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Oud Strings, Calculations and Tuning.

jdowning - 7-18-2012 at 12:23 PM

Thanks to Yaron Naor and Yair Dalal, I recently had the opportunity to examine some old gut and wound copper on silk core oud string fragments (see the topic 'Repairing a "Nahhat" Oud' on this forum).

In order to determine the physical characteristics of each string fragment, it was necessary not only to accurately measure the string outside diameters and the diameters of the windings (as well as confirming the materials of construction) but to undertake some basic calculations to determine string tensions and operating limits of pitch dictated by string breakage (upper limits) and inharmonicity (lower limits) for varying pitch standards (A440, A415 and A 392). This exercise confirmed that the strings (six double courses) were originally probably tuned to Arabic tuning CEAdgc'.

Forum members Alfarabi and Brian Prunka made the interesting observation that the old ouds (made before 1950 when gut and wound silk strings were the norm) generally are considered sound 'better' when tuned a semitone or a full tone lower than A440 pitch standard yet there seemed to be no reason given in the literature as to why this should be.

The answer - as far as the European gut strung lute is concerned - is that string breakage of the first course is the limiting factor. So, for example, one of my lutes - a copy of an historical original - with a string length of 60 cm, if strung with plain gut, can be taken no higher in pitch than f' if the top string is to last more than an hour or two before breaking. A lute of this string length is generally designated as tuned in nominal G tuning i.e. with the top string at g' which is possible with gut but not really practical due to the potentially short string life. With modern plastic strings it is possible to take the top string to g' without string breakage problems. However I have found that on this lute - although strung with modern strings - sounds more resonant and responsive when tuned a full tone below G tuning.

However, this is not the case with an oud of the same string length where the top string (Arabic tuning) is at a lower pitch of c' - no breakage limitations here. So, why do the old ouds fitted with modern strings sound better when tuned down a semitone or a full tone below standard A440 pitch?
I don't claim to know the answer - but I have a suspicion and would be interested in ideas that other members might have.

In the meantime, I thought it might be useful and of general interest, to go through - in detail - the procedures and tools that I used to calculate the string properties of the old oud string fragments (comparing gut with modern nylon stringing) so that anyone with a pocket calculator and readily available special slide rules might be able to undertake their own string calculations without having to delve into and understand all of the complexities of string design theory.

So that's the challenge - lets see how it goes!

ameer - 7-18-2012 at 12:28 PM

One possibility is tuning lower allows for thicker strings which sound warmer. This is particularly relevant if using lower tension. A related possibility is materials like nylgut or PVF develop a distinct twang at lower tensions which are made more practical at lower tuning; to my ear using nylgut or PVF at higher tensions makes them sound a bit choked and lifeless.

Jody Stecher - 7-18-2012 at 02:05 PM

Is it possible that older ouds sound better at lower pitch due to factors involving the wood, the neck angle, the resonating cavity, and other non-string factors? I am remembering my 1965 Khalife oud. I got it new in 66 or 67. For the first 25 years it sounded good at standard pitch Arabic tuning with the first course at cc. I used LaBella oud strings or different brands of classic guitar strings at appropriate gauges. No matter what strings I used the oud responded well at standard pitch. Occasionally circumstances dictated that it needed to be tuned a bit higher. No problem. But then, starting in about 1998 this oud not only sounded better at lower pitch but it if left unplayed for several days it would drift down between a half step and a step in pitch. The oud would be pretty well in tune with itself, just lower than where I left it. This oud simply wanted to sing lower. It's scale was 61cm by the way. I think that changes in the soundboard and neck angle might have been a factor.Once the oud began to show its preference for lower tuning if I would tune it up to standard pitch it responded less readily than when the strings were just a bit slacker.

ameer - 7-18-2012 at 02:14 PM

Interesting thought. Both the 2006 Sukar 213 I used to own and the 2011 Michael Moussa GamilGeorges replica I currently own resonate better at certain lower tunings e.g. the Sukar was great at anything below standard where as if you tune the Moussa anything more than a halfstep down the third course sounds unusually dead. So I think you could say the Moussa likes lower tuning a little bit less. What's interesting is both ouds' tuning will drift downwards but the relative tuning of each course will be correct and the strings in each course will almost always be perfectly in tune with one another. I never calculated the neck angle of either oud but they feel relatively similar for whatever that's worth.

farukturunz - 7-19-2012 at 03:02 AM

Quote: Originally posted by Jody Stecher

Is it possible that older ouds sound better at lower pitch due to factors involving the wood, the neck angle, the resonating cavity, and other non-string factors? I am remembering my 1965 Khalife oud. I got it new in 66 or 67. For the first 25 years it sounded good at standard pitch Arabic tuning with the first course at cc. I used LaBella oud strings or different brands of classic guitar strings at appropriate gauges. No matter what strings I used the oud responded well at standard pitch. Occasionally circumstances dictated that it needed to be tuned a bit higher. No problem. But then, starting in about 1998 this oud not only sounded better at lower pitch but it if left unplayed for several days it would drift down between a half step and a step in pitch. The oud would be pretty well in tune with itself, just lower than where I left it. This oud simply wanted to sing lower. It's scale was 61cm by the way. I think that changes in the soundboard and neck angle might have been a factor.Once the oud began to show its preference for lower tuning if I would tune it up to standard pitch it responded less readily than when the strings were just a bit slacker.

I was directed to this thread by Alfarby with these remarks "There's a hanging question in the forum about this issue here, : http://www.mikeouds.com/messageboard/viewthread.php?tid=13176 that has not been concluded yet. Please Master, tell us there what do you think about this issue.
Waiting to learn more about this question"

To me this issue is highly complicated. But I suppose there is one answer to be near the mark: We all know that inner force vectors of a construction at the shape of a dome are in the direction to the edges of the dome thus a dome is the most stable construction. You can not break an egg squizing it in your palm in the direction of the long axis.
As the oud gets aged a dome-like shape starts to buil up in front of the bridge and at the back of it. These areas traverse when the instrument is plucked and are dominantly responsible for the sound characteristics of the instrument. Having become dome-like these areas are now more stable than before. This is associated with a shift of the resonant areas. Formerly these back and front areas were more resonant against the fundamental frequencies of the higher pitche strings likewise of the higher pitche sounds but now they became more resonant against the second, third and fourth harmonics of the lower pitche strings likewise of the lower pitche sounds .
This is only a proposition to handle the issue and understand what is going on with the aged instruments. I think this approach of mine is worth to be discussed.

jdowning - 7-19-2012 at 05:12 AM

These are some interesting comments. I can confirm that the observation that ouds seem to automatically find their optimum pitch level has also been observed in lutes. This suggests that the instruments over time adjust to overstress due to string loading - local stresses that are relieved or redistributed not only in wood movement/distortion but also in glued joints temporarily relaxing slightly - particularly under conditions of high temperature and humidity.

Too high a stress may also result in an instrument sounding 'tight' or less free in its response as noticed by ameer. It is interesting that ouds are strung with heavier plain strings in the upper courses (compared to lutes) and no doubt this is to help give a 'warmer' sonority - thicker strings having less upper harmonics that are in tune with the fundamental. However, due to this inharmonicity, there is a limit as to how far one can go before a string becomes unacceptably dull in sound. This lower limit is dependent upon a number of factors - personal preference, the way a string is sounded, acoustic performance of the instrument etc. For a plain gut strung lute of 61 cm string length for example, with the strings sounded with the soft fingertips, the lower limit is around B at A440 standard pitch. For an oud this limit may be a bit lower due to the strings being plucked with a risha that increases somewhat the range of upper harmonics sounding.
Therefore, for an oud of 61 cm string length and Arabic tuning, plain gut trebles should serve for the first three courses (although they would likely not sound as bright to modern ears as modern plastic string equivalents).
The lower limitation may be increased by making a string more elastic (a higher degree of twist or use of roped construction) or by use of composite (wound) strings that are thinner and more dense than plain strings.

ameer - 7-19-2012 at 05:29 AM

This thread makes me want to try gut strings.

Alfaraby - 7-19-2012 at 11:51 AM

Very interesting . I knew we could count on Mr. Downing

I wrote to Master Faruk in his Jamil George's thread that:

"I have had tens of Arabic ouds during the last 35 years I've been in the oud world. Some were very old and some were newly born, Syrian, Egyptian, Iraqi, Moroccan, Palestinian .... Every single one of them sounded better to my ear when tuned F instead of G. This of course might be a subjective preference, not a scientific conclusion. "

Well, it's not only old ouds respond "better" in low pitch. All ouds, not only Arabic, resonate in a mellow & soft mode when tuned a whole step lower. even a Manol 1904 sounded even greater on F instead of G.

Does this remind of viola tuning compared with violin tuning ? Is there any similarity between the two cases ? I don't really know how to tune a viola nor a violin, but the warm sound of the first just crossed my mind, so excuse me if this question does not fit in. Just an idea/question !

To make this even more complicated: Why is a Turkish oud tuned a whole tone higher ?!
Oh yea, WHY ? Are we allowed to Turkish-tune an Arabic oud without replacing the strings ? This involves another issue: strings. The lower you go, the thicker you choose & vice versa !! Does this make any sense or am I just distracted/ing ?

Yours indeed
Alfaraby

ameer - 7-19-2012 at 01:39 PM

Tuning an Arabic oud to Turkish tuning (with appropriate considerations to tension of course) can be quite interesting. Two very different examples that come to mind are Amir Amouri's performances with Sabbah Fakhri where they use Turkish tuning and the recording of Farid El Atrash and Muhammad Abdel Wahab where they use Turkish tuning without the high D.

fernandraynaud - 7-19-2012 at 11:39 PM

To those contemplating different tunings, tuning an Arabic oud up two half-steps to Turkish tuning WILL develop more tension than on an Arabic tuning, that's obvious. Normal tension on an oud string is around 2.5-3.5 kg (on a Spanish guitar, stronger in build, double that is customary - roughly the same strings are pulled to higher pitch on a longer scale). Whether this is OK depends on the string set and the oud. If the oud has a relatively short scale, below 600 mm, it can be fine. But if the strings are relatively thick, or the scale is longer, or both, the tension (over 4 kg) may well damage the instrument. It is said that ouds cannot tolerate tensions over 5kg per string. But 4 kg is already a lot. Specifically what happens is that the tension will 1) stress the bracing and soundboard, braces may come unglued, the soundboard may distort/buckle 2) pop the bridge, which is normally just glued onto the soundboard. The higher the bridge, the more stress develops under tension (leverage). If the glue "gives" in a slingshot moment, the bridge goes flying. It's a disturbing event, but a simple repair. If the glue is very strong (Aliphatic glues are not a good idea for the bridge) and the tension very high, part of the soundboard can come off with the bridge. Catastrophic.

So, Alfaraby, we ARE allowed to tune a Turkish short scale oud either Turkish or Arabic. It may sound weaker down-tuned Arabic, or it could sound great. But it's a generally a very bad idea to tune an Arabic (longer scale) oud up to Turkish unless the string gauges are thinner than normal. In any case it takes some study and caution.

As to WHY Turkish ouds are tuned that way, it could be because the shorter scale Turkish ouds sound better with Arabic strings pulled up to the same tension (as on an Arabic longer scale oud) by raising the pitch, but they could have used thicker strings, so that's not much of an argument. That's like the theory that it's because their demented notation has confused them. Could it be because other Turkish instruments prefer to make D = Rast and not C?

Alfaraby - 7-20-2012 at 02:49 AM

Quote: Originally posted by ameer

Farid El Atrash and Muhammad Abdel Wahab where they use Turkish tuning without the high D

Without the high D, it would be AEBGD instead of CGDAF. This means 1.5 tone lower than A440 ! Sunbaty also played on this tuning, as much as I can recall, in several recordings.
Is this right ?

Yours indeed
Alfaraby

ameer - 7-20-2012 at 04:15 AM

You're right that it is 1.5 tones down from A440. As for Sunbati I'm not aware of him using this tuning but given that Qassabgy, Abdel Wahab, etc tried it it wouldn't surprise me.

jdowning - 7-20-2012 at 03:39 PM

Well, ouds (and lutes) may sound better at a lower pitch as they age but human beings may not hear better as time passes. Perhaps this is a reason why an instrument is perceived to 'improve' with time but at a lower pitch than original?
Loss of hearing (Presbycusis) - particularly of the higher frequencies - is an inevitable and irreversible part of the aging process which may be 'amplified' (!) by genetics,gender, race and ill health as well as exposure to loud noise (rock concerts and other over- amplified music etc.). The loss of hearing starts at about the age of 8 years and slowly becomes measurable by the age of 60+.

Interestingly the trend in lutes from the late 16th C was to add more bass courses to the nominal 6 or 7 course lute of the
16th C culminating in larger solo lutes with 13 or 14 courses by the mid 18th C at which time the lute went out of fashion in Europe. Yet the development of the oud never appears to have exceeded 7 courses throughout its history - perhaps compensating in some cases by working with relatively thicker strings and lower tunings than found on lutes with the same string lengths?

So when an oud appears to sound better at a lower pitch after a period of 25 years or so, is this due to physical changes within the instrument itself or due to a slow deterioration of the hearing ability of its owner the auditor?

So we will all suffer from this ailment to a greater or lesser extent as we age but - hopefully - not to the extent experienced by poor English lutenist Thomas Mace (Musick's Monument, 1676) who in 1672 at the age of about 59 found it necessary to invent a new, louder lute (the Lute Dyphone) with 50 courses so that he could continue to play despite a serious hearing loss. Even then his affliction was so severe that he could only hear all that he played by holding the edge of the sound board in his teeth - the sound being transmitted satisfactorily to his brain in that manner!! Now there is a dedicated musician!
Nevertheless the Dyphon never did 'catch on'.

Jody Stecher - 7-20-2012 at 05:03 PM

In my case it had nothing to do with hearing loss. Since the time I acquired that oud any number of string instruments of all vintages passed through my hands and none of them tuned themselves down to where they sounded better and none of them sounded better when I deliberately tuned them down. This was true for instruments that were both older and newer than this particular oud. Had it been a matter of subjective perception surely it would have applied to my guitars etc.

And since typical hearing loss is of high frequencies why would a lower pitch appear to provide the missing highs? I suppose one could argue that a lower fundamental creates lower harmonics and the upper partials become more audible when a string has a lower tension and therefore a lower fundamental. But I think the issue here is not string behavior but the behavior of the resonating box called oud.

Quote: Originally posted by jdowning

So when an oud appears to sound better at a lower pitch after a period of 25 years or so, is this due to physical changes within the instrument itself or due to a slow deterioration of the hearing ability of its owner the auditor?

jdowning - 7-20-2012 at 05:58 PM

My thought here was only another general theory to add to the others already expressed on this thread and was not intended to be directed at you personally Jody Stecher.
Rightly or wrongly my thought is that the lower pitch does not provide the missing upper harmonics - only that lower frequency harmonics remain audible.

Incidentally there are audio tests available online to check for the incidence of hearing loss over time for those curious about how well (or not) they can hear. One website even suggests that it is musicians who are most likely to suffer greatest from this ailment - not sure why.

So it was only the oud - among all of the other instruments that passed through your hands - that auto-tuned itself downwards to sing better at an optimum pitch level? Could it be that this phenomenon is only to be observed in ouds and other related instruments like the lute? If so, I wonder why?

Jody Stecher - 7-20-2012 at 06:37 PM

Hi jdowning,

I put my responses in italics between your lines in the quote box below. Hmmm, probably not an optimum way to respond to individual points. Oh well, I'll give it a whirl this one time.

Quote: Originally posted by jdowning

My thought here was only another general theory to add to the others already expressed on this thread and was not intended to be directed at you personally Jody Stecher.

Sure, I understood that.

Rightly or wrongly my thought is that the lower pitch does not provide the missing upper harmonics - only that lower frequency harmonics remain audible.

that's what I meant

Incidentally there are audio tests available online to check for the incidence of hearing loss over time for those curious about how well (or not) they can hear. One website even suggests that it is musicians who are most likely to suffer greatest from this ailment - not sure why.

perhaps they equate "musician" with "rock musician". In the latter category, hearing loss is a common occupational hazard

So it was only the oud - among all of the other instruments that passed through your hands - that auto-tuned itself downwards to sing better at an optimum pitch level?

yes

Could it be that this phenomenon is only to be observed in ouds and other related instruments like the lute? If so, I wonder why?

that's what we're all trying to find out here.

jdowning - 7-21-2012 at 04:21 AM

Your system of relating quote and answer seems to work well enough. Happy that we are all working on the same page!

I guess that they were referring not only to 'rock musicians' but also to all those who regularly listen to or play over amplified music. This of course is a fairly recent occupational hazard for musicians and music lovers of all stripes. However even if one manages to steer clear of occupational sustained high noise levels (by wearing adequate ear protection in a commercial woodworking shop environment or when operating heavy machinery for example) - the hearing loss due to aging for everyone would seem to be inevitable to a greater or lesser degree dependent upon a number of factors already mentioned - and would likely be the same (or worse) for a person living in earlier times.
Just a thought.

jdowning - 7-24-2012 at 12:21 PM

Out of the multitude and variety of Arabic tunings I am curious about the origin of the Arabic tuning F A d g c' f' (or F2 A2 D3 G3 C4 F4 in American nomenclature). I assume that the first course here is tuned to F above middle C - i.e frequency 349 Hertz and not down an octave (re-entrant tuning found on some of the longer string length European lutes)?

Does anyone know when this tuning first appeared historically? Does it only go back as far as the 1950's (Bashir style ouds) or is the history more ancient?
For a string length of 61 cm, f' would be the upper practical limit of a gut first course if frequent string breakage is to be avoided.

Jody Stecher - 7-24-2012 at 12:49 PM

You can hear that George Michel has a high f string on the oud he used on a 1960s LP recording "Authentic Instrumental Music by the Most Famous Arab Artists". This is certainly a fixed bridge oud. It seems to be a 7 course oud since low C is sounded several times.

Quote: Originally posted by jdowning

Out of the multitude and variety of Arabic tunings I am curious about the origin of the Arabic tuning F A d g c' f' (or F2 A2 D3 G3 C4 F4 in American nomenclature). I assume that the first course here is tuned to F above middle C - i.e frequency 349 Hertz and not down an octave (re-entrant tuning found on some of the longer string length European lutes)?

Does anyone know when this tuning first appeared historically? Does it only go back as far as the 1950's (Bashir style ouds) or is the history more ancient?
For a string length of 61 cm, f' would be the upper practical limit of a gut first course if frequent string breakage is to be avoided.

Jody Stecher - 7-24-2012 at 01:12 PM

Whoops, I posted before finishing my thought. Apologies if the following is too obvious:
My idea, which is meant to be descriptive, not historical, is that a high F course added to the tuning C F A d g c' yields C F A d g c' f.' And then dropping the low C yields F A d g c' f'.
Looked at another way it is simply moving up by an interval of a 4th the tuning of C E A d g c'. But perhaps by "origin" you mean "who did it first"? My guess would be that it was someone with access to nylon strings.

ameer - 7-24-2012 at 01:16 PM

According to Simon Shaheen Qassabgy introduced the high F some time around the middle of the century. I can't find the interview link, but it was the Afropop interview in 2003 I think.

jdowning - 7-24-2012 at 05:51 PM

Thanks Jody Stecher and ameer. Interesting.
Yes I am curious to know when this tuning may have first been introduced. So far around 1950 would seem to be the case?

For a 'standard' Arabic style oud of around 61 - 62 cm string length, this tuning would still be feasible with plain gut trebles so might still have been used before nylon strings became generally available for the oud.
The 'rule of thumb' upper practical limit for low twist plain gut strings (based upon frequency of breakage) is that the product of tone frequency (Hz) X string length (metres) should not exceed a value of about 210 (source E. Segerman, N.R.I. Instruments) So if a 61 cm long first course is tuned to f' - i.e. 349 Hz at A440 standard - then 210 X 0.61 = 213 or just about good enough. The string pitch can be increased above this limit but string life might then be only a matter of minutes or hours before breakage occurs.

Whoever introduced this high f' tuning was presumably seeking a brighter tonality - a move away from the traditional deeper, darker sound of the Arabic oud so another indication that this might have been a mid 20th C innovation perhaps - or does this tuning have its origins during a much earlier period for different reasons?

For the European 6 course gut strung lute f' first course tuning would have been the norm for a nominal tuning of G c f a d' g' i.e. tuned down a full tone. The top string was tuned as high as possible to help make the thicker, lower plain gut strings sound better (with the help of octave strings to brighten the sound or by employing greater string twist to improve string elasticity - that is before wound strings as we know them today (late 19th C) were invented). For a plain gut string 61 cm long this lowest acceptable limit (due to inharmonicity - out of tune upper harmonics causing the string to sound dull) is around B 98 Hz perhaps slightly lower for a string sounded with a risha.
So one has to wonder, therefore, what type of bass strings were used on five or six course ouds prior to the 19th C and what was the pitch of the first course as a consequence.

ameer - 7-24-2012 at 06:24 PM

In listening to the Qassabgy taqsims on this site you can hear him using a high F by 1939. See http://www.mikeouds.com/audio/qassab/qassab_taqsim_hijaz_kar_kurd_1...
Note that it is hard to determine the natural pitch of these recordings do the quality of the initial transfers, but given the time period and other Qassabgy recordings it is likely that he tuned a wholestep down making it a high Eb rather than F.

jdowning - 7-25-2012 at 05:51 AM

In addition to plain gut strings it is likely the oud was also fitted with wound basses - copper wire on silk filament core - giving a brighter more balanced sound overall.

On the string technology front I should mention that the useful frequency X string length product used for judging the upper limit of gut due to string breakage (210 or less) can also be used to determine the lower limit of a plain gut string the product f X L being equal to or above about 80 before the sound is so dull that a change to wound strings is necessary. This lower limit value is subjective and depends upon the how a string is sounded (plucked or bowed), the acoustic quality of an instrument and the tolerance of a listener to the sound quality.
The same limits apply to plain plastic strings except that they can be taken to a higher pitch limit than gut (the limit for nylon, for example, is about four semitones higher). At the other end of the scale, however, plain gut can be taken down to a lower pitch than plain nylon (about three semitones lower).

Gut strings can be taken to a lower pitch if they are made more elastic by providing more twist to a string or by making the string like a cord or rope by combining two or more threads together. By this means a plain gut string with higher twist,
61 cm long can be taken down to F and a roped gut string down to C (source E. Segerman, NRI). The downside is that the more highly twisted strings will break at a lower tension than those of low twist so should not be considered for use in the upper courses. Whether or not ouds in the 15th or 16th C used the same string technology as the European lute is, of course, not known (it is not perfectly clear in the case of lutes either).

The other possible historical method for bass string design is to add weight uniformly to a string without increasing the string diameter (achieved today by means of wound string design). Mimmo Peruffo of Aquila Strings has experimented with this possibility by adding metal powders or salts to gut strings pointing to evidence for weighted strings in the iconography (coloured strings - possibly loaded with mercury or iron salts/powders) and in the small diameter string holes measured in original lute bridges - too small to accommodate strings made from untreated gut.

I have never tried it but a few years ago some lute players advocated twisting plain nylon strings to improve elasticity. This was achieved by mounting a string on a lute, threading the string though a peghole and applying the required amount of twist before wrapping the string around the peg and bringing it up to tension. I suppose that these days the availability of denser PVF strings now makes this practice unnecessary. I must try it some time on my string test rig just to see how well (or not) it might work.

jdowning - 7-26-2012 at 12:21 PM

Given the upper and lower tonal limitations of gut strings, the way to go for greatest tonal range - given all gut stringing (i.e. prior to availability of close wound bass strings) - is to increase string length (to extend the lower range) and to pitch the top string as high as it will go practically.

Some of my recent studies of early oud and lute geometries (see 'Old Oud compared to Old Lute Geometry, page 2) suggest that the string lengths of some ouds (Al-Kindi, Kanz al-Tuhaf and Ibn al-Tahhan) and lutes (Laux Maler) dating from the 9th to 16th C may have had string lengths consistently measuring 67.5 cm in length - dependent upon the ancient standard of measure chosen (a 'finger' unit ranging in value from 1.875 cm (Egyptian) to about 2.25 cm (Persian). For a plain gut first course this would give a maximum practical pitch at A440 standard of about e' (330 Hz) or f' (349Hz) at A415 standard.

So this might suggest that the Arabic tuning F A d g c' f' historically came well before (centuries before) the current C F A d g c' with its lower pitched darker tonalities - achieved by adding the lower C and removing the upper f' - only made possible on a shorter 61 cm string length oud by the invention of close wound silk filament bass strings first available in the late 19th C?
If so, a 14th C 67.5 string length, gut strung 5 course oud might have been tuned A d g c' f' (i.e. in fourths) by eliminating the low F course - the A course then just about being the lowest pitch limit for a plain low twist gut string (at A415 standard) or a pitch easily achieved with a higher twist plain gut string (lowest pitch limit about E). So the next step historically from a five course low twist gut strung oud tuned A d g c' f' to six course F A d g c' f' would have been made possible by simply increasing the degree of twist of a gut string from about 25° (low twist) to about 45° or so (high twist).

Note that historically both plain gut and silk strings were used on ouds - silk being about the same density as gut but a bit stronger.

jdowning - 7-27-2012 at 12:03 PM

It is worth mentioning here in the context of pitch standards that the international standard pitch of A440 Hertz (i.e a') is relatively modern - established in 1939 - after several attempts at creating a universal standard made by countries like France, Britain and America during the 19th C. Until then it was pretty much chaos for the travelling musician and instrument maker with every country, province, city, town, church or opera house having their own preferred 'standard' over the centuries. Military bands preferred a higher pitch because it sounded better out doors, opera houses and churches preferred a lower pitch to avoid 'screeching' sopranos but not so low as to inconvenience the basses etc.
Pitch standards over the centuries - determined from old tuning forks, pitch pipes and organ pipes or calculated from data provided in theoretical works - ranged from a low a' of A373.1 Hz to a high of A567.3 Hz (source A.J.Ellis - translator of 'On the Sensations of Tone' by Hermann Helmholtz).

The pitch of A415 adopted in recent times (1960's?) by the early music fraternity as supposedly representing the pitch of Baroque period instruments has no historical basis. It is just a convenient full semi tone lower than the reference pitch A440. The same applies to A392 which is a full tone lower.

It would seem from earlier comments that early middle eastern musicians may have (and still do) instinctively tune to a pitch approximately a semitone to a full tone below A440 (presumably without any reference to a pitch standard tuning fork) just because an oud or other instrument happen to sound and feel 'right' at a lower pitch. Something to do with the peculiar sensitivity of human hearing to being receptive to a particular middle range of frequencies perhaps?

jdowning - 7-30-2012 at 05:06 AM

I have just found out that the international standard of A440 proposed in 1939 was only affirmed as an international standard in 1955 by the International Organisation for Standardisation. So that pitch standard, officially, is of even more recent history!

Forum member al-Halibi in the past topic 'Safi al Din 'Abd al-Mu'min aka: Kitab al-adwar" mentions that at the first Congress of Arab Music held in Cairo in 1932 it was decided that the note 'Rast' would be standardised as C (I assume that is c' or middle C (261.63 Hz) in Western nomenclature?) and that the Turks standardised the standard pitch in the modern period with the second open string of the oud (neva) equal to A220 Hz (i.e. a) which is in line with the Western standard.

jdowning - 7-30-2012 at 12:07 PM

The traditional Turkish oud tuning as I understand it is E A B e a d' for a traditional string length of 58.5 cm? This tuning would have been used prior to the availability of modern nylon strings so would have once have been used when the Turkish oud - like the Arabic oud - was gut strung.

For a gut strung Turkish oud of 58.5 cm string length the highest pitch that the first course can be taken to without problems of frequent string breakage is f X L = 210 - regardless of string diameter. So with L equal to 0.585 metre, maximum pitch would be about 210/0.585 = 359 Hz or about f'# at A440 standard (or a semitone below g' at A440 ).

So why is the Turkish oud not tuned to a nominal relatively high tuning of F B c f b g' as well as the lower pitch tuning E A B e a d' similar in concept to the longer string length Arabic oud which has a high pitch tuning of F A d g c' f' as well as the lower pitch tuning of C F A d g c'?
Could it be that this high pitch tuning was at one time in use for the Turkish oud - a tuning now long since forgotten? If so the instrument could have been entirely strung in plain gut/silk in the days when only plain gut and plain silk strings were the option - i.e. using high twist plain gut for the basses? A plain silk first course could have been tuned a bit higher than gut to g'.

Today, of course, with nylon strings and wound basses both high or low pitch tunings would not be a problem.

Also, if Arabic oud players instinctively tune to about a semi tone or tone lower than A440 standard pitch does this phenomenon also apply to Turkish oud players?

jdowning - 8-17-2012 at 12:20 PM

This thread was started with the intent to discuss some basic tools and methods available for string calculations - for plain strings as well as wound - to assist those who are generally unfamiliar with string technology and are not rocket scientists.

As the discussion about tuning systems etc. now seems to have fizzled out, this might be an appropriate time to talk about the available tools for string calculations.
Those handy with a pocket calculator can always work from first principles by applying the Mersenne-Taylor law for plain, cylindrical vibrating strings but others have developed calculators that make this step unnecessary. The following calculators have been designed for lute string calculations that will also generally apply to the oud except that for the oud the lower range limit of plain strings may be extended somewhat due to the use of a risha (rather than soft fingertips) to pluck the strings.

First there is the free on line calculator developed by Arto Wikla for plain string (not wound string) calculations handy for providing string tensions for particular string diameters (and vice versa) given known string length, pitch and string density.
JavaScript (free) must be enabled to view the calculator.

http://www.cs.helsinki.fi/u/wikla/mus/NewScalc

Northern Renaissance Instruments (N.R.I.) have developed a slide rule calculator that may be purchased (high resolution hard copy) for a nominal sum or downloaded free at :

http://www.nrinstruments.demon.co.uk/StrCalc.html

This slide rule supercedes an earlier version that was originally published in FoMRHI (Fellowship of Makers and Researchers of Historical Instruments) Quarterly 013, Comm. 162. I attach a copy here for those who who would like to print it up and make the slide rule by gluing the scales to a thick card, wooden or plastic backing. The attached image shows my version of the slide rule made up, ready for use. The two elastic bands are a rough and ready way to keep everything in place.
The advantage of a slide rule is that it enables many options of tension, diameter, string length etc. to be compared at a glance.

The low cost commercial slide rule available from Pyramid strings covers plain strings in gut, nylon and PVF as well as their range of wound lute strings and the range limits of the plain strings.
(Bernd Kürschner once produced a similar slide rule but this apparently is no longer currently available).

A very comprehensive string calculator has been developed by lutenist Paul Beier that does the plain string calculations and includes wound string data from Savarez, Pyramid, Kürschner and Aquila string catalogues. This software is shareware but is free to download and test at:

http://www.musico.it/lute_software/bsfc/index.html

Pyramid Calculator front (346 x 722).jpg - 79kB

Pyramid Calculator back (716 x 336).jpg - 64kB

Attachment: NRI String Calculator Old Version.reduced pdf.pdf (368kB)
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Aymara - 8-17-2012 at 11:17 PM

Hi J.D.!

Quote: Originally posted by jdowning

It is worth mentioning here in the context of pitch standards that the international standard pitch of A440 Hertz (i.e a') is relatively modern ...

... and was often critized by experts (composers and musicians).

This is an interesteing topic. So let me mention the so called "natural standard pitch" of C in 256 Hz / A in 432 Hz, also called the Verdi pitch.

Did you ever do a research on this pitch and it's theory? If not, THIS might be a good start.

It might also be worth mentioning, that medical tuning forks are based on this pitch, mainly tuned in 128 & 256 Hz.

I'm asking myself, if this might be related to the original question regarding old tone wood / old ouds.

What do the others think? Especially you J.D. and also Master Faruk.

Am I on the wrong track with my thoughts? The 8 Hz difference is much less, than what was discussed before. But maybe it's worth to try it out.

jdowning - 8-18-2012 at 04:27 AM

Thanks for the link Aymara - I look forward to reading the article when I get a moment.
Some connection with earlier beliefs in Celestial harmony or 'Music of the Spheres' perhaps? - once supported by some of the greatest scientific minds in history but now dismissed as a pseudoscience by modern scientists in their infinite wisdom (too much magic and mystery about it to be trusted or proven).

During the 19th C pitch standards were still 'all over the place' although certain pitch standards were becoming popular for a variety of reasons. At one time I had an active interest in restoration of mid 19th C American reed organs and brass wind instruments - working for a local museum. The pitch often chosen for 19th American keyboard instruments was the so called French or 'Flat' pitch A435 Hz or sometimes the English New Philharmonic Pitch of A439 Hz.
Brass instruments sound better outdoors at a higher pitch so the 19th C American instruments were often built to the so called 'Sharp' pitch of A452 Hz. Something to watch out for when purchasing original 19th C instruments if the intention is to perform with other modern instruments built to A440 pitch.

A435 is even closer to A432 so perhaps was chosen for similar reasons - because it somehow felt right?

Aymara - 8-18-2012 at 05:20 AM

Quote: Originally posted by jdowning

... too much magic and mystery about it to be trusted or proven) ...

Nowadays critics call it "new age nonsense", but it isn't because it has a long history, which shows THIS article for example.

If there wouldn't have been World War II, 440 Hz would never have become standard ... I found german sources proving this theory.

Quote: Originally posted by jdowning

A435 is even closer to A432 so perhaps was chosen for similar reasons - because it somehow felt right?

It seems very likely and is a good question, but I think we'll need a lot of research to answer this question correctly. Maybe it was just a measurement error?

jdowning - 8-18-2012 at 12:16 PM

At the '19th C living history' museum where I was once employed, two reed organs were regularly used as accompaniment for singers one pitched at the French pitch A435 the other at A439. I have to admit that the A435 Hz instrument felt much more comfortable for those of us who were not trained singers - which, no doubt, is the reason that this standard pitch was adopted by the French in the mid 19th C. However, a much greater influence over pitch standards was exerted by the military who had adopted a higher pitch and employed many of the instrumentalists at the time. They were therefore reluctant to change their position to a lower pitch standard not only due to cost considerations but because the sound of a military band instruments carried further at a higher standard pitch. So presumably this was the main reason for adoption of A440 as a compromise (close enough to the British Philharmonic standard of A439 as well). Also, from the German Reich viewpoint at the time (1939) the exclusion of the French in the proceedings and non adoption of their A435 standard would, no doubt, have had some considerable political significance not connected to any musical considerations.

As European military bands were originally developed from the Janissary military bands of the early Ottoman Empire one has to wonder what pitch standards the Turkish wind instruments were (are still?) made to? I would not be surprised if it was once significantly above A440.
Of course, the original purpose of military bands was to intimide the enemy and not for musical entertainment - loud noise that not only carried far but was psychologically disturbing. A432 pitch would probably not have been the way to go!

Aymara - 8-18-2012 at 01:54 PM

Quote: Originally posted by jdowning

I have to admit that the A435 Hz instrument felt much more comfortable for those of us who were not trained singers ...

I read somewhere, that a famous composer criticized A440 and higher pitches for being difficult even for trained singers.

Quote:

Also, from the German Reich viewpoint at the time (1939) the exclusion of the French in the proceedings and non adoption of their A435 standard would, no doubt, have had some considerable political significance not connected to any musical considerations.

Interesting theory! I read that this bastard Goebels was mainly influenced by Wagner regarding A440, but this political thing might have been a further reason.

Quote:

A432 pitch would probably not have been the way to go!

Mmh, this military theory is new to me, but it seems to make sense.

Back to the oud ... do we know anything about the "arabic pitch history"?

jdowning - 8-19-2012 at 05:13 AM

As previously mentioned in this thread, I suspect that the early ouds - like the European lute - strung as they were in gut or plain silk - tuned the first course as high as it would go without frequent breakage in order for the lower bass courses to sound well.
The string lengths would likely have been somewhat longer than found today for the same reason - to provide an optimum response from the basses.
My guess (from studies of the early descriptions on the oud as well as examination of the earliest surviving lutes from the early 16th C) is that string length would have been 67.5 cm or thereabouts. Given this assumption and assuming a plain silk first course (stronger than gut), the upper tonal limit would have been f' (at A415 Hz standard pitch) according to my Pyramid calculator for a tension of about 34 Newtons or around 3.5 Kg (9.8 Newtons = 1Kg although I usually just divide Newtons by 10 to convert to Kg).
This suggests to me - based upon these physical constraints - that the earliest tuning for a five course oud of the 14th C may have been AA dd gg c'c' f'f' - all courses then being tuned a fourth apart.
Again A pitch (at A415 Hz standard) would be about the lower tonal limit for a plain gut Bamm (fifth course) string so that all fits quite nicely - suggesting in turn that the modern (Munir Basher) high Arabic tuning of F A d g c' f' may be a lot more ancient in origin than might at first be apparent.

Aymara - 8-19-2012 at 08:48 AM

Quote: Originally posted by jdowning

Again A pitch (at A415 Hz standard) ...

It might be interesting to research, if this 415Hz assumption is correct.

A question that might help ... how about the Ney? How was it's pitch in history?

Wasn't the Ney played together with oud in history too? If yes, then the oud was tuned to the Ney in ensembles, wasn't it?

And we shouldn't forget, that e.g. in medieval times it wasn't possible to measure the pitch as good as it got possible in 19th C.
So your assumption regarding string brake makes especially sense for solo play, I think.

But what about singers? Might the oud have been tuned to the capabilities of the singers too?

jdowning - 8-19-2012 at 12:10 PM

The A415 (a semitone below A440 reference standard pitch) is just for our reference so that we know exactly what frequency we are all talking about. At this reference standard pitch, the note A (about 104 Hz) of the fifth course is the low pitch limit for a plain, low twist gut string which would have been much the same for the ancients who, of course, would have had no idea what frequency meant let alone how it might be measured in Hertz.

The physical properties of the strings dictates the maximum possible viable tonal range for a given string length - i.e. string life for the first course and acceptable dullness of sound for the bass strings.

To avoid any confusion - at A415 standard reference pitch the limiting string pitches for a plain gut strung five course 14th C oud of 67.5 cm string length might be

f' - 329 Hz
c' - 247 Hz
g - 185 Hz
d - 138 Hz
A - 104 Hz

The thinnest gut string available would have been about 0.40 to 0.45mm giving a tension for the first course of about 3.5 Kg to 4 Kg and for the fifth course, diameter around 1 mm about 2.5 Kg

Aymara - 8-21-2012 at 06:43 AM

Quote: Originally posted by jdowning

The A415 ... is just for our reference ...

I tried to find out more about the pitch in arabic history ... nothing so far ... I searched generell, for Oud and Ney ... all I could find was, that the pitch was based on oral tradition ... I hope that's the right word in English

And then the question is, about witch aera do we talk. I myself am very interested in Al-Andalus. How about Ziryab's time? Isn't it likely, that the pitch was lower in that time? Just a guess. And how about the string length ... isn't it likely, that it was shorter at that time.

I think you are the expert in this "business" ... and I'm curious

Quote:

f' - 329 Hz
c' - 247 Hz
g - 185 Hz
d - 138 Hz
A - 104 Hz

That makes sense, if our assumptions regarding string length and pitch are correct/close.

Aymara - 8-21-2012 at 11:05 AM

Quote: Originally posted by Aymara

I tried to find out more about the pitch in arabic history ... nothing so far ...

... but I found something interesting from Western history. Even nowadays some historic instruments are still tuned to the Versailles pitch A = 392 Hz as an alternative to 415. I saw that for example on the website of a flute builder, in this case Barock flutes. Even Barock concerts are sometimes played at that pitch nowadays.

Do you know something about it's "origin"? Might it be even older?

Might similar low tunings have been used in the arabic world?

That would fit to the findings reported about old ouds, doesn't it?

jdowning - 8-21-2012 at 12:19 PM

The standard pitch A392 - like A415 - has no particular historical significance either. It is just a full tone below A440 adopted for convenience by the modern early music fraternity.

For a comprehensive listing of historical pitches, check the Appendix added by Ellis to Hermann Helmholtz's 'On the Sensations of Tone' An English translation is available free from Internet Archive (Google search)

An abbreviated tabulation can also be found at:
http://www.dolmetsch.com/musictheory27.htm
(as well as a lot of interesting related information).

There is only one example of A392 (actually 392.2) said to be from a clavichord in St Petersburg, Russia, 1739 - so not sure where the 'Versailles' pitch comes from.

For the early oud - as with any stringed instrument strung in plain gut or silk - the max/min pitch range for a given string length is determined by the the life of the top string (breaking frequency) and the dullness of sound of the lowest bass string (a subjective judgement). In modern times we are able to give a maximum and minimum frequency value to this range. In early times no such standard measurement existed to my knowledge - the early theorists on the fretted oud only referring to tuning intervals for the open strings.

Ziryab (9th C) is said to have introduced a fifth string to the four course oud - but this additional string was placed between the third and second courses (Mathlath and Mathna) according to G.H. Farmer so I am not sure what the tuning of the open courses might have been. Ziryab is also said to have introduced silk strings as well as gut to the oud.

Farmer speculates (based upon the tuning of the surviving Tunisian oud arbi) that the tuning of the four course oud was originally c d g a but was changed to A d g c' (i.e. the open strings tuned a fourth apart, Bamm, Mathlath, Mathna, Zir) when the Arabs adopted the Persian system. The five course oud according to Al-Kindi added a high first course (Hadd) making the tuning A d g c' f' (in modern nomenclature).

The four course oud might also have been tuned d g c' f' (or somewhere in between the extreme limits of A and f' at A415 modern standard pitch) - for an assumed string length of 67.5 cm. Shortening the string length to say 60 cm would result in a max/min range for plain gut of about c to g'.
My guess is that the longer string length of 67.5 cm prevailed until wound strings became available to allow further lowering of the bass string tonal limit.

We have practically no idea about the construction of early oud strings (or for that matter, lute strings) made from gut so it is best to assume that the earliest strings were simply twisted without being 'weighted' or made of roped construction to improve performance (as proposed by some modern 'historical' string makers). For a plain, simply twisted string, some degree of improved elasticity (and consequent greater bass range) - at the expense of breaking strength - can be achieved by increasing the degree of twist of the gut fibres.

jdowning - 9-8-2012 at 11:08 AM

Wound string calculations.

In order to determine the string tension for a given pitch of a wound string - round section wire wound over a flexible core material - it is first necessary to establish an Equivalent Diameter for the string - as if made only from the core material. This then allows string tensions to be determined using the Mersenne-Taylor law applicable to plain cylindrical strings of uniform density.

The attached image shows a well established relationship used for close wound string design. Knowing the core diameter, wire diameter and densities of the core and wire materials the Equivalent Diameter may be determined.
Conversely, for a string of unknown manufacture, the Equivalent Diameter may be determined by accurately measuring the outside diameter and the wire diameter, knowing the core and wire densities.

For those of us who are not rocket scientists the calculations might seem forbidding but with a low cost ($5 - $10) scientific pocket calculator and a low cost (but good quality) Chinese micrometer (to accurately measure diameters) it is only a matter of punching in the numbers and letting the calculator do the work. No need even to understand how to calculate the square or square root of a value - just press the appropriate calculator button to get the answer.
Furthermore, once having calculated the Equivalent Diameter there is no need to have knowledge of the Mersenne-Taylor law to calculate string tensions. This has all been done by Arto Wikla - just use his free online calculator.

For information, I have used as an example the third course d string from a set of Pyramid #650 oud Orange label. The Pyramid catalogue gives the (average) string tension of each third course string as 4.0 Kg (string lengths between 60 cm and 62 cm).
The calculated Equivalent Diameter for a plain nylon string is 1.22 mm diameter. For a string length of 61 cm the Arto Wikla calculator gives a tension of 3.977 Kg. Pretty close!!

Next to further verify the results using my string test rig.

Wound String Calculation Example.jpg - 115kB

Calculator Buttons (768 x 576).jpg - 104kB

Aymara - 9-8-2012 at 11:27 PM

Thanks for your detailed explanation.

Quote: Originally posted by jdowning

... but with a low cost ($5 - $10) scientific pocket calculator ...

No need to buy one, if you have a Windows PC. Just use the Windows Calculator and switch to scientific mode in the View menu.

jdowning - 9-9-2012 at 12:02 PM

Thanks Chris. A useful alternative.

It is amazing how low in cost basic scientific calculators are now - compared to when they first came on the market in the 1970's costing hundreds of dollars. Here is what they cost now at a local store. Even when Sales tax is added $3.36 Canadian is a price hard to resist (no - I do not need another one!).
Lee Valley of Ottawa are selling a Chinese micrometer (Imperial units) that is said to be accurate to 0.0001 inch (0.0025 mm) costing only $18.50 Canadian. The tool does have insulated pads to reduce heat transfer from fingers that will affect accuracy - however, that level of accuracy is 'pushing it' a bit outside of a climate controlled tool/metrology room environment. Still excellent value and quality.

I prefer to work in metric for string dimensions so have a non Chinese micrometer (Starrett #436 )for that purpose inherited from my father. I hate to think what that might cost these days.

jdowning - 9-9-2012 at 04:25 PM

For information here are some details of a rig built to aid in the development of historical oud strings made from silk. It will be used here to verify string tensions of a set of Pyramid #650 orange label oud strings for comparison with calculated values.

The test rig is a simple affair consisting of a sound box on a frame with a fixed bridge at one end and a sliding bridge at the other to allow adjustment of string length.

String tension is applied directly by means of a suspended load attached to the string passing over a pulley. The rig is mounted at an angle of 45° in order to minimise friction losses at the pulley that might affect measured string tension (resulting in an overestimate of string tension).

Frequency of vibration for a given string length and load is measured with a low cost digital tuner calibrated with a standard tuning fork.

In operation a load is applied to a string with lead weights and steel washers added for fine adjustment until the required pitch is consistently measured. The total load is then measured using a digital scale accurate to 1 gram.

Next the test results.

fernandraynaud - 9-9-2012 at 05:29 PM

It's very useful to have this method. And at last we'll be able to confirm some of the calculations, and over a wider range!

But these simplified "effective" thickness models rely on the same equations as are used for a homogeneous material, right? So I don't see anything here any more nor less correct than the very simple working method I use. Can you look and tell me? Let me try to clearly state what is actually quite obvious.

The question most often asked is: what will be the tension if I tune this specific string to pitch X on scale Y? With a homogeneous material of known density, say Nylon or PVF, we can plug the material, length, thickness and pitch into Arto's, and get a tension. We can probably agree that Arto's is a good usable interface to the fundamental equations.

The first problem with wound strings is that density varies with "gauge" or thickness. Your method attempts to derive that, and then express it not as density, but as an equivalent thickness of the core material. I'm not sure we can assume the core is homogeneous, nor that the winding is always done the same way, but it seems irrelevant to me how we "virtualize" it. In the end we'll plug it into the same equation.

The non-rocket science way I've been doing it doesn't rely on measuring the winding wire and assuming the packing of the core or winding will be the same from one string maker to another. It just uses data provided by the string-maker. Like this:

We take the tension listed for the string under consideration at one length, gauge and pitch, and we use Arto's in reverse to hunt for the density that would give that tension. That is the "effective density" of that composite string. Say 5765 Kg per cubic meter for a 0.033" D'Addario wound string.

It's likely, though I've never tried it, that if we weighed a known length of a given string, we would come up with a similar density. And if not, then the "effective density" is more relevant, because it works in the equation.

We know, for sure, that if the maker's tension is accurate, that our "effective density" of the composite wound string works (if nothing else) at that specific length and pitch in Arto's.

At this step I just do the same thing you do, which is to assume the same equation will apply at different lengths and pitches, within reason of course, i.e. over a pretty narrow region. It's doubtful if Arto's calculator will give accurate tensions if you stretch the material to near the breaking point, or if it's very slack.

The advantage of this approach is that it requires no measurements, or extrapolations concerned with windings or core, we just use ONE presumably correct figure the maker has provided, and we extrapolate it to (slightly) different lengths and pitches.

The disadvantage is that we must have at least one tension figure from the string-maker, which is not always the case, so knowing if your method gives the expected results on known tension cases is essential.

Make sense?

=================
I would think that by compiling and graphing some effective densities for different maker's wound strings we can interpolate into a chart or nomograph that would give anyone a ball-park effective density for any given gauge of a "garden variety" silvered copper on nylon fluff wound string. Perhaps we can derive a correction for specific thickness of winding wire. If this works out, Arto's calculator might incorporate it.

=================
For instance, the "effective densities" of D'Addario wound strings, that I jotted down on one case I was working, were:

DD 0.024" 4540 kg/m^3
AA 0.029" 4890
FF 0.033" 5765
C 0.040" 6225

These might not be exact, they're from scribbled notes. I might have used slightly different gauges, or changed something, but they're in the ball park. Say on that 0.033" FF course at 600 mm scale, using the 5765 density, we get 3.6 Kg tension.

jdowning - 9-10-2012 at 12:16 PM

If you already know the manufacturer of a wound string and the code number of a string that is of interest then the best approach would be to simply ask the manufacturer for information about the tension/pitch relationship for that string for a given string length.
Pyramid once provided tension tables to enable customers to determine these parameters for both their plain and wound strings. I posted these tables (with the permission of Pyramid) some years ago on this forum. Can't remember when or the title of the topic but a forum search should be productive for those interested.

The approach of this thread is to determine the tension/pitch relationship of a string (or string fragment even) if the manufacturer is unknown. This cannot be determined by measurement of the outside diameter of a wound string alone. The previously posted, well established formula - although approximate - may be used by those brave souls who might be interested in designing and making their own wound strings.
I am using some known strings by Pyramid simply to be able to verify the results of this calculation method.

The design of wound strings compared to plain cylindrical strings (that obey the Mersenne-Taylor law) is complicated by the geometry - smooth cylindrical core and a 'bumpy' external 'sleeve' as well as the difference in the core and wire densities. So I doubt if your 'equivalent density' approach will prove to be valid - at least for the design of compound strings.
However - with open mind - once I have completed this part of the thread I shall examine and test some of the Pyramid lute wound strings that I have in stock to test if your approach is valid even if only approximately.

Concerning establishment of a possible chart or nomograph to compare wound strings between different makers based upon outside diameter measurements - that might be quite an exercise as the range of wound strings (for lute) from Pyramid alone amounts to over 90 different varieties. However, has this work in comparing wound string equivalents between some makers not already been done by Paul Beier and his string calculator?

fernandraynaud - 9-10-2012 at 12:34 PM

Great! And I in turn eagerly await the results of the "effective thickness" model! Thanks for all your great digging!

fernandraynaud - 9-12-2012 at 09:19 PM

I am actually excited by the measurement I just got on a piece of wound string. You may recall that, in order to estimate tension on a wound string of a given length at a given pitch, I was first deriving an "effective density" for such a wound string based on the tension provided by the manufacturer and plugging it into the equations used for a homogeneous material.

In other words I was considering the wound string to behave rather like a homogeneous string made of a hypothetical material with that density.

Thus for a 0.024" D'Addario silvered copper on nylon fluff core, I was deriving an effective density of 4,540 kg/m^3. I was suggesting that if we weighed a length of such string we might well find that its actual density was close to that.

So I measured the weight of a 6 inch length of 0.024" (0.60mm) Daniel Mari wound string, that being ~ 185 mg. The volume was then calculated as a cylinder's:
(Pi (0.30 mm)^2) (152.4 mm)
and the density worked out to 4309 kg/m^3, compared to the 4540 Kg/m^3 derived effective density on a D'Addario string. YES!!!

That makes sense to me, as a Mari string is a similar wound string, but of a slightly lighter construction than a D'Addario. I'm looking for some other spare string pieces to weigh, but this is more encouraging even than I expected.

To review, Nylon has a density close to 1040 Kg/m^3, PVF around 1800 and brass around 8600 Kg/m^3. It certainly makes sense that a composite wound string would have an intermediate density, and as the string gauge increases, the winding gets thicker, and the density of the whole increases.

It should be possible to use the packing geometry and density of the windings and core the way JDowning suggests to calculate a density to use in the equation, rather than expressing it as an effective thickness of a core material. In the end it seems like a different road to the same approximation.

How are the tension measurements coming?

jdowning - 9-15-2012 at 05:53 AM

The Arto Wikla calculator is for calculating tensions/diameter based upon the Mersenne-Taylor law where frequency of string vibration is a function of string length, string tension and the mass per unit length of a string (assuming the mass is uniformly distributed within a string and the string is a uniform smooth cylinder - neither of which applies to a wound string). Alternatively the frequency of string vibration can be expressed as a function of string length, string diameter, string density and string tension. If an 'effective density' is to be valid then an associated 'equivalent diameter' (which is not the measured outside diameter of a wound string) must also be determined in order to use the Mersenne-Taylor law to calculate string tension for a given string length and frequency of vibration.

For a wound string it is the core that carries the string tension - the purpose of the winding being to add mass without significantly increasing string stiffness in order to lower frequency of vibration. Provided the core can sustain the required string tension without breaking then there are a number of possible wire diameter/core diameter combinations (from thick core thin wire to thin core thick wire) all of which will result in the same measured outside diameter but which will result in different tensions for a given frequency of vibration.

As it is the core that is the structural component of a wound string, string makers design wound strings based upon the 'equivalent diameter' of a cylindrical smooth string of the same uniform density as the core material according to the relationship previously posted or similar. The final string performance will then depend upon a number of manufacturing related factors such as tension of the wire during winding causing stretching and compression of the wire etc.

For information I shall next post how the 'equivalent diameter' formula is derived from first principles by considering the mass per unit string length contributed by the wire and that of the core.

jdowning - 9-15-2012 at 02:53 PM

Attached is the derivation of the 'equivalent' wound string formula that enables string tension to be calculated according to the Mersenne-Taylor law for a uniform smooth cylindrical string of uniform density.

This is the well established basic formula used by string makers (not my idea!) as a preliminary step in the design of close wound strings made of round wire over a uniform density core.
There are other more complicated variants of the formula aimed at greater precision but the objective here is to test the level of accuracy and hence usefulness of this basic formula by comparison with some published string data (Pyramid oud string set #650 Orange label) and with data obtained from load testing strings on a simple test rig.

The most direct information may be obtained from testing a string on the rig as it does not involve any indirect computation.
However, faced with a string fragment of unknown manufacturer - knowing the core and wire diameters and the densities of of the wire and core materials - the string tension for a given tone frequency may be calculated using the
'equivalent diameter' formula. How accurate? That is what this current investigation is all about.

fernandraynaud - 9-15-2012 at 04:52 PM

Yes, of course I understand. But you alone are making that most useful test fixture, and since this is all on the topic of estimating wound string tensions, is it necessary to open a new thread? If you would be so kind as to also test my approach on your test rig with an open mind, I would really appreciate it, as I have no such setup.

Since my proposed method begins at one known point supplied by the string-maker and extrapolates, over a narrow range, using the effective density that fits the known point, I find it quite interesting that the actual measured density of a wound string composite seems to be close to that reverse-derived in that way.

If we accept that the equation is correct at the point supplied by the manufacturer, the question is just how far we can extrapolate, and how well the "effective density" helps in predicting other points on the scale/pitch lines.

I eagerly await what you will find with respect to both methods.

jdowning - 9-15-2012 at 06:05 PM

As I do not understand why you would want to try to guess the tension of a string by a known manufacturer who can supply that information on request I feel that I really cannot usefully contribute to the development of your idea about 'effective density'.

On the other hand - if you feel that a string test rig will enable you to progress with your investigation - a rig like the one that I have created is simple enough and cheap to make yourself. If you (or other interested forum members) cannot figure out how to do it from my posted images and description of my rig I would be more than happy to provide any further detail on request.

Under the circumstances, however, I do think it would be a good idea if you were to develop your proposal under a new topic on the forum.

As you will have to analyse and compare a multitude of wound strings by various makers (Pyramid alone make over 90 varieties of wound lute strings) rather than depend upon just one or two tests to convince yourself and others, one way or another, it could take some time and effort (not to mention the expense of purchasing the strings for testing).

Good luck with your project.

fernandraynaud - 9-16-2012 at 01:27 PM

Gee, thanks. You don't have to share your toys. A few measurements wouldn't take much time, but obviously too much.

But as to the why, isn't it obvious that many strings come with maybe one tension point defined? If we had a method to extend the data to other scale-lengths and tuning, the value seems obvious. And if we could perhaps just weigh a piece of string with no tension data supplied, and derive some tension data from that, isn't the usefulness of that obvious? But OK, ciao.

jdowning - 9-16-2012 at 02:54 PM

Bye bye

jdowning - 9-16-2012 at 04:08 PM

Moving on.

The published data for the Pyramid Orange #650 oud string set - subject of this comparison - can be found in the 2012 Pyramid catalogue and from an 'on line' American source - see attached images.

The 'on line' source gives string gauges in Imperial measurement for Arabic tuning C F A d g c' - string length 60 to 62 cm - 'average' string tension 3.7 Kg. for a total of 40.7 Kg at A440.

The Pyramid catalogue data for Arabic tuning C G A d g c' - string length also 60 - 62 cm - is a bit more useful giving string outside diameters as well as each string tension (which must be an average value given the range of string length) for a total of 45.7 Kg (average 4.15 Kg. at A440)

Comparing the published string diameters:

Converting the Imperial dimensions to the more practical Metric for the online source gives :

1st 0.71 mm (plain nylon)
2nd 0.79 mm (plain nylon)
3rd 0.61 mm (wound copper)
4th 0.74 mm (wound copper)
5th 0.81 mm (wound copper)
6th 1.09 mm (wound copper)

For the Pyramid catalogue data:

1st 0.70 mm (plain nylon)
2nd 0.80 mm (plain nylon)
3rd 0.60 mm (wound copper)
4th 0.73 mm (wound copper)
5th 0.82 mm (wound copper)
6th 1.10 mm (wound copper)

My measurements of a Pyramid #650 Orange string set using a 'Starrett' precision Metric micrometer are as follows:
1st 0.70 mm - 0.72 mm
2nd 0.81 mm - 0.83 mm
3rd 0.63 mm
4th 0.74 mm
5th 0.83 mm
6th 1.10 mm

From these measurements it can be seen that the plain nylon strings are not perfectly cylindrical but are also larger in diameter than the catalogue values - which will in turn result in higher tensions than published.
The other variations in dimensions may be due to manufacturing tolerance, translation losses between Imperial and Metric measure or differences in accuracy between my micrometer and Pyramid's.

Calculating the 'equivalent diameters' of the wound strings according to the previously posted formula, for a plain nylon string in each case:

3rd - d pitch - 1.22 mm
4th - A pitch - 1.56 mm
5th - F pitch - 1.85 mm
G pitch - 1.85 mm
6th - C pitch - 2.67 mm

Using the Arto Wikla strings tension calculator for plain nylon 'equivalent diameter' strings at A440 - string tensions for 60 cm to 63 cm string length range and assuming a density of 1.04 gm/cc for nylon and 8.8 gm/cc for copper are:

3rd - d pitch -3.85 Kg to 4.11 Kg (cf 4.0 Kg avg. Pyramid data)
4th - A pitch - 3.53 Kg to 3.77 Kg (cf 3.8 Kg avg. Pyramid data)
5th - F pitch - 3.13 Kg to 3.34 Kg ( no Pyramid data)
5th - G pitch - 3.94 Kg to 4.21 Kg (cf 4.0 Kg avg. Pyramid data)
6th - C pitch - 3.66 Kg to 3.91 Kg ( cf 3.7 Kg avg. Pyramid data)

So far so good. Next to directly measure string tensions on the test rig for comparison.

jdowning - 9-18-2012 at 09:03 AM

The string test rig has been modified so that it is now set up at 10° from the vertical (rather than 45°) in order to minimise any friction at the lower pulley that would tend to give higher string tension readings than actual.

The following results for my Pyramid #650 set are as follows. As this is a particularly busy time of year for me there has only been time to measure tensions for 61 cm string length - representing the average values published by Pyramid. Both plain nylon and wound strings have been tested:

1st Nylon c' pitch - 4.5 Kg (cf 4.4 Kg Arto Wikla calculator)
2nd Nylon g pitch - 3.3 Kg (cf 3.2 Kg ditto )
3rd Wound d pitch - 4.0 Kg (cf 4.0 Kg from equivalent diameter calculation)
4th Wound A pitch - 3.8 Kg (cf 3.7 Kg ditto )
5th Wound F pitch - 3.3 Kg (cf 3.2 Kg ditto )
5th Wound G pitch - 4.2 Kg (cf 4.1 Kg ditto )
6th Wound C pitch - 3.8 Kg (cf 3.8 Kg ditto )

So it can be seen that the test and calculated values correspond closely. If anything the test results may a little higher by about 0.1 Kg.

So how do the published Pyramid values compare? For the set advertised as C F A d g c' the average value of tension given is 3.7 Kg per string.
Taking the adjusted test tensions (measured values minus 0.1 Kg) the total tension for the 11 string set is 40.5 Kg which works out to 3.7 Kg per string.

This result tends to verify that the first and second nylon strings have an average tension of about 4.4 Kg and 3.2 Kg respectively.

This contrasts with the respective average values of 3.9 Kg and 2.8 Kg published in the 2012 Pyramid catalogue (tuning C G A d g c') for the smaller string diameters of 0.7 mm and 0.8 mm. However using the Pyramid Lute string calculator gives a value from 4.1 Kg to 4.4 Kg and 3.0 Kg to 3.2 Kg respectively for these diameters for string lengths ranging from 60 cm to 62 cm.

The values for plain nylon lute strings from the attached tension tables, published by Pyramid in the 1970's, give intermediate values to the above.

It should be born in mind that the Pyramid #650 is a relatively low cost string set compared to the Pyramid custom design string sets costing about 3 or 4 times more so perhaps these apparent discrepancies in tension are due to wider manufacturing tolerances (particularly for the plain nylon strings) that may may be applicable only to the lower cost strings?

Time permitting some Pyramid wound lute strings that I have in stock will be analysed according to the basic equivalent diameter formula and tension then verified on the string test rig.

Brian Prunka - 9-19-2012 at 11:55 AM

John, thanks for undertaking this very interesting research.

I confess my math skills a a little rusty, and I am confused by something that I am hoping you could help clarify.

D'addario defines their density by "Unit Weight" aka "Specific Weight", while others use "Linear Density". As far as I can tell, the distinction is generally that Unit Weight is by volume, not length. I surmise that in an effectively one-dimensional object like a string, that we regard linear density and unit weight as the same for our purposes? Are there any ramifications to the fact that UW should be in cubic measurements rather than linear measurements?

D'addario describes their Unit Weight numbers as being in "pounds per linear inch", which sounds like they are really describing linear density.

Also can you clarify to me the role of the constant in the formula supplied by D'addario (clearly based on the Mersenne-Taylor Law)?

T(Tension) = (UW x (2 x L x F)2) / 386.4
(Tension in lbs., UW in lbs./in., L in in., F in Hz)

Where is the 386.4 coming from?

I don't wish to derail your thread now that it is back on track, but you clearly have a lot of experience in this area.

Is there any drawback to calculating the effective linear density using the M-T Law, given T, F, and L info from a manufacturer?
I understand that you are undertaking to extrapolate that information in the absence of such data, so not exactly relevant, but other than imprecision or inaccuracies on the part of the mfr., such an approach should work, correct? Or am I missing something?

thanks
Brian

jdowning - 9-20-2012 at 06:01 AM

Hi Brian
I am not familiar with D'addario strings or the procedures, assumptions and definitions that they use in the design and measurement their strings.

First observation from what you say is that a practical instrument string (of say 60 cm or so in length) is not effectively a one- dimensional object at all. It has a measurable diameter and volume and so is a three-dimensional object i.e. a smooth uniform homogeneous cylinder for plain strings.
I am not sure but imagine that the concept of 'linear density' originated in the textile industry where accurate measurement of the diameter of a soft irregular twisted yarn or thread is impossible. Their solution is to 'weigh' a very long length of yarn (1000 to 9000 metres is the standard). For these lengths the yarn may be considered to be approximately one-dimensional and so the mass per unit length is deemed to be equivalent to 'linear density'.
It is perhaps possible that a string manufacturer like D'addario uses this definition for convenient measurement of their plain strings in bulk (i.e. coils of several thousand metres in length)?

For a short instrument string the situation, as I see it, is different in that the mass per unit length is not the same as the density.
Density is an absolute value defined as mass/volume. Therefore, mass per unit length = density per unit length x volume per unit length. The volume = cross section area of a string x length. so for a unit length volume can be expressed as the cross section area = Pi/4 x D²(Pi being the constant = 3.142). So the mass per unit length, in this case = 3.142/4 x D² x 'linear density'.

You mention that D'addario measure their 'linear density' (Unit Weight) for plain strings in Imperial units 'pounds per linear inch'.
and the formula they use for calculating the tension of strings is:
T = (UW x (2x L x F)²/ 386.4. As you observe this expression is non other than the Mersenne - Taylor law for vibrating strings (smooth, uniform homogeneous cylinders). But where does the 386.4 constant come from you ask?

Weight = mass x g (where g is the acceleration due to gravity). The acceleration due to gravity varies depending on geographical location on the earth - but for America and Europe an approximate value of 32.2 ft/sec² in Imperial units is usually considered to be close enough.
D'addario uses Unit Weight as a measure so the unit mass = UW/32.2. However as D'addario also measure lengths in inches (not feet) there being 12 inches to the foot, g becomes 32.2 x 12 = 386.4.

What do you mean by 'effective linear density' applied to the Mersenne - Taylor law which is valid for plain cylindrical uniform homogeneous strings?

The above are my preliminary thoughts. My memory of physics and maths is also pretty rusty so will check out what I have written here for any obvious errors later on when I can find the time - and add any further observations that may come to mind.

Brian Prunka - 9-20-2012 at 07:18 AM

Thanks for the response.

The equation you gave:
mass per unit length = 3.142/4 x D² x 'linear density'

Seems to give me strange results when I try using D'addario's values.

For example,
D'addario gives a UW of .00004751 lb/in for a rectified nylon string .041in in diameter.

(.7855) x (.00161 in2) x (.00004751 lb/in) = .00000006273...lb-in

Shouldn't we get lb/in3 instead of pound-inches? So shouldn't we be dividing by diameter instead?
(.7855)x(1/.00161 in2) x (.00004751 lb/in) = .023lb/in3

.023lb/in3 = 641.6Kg/m3

Trying with another string:

UW = .00003078, diameter .033in rectified nylon:
(.7855)x(1/.001089 in2) x (.00003078) = .022lb/in3

.022lb/in3 = 615.5Kg/m3

Now I would think that two rectified nylon strings would have equivalent densities. These are close, but not the same, and are far less than the 1040Kg/m3 that is suggested for nylon by Arto's calculator. I'm sure I did something wrong here, but I am having trouble figuring out what is going on . . . it's been almost 20 years since physics . . .

D'addario's equation seems to work for its own values but I am having trouble figuring how to relate that to more standard calculations.

jdowning - 9-20-2012 at 04:00 PM

O.K let's try this approach.

D'addario weighs a bulk quantity of rectified nylon string (i.e. string that has been ground perfectly round in section) of known length to determine the weight per inch of the string - this is their Unit Weight value for that particular batch.

Specific Weight is the weight /unit volume this is the 'linear density'

The unit volume of a string (i.e. volume per inch) = Pi x D²/4 = 0.7855 x D² where D is the diameter of the string.

So, for the first example linear density = UW/ 0.7855 D² = 0.00004751/0.00132 = 0.036 lb/cu inch = 1.0 gm/cc or 1000 kg/m³

and likewise for the second example 'linear density' = 0.00003078/0.00086 = 0.036 lb/cu inch = 1000 kg/m³

so the nylon is of the same 'linear density' for both strings.

The slightly low value (compared to 1040 kg/m³) for the nylon may be due to the rectifying process that removes material from the string in order to make it perfectly round in section at a smaller diameter and so affects (compared to the original plain un-rectified nylon) both the weight per inch component as well as the diameter component of the 'linear density' relationship.

Brian Prunka - 9-20-2012 at 04:43 PM

I see, I was mixing up mass per unit length and mass per volume so instead of :

UW (m/L)= volume[(3.14/4)xD2] x density (m/V)

I was doing:

density = volume x UW

Which of course, makes everything wrong.

The calculation
Density = UW/V

then makes sense.

The confusion comes from the definitions I was getting from Wikipedia, that UW is weight per volume and Linear Density is mass per unit length, where you are saying it's the opposite in your equation, I think. Right?

Aside, do you have a guess as to what the units of measurement might be in Savarez's linear density figures? All they say is that the wound strings range in linear density from 70 to 560, but they don't mention the units.

fernandraynaud - 9-20-2012 at 11:29 PM

I get a density of 995 Kg/m^3 for D'Addario's nylon. Their wound silvered copper on nylon fluff strings have a density between 4700 and 6700 kg/m^3.

Brian, if you download the first spreadsheet in my Oud .. Tensions thread, you can copy the calculation in the densities E column to compute any density in kg/m^3 from a diameter in mils or mm (D) and linear density (C) (in lbs/inch), by placing them in the appropriate columns, or changing the cell references in the formula. It's not elegantly expressed, but it works. Just use it in another spreadsheet. The .453.. is the lb->Kg conversion.

=((C12/(((((D12/2)^2)*3.142)*25.4)/1000000))*0.45359237)*1000

jdowning - 9-21-2012 at 05:40 AM

I get a 'linear density' of 0.99627 gm/cc and 0.99068 gm/cc for the two cases respectively which I have rounded up to 1.0 gm/cc for practical simplicity translating to 1000 kg/m³ if preferred. Close enough for all practical purposes (for these two cases at least).

This seems to be a lower value of density for the grade of nylon often used by string manufacturers for instrument strings that might fall within the range of 1.04 to 1.08 gm/cc (or 1040 kg/m³ to 1080 kg/m³ if preferred).

Brian - I have no information about Savarez strings but perhaps fernandraynaud (who is currently investigating the possible application of the 'linear density' concept derived from known data published by some string manufacturers on his 'Ouds ... Tensions' thread) could take the time to look into it and include the findings on his thread?

Unit Weight (or Specific Weight) is defined as the weight per unit volume of a material (mass per unit volume is the definition of Density) - but weight per unit volume is sometimes conveniently referred to as 'density'.
D'addario appears to define UW as the weight per unit length (i.e. it is a weight not a 'density') of a string - and so creates some confusion. To obtain the 'density' of a unit length of string it is necessary then to divide the D'addario UW by the volume of a unit length which is the cross section area of a string calculated from the outside diameter.

Note that if a homogeneous string is not a perfectly round smooth cylinder then a diameter - which is less than the measured outside diameter - would have to be determined in order to calculate the density per unit length.

Brian Prunka - 9-21-2012 at 06:57 AM

Thanks. I guess to be accurate, it is D'addario who has confused the terms in referring to their measurement as unit weight.

jdowning - 9-21-2012 at 11:44 AM

Well I find the terminology confusing but of interest - so thanks for raising the question Brian.

Just to recap, the purpose of this part of the thread is to investigate the mathematical model used to calculate the design of close wound strings from first principles and to verify how accurate (or otherwise) the formula may be by comparison with published data and measurement of some actual strings made by a known manufacturer (Pyramid).
This information might prove to be of general interest to those curious about the physics of wound strings or useful to those who might even be interested in making wound strings (at least as a starting point!). Working in reverse the application may also be of value in assessing and estimating the tension/pitch characteristics of a wound string or small fragment of string of unknown manufacture (for example see 'Repairing a"Nahhat" Oud' by Yaron Naor on this forum)

At this point, the results show quite a close correlation with the strings currently examined and tested and there will be more to follow as opportunity is taken to examine and test more string samples.

The derivation of the wound string formula previously posted makes some simplifying assumptions in order to arrive at an 'equivalent diameter' string of the same homogeneous density as the core material of the wound string (the component subject to string tension) allowing use of the Mersenne-Taylor law to calculate string tension. The resulting formula is in agreement with the 'equivalent diameter' formula proposed by other researchers.

A more accurate derivation is to follow - avoiding the simplified assumptions of the basic mathematical model (but still incorporating some simplification) - in order to evaluate if the resultant, more complicated, formula is worth the extra computational effort in order to achieve a small increase in accuracy of the final result. A case of diminishing returns perhaps?

jdowning - 9-22-2012 at 04:41 AM

In trying to get a better understanding about the design of wound strings I came across the attached brief article "Overspun Strings" from the pages of FoMRHI published back in January 1978 that I thought might be of general interest. (All of the past articles from FoMRHI, dating from 1975, are now freely available 'on line').
The author, Dr Cary Karp, is Associate Professor of Organology at Uppsala University, Sweden, past curator of the musical instrument collections at the Music Museum in Stockholm as well as having been Director of Internet Strategy for the International Council of Museums and member of various museum related IT development committees and working groups

Dr Karp proposes two formulae for determination of the 'equivalent diameter' used for the design of wound strings. The first is derived from a simplified mathematical model and the second is a more accurate corrected version.
As Dr Karp does not take the reader through the steps to arrive at the formulae, it has been necessary to figure it out from first principles (always a good idea when trying to understand a concept!).
The derivation of the first formula, as I understand it, has been previously posted and is currently being investigated to determine how accurate a design tool it might be.

The first formula assumes that the winding is a series of rings or toroids on the core. The second formula assumes that the winding is wound on spiral fashion as it would be for a real wound string.
However, is the more accurate formula worth all of the extra computational work compared to the original simplified version?
The author briefly discusses the relative accuracy implications as it applies to the practical manufacture of wound strings as do the authors of the commentary attached to the article - by the pioneering historical string makers of 'Northern Renaissance Instruments'.

But how the heck did Dr Karp arrive at formula #2?

To follow - for general information - is my Derivation #2 for the corrected 'equivalent diameter' - which, as it happens, agrees with formula #2 proposed by Dr Karp.

Attachment: wound strings comm 102. reduced pdf.pdf (944kB)
This file has been downloaded 484 times

jdowning - 9-22-2012 at 12:19 PM

Derivation #2 for the equivalent string diameter is attached for information.

This model assumes that a single spiral turn of wrapping wire is cut and the two open ends are squeezed together to form a ring of a slightly larger diameter than the core. This trick allows the modified (slight increase) in the mass of the wire to be substituted in the equation of the first derivation for the mass of the wire component without affecting the two other components of the equation. the increased diameter of the ring (Dm) is the hypotenuse of a right triangle one side of length Dw/2 and the other (Dc + Dw). According to the rule of Pythagoras (that the square on the hypotenuse is equal to the sum of the square of the other two sides):
(Dm)² = (Dw/2)² + (Dc +Dw)²
and so Dm is the square root of the other two components and the length of one turn of wire is 3.142 X Dm.

The cross section area of the wire = 3.142/4 x Dw²

and the volume of one turn = length X cs area

and the mass of one turn of wire = length X cs area X density of wire

all reducing to the corrected 'equivalent diameter' shown in the attached derivation.

This derivation makes the same simplified assumption as the first derivation i.e. that there is no compression of the core or distortion of the wire (that occurs during manufacture of a string). So core diameter is determined from the outside diameter of the string assumed to be = Dc + 2 Dw.

The correction factor in practice is small and so has little measurable significance until the wire diameter approaches that of the core (as it does for the 6th course string of the Pyramid #650 orange label oud string set). Otherwise use of the corrected equivalent diameter equation would generally appear to be an unnecessary complication.

jdowning - 9-23-2012 at 04:29 AM

Note that a string designer when using the equivalent diameter formula must make allowance for the stretching and deformation of the wire and compression of the core (it is necessary for the wire to be wound onto the core under tension so that it remains tight against the core to avoid future 'buzzing' of the windings). So the finished diameter of a wound string will always be slightly less than the calculated value. A string maker must make adjustments (by experimentation) to allow for this.
A further constraint on a string maker is that in practice (in the interests of economy) only a limited range of wire and core diameters will be available from which to make an appropriate selection of core/wire diameter combinations.

The starting point for a string designer of a wound string would be the determination of a minimum core diameter capable of withstanding the anticipated string tension without breaking. The stress on the core (not the windings) is the tension divided by the cross section area of the core (as the wire winding adds mass to a string and does not - or should not - carry a significant proportion of the string tension). The maximum stress that a string material can take before breaking is defined as its maximum tensile strength which is Tension/ A where A = 3.142/4 x (Dc)² or 0.7855 (Dc)² from which a minimum core diameter can be determined.

The above applies less to the reverse situation where the outside diameter and wire diameter of a finished string are measured directly in order to determine string tension for a given pitch (using the equivalent diameter equation) as all reductions in diameter of core and wire are already accounted for. The measured outside diameter of a finished wound string can, therefore, be assumed to be Dc + 2Dw so the resultant calculated tension should be quite close to the measured tension on a test rig as appears to be the case so far from these trial results.

Note also that only close wound strings are under consideration here. Open wound strings have an application on lutes (plucked with soft finger tips) but would likely not be practical on ouds or other instruments sounded with a risha, pick or plectrum.
Derivation #2 previously posted may be modified further for open wound string design calculations by increasing the (Dw/2) component of the right triangle geometry accordingly.

jdowning - 9-24-2012 at 12:17 PM

From a practical point of view, the strength of the wire must also be taken into account by the string maker so that the finer wires (0.1 mm diameter or less) do not break under the tension of being wound on the core during string manufacture.
Pyramid make some of their smaller diameter wound strings with two strands of wire wound together (side by side) on the core - presumably to avoid problems of wire breakage?

jdowning - 9-28-2012 at 11:43 AM

Here is another old article from FoMRHI (Comm 163, October 1978) concerning wound string calculations by Eph Segerman and Djilda Abbott of 'Northern Renaissance Instruments' of Britain.

The authors go into greater mathematical complexity in deriving an equivalent diameter formula that covers both close wound and open wound cases that is independent of the cross section shape of the wire. As the authors are string makers they also provide some hands on commentary about the deformation of core and wire observed during the string manufacturing process.

Although a more rigorous approach than the basic formula that is subject of the investigation reported in this thread - the article might be of some value to those who may be contemplating making their own strings as well as being of general interest to those interested in delving further into the detail of string design theory.

String diameters in the article are in thousands of an inch so I have added the more convenient metric equivalent.

[file]24232[/file]

fernandraynaud - 9-28-2012 at 06:25 PM

What am I missing here ... the core of most wound strings is not a nylon monofilament, but a bundle of nylon fibers, with a lot of air in between the strands, more or less squeezed together. So how can we be talking about the core as having a fixed diameter, with the density of a solid nylon cylinder, and then extending this metaphor to a wound string having a "virtual diameter" of nylon?

jdowning - 9-29-2012 at 06:17 AM

Well there is not 'a lot of air between the strands' - in fact likely none at all - so density is not compromised.

The core is not a fairly loose bundle of fibres with air trapped between but is highly compressed by the winding so much so that the wire stretches and reduces diameter when a wound string is made (about 3 -6 % 0f original diameter according to N.R.I.) and the string (core) stretches in length under the compression load (about 0.5% according to N.R.I.). If this were not the case the winding might slip on the core in use causing 'buzzing' due to a loose winding.
Nylon is a relatively soft material and so will deform under compression loading producing a close packed homogeneous material of uniform density.

Both Nylon floss (and silk filament) used for wound string making are extremely fine with filament diameters that can be 0.015 mm or less in diameter further ensuring close packing of the core filaments. For example a Pyramid wound #906 lute string measures 0.47 mm outside diameter, wire diameter 0.06 mm diameter so the compressed core diameter of the finished string is 0.35 mm diameter. The wire probably started life with a somewhat larger diameter before being wound on to the core and stretched under tension.
I have compared the diameter of the wire and nylon filament, side by side, under a microscope and estimate that the filament is about 1/4 to 1/5 the diameter of the wire.

As previously mentioned in this thread the equivalent diameter formula is just a starting point for the string designer. Factors such as wire stretching and core compression during manufacture being established empirically (for a particular manufacturing procedure) by actually making strings and measuring the result.

fernandraynaud - 10-2-2012 at 06:54 AM

Or weighing a piece of the finished string? ... ;-)

jdowning - 10-2-2012 at 11:24 AM

..... or not.

jdowning - 10-7-2012 at 11:54 AM

In order to further evaluate the accuracy (or otherwise) of the basic 'equivalent diameter' formula, some of the 'Pyramid' wound lute strings that I have in stock were measured, the 'equivalent diameter calculated and then compared with the tension derived from the Mersenne-Taylor law (using the Arto Wikla calculator), from the 'Pyramid' tension tables, the 'Pyramid' slide rule calculator and measured directly from the string test rig.

'Pyramid' make a range of 91 different wound lute strings covering a range of almost 3 octaves for a given string tension - from the thinnest #809Al to the heaviest #3272.
The strings subject to testing range from #906 to #1027 covering the middle range of an octave plus 3 semitones for a given string tension. A total of 12 strings have been tested - all copper wound on nylon filament.

Wire diameters measured ranged from 0.06 mm to 0.27 mm and calculated core diameters from 0.35 mm to 0.39 mm.
Calculated 'equivalent diameters' (for monofilament nylon) ranged from 0.88 mm to 2.08 mm assuming a core density of 1.04 gm/cc (or 1040 kg/m³) and wire density of 8.8 gm/cc (or 8,800 kg/m³).

For information, the most common commercially available wire diameters range from 0.02 mm (metric gauge #2), 0.0254 mm SWG #50 equivalent (Standard Wire Gauge - British Imperial) and 0.0787 mm #40 AWG equivalent (American Wire Gauge). For the string maker there is likely a practical lower limit to wire diameter in order to avoid wire breakage during winding the core.

More to follow

jdowning - 10-8-2012 at 04:40 AM

Here are the first of the 12 test results: All strings are round (silver plated) copper wire on nylon filament core. String length 61 cm. Equivalent diameter calculated for nylon core material derived from measurements of string outside diameter and wire diameter according to the basic formula for close wound strings previously posted.
Note that tensions on the Pyramid calculator are in Newtons so have been converted to Kilograms force (9.8 Newtons = 1 Kg)
Some strings tested are used (but not worn) others are new.

Pyramid #906 at f pitch (175 Hz) - used string
Calculated equivalent diameter = 0.88 mm.
Tension Arto Wickla calculator = 2.93 Kg
Tension Pyramid calculator = 2.96 Kg
Tension Pyramid tables = 3.04 Kg
Tension test rig = 3.09 Kg
Range 2.9 Kg to 3.1 Kg

Pyramid #908 at e pitch (165 Hz) - new string
Equivalent diameter = 1.01mm
Tension A.W. calculator = 3.37 Kg
Tension Pyramid calculator = 3.27 Kg
Tension Pyramid tables = not available
Tension test rig = 3.47Kg
Range 3.3 Kg to 3.5 Kg

Pyramid # 1010 at c pitch (132 Hz) - used string
Equivalent diameter = 1.15 mm
Tension A.W. calculator = 2.80 Kg
Tension Pyramid calculator = 2.96 Kg
Tension Pyramid tables = 2.90 Kg
Tension test rig = 2.92 Kg
Range 2.8 Kg to 3.0 Kg

Pyramid #1011 at B pitch (124 Hz) - used string.
Equivalent diameter = 1.18 mm
Tension A.W. calculator = 2.63 Kg
Tension Pyramid calculator = 2.91 Kg
Tension Pyramid tables = 2.80 Kg
Tension test rig = 2.90 Kg
Range 2.6 Kg to 2.9 Kg

Pyramid #1012 at B pitch (124 Hz) - new string.
Equivalent diameter = 1.27 mm
Tension A.W. calculator = 3.05 Kg
Tension Pyramid calculator = 3.27 Kg
Tension Pyramid tables = 3.04 Kg
Tension test rig = 3.29 Kg
Range 3.0 Kg to 3.3 Kg

Pyramid #1013 at B pitch (124 Hz) - new string
Equivalent diameter = 1.29 mm
Tension A.W. calculator = 3.14 Kg
Tension Pyramid calculator = 3.47 Kg
Tension Pyramid tables = 3.30 Kg
Tension test rig = 3.50 Kg
Range 3.1 Kg to 3.5 Kg

Pyramid # 1016 at G pitch (98 Hz) - new string
Equivalent diameter = 1.49 mm
Tension A.W. calculator = 2.64 Kg
Tension Pyramid calculator = 2.76 Kg
Tension Pyramid tables = 2.77 Kg
Tension test rig = 2.61 Kg
Range 2.6 Kg to 2.8 Kg

More to follow.

jdowning - 10-8-2012 at 11:57 AM

Pyramid #1017 at G pitch (98 Hz) - new string
Equivalent diameter = 1.55 mm
Tension A.W. calculator = 2.86 Kg
Tension Pyramid calculator = 3.01 Kg
Tension Pyramid tables = 2.87 Kg
Tension test rig = 3.03 Kg
Range 2.9 Kg to 3.0 Kg

Pyramid #1020 at F pitch (87 Hz) - new string
Equivalent diameter = 1.71 mm
Tension A.W. calculator = 2.76 Kg
Tension Pyramid calculator = 2.83 Kg
Tension Pyramid tables = 2.79 Kg
Tension test rig = 2.81 Kg
Range 2.8 Kg

Pyramid #1023 at E pitch (82 Hz) - new string
Equivalent diameter = 1.81 mm
Tension A.W. calculator = 2.76 Kg
Tension Pyramid calculator = 3.06 Kg
Tension Pyramid tables = 2.91 Kg
Tension test rig = 2.98 Kg
Range 2.8 Kg to 3.1 Kg

Pyramid #1025 at D pitch (73 Hz) - new string
Equivalent diameter = 1.95 mm
Tension A.W. calculator = 2.54 Kg
Tension Pyramid calculator = 2.60 Kg
Tension Pyramid tables 2.62 Kg
Tension test rig = 2.71 Kg
Range 2.5 Kg to 2.7 Kg

Pyramid #1027 at D pitch (73 Hz) - used string
Equivalent diameter = 2.08 mm
Tension A.W. calculator = 2.89 Kg
Tension Pyramid calculator = 2.96 Kg
Tension Pyramid tables = 2.86 Kg
Tension test rig = 2.83 Kg
Range 2.8 Kg to 3.0 Kg

jdowning - 10-9-2012 at 12:02 PM

Taking a closer look at the results there appears to be quite a reasonable consistency between comparison of tensions from the four sources.
Comparing tension results:

The test rig results compared to the Arto Wickla values calculated from equivalent diameter agree within in a range from 0.03 Kg to 0.36Kg maximum.

The tensions from the Pyramid calculator compared to the A.W. calculated values agree within the range 0.03 Kg to 0.33Kg - quite consistent with the results from the test rig.

Tensions from the Pyramid tension tables compared to the A.W. calculated values agree within the range 0.01 Kg to 0.17 kg

Tensions from the test rig compared to the Pyramid tables (presumably measured on a similar test rig) agree within the range 0.02 Kg to 0.25 Kg.

Comparing tensions between the Pyramid Calculator and the Pyramid tension tables agree within the range 0.02 Kg to 0.23 Kg

So it can be seen that there is even some degree of discrepancy of values between the Pyramid data - probably due to manufacturing tolerances of dimensions and material densities in the string manufacture.

For all practical purposes tensions may be quoted to the first decimal place it being pointless to assume any greater degree of accuracy given the normal variables.
The results of these test confirm that application of the basic equivalent diameter formula can provide a pretty accurate assessment of string tensions based upon accurate diameter measurements of string outside diameter and winding wire (using only a well made and accurate Chinese micrometer costing under $20).
For copper wire wound on nylon filament core - densities within the range of 1.4 gm/cc to 1.8 gm/cc for the nylon filament and 9.8 to 9.9 gm/cc would appear to be acceptable. For gut and silk filament cores assume a density of 1.3 gm/cc.

jdowning - 10-10-2012 at 11:33 AM

Found three low cost unused wound guitar strings from times past so measured the strings to see if they might be be viable for a 61 cm string length oud in Arabic tuning.
The string makes are unknown so there are no convenient manufacturer's tension tables to refer to. The cores are nylon filament with silver plated copper wire winding.

String #1
Outside diameter = 1.07mm
Wire diameter = 0.28 mm
Calculated equivalent diameter = 2.478 mm
Tension for oud 6th course C (65 Hz) tuning - Arto Wikla = 3.3 kg

String #2
Outside diameter = 1.17 mm
Wire diameter = 0.28 mm
Calculated equivalent diameter = 2.645 mm
Tension for oud 6th course C (65 Hz) tuning - Arto Wickla = 3.7 Kg

String #3
Outside diameter = 0.89 mm
Wire diameter = 0.019 mm
Calculated equivalent diameter = 1.95 mm
Tension for oud 5th course G pitch (98 Hz) tuning - Arto Wikla =4.5 Kg
Tension for oud 5th course F pitch (87 Hz) tuning - Arto Wickla = 3.6 Kg
Tension for oud 5th course E pitch (82 Hz) tuning - Arto Wickla = 3.2 Kg

Verifying string tensions directly using the string test rig gave the following results:
String #1 - 3.3 Kg
String #2 - 3.8 Kg
String #3 - G - 4.8 Kg
F - 3.8 Kg
E - 3.4 Kg

So again these tests confirm that the equivalent diameter calculation gives reasonably accurate results for close wound string tensions - in the absence of any manufacturers string data whatsoever - given only the measured string outside diameter and the winding wire diameter (and knowing the core and wire densities within a reasonable tolerance).

For string fragments too short (a centimeter or so in length) to allow direct measurement of string tension on the rig the calculated results are, of course, just as accurate as for a complete string.

jdowning - 10-13-2012 at 12:18 PM

As 'classical' guitar strings the previously tested wound strings at guitar pitch gave a tension on the test rig (at 61 cm string length) as follows:

String #1 - low E, sixth string - tension = 5.3 Kg

String #2 - low E, sixth string - tension = 6.1 Kg

String #3 - fifth string, A pitch - tension = 6.0 Kg

I am not familiar with classical guitar tensions (they usually are vaguely designated as 'light', 'medium', 'heavy' tension etc) but assume that the above tensions are not out of line.

jdowning - 10-14-2012 at 05:43 AM

Interesting to speculate here (on the basis of the tests on the three guitar strings) that oud players of the late 19th early 20th C may have switched to using European guitar strings (then gut trebles and close wound copper on silk filament - later nylon trebles and copper wound on nylon filament) from the all gut/silk strings of earlier times. This would allow use of an oud design with a more convenient shorter string length of say 60 to 63 cm and enable a lowering of the pitch to the current Arabic tuning of say C E A d g c' (rather than high f' gut tuning) still in general use today. If this is the case then the ouds of this string length and size seen today may be a relatively 'modern' innovation (late 19th C).
Indeed, forum member Jameel reported in the topic 'Oud or lute?' on this forum that kanun player Dr George Sawa once told him that "all old ouds before the turn of the 20th C had this (elongated) shape and that the wider bigger ouds are relatively new."
The large old oud - subject of the discussion on the topic 'Oud or lute?' - is currently owned by Richard Hankey awaiting restoration. The sound board is missing but it may be speculated the dimensions conform to a string length of about 67.5 cm which fits nicely with the estimated size of both early (gut strung) ouds and lutes. The extra string length was necessary to provide adequate performance of the gut basses but in turn limited the highest pitch of the gut or silk trebles to about f' (A440 pitch standard).

As mentioned earlier in this thread little is known about the early gut basses and there is much speculation about how the strings were made so that a lute of the 17th C (10 or 11 courses) could have a range of the open strings of 2 1/2 octaves without adding extra weight to the strings or making them like little ropes when writers of the period makes no mention of the strings being other than just being plainly twisted in construction.
Marin Mersenne in his monumental theoretical work 'L'Harmonie Universelle', Paris, 1636 in book three page 167 writes that "le son des grosses cordes de luth est appercue de l'oreille durant la sixiesme partie ou le tiers d'une minute". As if to confirm performance to future doubting readers of the 21 st century, Mersenne goes on to state that " ... c'est à dire pendant que artere (?) du poux d'un homme sain,& sans emotion bat dix , ou vingt fois".
(.. that the sound of the largest (diameter) strings of a lute is perceived by the ear to last for a sixth or a third (part) of a minute - that is to say while the pulse of a calm healthy man beats ten or twenty times.)

Here Mersenne confirms that the gut bass strings of his day had a performance that no modern made gut string can come close to matching (as well as confirming that a healthy Parisian male had a pulse rate of 60 beats per minute). Indeed, string sustain of the previously tested wound strings (61 cm string length) has been measured to be between 15 seconds and 20 seconds on the string test rig - similar to Mersennes gut basses!

No wonder that there was a general switch from plain gut to wound basses as these strings became more generally available. We have some idea as to where some of the early lute strings in the European centres were made but what about oud strings? I suspect that many of the early gut and silk oud strings were made in the Moorish controlled Iberian Peninsula where silk and wool production were the primary agricultural industry (until destroyed by punitive measures in the late 16th C after the Christian conquests). Persia - an early centre of the silk trade may also have been a candidate.

But what about the wound strings? Were these made in the Middle Eastern regions or just conveniently imported as guitar strings from Europe? If the strings were not imported where were they made and by whom? Is there any information out there about this?

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