the pitch can be varied rather more on them than on recorders by the way it is blown.
An instrument for which there are no uncertain pitch-affecting factors, except for how it is blown, is the trombone (or sackbut). From measuring the lengths of the vibrating air columns in Praetorius's drawings of the trumpet and 5 sizes of trombone, Steve Heavens and I have shown that, as expected, they played at the same pitch standard, if the method of blowing was the same.29 We then showed that the pitch reported by modern blowing of a surviving Nuremberg trombone contemporary with Praetorius (who preferred such a trombone), when scaled to the length of Praetorius's trombone, would sound just over a semitone higher than it would sound if a' = 430 Hz.30
Assuming the theory that the pitch deduced from the pitch-pipes is true, the only explanation for this result is that the modern style of trumpet and trombone blowing (the same in modern and early music ensembles) produces a pitch about a semitone higher than in the blowing style at Praetorius's time. Modern blowing technique is characterised by what has been called the 'keyhole principle', in which the vertical direction in an old-style keyhole (having a vertical slot with a wider round top) represents the possible pitches, and one blows to pitch at the round top. Above the round top, the pitch breaks into the next higher harmonic of the vibrating tube. At the round top, the sound is richest (with more contribution of higher harmonics to the sound quality), is the most resonant, and it is easiest to blow a stable pitch. This can be called 'playing on the resonances'. In that explanation, in the early style of playing the trombone and trumpet (in non-military circumstances), they were played about a semitone lower than at the resonances. The softer sweeter sound of playing lower than the resonances could well have been considered to confer a more vocal quality.
There is early evidence that supports the hypothesis that wind instruments that could be played off the resonances often did. The virtuoso music for 17th century trumpet includes short ornamental notes that could only be played by lipping both a tone above and a tone below their normal notes. The evidence on early reeds indicates that they were much stiffer than the reeds that modern players use in both modern and early music. Stiffer reeds transfer much of the control over pitch from the fingering to the lips, with more effort in playing and more concentration needed to play in tune. A good reason for normally lipping a semitone lower than the top of the pitch range for a note is that instruments that could imitate the vocal appoggiatura strove to do so (a modern equivalent is that instruments that can imitate the vocal vibrato, usually do so). According to Tosi31, the appoggiatura was a continuous slide in pitch. The slurring between two fixed pitches on keyboard and fretted instruments would be an inferior imitation. A practice of normally lipping below the resonance would give the continuous pitch range for lipping that accommodates the appoggiatura from above. This appoggiatura was a very important component in music performance from the middle of the 16th century onwards through the baroque and later.
If we allow ourselves the subjective luxury of judging the trustworthiness of the evidence, without accepting responsibility for having to present a reasonable case (based on other evidence) for what could be wrong with what we do not trust, then of course, we would prefer Praetorius's pitch-pipe evidence to be wrong, rather than modern lipping on the trombone to be wrong. We enjoy the music that modern early music groups produce. We want to trust that the wind players are playing in a reasonably accurate simulation of the original style, and would prefer to avoid considering that this may not be true. But the only admissible evidence for evaluating an historical theory should be historical evidence, and we should maintain the scholarly discipline of considering this modern evidence to be historically irrelevant. The very popular expectation of early musicians that an instrument of authentic design will automatically lead the player to authentic performance practices is pure fantasy.
A majority of the people presently interested in Praetorius's pitch are organ and wind-instrument specialists, who make broad generalisations about original pitch standards from pitch evidence collected within their specialisms, ignoring other kinds of evidence. They believe that it was a semitone higher than modern. That theory remains falsified by Praetorius's own way of communicating that pitch, the pitch-pipe evidence. The evidence of the limits on the relationship between string-length and pitch (the theory of which is given below), as given for gut strung instruments in this book, is consistent with the pitch given by Praetorius's pitch pipes, and not with the semitone-higher theory.
Pitch and string-length limits from string properties and Praetorius's evidence
String physics can relate the vibrating string lengths of instruments to the range of pitches that the strings can be tuned to. The highest string has to last long enough for the musician to get on with making music, and the lowest string needs to sound well enough to be musically useful. The breaking stress (i.e. tensile strength) of the string material is closely related to string longevity, but how close to the maximum stress that a string can 'safely' be tuned to is a matter of judgement, which could (and has) varied in different historical circumstances. The deterioration in the sound of strings made of any particular material as they get thicker and are tuned to lower pitches is largely understood in terms of inharmonicity (loss of harmonics, leading to loss of pitch focus and dullness of sound), pitch distortion (sharpening on fretting) and pitch instability (the variation of pitch with changing vibrating amplitude), but again, how bad is too bad is a matter of judgement in the culture of the time. I will quantify what these judgements were for gut-strung instruments by analysing historical evidence, and then present a table of acceptable ranges. Rather rougher estimates of the ranges of metal strings will also be made.
The highest-pitch longest-length limit
For a uniform string, according to the Mersenne-Taylor Law, the fundamental pitch frequency (f) of a string times the vibrating string length (L) equals half the square root of the string stress (S) divided by the density (r), or fL = (1/2)sqrt(S/r). Stress in a string is defined as the stretching force (tension) divided by the cross-sectional area. The tensile strength of the string material is defined as the stress at which breaking occurs. The tensile strength of plain metal strings can depend on diameter since the process of drawing a wire through successively smaller die holes introduces dislocations in the structure that inhibit the crack propagation that is necessary for breaking. In fresh well-made gut strings, the tensile strength depends mainly on the average angle between the gut fibres and the string axis. That angle results from the twist that is put into the string when it is made. For maximum strength in thin treble strings tuned near the breaking stress, they have normally been made with the minimum twist necessary to produce cylindrical strings out of the few membrane-like pieces of gut each is made from. These are called 'low-twist' strings.
We can then consider that there is a maximum working stress for a treble (low-twist) gut string that represents the stress at which the rate of string breakage of the highest-pitched string is just tolerable. With strings of the same material, density is constant, so we can consider that there is a highest acceptable product of the frequency and the vibrating string length, or 'fL product', which is proportional to the square-root of the highest acceptable stress. Some musicians find it difficult to accept that gut string breakage depends only on the string length and frequency, and not on the diameter and tension. They associate higher pitches with thinner strings and expect that a thinner string can go to a higher pitch. But if they did the experiment of tuning a low-twist gut violin 1st string until it broke and then did the same with a low-twist gut violin 2nd, they will find that the 2nd will break at a much higher tension, but the pitches at breaking would be as close to the same as can be expected from the variability of a natural product.
To determine the maximum fL product tolerable in a historical period, we need to consider instruments that push the pitch limits by having an exceptionally large open-string range, and for each we need to know simultaneously its vibrating string length, the nominal pitch of its highest string and
the pitch standard that applied to that nominal pitch. A source that provides all of this information is the book Syntagma Musicum II by Michael Praetorius.32 Scaled drawings of most of the instruments discussed in the text provide the vibrating string lengths, tables of tunings provide the nominal pitches, and the basic pitch standard used is defined by the speaking lengths and cross-sectional dimensions of diatonic octave sets of cylindrical and square pitch pipes, indicating that his standard was about a' = c. 430 Hz33.
The gut-strung instruments in the book with the large open-string ranges are the lute in chorthon (2 octaves + 5th on 61.8 cm), the short neck of the Paduan theorbo (2 octaves + 4th on 97.2 cm), the large 5-string bass viola da braccio (2 octaves + major 3rd on 75.0 cm) and the viola bastarda type of viol (2 octaves + 4th on 72.9 cm).34 The fL products calculated for the highest strings on these instruments are respectively, 211, 209, 207 and 209 metres/sec, indicating that a good estimate of the maximum fL product acceptable in the early baroque was about 210. In the middle of the 19th century, when orchestral woodwinds were asserting their power by pushing pitch standards up to sound more brilliantly, many violinists had to live with an fL product of over 220. It was mainly pressure from the rate of breaking of violin 1sts that lowered the pitch standard, as a compromise, to a' = 440 Hz later in the 19th century. At 440 Hz, the violin 1st fL product became about 216.
The metal-strung instruments tell us about the highest fL products of some of the metals involved. The fan-shaped fretting of the bandora suggests that the string length of the top course was at its maximum when the design was developed in the 3rd quarter of the 16th century. For the iron of that time, the top course was 5 semitones lower than it could be with gut. By 1580, when the orpharion was invented, much stronger ferrous metal was available from Meuler in Nuremberg, and this was reflected in the top course being 1 semitone higher than it could be with gut. After 1600, Meuler apparently perfected his process and the top course of the theorboed lute was almost 5 semitones (and of the gittern-tuned small English cittern over 4 semitones) higher than it could have been with gut.35
There are indications that after Meuler's success in achieving dramatic increases in tensile strength of ferrous wire, the other Nuremberg wire drawers improved their processes so that the subsequent highest fL product for iron was increased by about 2 semitones, being about 3 semitones lower than that for gut. They apparently did the same for brass, resulting in a highest fL product about 6 semitones below that for gut.36
The lowest-pitch shortest-length limits - pitch instability and pitch distortion
In pitch instability, the pitch sharpens in strong playing. The frequency changes because the string length and the string tension changes while playing. When that is because of the high amplitude of vibration, the frequency change (Df) divided by the frequency (f) equals a quarter times the ratio of the elastic (or stiffness or Young's) modulus (E) divided by the string stress (S), times the ratio of the maximum stretch of the string due to strong playing: (DL) divided by the vibrating string length (L). In symbols only, Df/f = (1/4)(E/S)(DL/L). The maximum stretch divided by the vibrating length (DL/L) for a plucked string is [1/(2(r-r2)] times (d2/L2), where r is the fraction of the vibrating length that the distance of the plucking point from the bridge represents, and d is the initial displacement of the string at that plucking point. For the bowed string, r is 1/2 and d is the displacement at the mid-point of the vibrating length. The Mersenne-Taylor formula can substitute for the stress, S = 4rf2L2. Then the pitch instability (Df/f) equals the product of a constant (1/32), times a term of properties of the string material (E/r), times a term of how the string is used on the instrument [1/(f2L4)], times a term of how the string is played [d2/(r-r2)].
To modern ears at least, the maximum tolerable pitch instability is about a third of a semitone, or Df/f = 0.02 (2%). On plucked instruments, the maximum amplitude occurs at the pluck, after which the pitch decreases as the amplitude dies away. If the ear's initial judgement of pitch is not confirmed immediately afterwards, the perception is of a twang with only an impression of pitch. This happens mostly with low-tension iron or steel stringing. On such ferrous metal strings, E/r is very high (about 25 Km2/sec2), but on gut strings, with E/r less than 5 Km2/sec2, inharmonicity becomes serious well before pitch instabi