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# String Instruments

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Hello everyone and everybody!

Some string instruments have parts, notably a fingerboard, commonly made of ebony: some Diospyros species, sometimes a Dalbergia species
Fingerboard , Ebony , Diospyros , Dalbergia on wikipedia

Ebony has drawbacks: the trade and travel of many Diospyros and Dalbergia species is restricted, even as components of an instrument; it takes many years to dry before processing; and it's expensive.

Replacements were proposed, including hard rubber "ebonite", which has only 1/10th the stiffness of ebony and maybe not the resistence to abrasion.

My suggestion is a polymer loaded with short graphite fibres with random orientation. They are hard and stiff (1/2 to full ebony lengthwise Young's modulus, transverse outperforms), some resist abrasion in plain bearings, they slip well and feel soft. Many are readily available, like POM and PEEK.

Turning and milling tend to blunt the cutting tools quickly, faster than metals but slower than aramide fibres do. I suspect sanding roughens their surface, while planing and scraping have better chances.

Worth a try?
Marc Schaefer, aka Enthalpy

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Guarnieri, Stradivarius, Guadagnini... Soloists prefer old instruments from these luthier for the strong sound, shrill timbre that gives "projection", and easy response, while orchestra musicians may prefer a warmer sound.

Many theories have been proposed about these old instruments, the lacquer is less in favour now, the wood is more fashionable
pnas.org , nagyvaryviolins.com , (in French) guillaume-kessler.fr
where the decomposition of hemicellulose is considered, as well as rotting in a lake before processing, and more.

These suggest a loss of material that would make the wood lighter, propagating flexural waves faster, or rather allowing for the same resonant frequencies lighter soundboards with an easier response.

At least lighter tables and backs would be possible with usual luthier wood while keeping the resonant frequencies. Just make them thinner, and stiffen them with more bracing like the guitar has, instead of the sole bass bar on bow instruments
violin construction and guitar bracing on wikipedia
Of course, this needs a complete redesign, at least of the bracing. Maybe one bass bar (at the back too), taller than presently, and several transverse bars, not as tall as the bass bar but linked firmly to it.

A guide would be to compare the resonant frequencies of the table and back with a good instrument and adjust the bracing to match them. Immediate display of the nodes and antinodes would help a lot, since adjusting and replacing a bracing element is fast. For that observation, I believe the rim should better carry some reasonable extra mass, rather than floating free.

Marc Schaefer, aka Enthalpy

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• 2 weeks later...

Many string instuments use Picea abies and Acer pseudoplatanus (Norway spruce and sycamore)
Picea abies and Acer pseudoplatanus on wiki
for which Voichita Bucur and other sources measured the elasticity:

Data spreads much, as expected from a natural material. The experiments need careful interpretation, as for instance the small shear modulus can reduce a beam's flexural stiffness. Elasticity must also depend on the frequency, so static or ultrasonic data may be inaccurate for music instruments. Good news: all the sources I've seen agree on the axes L, R, T.

Attempts are reported from time to time to replace wood by man-made materials: aluminium, graphite fibres, 3D printed polymers... Wood outperforms them by far because it propagates flexural waves faster, as is known to instrument makers and to academic researchers. For a given resonance mode, faster waves enable a bigger soundboard that radiates better. Or at identical wave speed, the soundboard can be lighter to couple better with the strings. Also, when flexural waves are faster than pressure waves in air they pass efficiently to the air; this happens above a frequency reached earlier with wood.

The equation for 1D flexural waves is
(2piF)2*rho*e = k4*E*e3/12
where the wave speed and the soundboard's mass depend on
E*e2/rho and e*rho
for which Picea abies brings EL~10GPa, ER~1.6GPa, rho~400kg/m3.

Compare with aluminium alloy: E=72GPa rho=2740kg/m3. To match spruce's flexural wave speed in the L direction, aluminium would be as thick and 6.5* as heavy, ouch. To match the R direction, aluminium would be 0.4* as thick and 2.6* as heavy, yuk. So what about graphite fibre composites? They can achieve rho=1550kg/m3, EL~150GPa, ER~20GPa: to match spruce's flexural wave speed, graphite would be 0.5* as thick and 2.0* as heavy.

So while graphite fibres may compose some day the radiating body of a good string instrument, they can't mimic a spruce or sycamore soundboard. Sandwich construction may be the path to fast flexural waves. Or the radiating body better uses compression waves somehow.

Marc Schaefer, aka Enthalpy

Edited by Enthalpy
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I suggested in this thread on December 16, 2018 06:30 PM to add bracings to the tables and bottoms of the violin family, to make the plates thinner, lighter, and hopefully lounder and more responsive.

The thinner shell adds resonance modes between the bars, so these must be close enough to eject to modes to high frequencies. A violin can play more or less a B at 3951Hz on the fingerboard (a bit higher by playing so-called harmonics): the lengthwise flexural half-wave in 1.5mm thin spruce is 40mm, and this shall be the distance between the bars. Irregular spacing helps further. For comparison, a guitar table is 1mm thin.

This spreadsheet computes the EI/rho of plain wood (spruce for the top plate) and a thinner shell with bracings. To obtain the resonant frequencies of 2.5mm plain spruce from a 1.5mm shell, 1.1mm thick and 3mm wide spruce bars spaced by 40mm suffice in the R direction (they ar cut from the stiffer L direction), while those stiffening the L direction are 2.4mm thick. The resulting table would weigh 0.70* as much as plain 2.5mm.

The plates' curvature adds stiffness, hopefully in the same amount with thinner shells and bracings.

The bars stiffening the R direction could be cut thicker from the R direction to keep its damping. A CNC milling machine could carve them from thicker wood together with the shell in one part, much like isogrid construction in aluminium.

A few preimpregnated graphite fibres laid on the shells' inner faces might replace the added wood bars, but their damping differs and they seem more difficult to adjust.

The sketched bracings are by no means an optimum nor the only possibility, as guitars show, and they will need adjustments beyond a spreadsheet's possibilities. At best, they may guide the first experiment, to check if the idea has potential. Since violin-like instruments alternate the resonances among the top and bottom plates, both plates should be modified to keep the balance.

Marc Schaefer, aka Enthalpy

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Instruments of the violin family have arched tables and bottoms. This brings stiffness at precise locations with no added mass.

Could bracings replace the arched shape?. This works for the guitar, whose table is flat and 1mm thin, a value that would be difficult to attain by carving arched plates but is usual for flat material using industrial tools.

The higher pitched violin would need bracings stiffer than the guitar, and stiffer than in the previous estimate whose plates are arched. If the spruce table is 1mm thin, a typical distance between bars drops to <30mm, and the bars are taller - to be experimented, for instance with Chladni patterns, possibly after varnishing.

Reinforcements seem necessary at the plates' rim, continuous but possibly of several overlapping glued parts - or shall all bars and the rim but shaped from a single plate? The instruments needs also taller ribs to keep the volume, and a taller bridge to give the bow a way.

Marc Schaefer, aka Enthalpy

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Do a violin's table and bottom radiate efficiently?

I try to estimate the radiated power from the vibration pattern on Martin Schleske's fantastic website:
schleske.de

For instance at the 409Hz resonance, the vibrating zones are ellipses about 100mm*50mm. Lambda/4=209mm, so the 5.6mohm*F2=0.9kohm radiation resistance of a small source isn't too wrong here. Average arbitrary 1m/s rms in one such zone pushes 4dm3/s air to radiate 14mW rms. The zone's mass, about 3.5g, takes 9N and reactive 9W to accelerate. Radiation contributes 0.15% to 1/Q while losses are 1.5% in Schleske's measures.

At 884Hz, the small source resistance would be 4.4kohm but the observed individual zones are about 100mm*60mm while lambda/4=97mm so the resistance would be bigger. Though, the pattern is a quadripole, 75mm=0.19*lambda and 160mm=0.41*lambda, and the 0.19*lambda reduce the individual resistance. I just keep the 4.4kohm per zone, should be good enough for the qualitative conclusion. Again 1m/s lets each zone radiate 98mW and absorb reactive 30W so radiation contributes 0.3%.

At 2060Hz, the zones are about 60mm*45mm but lambda/4=42mm so the small source's 24kohm and 0.1W are less wrong than a wide piston's 0.9W, while the multipole's spacing of 70mm=0.42*lambda changes little. Each zone's 1.9g absorbs reactive 25W so radiation contributes 0.4% while losses are 3% as deduced from the resonance peak width.

Picea abies' (spruce) reported losses are typically 0.8% lengthwise at acoustic frequencies. Luthiers seek exceptional wood and excel at avoiding other losses.

The violin's table and bottom radiate rougly 0.1* the power received from the strings. That's better than expected but it would usefully improve. It's my reason to seek lighter tables and bottoms, with bracings.

A pizzicato sound is much shorter when pressing the string against the fingerboard than with an empty string. This tells that the finger (and fingerboard) absorbs most of the string's power, even before the power has a chance to reach the table and bottom. But an instrument with frets would not be a violin.

Marc Schaefer, aka Enthalpy

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Here on December 30, 2018 05:13 PM I suggested to lay a few graphite fibres on the plates as bracings. I suppose the composite damps less than spruce does, but then a well adjusted mix of aramide and graphite fibres would tune the damping at will.

Aramide fibres are horrible to cut. The mix offers more tuning possibilities than wood, nice.

Marc Schaefer, aka Enthalpy

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Graphite fingerboards exist already for guitars, cited there
MJ6iyWSEzIs on Youtube, at 1:32, no other detail about the exact material nor how it performs
this task is easier at a guitar, where the strings rub on the frets, than at a violin. But at least, experience exists.

==========

Graphite top and bottom plates were tried on violins in the mid-70's. The caring commentary is "interesting then, but drifted over time, abandoned". Ah, OK.

Graphite sound plates exist at guitars, example in the same video
MJ6iyWSEzIs on Youtube, comparisons with wood at 3:06 and 3:38, 4:14 and 4:52
and they sound just like one expects. Possibly resonances closer to an other due to the low flexural speed, and stronger as a result of small damping.

While I have no opinion about guitars, I would not play a violin that sounds like this. But sandwich construction and some aramide fibres may improve the instrument.

Pianos with a graphite soundboard exist presently: the Phoenix 232 for instance, hear there
EBvrQlZYS-o on Youtube
while it sounds generally good to my ears, some notes don't, around 0:44  (octave above A=440Hz) and 1:07(two octaves higher). Sandwich and aramide? At least their bracings look like wood.
I don't notice these unnice notes at the more recent Steingraeber-Phoenix E272
rCqgpt3KGEc on Youtube, music begins at 0:07
which sounds nicely to me (and differently from Kawai and Steinway) except for the lowest notes, but that's a matter of taste.

The manufacturer, Phoenix, is a partner of Steingraeber, and has a Youtube channel too
phoenixpianos and their Youtube channel

==========

I wanted to suggest metal agraffes on piano bridges to sustain the highest notes longer, but Phoenix has them already
Phoenix agraffes
and I hear no longer sustain. Saved my time.

Edited by Enthalpy
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The WNG company produces piano actions made of graphite fibers instead of wood
wessellnickelandgross
they claim graphite improves stiffness and mass.

Their argument is simplified. Laying graphite filaments or fabric for each of the 88 piano keys is unaffordable, so instead they inject a thermoplastic loaded with graphite choppers. Very nice idea for mass-production - but wooden parts too are profiled collectively by shaped tools, then just separated by a disk saw. For stiffness, 30% choppers bring only E=18GPa, far behind oriented long fibres.

This table compares the stiffness of parts if dentical mass, for beechwood, thermoplastic with 30% graphite choppers, and electronickel: CFRP has no magic property. The trick, too long for commercial explanations, is that wooden parts are plain while CFRP parts are lattices or tubes as injection allows it. This can outperform wood.

                    E       p      E/p3    E/p2     E/p
=========================================================
Beechwood       |  10G  |  720  |  27   |  19k   |  14M
CFRP            |  18G  | 1200  |  10   |  13k   |  15M
Electronickel   | 209G  | 8900  |   0.3 |   2.6k |  23M
=========================================================
Pa     kg/m3   Plate     Rod    Tube

Piano actions might also be electroformed. Metals outperform injected CFRP at hollow parts and lattices. Ni-Co is a vibration damper, Sn and maybe Mo can alloy Ni. Electroformed parts can be thinner and more accurate than injected ones, Ni and Co can be brazed. Time, air, light, humidity, temperature do nothing to Ni and Co parts. Fast production is less clear, but pianos don't sell like candies neither.

Marc Schaefer, aka Enthalpy

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The piano's highest notes are extremely short, (perceived) too low, and just bad. As opposed, the celesta, glockespiel, metallophone and other percussions have decent high notes, with agreable sustain, and are perceived accurately and in tune. Whatever the reason for the bad piano notes, I propose to replace the piano's highest strings with percussion elements to solve both issues. Success would even allow to extend the piano's high range.

The sound may differ from the lower notes' strings, but the highest strings differ already, and they're so bad that an improvement is likely. Keeping hammers and their felt should help mimic the sound of the lower strings.

The controlled dampers become necessary again. A piano doesn't have any at the treble strings because these decay too quickly anyway. A switch or pedal could shorten intentionally the sustain to play pre-2019 music.

Tuning differs from strings. If it's needed, it may result from drops of solder added or filed away. Detuning is expectedly small, as for most alloys Young's modulus drops by 0.2% and the resonance by 0.1% over 5K temperature variation, and Elinvar would drift even less. Strings tend to drift 10* faster, and the percussion elements won't follow the strings. The ideal procedure is hence to leave the percussion elements as they are and tune the strings as needed.

==========

Simple tempered carbon steel is a good material for high percussions. Bell bronze Cu81Sn19Pb0 (uns C19300) and Invar Fe64Ni36 prolong the sustain. Superinvar Fe63Ni32Co5 may be worth a try.

One possible shape is a superimposed pair of bars suspended at the ends.

• They can hold at the frame at one end, at the bridge at the other, be struck from below (in the case of a grand piano) and damped from the top. Small or no modification to the piano. Retrofit?
• The choosable dimensions can resemble the strings, reducing the modifications to the piano.
• The bars are much heavier than the short strings. This matches the hammers better, and whatever the damping process, it helps a longer sustain.
• When only the lower bar vibrates, it transmits much force to the bridge. Soon after, the upper bar vibrates in opposition, and the pair transmits little power to the bridge, prolonging the sustain. Same process as with 2 or 3 strings per note, and this hopefully produces the same sound envelope and perception.
• The symmetry of the bars and the rigidity of their connection to the bridge can be chosen for longer or for stronger sustain.
• Bars suspended at the ends have their partials in tune (like N2), in case someone believes it's useful.

A drawing should follow.
Marc Schaefer, aka Enthalpy

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Here's a sketch of a resonator for the piano's highest notes. The example consists of a superimposed pair of bars suspended at the ends, machined in on part.

Data about bell bronze is scarce and very inconsistent. Taking E=117GPa and rho=8600kg/m3 for the uns C91300, ideal bars with L=40mm and e=4mm would resonate at C=4186Hz, the piano's highest note. This needs adjustments, because the speaking length isn't so simple, and G, nu and the end stiffness mess up. Possible replacement anyway for the strings with 48mm speaking length. The width changes the direct radiation and the vibrating mass: 6mm let two bars weigh 16.5g, versus 0.7g for three D=0.9mm strings, while a usual treble hammerhead may weigh 2g. The bars can be reasonably thicker and proportionally longer at constant frequency.

Where the soundboard and the frame hold the resonator, vertical stiffness lets the element transmit more energy at the note's beginning and leave less in the sustain.

As the resonator is sketched, the ends move horizontally a bit. Design can improve that, or maybe the fastenings can be made to dissipate little.

Partials are aligned like N2 in an ideal case, but G, nu and the end stiffness mess up. If someone believes it matters, thickness adjustments at the proper locations align the partials, even at 1:3 or 1:2.

At >3kHz we're nearly deaf to harmonics and I'm pretty sure that the piano's felted hammers create none. The piano sound must result from the felt hardness, the hammer impact transmitted to the soundboard, the initial strong radiation followed by a slower decay, and a double bar can mimick all that.

Marc Schaefer, aka Enthalpy

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The concert harp has a magnificent sound, rather loud, harpists are agile, and the instrument is irreplaceable
Harp and Pedal harp at wiki
but it badly needs improvement:

• It de-tunes very quickly
• It is fragile but would better be louder
• The concert harp is still a diatonic instrument, yessir, with 7 strings per octave and pedals to raise them from flat to natural to sharp. Little hope for improvement as chromatic harps were introduced and are but abandoned. They can't play the glissandi of existing scores.

==========

New strings de-tune even faster. Harpists claim that the setting of the knot they make at the lower end is an important cause. The knots take also time to make. So here's my suggestion to hold a string without a knot:

(I suspect the bridge is in one part and interrupts the sound plate)

To grip the string, the inner part must rub on it but glide against the outer part, which might consist of

• Materials produced for plain bearing bushes. They contain graphite fibres, graphite powder, Ptfe powder... to glide easily. Among them, polymers are light.
• A material like aluminium alloy covered with a nickel layer that embeds Ptfe powder.

To help match the string's diameter, the inner part can have alternate slits, like machine tools hold some tools. A laser can cut narrow slits. All edges must be smoothened, as some strings are made of gut. Pom is easily machined and its coefficient of friction is higher, while Pa11 and Pa12 accept a big elastic deformation.

One place may need several parts to accommodate varied string diameters. Alternately, several separated subparts may replace the slit one, but they are less caring with the string.

Instruments use to live over a century and play with many musicians, so something is needed to avoid losing the parts, better at the instrument than separately. Some tool may help push the inner part up before stretching the string and later eject the part.

Marc Schaefer, aka Enthalpy

Edited by Enthalpy
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This variant shall hold a string without a knot too.

In one part, it seems easier to use and harder to lose. Maybe it accommodates more varied string diameters, but then it's less caring with the string. The part seems heavier, undesired for the inertia and damping of the soundboard. The part must not spread against the soundboard, so graphite choppers reinforcement may be needed, or some outer ring, possily of graphite fibres. A laser can cut narrow slits.

Marc Schaefer, aka Enthalpy

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Did I see that mammoth ivory makes string supports over bridges, ornamental parts of violin bows, faces of piano keys, and more? That would be the answer to the ban on elephant ivory trade. But:

• To my opinion, mammoths belong in museums and to science, not to sources of raw material, even if we have many mammoth remains.
• Ivory yellows over time and doesn't resist abrasion so well.
• If law changes, musicians will get trouble at the borders. Annoying now with precious wood used under older law.

Ceramic can replace it in many uses. Alumina is white, comfortable to touch, hard, extremely resistant to light and abrasion, lightweight, extremely stiff, and a 30mm*30mm*1mm plate can fall on a tiled floor without breaking. Zirconia is even harder and stiffer. Parts can be sintered to accurate final shape in many cases or shaped by grinding, including with hand-held machine tools.

To support a steel string over a thin wood violin bridge, a ceramic insert, sintered or ground to smooth shape, would outperform ivory.

Marc Schaefer, aka Enthalpy

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At a harp, the strings pull about 10kN directly on the soundboard.

• The soundboard must be thick, which loses sound strength.
• Nevertheless, soundboards sometimes break.
• Could the soundboard's deformations cause most detunings at the harp? Gut strings don't detune so badly at violins, but strongly bent wood may be more sensitive to humidity and temperature than stretched gut.

The other instruments do it differently. The violin family only bends the strings by roughly 1/10th over the soundboard, so the force is much smaller, and only 1/100th of the soundboard's compliance acts on the strings' tension. Strings make just a sidejump at the piano's bridge but exit parallel to their arrival, so no net force results, only a torque. I suggest to copy that.

Maybe the soundboard could be parallel to the plane of the strings, but this seems inconvenient at the treble strings at least, which are very short and played with both hands.

On the sketch, the strings pass through the soundboard, making just a small zigzag there, and they hold at the bottom of the soundbox.

A small tube, of alumina or other hard material, probably notched, can protect the soundboard (including the bridge) from the string fed through and transmit the vibrations. A small tilt suffices, as the maximum deflection of the string suggests. Powder of graphite or MoS2 would linder the sudden tension releases, and piano bridges have a special black material for that purpose.

The instrument's foot may need modifications, possibly more depth. If an added frame holds the strings' bottom, it might pass between the left and right pedals towards the column. A full-loop steel frame gives the steel-stringed piano stable tuning.

I have a feeling (and didn't check as usual) that this idea was already tried and abandoned, possibly because the extra portion of the many strings (which must be precisely parallel) is too stiff and hinders the soundboard's vibrations. Though, this construction works at a piano which has more strings and where the extra portion is short too. I should come back with proposals for fast added compliance in some directions.

Marc Schaefer, aka Enthalpy

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Violins and their relatives get a "purfling" at the rim of their table and bottom
Purfling at Wiki
to protect the table and bottom against cracks. Some sources add "and to ease the vibrations" despite the purfling is glued in the groove.

Maybe preimpregnated graphite filaments could replace the sandwich of precious wood there. They would be stronger and much faster to install, if possible on the wood without a groove, which would ease modifications.

Some clarinet makers reinforce already the ends of the grenadilla joints with graphite filaments instead of metal rings.

The matrix shouldn't absorb vibrations: no polyester, but epoxy maybe, preferably warm hardened. Or can hide glue, ubiquitous at luthiers, impregnate graphite fibres?

Several continuous turns of thinner filament at the table and bottom's rim would leave no weaker zone. I believe it's optically acceptable to run straight where the table and bottom have angles over the glued corner blocks; add black paint if preferred. Or cut the filament there.

Marc Schaefer, aka Enthalpy

==========

In the messages on January 20, 2019, I drew the cones (displayed green) that shall hold a string much bigger for visibility. They would be as small as possible to reduce the inertia and damping. The second variant, with a single part displayed green, seems to need more mass.

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I confirm that strings fastened directly at the bottom of the soundbox are too stiff, hindering the soundboard's vibrations. Good reason not to do it up to now.

The tension of the metallic bass strings bends present soundboards visibly, by >1mm, but 600N on 250mm extra length of a string with D=1.1mm steel core stretch it by 0.8mm only. This vertical stiffness added to the in-plane stiffness of the soundboard would hinder the perpendicular vibrations. The extra string length needs elasticity.

I didn't check if the gut treble strings bring enough elasticity naturally.

==========

This spring example gives elasticity to the string extra length and adds little mobile mass. Meant for an Eb=78.1Hz string, 1.27m spun metal, 770N tuned, 19.6g/m.

3850N tension in 400µm*10.5mm L=80mm cold-rolled steel (possibly split in two to pass the string, or replaced by steel wire(s), D=2.2mm cold-drawn) strain it by 0.35mm, contributing 2*3.5mm at the string. The mass is 4*2.6g, acting as 3.5g at the string. If someone can solder high-carbon steel, fine; if not, the band ends can be clamped together with many screws, or maybe a wedge.

7700N compression in Do=14mm e=1mm L=160mm AA7075 tube strain it by 0.42mm, contributing 4.2mm at the string. The mass is 4.1g, acting as 1.0g at the string.

11mm strain is more than a soundboard's deformation presently. 4.5g mobile mass is less than 40g per string for a present soundboard that needs 20mm wood to stand the traction. With that spring, a lighter soundboard looks possible.

==========

Can the spring be lighter?

• The Lambda shape of strong material is a good start: as in a longbow, it's light and uses at a lower speed an elastic material that can be heavier.
• The Lambda is lighter if less flat. Cascading several Lambda brings the slowing factor.
• If the Lambda don't bring enough elasticity, the added elements (a tube previously) can be nearly immobile to add less inertia to the string.

==========

I had imagined that the soundboard's deformation could cause most detuning, but after seeing experimental data about polyamide and gut strings, it's less clear.

Elasticity created by metal parts could maintain a more constant tension in polyamide and gut strings when these lose stiffness at heat. The tension drop might even be adjusted to compensate the reduced mass of the speaking length. Springs are then useful for medium and treble strings too, even if the extra length is elastic enough, in which case the springs can be heavy.

Marc Schaefer, aka Enthalpy

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Strings can be fastened at the bottom of the soundbox, and the extra length be stiff, if the contact with the harp's soundboard decouples the vertical movements and transmits the horizontal ones.

The left attempt uses rolls. These must move with the slightest force, which isn't trivial and may exclude any sliding guide. It also demands a smooth sleeve around the spun strings.

The right attempt has flexible parts. These must stay horizontal when tuning the string: glide easily, small zigzag.

Notches guide the string transversally in both cases.

Marc Schaefer, aka Enthalpy

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Guitars have an ultra-thin soundboard, around 1mm, made possible because their strings pull parallel to it, not perpendicular. So here's a harp whose strings pull parallel to the soundboard. 20mm thickness resist (often) the perpendicular pull over some 250+250mm width, so for a parallel pull, a 2mm thin soundboard should suffice even at the bass strings: we're getting somewhere. Abundant bracings are necessary to keep decent resonant frequencies.

The strings pull at the upper end of the soundboard, at least at the treble end, so that both hands reach the strings.

The soundbox' shell doesn't touch the soundboard near the strings, but it passes close to the bar there to reduce sound leaks. No access is needed through the shell, so acoustical experiments shall decide if the shell has openings.

I'd have a truss of non-resonating material over the radiating side of the thin large soundboard as a mechanical protection, and to make stronger and stiffer this part of the harp's frame stressed by the strings. Its holes let insert the strings. Closing the cross-section between the strings would help mechanically. A cover, removed to play, would protect the soundboard further.

The soundbox can be narrower than presently, which helps the transport. Consider foldable or removable pedals and feet. Inverting the neck, with the strings at right side, may make the harp even thinner.

Marc Schaefer, aka Enthalpy

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The next step is obviously to design a harp inspired by the piano arrangement: the strings run next to the soundboard parallel to it and make only a zigzag at a bridge to and end at a frame without pulling the soundboard. The arrangement resembles the previous message, so much of the description and the many advantages applies. Better, the soundboard can more easily extend higher than the end of the bass and medium strings to be bigger and louder.

Details should come.
Marc Schaefer, aka Enthalpy

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The arrangement of symphonic orchestras fooled me, but harpists show their front to the public, slightly from the right side. So, the soundboard pulled at its edge by the strings, as depicted here on January 27, 2019 07:39 PM, shall better radiate to the left and have its soundbox at the right.

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Here's a glipmse at a harp with unstressed soundboard inspired by the pianoforte, whose strings make only a zigzag at a bridge and continue to a frame.

From descriptions of playing techniques, the musician must access the strings near the bridge too
composingforharp.com
but only the left hand reaches the lowest 10th, and the right hand stops even earlier near the soundboard, so I believe the soundboard can extend over the bridge at the right side of the two bass octaves; where it extends much, its rim can be attached. The soundboard can also extend deeper below the bridge in the medium, as depicted. Could the bridge be curved to run higher in the medium? This would put the strings' middle on a more regular curve, possibly ease playing, but change the musician's habits.

The unstressed soundboard is much thinner than at present harps. It can be fastened at a position different from the strings. Abundant bracings, plus the bridge and possible bass bar(s), keep the resonances high enough. The bridge is a copy from a piano with the gliding layer and precise nails and wedges, reduced at the bass and much more at the treble.

The bass strings too must extend below the bridge for compliance, hence between the left and right pedals. Opportunity to extend the soundboard too.

A truss can improve the protection of the thin soundboard, protect the musician's legs against the nails, and close the traction-resisting section. Besides wood, fibre composites may perhaps constitute the soundbox' shell and the truss, or even cast metal, especially magnesium alloy, thin and with ribs - not only for this instrument.

The soundbox being as narrow as the neck and pillar, folded or disassembled feet and pedals would make an instrument thin and easier to transport.

Marc Schaefer, aka Enthalpy

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The chromatic harp is nearly extinct: it isn't taught for decades, the last instruments were built three-quarter century ago, most remaining ones are in a bad state.

As one drawback, it is less loud than the concert harp, for which some simple reasons exist. The added strings add mass, stiffness, and they pull stronger at the soundboard. The strings pull excentred at the soundboard, which may or not need extra thickness, but for sure the soundboard is harder to move from an excentric point.

My two arrangements solve that, with strings pulling in line with the sounboard or not at it. They need pairs of soundboards, one for each plane of strings (which I possibly swapped). Then a chromatic instrument would be as loud as the diatonic one which hopefully improves over the present design with flexed soundboard.

The soundboards have some directivity at high notes and harmonics, so tilted reflectors that double as protective covers shall direct all notes in the same direction, essentially forward. The (cover-) reflector may apply to the diatonic versions too.

Marc Schaefer, aka Enthalpy

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Harps have presently big holes in their soundbox that let mount the strings from behind the soundboard. Though, the first role of a soundbox is to enclose the rear wave of the soundboard so it doesn't cancel the front wave. Many people are concerned with these holes, some proposed covers, mobile to access the strings, or even controlled by a pedal, but all this is abandoned. So how critical is the leak?

At high frequencies, the soundbox' volume and capacitance builds little inside pressure, and the holes' inductance is big, so the rear wave doesn't exit, and the holes have no effect. Near the frequency where the holes' inductance resonates with the box' capacitance, the rear wave can be stronger than the front one and produce a stronger net sound, as for a violin's Helmholtz resonance. At lower frequencies, the inductance doesn't stop the rear wave which weakens the front wave - here the holes could be detrimental.

A harp can emit a Cb=30.9Hz but we don't perceive fundamentals so low, only the harmonics above roughly 80Hz, and a soundboard also radiates the low frequencies badly.

As grossly estimated from pictures - an acoustic measurement would be better:

• The soundbox is 1.6m long while lambda/4=1.1m at 80Hz, so it's not far from a lumped constants case: tweaking a bit suffices.
• The soundbox has 0.2m3 = 1.4µF, mostly near the bass strings, so I keep it fully.
• The 5 holes are 0.25m high each but I cumulate only 1m as a lumped constant.
The walls are thin so the inductance results from cylindrical spreading between mean w=120mm to W=500mm:
L = 1.225kg/m3*(2/pi)*Log(0.5m/0.12m)/1m = 1.1H (inside+outside). As much as 0.11m air thickness.
• The resonance is 128Hz very approximately. The loss is 4dB at 80Hz, not much.

It worsens at lower frequencies that we perceive less. But around 128Hz, the resonance by the soundholes amplifies the sound (I didn't estimate the Q factor). So there's nothing damning in these convenient openings.

Can we improve this?

Doors strengthen the radiation at 80Hz but waste the resonance at 128Hz, possible explanation for the mixed opinions when they were tried. Shifting the resonance to 80Hz or slightly lower should be better - maybe.

• Make the soundbox bigger. Under the medium strings, it could be reasonably wider and deeper. Significant contribution, but it won't double the capacitance.
• Make smaller holes? No, harpists need big hands.
• Close every second hole with a door. Their seals shall absorb the vibration, but not too much.
• Put membranes on the holes. A 950kg/m3 material, 140µm thin, would double the inductance of 0.11m thick air. I doubt they can be silent (hear a sudrophone...) and durable too.
• Add chimneys to the holes in the box. If 0.11m long, they double the inductance and shouldn't hamper the hands' activities. They can broaden at depth. They would also strengthen the Helmholtz resonance, for good or bad, more so with rounded edges.
• Close some holes with resonating doors. Keep 3 from 5 effective holes open, the Helmholtz resonance drops to 100Hz. Let a door resonate mechanically at 80Hz, the other at 65Hz. Made flat of Acer pseudoplatanus, they must be around 2.3mm and 2.8mm thick, to be experimented, then adjusted individually at production. The doors can be bigger than 250mm*120mm and thicker.
• The holes are small at celtic harps. Chimeys or resonating doors would keep the resonant frequency and allow bigger holes.

Marc Schaefer, aka Enthalpy

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A sheet of elastomer can be a door for a harp's hole. Very few mm suffice to stop the air by inertia. It's easily silent and airtight. Strings could hold the door (or patch) at round hooks at the soundbox. The patch could even hold in the hole by a firm fit.

I suppose the viscoelastic properties don't matter as the patch moves little. Polyurethanes resist wear best, while perfluoroelastomers are immune to chemical degradation. Of course, the finished patch must be nice.

==========

Strings can also go through the soundboard of a loud chromatic harp as suggested here on January 20, 2019. The springs of January 27, 2019 02:30 PM apply, oriented stiffness as on January 27, 2019 04:32 PM too.

Marc Schaefer, aka Enthalpy

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