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jdowning
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[*] posted on 5-2-2014 at 06:32 AM
The Question of Soundhole Acoustics


It is an established fact that musical instruments like ouds, lutes and guitars with soundholes behave like Helmholtz resonators - an essential feature of instrument design that 'amplifies' the sound of the lower pitched strings due to the fundamental resonance mode of the air in the bowl cavity.

Although this phenomenon has been known and used in ancient times, Herman Helmholtz in the 19th C (before the invention of fancy modern electronic audio analytical equipment) used different sized spherical resonators to analyse the frequency of soundwaves - hence the name Helmholz Resonator. For a smooth, non flexible, spherical resonator of volume V with a long vent of length L of cross sectional area A, the resonant frequency of a Helmholz resonator is given by the speed of sound C divided by 6.28 multiplied by the square root ofthe product of A divided by V x L.

Unfortunately this straightforward equation cannot be used to accurately calculate the air resonance of a guitar, lute or oud as these instruments are clearly not rigid spheres and do not have long vents - the typical soundhole vent being the thickness of the soundboard itself. To compensate, the Helmholz equation is usually modifed by introducing an equivalent length L.equ representing the column of air oscillating through the soundhole where L.equ is equal to the soundboard thickness h plus 1.7 times the soundhole radius.
However, due to the complexities of a musical instrument this equation still overestimates the resonant frequency (i.e. higher than it should be). In particular, the flexibility of an instrument soundbox/bowl results in a lower resonant frequency than that predicted by the above equation.

Knowing little about instrument acoustics, in 2009 on this forum I ran some trials using a classical guitar as a test bed to gain a better understanding of the relationship between the air volume of the guitar body and soundhole diameter as it affects resonant frequency as well as the influence of a rosette pattern compared to an open soundhole. The procedure followed was reported here:

'Old Oud - New Project' starting on page 9.

http://www.mikeouds.com/messageboard/viewthread.php?tid=8488&pa...

The results of the test confirmed that a smaller soundhole diameter lowered the air resonant frequency, that the open soundhole of the test guitar produced maximum amplification of the 5th (or A) string, that the Helmholtz equation overestimated the measured resonant frequency by about 25%. The results also suggested that a rosette did not seem to significantly affect the measured air resonant frequency compared to an open soundhole of the same diameter. In other words for a rosette it is the area A calculated with reference to the outer diameter of the rosette (i.e. as if it were an open soundhole of that diameter) not the open areas of the rosette that might only constitute about a third of the open area in total. This observation - if valid - would help simplify attempts to calculate the air resonance values of ouds and lutes.
Also - interestingly - it was observed that the presence of a rosette increased the sustain of the sound of the resonance compared to an open soundhole by about 100%. This, together with the resonant frequency amplification might have been an important acoustical feature at a time when duller sounding plain gut strings were used for the lower courses of a lute or oud.

More to follow!
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[*] posted on 5-4-2014 at 09:20 AM


My renewed interest in the acoustics of sound holes has come about after reading, a few days ago, the research paper 'Acoustic Function of Sound Hole Design in Musical Instruments' by Hadi Tavakoli Nia submitted as a partial requirement for the degree of M.Sc. in Mechanical Engineering, M.I.T.
A free to download copy of the paper is available on line here :

http://dspace.mit.edu/bitstream/handle/1721.1/61924/707340180.pdf?s...

Of particular interest is the attention given by the author to investigating the influence that complex soundhole designs, found in ouds and lutes, might have in affecting the acoustic response of an instrument. The approach to the subject is both theoretical and experimental. As my knowledge of fluid flow, gas thermodynamics, calculus and all of that good stuff (such as it was) is now a dim and distant memory of 50 years ago, I am unable to follow or usefully comment upon most of the theoretical work presented. Of greater interest are the reported observations of the experimental work which - if I understand correctly - include:

- The complexity of rosette design has no significance on the air resonant frequency of an oud or lute.

- The oscillating air mass through a circular sound hole at resonant frequency is concentrated close to the outer edges of a sound hole.

- The inner part of a sound hole, having little influence in the resonance response, may be blocked in completely without significantly affecting acoustic response.

- For purposes of calculating the air resonance frequency, the total open area of a rosette (about 0.3 to 0.45 of the total rosette area) does not provide the correct frequency whereas the total rosette area will (to a lesser degree of error). In other words - for the purposes of calculation - any influence that a rosette might have can be ignored.

Interestingly the latter observation confirms my conclusions based upon my earlier trials on the guitar test bed reported in the previous post.

The research paper is of necessity a summary of the experimental work so there appears to be a few missing details necessary for my complete understanding of the results. With this in mind I thought that it would be interesting to undertake some experimental work of my own in the hope of discovering a greater understanding of the subject.
Preliminary trials with my low cost experimental apparatus have already provided some promising results so to start the ball rolling I shall next describe my acoustic chamber apparatus and methodology for the information of all those interested.




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[*] posted on 5-7-2014 at 04:47 AM


The resonance chamber that will be used for part of these trials is a scrap welded steel pressure tank from a domestic water system (I never throw anything away that looks as if it might have a useful future purpose!).
The tank has a convenient mounting flange at one end with an outlet measuring 9.9 cm (3.9 inches) in diameter and a small hole at the other end (once used to hold a valve - part of the original system to pressurise the tank). The tank measures about 25 cm (10 inches) in diameter and about 35 cm (14 inches) overall length - almost custom made for the job! The small hole in the tank has been sealed with a bolt and washer and the exact volume of the tank determined by filling with water. The measured volume of water is 17,500 cc.
The tank is secured to a wooden cradle with a hose clamp and mounted in a vice to minimise any movement. Test sound holes made from 3 mm thick fibre board are mounted over the tank outlet flange with strong spring clamps.

After several trials, the air in the resonance chamber was simply set in motion with a pressure pulse generated by tapping the end of the tank with a rubber mallet.The resulting sound wave pattern was then recorded on an H2 Zoom digital recorder placed 10 cm distance from the sound hole and the sound recording subject to a spectrum analysis to determine the fundamental resonant frequency using a computer and the excellent 'Audacity' (free) software.

After a few more tests to determine the optimum position of the recorder in front of the test sound holes. A series of sound holes of different diameters will be subject to test and analysis - the results to be reported next.



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[*] posted on 5-7-2014 at 11:13 AM


The open circular sound hole trial has been undertaken with sound hole diameters of 3.2 cm, 3.9 cm, 7.1 cm, 8.7 cm and 9.9 cm (open resonator vent) giving peak air resonance frequencies (Audacity Spectrum Analysis) of 67 Hz, 75 Hz, 88 Hz, 105 Hz, 114 Hz and 124 Hz respectively. Sound hole areas are 8 cm, 11.6 cm, 20.4 cm, 39.6 cm, 59.4 cm and 77 cm respectively. Ratio of sound hole areas relative to the open resonator vent are 10%, 15%, 27%, 51%, and 77% respectively.
See attached image of the spectrum analysis for the 7.1 cm diameter sound hole. Lots of higher frequency acoustic 'noise' but the fundamental air frequency at 105 Hertz (cycles per second) is clear enough. The spectrum plot is loudness (in decibels) against frequency

For open circular sound holes of short length (thickness of test pieces is 0.3 cm) an end correction must be applied to determine the equivalent length (Lequ) of the slug of air oscillating through the sound hole. The empirical value of the correction factor has been determined by others to be 1.7 x sound hole radius. See here for more detail.

http://www.phys.unsw.edu.au/jw/Helmholtz.html

For comparison, calculating the air resonance frequency using 1.7R as the correction factor added to the test piece thicknesses of 0.3 cm for the above sound hole diameters works out as 66.4 Hz, 73.6 Hz, 85.6 Hz, 102.0 Hz, 113.4 Hz and 121.4 Hz (for the resonator vent) respectively. All in pretty close agreement so confirming the validity of correction factor for this particular experimental set up. The speed of sound in air was corrected for air temperature of 13 C but not for relative humidity.

The attached sheet shows the method of calculating the resonant frequency using the Helmholz formula modified for sound hole end correction. Note that if the thickness of the sound hole is ignored in the end correction (it being small compared to the sound hole radius) then - as the volume of the resonance chamber is constant and if the speed of sound at room temperature is taken as constant then the resonant frequency is proportional to the square root of the sound hole diameter or is equal to 39.2 X squ. rt. of the diameter.
So, ignoring the sound hole thicknes for the above sound hole diameters gives calculated resonant frequencies of 70 Hz, 76.9 Hz, 88.5 Hz, 104.5 Hz 115.7 Hz and 123.6 Hz (for the open resonator vent). It can be seen that these values are even closer in agreement with some of the recorded values above (all within 3 Hz or less.

These tests confirm what is already well known - that for a fixed volume of resonance chamber - reducing the sound hole diameter (and so area) reduces the resonant frequency.

Next to use the rig to measure the resonance frequency of a 'real' historical lute rosette compared to an open sound hole of the same outside diameter.

(Posting revised 09 May 2014)

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[*] posted on 5-9-2014 at 12:04 AM


May I add something?

What your setup resembles to me, is an udu :)

African Udu is an almost "purest" Helmholtz resonator - has chamber and neck with hole. Tapping with open hand on the other hole "pushses" air into chamber forcing it to vibrate.

Some time ago I picked this cheap, about 30cm height jug and drilled an extra hole (but the jug is still usable for flowers) to produce simple udu :) While not really best udu in the world, it allowed me to do some observations:

- it is somehow fragile and has too high sound (compared to professional ones) - probably neck lenght/width not matched to produce good sound,
- covering (reducing it intersection) the neckhole drastically lowers the sound (a fifth max or more), the sound can be adjusted,
- it also works when you cover mid-hole, but there the sound is slightly higher, and can be also "adjusted" by covering by hand,
- the mid-hole must be covered (partially) just from the edge. I mean, if you simply keep something in the mid-hole "goalmouth", it doesn't change the sound (?!). This trick - keeping something in the neck - actually works.

It looks like the like the mid-hole must be covered, as if it is a kind of "elongation" walls of the jug. I think that this observation applies to rosettes of lutes. It is mentioned in that MIT thesis, something jus like the vibrating air travels just near outer edge of soundhole.

I can remember, once one violin luthier said to me, that edges of the f-holes of the violin must be definitely straight (vertically), and have nice, sharp corners (may not be rounded). I think it creates even very small, but present, "necks" of the resonator. It is not mentioned in the thesis. Also - do it apply (and how much) to thin "walls" around lute rosette?

I haven't got a stuff to record even sound of this udu for a now, but would try to find some and demonstrate it a little.

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[*] posted on 5-9-2014 at 05:20 AM


The ancient ceramic 'flute' - the ocarina - works the same way but has more vent holes to adjust the resonant frequencies to enable production of a tune.

The MIT paper does go into a lot of detail concerning violin sound holes and how and why they appear to have developed from circular sound holes to the current narrow 'vertical' f holes. They also derive a more precise formula for calculation of the violin family sound hole (confirmed by experiment) that takes into account the narrow sound hole geometry and includes the resonance edge effects, increased bandwidth and the improved sound radiation due to the narrowness of the sound holes running roughly parallel to the sound board vertical axis that results in increased sound board vibrating area.

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[*] posted on 5-9-2014 at 11:54 AM


The next trial is to test a lute rosette taken from a scrapped soundboard. This is from a copy of a lute by Giovanni Hieber, Venice, circa 1570 that I made some years ago. The rosette complete with bracing has been cut from the soundboard and clamped to the resonance chamber. The rosette measures 8.7 cm in diameter (to the outer edges of the holes or voids in the rosette pattern) and is about 1.5 mm thick across the rosette.

The resonance frequency of the rosette on test is 108 Hz. This compares with the measured resonance frequency of an open test sound hole of 8.7 cm diameter of 3mm thickness of 115 Hz - so is about a semitone lower than the resonant frequency of the open sound hole. Nevertheless pretty close.
This is not quite as near to being the same frequency as the open sound hole as expected but may be explained by the fact that the open area or void area of this rosette pattern is estimated to be only about 17% of the open sound hole area - much more congested than the rosettes under test by the MIT project that range in void area from 31% to 45% of an equivalent open sound hole.
However, this does support the MIT conclusions that confirm that the using the void area of a rosette to calculate the resonant frequency is not valid - the maximum diameter of the rosette must be used and treated as if it were an open sound hole of the same diameter - agreeing with my observations on the guitar test in 2009.

In this case calculating the equivalent frequency for an open sound hole of 8.7 cm diameter 1.5 mm thick gives a resonance frequency of 113.6 Hz - a little closer than the measured experimental value.

It was also observed from the Audacity wave form that the measured sustain of this rosette pattern was about 60% less than that of the open sound hole - contrary to the observations made on the guitar test bed back in 2009 where the sustain of the rosette pattern tested was about twice that of the open sound hole. However, it must be remembered that the geometry of the resonance chamber used for these trials is quite different from that of a guitar body so is likely to provide somewhat different results for resonant frequency under comparative test conditions.

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[*] posted on 5-9-2014 at 04:07 PM


An observation expressed in the MIT paper and confirmed both theoretically and experimentally is that most of the air mass oscillating through a circular sound hole - with or without rosette - is concentrated around the outer edge so that resonance frequency changes less than a semitone with the central area of a sound hole completely blocked by 50% - a steep drop in resonance frequency occurring only when the blocked diameter exceeds 80% of the sound hole diameter.

I presume that the experimental results reported in the MIT paper result from tests of an open sound hole blocked at various inner diameters but am not sure. So two tests are next to be carried out the first with the Hieber rosette and secondly with an open sound hole both blocked with various inner diameters to compare results.

To block the Hieber rosette thin tinplate discs of relative diameter to the rosette outer diameter (d/D) were bolted to the rosette and the resonant frequency measured on the test rig. Disc relative diameters were 0.5, 0.6, 0.7, 0.8 and 0.9.
The measured resonant frequencies were - uncovered rosette 8.7 cm diameter, 108 Hz and 105 Hz, 103 Hz, 101 Hz, and 97 Hz respectively. The frequency for the 0.9 d/D measurement was difficult to be certain about as at this point the open area remaining was small - almost completely blocked. The most prominent lower frequency resonance - albeit small in amplitude - was 71 Hz. The attached plot of frequency f/fo normalised for maximum resonance of 108 Hz against d/D ratios of 0.5, 0.6, 0.7, 0.8, and 0.9 - follows the presentation of this data in the MIT report for direct comparison.

Although the MIT experimental apparatus was more sophisticated, it can be seen that the results obtained here are quite similar and also confirm that frequency drop remains at about a semitone or less with 50% of the rosette area blocked (d/D = 0.7).

One other interesting observation is that the rosette design incorporates a ring that measures about 0.65 d/D - an indication perhaps that the early luthiers knew more about sound holes and air resonance that might be first imagined?
I have also noticed similar central rings in the rosettes of ouds. I must check out the lute rosette patterns that I have on file to find more examples for examination.

Revised 22 July 2015. The Hierber rosette curve does not match the MIT resonance chamber curve exactly because my carving of the rosette pattern is not as fine as on the original so that at d/D of 0.9 there is very little open air space remaining.

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[*] posted on 5-10-2014 at 12:11 AM



I was a bit sceptical that the pronounced peak in the spectrumof your figure is indeed the Helmholtz resonance without further ado.
But I checked the numbers you give and they look plausibly in the right ballpark. Some more extensive data for error analysis would be nice, but otherwise this is really a very nice demonstration you have achieved here!

Reading through the interesting Nia preprint you are focusing on (thanks for the link), it looks as if for the case of a lute the 1/C term of his Eqn (2.3) will always dominate over the the h/A term (since h/R<<1 for rosettes)

So perhaps of the several directions you could now take with the data you have, one avenue would be to drop the h/A contribution, and focus instead on improving the C=2R approximation

My suggestion would be to try to fit an empirical scaling dependence on the void density n,
i.e.
C=f(n)R with f(n)=const x n^exponent


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[*] posted on 5-10-2014 at 05:22 AM


Thanks for your observations and suggestions narciso.

I hope - albeit with this very basic resonance chamber test rig and essentially quick preliminary testing just to get a feel for what is going on - to eventually be able to fine tune the empirical side on the basis of more extensive data if all looks promising enough to warrant the extra effort involved.

The problem with the empirical adjustments to sound hole end corrections is that they likely vary quite a bit on a case by case basis (i.e. from instrument to instrument) making it problematic to develop an accurate formula for luthiers to use in order to refine their instrument design. However, it may turn out that if the resonance predictions for an oud or lute can be limited within say, a couple of semitones accuracy, together with the frequency tolerance (band width) on either side of the peak resonance, this might be perfectly adequate for a luthier to come close enough to predicting - by straightforward calculation - the optimum value of sound hole diameter to bowl air volume.
The MIT results suggest that this approach may well be possible provided the proper value for sound hole area is chosen e.g. based on the maximum diameter of a rosette and not the total area of the holes in a rosette.

The air resonant curve (with the aid of the cursor in the Audacity Spectrum analysis) can be used to determine frequency bandwidth at 3dB less than peak resonance. This is the point of half maximum loudness (on a logarithmic scale) that I understand is the loudness limit that can be determined by the human ear. So the bandwidth (frequency range) measured on either side of the resonance curve at this point gives the frequency tolerance. Nia of MIT uses the minus 3dB in his paper as well as the minus 10dB level (100% reduction in volume?) that he applies to the violin family sound hole resonances - perhaps because violins are naturally much louder instruments than lutes?

I have just completed tests on ring sound holes (also part of the MIT study) and appear to have quite consistent results in comparison. To be reported next on this thread.

There is, of course, a significance difference between resonance values obtained on a Helmholtz type resonator compared to a real instrument like an oud, lute or guitar - requiring comparative studies to be undertaken if some kind of reasonably valid general empirical formula can be developed.
The MIT paper reports on trials to determine the air resonance peak of a 17th C lute but unfortunately did not try to estimate the air volume of the bowl in order to predict how close the calculated value might be.
Later in this thread I plan to estimate the bowl air volume of this lute (as well other lutes that I have to hand) and measure the air resonant frequency of my lutes to try to determine the validity or otherwise of empirical calculation in predicting the resonant frequency.
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[*] posted on 5-10-2014 at 12:09 PM


The open 'ring' sound hole arrangement was tested this morning to see if the MIT results could be more closely replicated without the presence of the rosette (see previous tests).

The test arrangement was an open sound hole made from 3 mm thick fibreboard and 8.7 cm diameter (same as that of the rosette previously tested) with a wooden bar glued across the opening for the mounting of central discs with d/D ratios of 0.5, 0.6, 0.7, 0.8 and 0.9.
The measured resonance frequencies at 17C ambient temperature were 115 Hz (open soundhole) and 113 Hz, 112 Hz, 109 Hz, 105 Hz and 95 Hz respectively. The resonant peaks in the Audacity spectrum analysis were clear throughout.
Plotting the frequency ratios against d/D it can be seen that the results agree quite closely with the MIT data. It will be useful to make a d/D disc = 0.95 to obtain a reading for the region where the resonance frequency drops dramatically dramatically. So quite happy with these results so far.

Lute rosettes were standardised - printed on paper for distribution to the luthier trade in the 16th and 17th C. Luthiers might use the full rosette pattern or a reduced diameter rosette made by only cutting the inner circle of the pattern. On some of the very large lutes, the sound hole area might be increased by cutting extra 'holes' around the outside of the standard rosette pattern.
I have never seen a lute or oud with the centre section filled in completely (covering up to 50% of the sound hole area) which would increase the total area of the sound board and total sound radiation. I wonder if at one time this was the practice among luthiers so that the open inner rings seen in rosette patterns today were at one time completely filled in?

There are a number of examples of this type of lute rosette (with inner rings in the pattern). Attached is an example of a similar oud rosette of this kind.

So luthiers, if you want to add a fancy engraved central solid plate to your rosette just go ahead - it will not significantly affect air frequency response of the instrument provided the d/D ratio is not more than 0.7.

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[*] posted on 5-11-2014 at 12:04 PM


Here are another two 16th C full size lute rosette patterns that I have on file - both with inner circles incorporated into the design.
The earliest rosette (circa 1530) on a lute by Laux Maler measures 86 mm diameter. The smallest inner ring measures 44 mm outside diameter and the largest ring measures 80 mm outside diameter. So if the smallest ring was 'filled in' (solid), d/D ratio is 0.51 and for the large ring d/D is 0.93. These also might be guides to allow a luthier to make smaller diameter 'cut down' rosettes measuring either 80 mm diameter or 45 mm diameter.

The second example by Michielle Harton 1599 measures 92 mm diameter with inner ring outside diameter of 45 mm - d/D being 0.49.
This basic rosette has been enlarged (not shown here) by the addition of small square holes cut around the perimeter of the rosette increasing the diameter to 100 mm modifying the d/D ratio of the inner ring to 0.45.


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[*] posted on 5-13-2014 at 11:56 AM


Part of the MIT study includes testing the bowl air resonance response of a surviving original 11 course lute (converted later to 13 courses) of the late 17th C by Andreas Berr in the Fine Arts Museum, Boston here:

http://www.mfa.org/collections/object/lute-51267

Unfortunately, it was not possible for the experimenter to directly measure the volume of the bowl in order to determine how close the calculated air resonance was to the measured value. The paper therefore only reports the measured resonance frequency (124 Hz) as well as the recorded plot of the normalised sound level against frequency. The rosette diameter of this lute is only 59 mm

I thought that it would be an interesting exercise to estimate bowl volume in order to calculate the air resonance frequency for comparison with the measured value.

The lute in question started life as an 11 course 'French' style lute, similar to the lute depicted in this portrait of the famous French lutenist Charles Mouton. Note the relatively small diameter rosette.

The rosette on the Berr lute is another example of a rosette cut down from the central part of a larger diameter design as shown in the attached example typical of the late 17th C early 18th C period. The cut down rosette measures about 60% of the larger diameter rosette design.
For comparison an image of the Berr rosette is posted on the Boston Museum website.

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[*] posted on 5-14-2014 at 12:05 PM


I do not have a full size drawing of the Berr lute bowl or precise details of the bowl sections. From a full face image (but of rather low resolution) of the lute I was able to draw a reasonably accurate sound board profile. Knowing that the bowl is of slightly 'flattened' cross section, a volume was calculated by dividing the bowl into 1 cm thick slices of semicircular section and adding together the volume of each slice. An adjustment was then made for the flattening by deducting the volume of a wedge shaped section of the bowl (again determined by adding the volume of individual strips together). See attached image.
Making allowances for sound board, rib and neck block thickness, total air volume of the bowl was estimated to be around 8,000 cc

Assuming an air temperature of 15 C (museum conditions?) when the lute was tested gives a speed of sound - corrected for temperature = 340.4 m/sec
Sound hole area at 5.9 mm diameter is 27.3 cm - i.e open sound hole ignoring the rosette.
Assuming a sound hole thickness of 1.5 mm the end correction factor is 1.7R plus thickness and so equals 5.17 cm.

Plugging these numbers in to the established corrected formula for the air resonance frequency gives 139 Hz compared to the measured frequency of 124 Hz. However the frequency response curve included in the MIT paper shows a band width of about 26 Hz at the 50% normalised sound level (i.e. at minus 3dB) so the tolerance on each side of the peak frequency of 124 Hz is about 13 Hz within which range the human ear can not distinguish a change in sound level from the peak value (if I understand things correctly!).
Therefore , the maximum value of the frequency response might be taken as 137 Hz - close enough to the calculated value of 139 Hz. The minimum value would then be about 111 Hz.

At A392 pitch standard (an historically appropriate two semitones below modern concert pitch standard of A440) eflat tone is 139 Hz and B is 110 Hz.

There are a number of possible tunings for this lute in both the 11 course and current 13 course configuration.
For both Baroque 11 and 13 course lutes, a tuning (Dminor) of A' B' C D E F G A d f a d' f' ( at A440 standard pitch) would be appropriate for the string length of 651 mm (gut stringing throughout). So it can be seen that the air resonance frequency would reinforce primarily the pitch of the fifth course and (partially) the sixth course - just as might be expected.

Next to test other instruments that I have for measured compared to calculated air resonance frequency response.



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[*] posted on 5-15-2014 at 02:06 AM



Inspirational stuff! Amazing that you are able to reverse engineer this sort of fundamental technical info using just museum exhibits and paintings!

As regards the volume calculation of the type you have here, rather than doing it longhand, I can recommend from agreeable experience the general purpose maths packages like Mathematica, Matlab etc.
You can use them to fit a cubic spline to profiles like the one you have drawn. then computationally integrating over the volume swept out by revolution is a doddle.

I am a bit unclear on where your bowl profile came from. Is it from the Charles Mouton portrait? Or were you able to photograph the Berr under glass in Boston?

Anyway it certainly looks more or less frontal in the Mouton jpeg you show. So I had a go at matching your plan to the portrait with the help of the Lute design program discussed elsewhere in this forum
(http://www.mikeouds.com/messageboard/viewthread.php?tid=13881).

In the attachments I show my reconstruction superimposed against your Berr bowl profile, along with the 3d visualisation as it looks when superimposed over the portrait.

The program calculates 10,400 cc for the volume of this reconstruction, so a bit more than your estimate; but this probably does not qualitatively change the conclusions you reach.



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[*] posted on 5-15-2014 at 05:38 AM


Thanks narciso for your helpful constructive comments and calcs.

The image that I used to create the profile of the Berr lute is posted in several locations on the Internet but I am not sure of the original source. See attached.

I will take a closer look at the programs that you mention that may save my current long winded 'macro integral calculus' approach to determine lute (or oud) bowl volumes. Note that few if any surviving lute bowls are perfectly semicircular (or elliptical) in cross section as can be seen from my current thread on this forum 'Old Oud compared to Old Lute Geometry'. They are either flatter or deeper than a semicircle which, of course affects the precise volume calculation compared to that calculated if a semicircle section is assumed.
However, for all practical luthier purposes this may not be a critical consideration - the volume contribution being the square root of the reciprocal of air volume for the purposes of calculating resonant air frequency.

It is not possible to tell if the bowl of the Mouton portrait lute is the same size as the Berr lute. It may be proportionally larger with a longer string length.However, if a volume of 10,400 cc is assumed then the calculated value of resonance air frequency is 122 Hz so - as you say this difference in estimated volume (8000cc compared to 10400 cc) should not significantly affect the practical outcome - both being in the same 'ball park' given the bandwidth frequency tolerance. So for a peak frequency of 122 Hz the bandwidth tolerance range would be from about 109 Hz to 135 Hz.

I have a number of full size drawings of original lutes on file so it would be interesting to see what the calculated resonant air frequencies for those might be. It would not, of course, be possible for me to verify the results with measured values on the original instruments but could be compared with the known tuning of the instruments with gut strings.

[file]31431[/file]
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[*] posted on 5-16-2014 at 04:35 AM


My interest in ancient technologies unfortunately also extends to modern computer systems! My two PC's of 2004 vintage with on board graphics and 32bit processors do not work with the Linux Ubuntu version of the Sharpitor Lute CAD software requiring a 64 bit processing capability. I still use Windows XP as an alternative operating system but since support for XP ended last month I only use XP for off line work. The Sharpitor version that works with Windows 32 bit systems also requires on line communication with the Sharpitor site in order to function so that alternative is now not an option. In any case, in my recent communications with Nick Braun - the developer of the Lute CAD program - it turns out that my computer's graphics capabilities are also inadequate for the job (needing 'power of two' dimensional capability such as 256x512, 512x512 etc.) so although I can take advantage of the volume calculation facility introduced in version 1.8 I could not (when last tested in April) create the required image overlay of an original lute sound board profile for analysis.
It should be noted that the Lute CAD program will not precisely give the volume of a lute bowl as most if not all lutes are not semicircular in cross section. The program also does not take account of volume taken up by internal bracing, sound board and rib thickness (although these factors can be estimated and deducted from the initial volume given by the program). Nevertheless - as has just been demonstrated - the volume based on a semicircular section and the outside sound board profile provided by the Lute CAD software should be close enough for all practical purposes in calculating resonance air frequencies and so in determination of the optimum sound hole diameter versus bowl air volume.

The alternative to using the Lute CAD program for me would be to adapt a CAD program for the purpose (I have several of these that will work off line on Windows XP). However, as I do not have the time or willpower required to learn even the basics of these programs, I shall for now stick to my more tedious ancient technology for estimating bowl volume which is probably not a bad thing as the method it is easily understood and requires only basic mathematical/geometrical knowledge to implement, so may be used by most luthiers - which is part of the objective of this topic.

Next to test the resonant air frequency of the classical guitar that was subject of the trials reported on this forum over four years ago (see link previously posted) to determine if the calculated resonant air frequency (based upon the formula that works with my experimental resonance chamber) is valid in predicting the measured frequency.
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[*] posted on 5-16-2014 at 11:42 AM


In Sept 2009 I carried out tests on a guitar and established that the sound hole diameter/air volume resonance coincided with the fifth string tone at A 110 Hz pitch. However, the calculated value for resonance frequency (applying the same formula used here for the resonance chamber trials - modified for end correction) was way off - 136 Hz compared to 110 Hz - nearly four semitones. I thought at the time that this was because the guitar was not a perfect Helmholtz resonator having - among other things - a flexible body that would reduce the 'spring' effect of the trapped air and result in a lower frequency than calculated. I suggested that - according to theory - the sound hole should have measured only 5.7 cm in diameter compared to 8.7 cm but could not otherwise understand what might account for the significant difference between the measured and calculated results. Clue - note that in this case the ratio of d/D is 5.7/8.7 = 0.66. Sound familiar?!

Re-testing the guitar yesterday with all of the strings damped against vibration with a cloth and recording the pressure wave at the sound hole with the recorder held about 10 cm distance while tapping the end of the bridge to initiate resonance, a peak frequency of 103 Hz was measured using Audacity Spectrum analysis.
The calculated value, however, was 135 Hz - again way off by over 4 semitones.

The answer to this discrepancy may have been provided by Nia in his MIT research project in demonstrating that most of the air flow activity at resonance occurs at the perimeter of a circular sound hole and that much of the central area (d/D = 0.7) contributes little and so might be ignored. Tests with my resonance chamber, reported earlier in this thread, support this finding.
Instead of using the entire sound hole area to calculate resonant frequency what happens if only the area out side the d/D = 0.65 'dead zone' is used? The open sound hole area D at 8.7 cm diameter is 59.5 cm and the 'dead zone' area diameter d = 5.7 cm is 25.1 cm.
59.5 minus 25.1 gives an 'active' area at the perimeter of the sound hole measuring 34.4 cm.

If this area 34.4 cm is used to calculate the air resonance frequency instead of 59.5 cm for the open sound hole then the calculated resonant frequency is 103 Hz - spot on!

Why is the formula valid for the test resonance chamber using the full open sound hole area and not for the guitar? Probably because my resonance chamber - made from steel with dome shaped end is much closer to being a true Helmholtz resonator than the guitar.

The other interesting observation is that for the Berr lute previously reported I had estimated that the rosette diameter had been cut down to 0.6 of the diameter of the original pattern used. I also observed earlier that the Hieber model rosette subject to testing on the test rig has an inner ring incorporated in the rosette pattern measuring d/D = 0.6+.

So that is one test on a real instrument so let's next test one of my lutes to see if the results might be reasonably consistent.
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[*] posted on 5-16-2014 at 12:23 PM


I'm attaching a couple of equationsI've written out to help with the bowl volume calculation, taking the flattening into account.

Hope they come in handy!


[file]31435[/file]
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[*] posted on 5-17-2014 at 04:34 AM


I have no plans to use integral calculus as a tool in this investigation but I have no doubt that it would be possible to define the entire bowl geometry using integral calculus - given that the geometry of lute and oud sound board profiles were once created from conjunct arcs of circles (using dividers, straight edge and 'divine' proportions).

In the meantime my estimate of lute bowl volumes will be undertaken working with full size museum drawings of original lutes and calculating volume 'slice by slice'. This approach is likely to produce accurate enough results free of scaling errors inherent in the Sharpitor Lute CAD approach and/or when working from small sized images that may also be subject to optical distortions.

I would be interested to know, for comparison, the estimated volume of the Berr lute bowl that you now arrive at, narcisco, by applying your derived calculus formula to the full face Berr lute image.

Adding to the observations about the possible 'hidden' importance to luthiers of rosettes with inner circles incorporated in the design, similar inner circles can be found in some traditional oud rosettes. For example the attached image of a Nahat (?) rosette from an old oud recently restored by Yaron Naor has such an inner circle. However - unlike the lute rosettes where d/D seems to be between 0.6 and 0.7 - the outside diameter of this oud rosette inner ring measures d/D = 0.44. This oud has a single sound hole but many ouds have a triple soundhole arrangement - one large diameter and two small (unlike any surviving lute that I know of). The MIT paper addresses the question of air resonance with triple sound hole arrangements. I have yet to look into the implications but it is interesting to note that a 'typical' relative size of the small sound hole diameter to large sound hole diameter used in the MIT investigation is d/D = 0.4.
For comparison, measuring the relative sound hole diameters on my old Egyptian oud I find that d/D = 0.42.

Too early to say without more data but could it be that this type of oud rosette design incorporates the proper proportions as a guide for a luthier to create a triple sound hole design as an alternative to a single sound hole - for whatever the reason might be?

As an aside for luthiers - note that Yaron, in rebuilding the damaged rosette, has incorporated a slot in the outer perimeter of the rosette. This useful feature allows the rosette to be installed or removed without need to remove the sound board.
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[*] posted on 5-19-2014 at 05:14 AM


Thanks for posting above the Boston museum frontal image of the Berr lute I asked about
I found another slightly rotated view of the same original instrument on the web which I used to visually tweak the flattening, again using the LuteCAD program.
The revised result calculated by the program for the Berr volume is 10,300 cc

I probably ought to clarify by the way (since I suspect that there was a misunderstnading) that my equations posted above were intended as a basis for computational calculation, not as a purely analytical alternative.
ie., I wasn't implying these equations would lead to significantly different results from the existing LuteCAD approach
just that the Pappus-Guldinus theorem gives a neat way to think about the flattening correction

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[*] posted on 5-21-2014 at 10:14 AM


We are both working from the same low resolution image of the Berr lute narcisco yet our bowl air volume calculations are so far apart - mine about 8000 cc and yours, using the LuteCAD program, about 10,000 cc. I have rechecked my measurements and calculations and arrive at a total volume - excluding the neck block volume but without flattening compensation etc. of about 8,800 cc which reduces to about 8,000 cc with flattening for the air volume of the bowl (and we want air volume not the exterior volume of the bowl.
As I understand it the LuteCAD program calculates the volume by rotating the lute half sound board profile 180 around a central axis to generate the bowl volume - which is the exterior volume of a semicircular bowl without flattening - presumably including the volume of the neck block and part of the neck?. This will overestimate the air volume unless appropriate manual corrections/deductions are made.
When I look at the tabulation of dimensions that you posted for the initial volume calculation using LuteCAD, the bowl width used is 30.2 cm and bowl depth is 14.3 cm whereas the maximum width of the Berr lute is 28.3 cm and depth 13.5 cm - the latter being the dimensions that I used as the starting point for my calculations.
Perhaps this dimensional error might account for the difference in our bowl volume calculations?
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[*] posted on 5-21-2014 at 10:37 AM


Before leaving the Berr lute I thought that it would be interesting to make a preliminary attempt at predicting the original geometry of the sound board - albeit based upon a low resolution image of the lute. The attached image of the proposed geometry - created using dividers and 'finger' units - is largely self explanatory and follows the geometrical proportions found in other surviving lutes that I have examined to date.
Note that the finger unit in this case measured the equivalent of about 17.5 mm which may be the actual index finger width of A. Berr measured just below the finger nail (if so he had more slender fingers than an average male).

Note that the sound hole diameter is a quarter of the sound board width measured at the sound hole centre position. This may be an important proportion in lute design. Arnault de Zwolle in the 15th C gave the same proportion of sound hole diameter to sound board width as 1/3 - found on some surviving lutes. It is unlikely that early luthiers used mathematics to calculate air resonance but may have used rule of thumb methods of which the sound hole to sound board width may have been a part.
The easiest way to accurately determine the air volume of a finished bowl before fitting the sound board is to fill it with seed and measure the dry volume. Interestingly there is a connection with the ancient volume measures and finger units that were based upon equal sided cubes with sides measuring 15, 16, 17 or 18 finger units - all related according to the material being measured - water (16), wheat (17) or barley (18) all being of equivalent weight. Could the old luthiers have used these old measures as part of some empirical method to arrive at the correct sound hole diameter for a bowl volume measured according to these ancient standards? Just a thought!



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[*] posted on 5-22-2014 at 12:15 PM


The first lute to be tested is one that I built 25 years ago a 'copy' of a seven course lute by Giovanni Hieber (circa last quarter 16th C) - from a full size working drawing of the instrument. String length is 60 cm.
My version of the lute has been repaired (after being dropped on a concrete floor!) and the sound board (that later split under dry winter conditions) replaced with a new one with a slightly larger rosette design than on the original (8.9 cm diameter compared to 8.6 cm). There currently exists a small variation in sound board profile compared to the original lute but the estimated bowl air volume based both upon the working drawing and the lute copy are the same at about 10,000 cc. ('slice by slice' method of calculation)

With the lute up to full string tension the strings were damped with felt strips and the air resonance recorded (tapping on the end of the bridge to set things in motion). The Audacity spectrum analysis gave a peak resonance reading of 133 Hz. I am not sure how (or if it is possible) to accurately measure the band width tolerance at half loudness (minus 3dB) of the response curve using Audacity but if a value of 10 Hz either side of peak frequency is assumed to be reasonable, then the tolerance range would be from 123 Hz to 143 Hz.

With modern nylon strings this size of lute could be tuned to G tuning at A440 pitch standard i.e. g' d' a f c G D. However with all gut stringing the original lute would have been tuned a semitone or tone lower in pitch to avoid frequent top string breakage. So the fifth course pitch at A440 standard would be c = 131 Hz and a semitone lower 124 Hz or full tone lower 117 Hz.

Calculating air resonance frequency - assuming that the full sound hole area (diameter 8.9 cm) is used in the calculation - gives an air frequency of 156Hz - 3 semitones too high.
However, if the 0.6 D sound hole 'dead zone' area is deducted in the calculation (to give an 'active' area of 39.9 cm) the calculated resonant frequency is 125 Hz.

Time out more to follow in conclusion.
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[*] posted on 5-22-2014 at 03:40 PM


...... calculating the air resonance frequency from the original lute dimensions - active rosette area = 37.1 cm - gives an air resonance frequency of 123 Hz. Both calculated frequencies are just within the lower bandwidth tolerance of the measured frequency for the fifth course (123 Hz to 143 Hz)

Interestingly - although the measured air resonance frequency of 133 Hz appears to favour the lute being tuned in G tuning (at A440 standard), to my ear, the lute sounds best (somehow less stressed) tuned a full tone below using modern plastic strings rather than gut - i.e. with the fifth course at about 117 Hz.

On the other hand if the diameter of the rosette 'dead zone' is taken to be 0.66D (2/3) the active area of the original lute rosette is then 32.7 cm and the calculated air resonant frequency is 115Hz.

Next to test another of my lutes.

[file]31483[/file]

[file]31485[/file] [file]31487[/file]
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