# String Instruments

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A 237th check tells me eventally that the volume of the harp's soundbox isn't 0.2m3. It's nearer to 0.03m3, depending on the model, giving it 210nF capacity. This needs updates to my February 03, 2019, 11:33 PM message.

Acoustic measurements of a harp exist there
Le Carrou's thesis (mostly in French)

The measured soundbox has 5 elliptical holes (table 3.1), of which I keep the 3 lowest. I assimilate their inductance to a disk of same area, which acts as a cylinder of length (0.3+0.3)*D:
D131 (7.1H), D120 (7.8H), D111 (8.4H) total 2.6H
to estimate the Helmholtz resonance at 216Hz. Le Carrou attributed it 172Hz rather, after subtle arguments since his fig 3.8 provides no obvious logic, probably because the soundbox isn't short. For instance, the strong resonance that appears at 190Hz with holes closed has lambda/2=0.9m, shorter than the soundbox.

Can soundbox' resonances be brought usefully below 154Hz, the measured lowest soundboard resonance?

I suggest resonating doors tuned to 123Hz, 99Hz, 79Hz, 63Hz. Of Acer pseudoplatanus, they could measure approximately 170mm*60mm*1.3mm, 190mm*70mm*1.4mm, 200mm*90mm*1.2mm, 210mm*100mm*1.1mm - or rather thicker with an adjusted mass in the middle. The last hole would resonate at 50Hz in Helmholz mode with 48H inductance resulting from an 185mm*85mm*0.8mm elastomer membrane.

...Maybe. The resonating doors need a non-absorbing airtight fastening. A harp that radiates low frequencies like a plucked contrabass may sound denatured.

Marc Schaefer, aka Enthalpy

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Le Carrou used already a shallow chimney at one hole to indentify the Helmholtz resonance. How much would tall chimneys at harp holes bring?

I take 100mm height at the holes Le Carrou measured. The narrower soundbox end isn't that deep, but it adds its own inductance.
D131 (16.2H) // D120 (18.6H) // D111 (21.1H) // D89 (30.2H) = 5.1H
which resonates the soundbox' volume at 154Hz, same as the soundboard at this harp model hence useless.

This improves if doors shut some holes. 3mm elastomer are worth 2.2m air. The soundboard's compliance contributes too. With chimneys at the lowest holes (could be elsewhere), 2 holes resonate at 118Hz and 1 at 86Hz.

Arbitrary 1Parms at 118Hz in the box would radiate 1.9ÂµW, conduction would waste 0.1ÂµW and viscosity >0.1ÂµW, elastomer doors contribute, for Q<68. A rosace or narrow F-holes would increase the viscosity losses, as would leaving a single hole open with a shallower chimney.

Fluffy material, as in loudspeakers, can dampen too strong resonances of the long air column in the almost-closed soundbox.

Marc Schaefer, aka Enthalpy

Edited by Enthalpy
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The soundboard of the usual concert harp, 8 to 10mm thin (my mistake) and 580mm wide, can't resist alone the traction of the low strings. The midrib (=bridge at present harps) does it there by holding at the pillar, but this makes the soundboard very stiff under the bass strings. The bass strings also resonate longer than needed, so a more compliant soundboard could be louder.

Imagine that the soundboard flexes by 0 to 10mm under the 15 lowest strings that pull each mean 500N, that's roughly 1.5MN/m, neglecting all angles. Badly stiff.

==========

Tone wood isn't flexible at identical bending resistance. Accordingly, the luthier Camac replaced at least the lower end with an aluminium bar.

Material   Pedantly        Resistance    Young    Merit
---------------------------------------------------------
Spruce     Picea abies          70         12       49
Sycamore   Acer pseudopl.       95         10       93
Beech      Fagus Sylvatica     115         12      103
Yew        Taxus baccata       105          9      120
Aluminum   AA7075              480         72      146
Titanium   Ti-Al6V4            830        114      210
Steel      NiCoMoTi 18-9-5    2000        190      471
---------------------------------------------------------
R MPa      E GPa   R^1.5/E

Steel would give more flexibility than aluminium. This lowers the resonances consequently. Thickness, and optionally profile, that vary with the position, can increase the soundboard's flexibility only at its wide but underused lower end. Or if keeping wood, a wider thinner end of yew (it made longbows and mandolines) should outperform spruce and sycamore.

Additional parts can resist the force and give more flexibility than a straight bar, for instance a transverse bar.

• The soundboard must be thin to accept the deformation.
• The midrib's end can pull the soundboard low until the strings pull it up.
• The position of the midrib's end can be adjustable, at the factory or while the musician tightens the strings.
• I'd have stops at the midrib's end to protect the soundboard.

==========

decouple the soundboard from the column, build it from composite materials to resist the string's pull but be flexible, prestress it, among others.

==========

My two versions of vertical soundboard are far more flexible. Over 180mm for the same 15 lowest strings, spruce 3mm thick and 200mm high contributes only 2kN/m bending stiffness, and 40MPa allow 27mm deflection. If fastened 200mm lower, the 7500N cumulated tension contribute 38kN/m, whether this tension is in the string extra length or in the soundboard.

This oriented compliance lets a string swing slower, but only in the transverse mode. For a string tightened with 770N, this acts like 20mm extra length over 1.27m or 0.8% pitch mismatch, so the beat half-period is 0.8s, shorter than the exponential decay time. Around 5* stiffer, or 200kN/m, would be better in this register: fasten the strings 40mm below the bridge rather than 200mm, or add wood springs at the bridge.

The unstressed design needs abundant bracings for adequate resonances. +-45Â° orientations may protect the soundboard better against in-plane traction by the musician.

The tensile soundboard has a big wave speed parallel to the strings. 14MPa tension and 400kg/m3 give it 190m/s, so a half-wave in 200mm height give a lowest resonance at 470Hz without bracings. Resonances need only bracings perpendicular to the strings. But since this soundboard moves like a flat sheet, its base concentrates the bending stress and may demand some protection.

My two designs seem to have design margins everywhere, including for thicker soundboards. With a radiating area similar to the present harp, but movements about 7.5* bigger, my designs should be 17dB louder, as much as 50 present harps. Could that be a first step towards the gaffophone?

Marc Schaefer, aka Enthalpy

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Estimated bridge stiffness required by my two harp designs with vertical soundboard.

From the previous message, the bass strings should feel about 200kN/m, and if the bridge is to spread the side movements over +-0.1m, R~1MN/m2. The bridge must be stiff enough for that: EI~100N*m2.

Beech (E=12GPa) needs W=e=18mm. If it sounds decently, 1D graphite (170GPa) on wood needs e=1+12+1mm W=7mm, a bit lighter.

Medium and trebles need different dimensions.

==========

At its column end, the bridge could be anchored with elasticity so the lowest strings feel a good stiffness and move the soundboard at the higher strings too.

The unstressed soundboard can hold at its top ridge under the bass strings, and be free at the bottom.

==========

Imagine that the narrow tall soundbox contains 0.03m3=210nF with the unstressed soundboard. The lowest H2 has 62Hz and we don't hear fundamentals lower.

For arbitrary 1Parms in the box, the power radiated by the small source is 0.15ÂµW while conduction to 0.6m2 box wastes 0.03ÂµW, so it's big enough for that.

Air elasticity pushing on equivalent 0.2m2 at the bass bridge portion adds 200kN/m stiffness, the full stiffness goal, so the box could be slightly bigger or the soundboard smaller. If the equipped soundboard brings 150g equivalent inertia and the bass strings too, air elasticity resonates them near 130Hz. Fluffy material in the box can dampen this resonance.

My designs have leaks around the soundboard, say 1mm*1.6m wide and 15mm long. At 62Hz and for 1Parms in the box, inertia limits them to 0.14m/s and 0.2dm3/s compared with radiated 0.08dm3/s. The leak intensity improves with the frequency squared and the box volume, and it's nearly in phase quadrature anyway, resembling more a Helmholtz resonance around 100Hz, combining with the previous 130Hz to make 160Hz.

The leaks waste power by viscosity. For 1Parms at 62Hz hence 0.14m/s it's 16ÂµW. This reduces the strings' decay time. The box volume improves this loss, holding the soundboard where possible too.

Frequency improves all this quickly. At 140Hz, radiation equals viscous losses.

Marc Schaefer, aka Enthalpy

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The wolf tone is a sound instability that can appear on celli and double basses, rarely on violins
de.wikipediaÂ (audio) en.wikipedia - schleske.de/de - schleske.de/en

theories exist, essentially a strong body resonance that couples too much with the string. These theories match some observations but fit others imperfectly.

A string can and does vibrate in any perpendicular direction, plus all the combinations, which includes elliptic modes. If the bridge is stiff, all modes have the same frequency. But if the soundbox resonates strongly, the bridge is more mobile, which lowers the string's frequency, and more so in one direction decided by the soundbox' behaviour. The string modes split in two that have different frequencies and can beat.

The split may be more common at celli and double basses because their bridge is tall and narrow, so body resonances matter more to the string in the transverse direction.

I suggest to inject this mode split in the current theories.

Some experimental checks:

If the wolf tone persists when a single string remains on the instrument, try unusual bowing directions, observe if they have an influence. Will that be convincing?

On a hauling cello, use a capodastro, check by an actuator if the string has split modes and if their frequency difference matches the beat when bowing.

Build a pseudo-instrument with a string but no soundbox, where the bridge is stiff in one direction but flexible in the other, for instance steel wire in V shape, or flat wood aligned with the string, preferably at 45Â° with the bow. Check if the wolf tone appears with the mode split but without any body resonance. Measure both modes, check if the instability's frequency is the difference of them. Pluck the string, compare with the bow.

Marc Schaefer, aka Enthalpy

Edited by Enthalpy
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To check the explanation I proposed for the wolf tone, the experimental setup could look like this. At left a bridge is flexible laterally, at right it's a steel string.

Here at least the vertical modes of the string are harmonic thanks to the boundary conditions and the uniform lineic mass. Horizontal compliance lowers the string's horizontal mode by adjustable 4Hz from E=165Hz. This results from the equivalent of 20mm extra length, that is horizontal 6.0kN/m side stiffness of the imperfect node.

The string's non-speaking length keeps its damping yarn and it can be 75mm to have no common low harmonic with the speaking length (I checked only the vertical modes). The stiffness of these 75mm with 120N string tension leaves horizontal 4.4kN/m obtained from the tweaked bridges. 1mm is the maximum lateral deviation of the non-speaking length of the string at the tweaked bridges.

==========

The flexible wooden part (left on the sketch) uses stiff glue. Mind the wood's orientation. The height of the thin section adjusts the frequency drop of the string's horizontal mode.

At right on the sketch, a violin E string of 0.25mm unspun steel serves as a pseudo-bridge. Some 12.6N tension would resonate the 100mm at 902Hz to avoid common harmonics with the cello string, but more tension may be better, and additional reasonable damping looks useful. The violin string is bent sharp pemanently. The mere tension of the four 100mm sections brings 0.5kN/m horizontal stiffness, and the adjustable 2*14mm width of the Lambda shape 4.0kN/m more. A reasonably sturdy wooden frame, not displayed on the sketch, holds the upper V made of violin string.

==========

If a wold tone appears in this setup with no soundbox resonance, it will favour my explanation.

Many cello strings are ferromagnetic, useful to excite each mode separately. A repetition rate of the wolf tone near the frequency difference between the modes would be a further argument. Both variants of the setup let adjust the frequency difference.

Marc Schaefer, aka Enthalpy

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The other way to do it is add weight Â to the bridgeÂ  so that it the topsÂ resonance differs from the offending note.

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Luis & Clark manufacture graphite fibre cellos and other instruments
luisandclark.com
as do some competitors.

One amazing record by Scott Crowley
the dÃ©tachÃ© starts quickly and neatly, without the noises so common with celli. The musician and the strings matter a lot, but the instrument too.

The timbre is extremely clear. This strikes me less at a cello, which I don't play, as wooden instruments too have quite a clear sound. The timbre is also hollow. Most musicians owning a graphite cello comment "useful under temperature contrasts" or "sturdy and easy, nice for students" but "won't give up my wooden instrument".

Records of (carbon) graphite fibre violins exist too on Youtube, and they sound just like one expects: badly clear, hollow, with very uneven intensity. No, thanks.

From manufacturing videos, the body is just a couple layers of fabric. Then graphite can't compete with wood, as explained here on December 30, 2018. To the very least, it would need a sandwich, for instance with a balsa core, to achieve a decent velocity for flexural waves. Copying a violin's dimensions with an isotropic fabric isn't reasonable neither.

On 2/14/2019 at 3:30 AM, StringJunky said:

The other way to do it is add weight Â to the bridgeÂ  so that it the topsÂ resonance differs from the offending note.

Hi StringJunky, thanks for your interest!

This works. Several ways exist to kill the wolf tone, with varying selectivity. Some instruments exhibit the instability over 3-4 semitones, which prevents tuning the offending frequency between two semitones. Then you have the worry of unusual tunings (for baroque music, or to play with some historic or detuned instrument), of glissando, portamento...

A more selective approach puts an extra mass at the best place on the table. It's also a shift of the offending frequency, but it doesn't affect all the notes.

The more common approach puts a damper on the string, between the bridge and the string holder, where the string isn't supposed to resonate. This one reduces the resonance instead of shifting its frequency. But it acts on all notes.

Martin Schleske claims to taylor a resonator that dampens only the instrument's offending resonance
schleske.de/en
his curves support the claim. This would be the best targeted intervention, working for all tunings and leaving intact the rest of the response.

The setup I propose is more for research than for an actual instrument. It aims to reproduce my claimed mode split without using a resonance, so if a wolf tone is observed, this will favour my explanation. Or disprove it.

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What would the damper be made of and how does it work since that part is not active, is it?

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14 hours ago, StringJunky said:

What would the damper be made of and how does it work since that part is not active, is it?

Usual wolf killers seem to use banal elastomers between the string and a metal mass, so they would dampen all frequencies, their relative effect being best felt at the strongest resonance.

Schleske's damper is allegedly tuned, and his measured response curves support the claim. He doesn't tell on his website how the damper is built despite having sold several, secretive thing. Just elastomer and a mass is conceivable, but for a stable resonant frequency, I'd prefer an all-metal design which looks easy at 100Hz using small parts in flexural mode.

Strings vibrate between the bridge and the holder. They shall not resonate there and get some damping organic wrap from the manufacturer, but they receive movement from the speaking part of the string, over the bridge sitting on the table and the bottom, whose stiffness is limited as they shall vibrate.

Some violin workshops even let musicians pay to remove the damping material from the strings there. This changes the sound, and some customers even believe it improves.

==========

Erratum to the figures in my message of February 14, 2019 03:09 AM.

Was       Now
------------------------
50mm      36mm (drawing)
14mm      22mm (drawing)
------------------------
20mm      10mm
6.0kN/m   12kN/m
4.4kN/m   10.8kN/m
50mm      36mm
14mm      22mm
4.0kN/m   10.3kN/m
------------------------

Â

Edited by Enthalpy
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Musical strings stretch the strongest materials to their limit. The string sound speed sqrt(sigma/rho) is 300 to 500m/s in music instruments, and where a string must be shorter, it is spun with metal wire over a thinner core that is still extremely stressed. Examples:

• Violin E. 662Hz, 325mm, 430m/s. Plain steel, 7850kg/m3 needs 1455MPa tension, and many E strings are overspun with aluminium wire.
Was gut in the past, then estimated 1000kg/m3 needed 185MPa.
• Harp Eb. 625Hz, 287mm, 359m/s. Plain gut, 1320kg/m3 needs 170MPa.
• Harp Gb. 2973Hz, 78mm, 463m/s. Plain polyamide, 1040kg/m3 needs 223MPa.
• Piano C. 4186Hz, 48mm, 402m/s. Plain steel, 7850kg/m3 needs 1268MPa.

Plucking or striking the string increases the stress further, in addition to bends at a knot, bridge or nut.

For strength, polymers are drawn to wires, which stretches the macromolecules. Hardened high-carbon steel is cold-drawn to harden further.
roeslau-draht.de
>1720MPa for D=5mm to >2790MPa for D=0.28mm. I mean, wow.

========== Is different steel possible?

Austenitic stainless steel exceeds 2000MPa by cold-working. Quality Strings alleges it's abandoned because it cracked more easily when flattened
qualitystrings.com
but I experienced the opposite with 2000MPa cold-laminated band: notches kill carbon steel band while the 17-7 alloy can be bent flat with a hammer after short tempering around 180Â°C which improves both the resilience and the proof stress. Tempering uses also to reduce the vibration losses, which I suppose were the real disadvantage. I doubt 17-7 attains 2700MPa but it retains more strength at bents and knots than carbon steel does.

Duplex stainless steel behaves much like austenitic.

Precipitation hardening austenitic stainless steel hardens by aging after cold-working, easing the effort. The PH 15-7 Mo spring alloy is documented to 1800MPa only but mechanical uses probably didn't exaggerate the cold work enough.

Martensitic and ledeburitic stainless steel behaves much like carbon steel. PH 13-8 precipitation-hardens to 1400MPa, so prior cold-drawing may give a good hardness.

Maraging steel is seducing. 18Ni12Co5Mo1Ti bring 2400MPa by aging, even at big diameters, with much resilience worth more than brittle 2800MPa. 50% reduction hardens the 18Ni9Co5Mo1Ti from 1900MPa to 2400MPa for instance
diccism.unipi.it
A violin or a piano afford easily the 50â‚¬/kg. Maraging would not rust, even at finger contact, but it can trigger allergies if bare.

========== Other alloys?

The cobalt alloy CoCr20Ni16Mo7 similar to maraging steel resists corrosion better than needed. It can trigger allergies if bare. Its strengthening by cold work is documented
matthey.ch
1920MPa @60% reduction, 2290MPa @90%, can increase further.

Thicker strings of lighter metal may sometimes be better. A violin E string thicker than 1/4thmm would be more comfortable, it might be less prone to hiss and stick better to the bow. Thicker piano bass strings would carry the heavy copper wire in a single layer, which some manufacturers prefer
pianopricepoint.com

Titanium alloys resist corrosion. Ti6Al4V, Ti6Al6V2Sn, Ti10V2Fe3Al attain by ageing 1100MPa, or the same sound speed as 1950MPa steel, and the same elongation as 2050MPa steel. The equivalent of 2600MPa demands 1470MPa from titanium, hopefully obtained by cold-working. A titanium core of identical mass would be 1.7* stiffer than steel against bending, which has no consequence at a piano bass string.

Exotic aluminium alloys attain 810MPa, for instance the RSA-707 made by RSP by rapid solidification and sintering. Same sound speed as 2200MPa steel. Maybe this one, or more common ones like AA7075 (480MPa), attain by cold-working 950MPa, the equivalent of 2600MPa steel. Cold-rolling brings the AA5456, which would resist finger corrosion, to 432MPa at 60% reduction and 487MPa at 80%, so more is possible.

High-Pressure Torsion brings AA7075 to 1000MPa and the corrosion-resistent AA5083 to 900MPa
nature.com
while High-Pressure Sliding, better suited to wires, brings AA7075 to 700MPa
researchgate.net
they apply to titanium alloys too, but I've seen only the superplastic properties.

Metal matrix composites improve the strength-to mass ratio of metals, but they tend to increase the E modulus too, and I suppose they dampen more.

========== Polymers?

Polymer ropes of aramide, polyester or polyethylene are lighter than steel at identical resistance
cousin-trestec.com
exceeding 1000 or 1500m/s sound speed, equivalent to 18 000 MPa steel, but they sound "poc" when plucked.

I suppose that braiding, impregnation and cover create damping by friction. Just twisting, possibly twice as in a steel rope, must be better. Polyamide musical strings are monofilament (and don't equal gut sound by far). Polyester and polyethylene get strong by fine extrusion, so quite possibly they must stay multi-filament and keep lossy. How would metal-spun Dyneema sound, properly assembled and stretched, no idea.

Ropes thrive to minimise the strain, but musical strings need elastic elongation. That's one simple property where gut outperforms polyamide.

The stiffer para-aramide uses to make ropes and meta-aramide fluffy heat-insulating material, but yarn exists too
teijinaramid.com
Meta-aramide has 1/4th the strength of para-aramide as a fibre. If it retains that factor as a string, it attains 500m/s, and more if twisted rather than braided. So meta-aramide strings can be worth trying.

If needing an impregnation, natural rubber is the elastomer with smallest losses.

Tennis rackets and other sport goods need strings with similar qualities as music instruments. Meta-aramide may improve them.

Marc Schaefer, aka Enthalpy

Â

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The best musical strings are still made from gut, possibly spun with metal wire. Gut is often replaced with PA11 polyamide or with metal, but nothing provides the crispy, profound and long sound of gut, for reasons not fully understood. Strength per mass unit is mandatory, very low mechanical losses too, density and bendability are useful, and I believe elastic strain matters.

"Catgut" is one sheath of the lower part of the intestine of sheep, sometimes goats or cows, after mechanical and chemical processing which I understand leave only the collagen, in fibres oriented essentially lengthwise
gamutmusic.com and web.mit.edu

The upper part of the intestine made sausage casings, but for decades collagen widely replaces it because the process is simpler
wikipedia

Similarly, it would be nice to make musical strings of collagen, where at some process step collagen would be a homogeneous melt or solution, to obtain more easily strings of repeatable properties. The cited Wiki paragraph, brief and not quite clear about it, mentions:
"It is widely used in the form of collagen casings for sausages, which are also used in the manufacture of musical strings."
but I've never heard about a musical string made of collagen, far less a good string, so there must be hurdles.

Yarn from collagen exists already and serves for medicine. Citing subchap 2.4 of:
Â  Â Biomaterials Science: An Introduction to Materials in Medicine
Â  Â By Allan S. Hoffman, Frederick J. Schoen, Jack E. Lemons
"Reconstituted collagen is obtained by enzymatic chemical treatment of skin or tendon followed by reconstitution into fibrils. These fibrils can then be spun into fibres..."

Gut is a raw material long enough for strings, but to spin fibres, tendon seems an interesting alternative. Or continue with gut if the strings are better.

Wiki suggests that the exact spinning method is paramount to stretch and orient the macromolecules and transform weak polyethylene into ultra-strong Dyneema and Spectra
Polyethylene fibre and Gel spinning at Wiki
it seems logical: the lower exit temperature in gel spinning keeps the order acquired by the macromolecules in the spinneret.

Whether this achieves strings as good as gut?
Marc Schaefer, aka Enthalpy

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If a harp's string plane is horizontal and the musician plays from the top, the lower side is available for a big, unstressed, lound soundboard.

A double-action diatonic concert harp could resemble a qanun and other zithers:

14 to 16mm spacing spread the 47 strings over 0.7m, letting small harpists lean forward or backward a bit. Both hands access the full range if needed but risk to collide at high notes. The playing technique is the same as on a vertical harp, excepted the less comfortable hands position, and I expect harpist to adapt swiftly. The equivalents of prÃ¨s de la table, sons xylo, harmonics and others look possible.

The space between the natural and sharp disks can be straight, as on the sketch, for better action designs. Or the sharp disks could be aligned, or the soundboard's edge straight, and more - the strings middle line seems decent on the sketch, but the hands could look more to the right at high strings. Alternately, the action parts could reside below the soundboard, with plenty of room for a better design. A post for the pedals would usefully be removable or foldable, making the thinner instrument easier to transport than the vertical harp.

The construction resembles a grand piano, notably the bridge where the strings can zigzag at two pins to inject no force in the soundboard, though a single downward bend might be good too. Stiff bracings and a closed box would give a deep sound; this can be adjusted to differ less from the vertical harp. An optional sound hole would reinforce low notes down to a cutoff; I'd put it at a side, not at the soundboard, and a rosace or many small holes would tame the resonance. Resonating doors, as described in this discussion on 03, 05, 07, 09 February 2019, apply here too. To be easily strong and stiff as on a piano, the frame can run under the soundboard rather than around it, best through the soundbox, and be of metal, wood, fibres... A removable hard cover, or a complete box, must protect the soundboard and strings.

==========

A chromatic version would be simpler and could have two bridges, but as usual, it can't play the existing harp scores, and low strings must keep the 16mm spacing to avoid collisions, so the range takes an impractical span.

The soundboard could carry several string groups side by side as on the cimbalom. This limits the playing technique. Or the individual range must be reduced and spread over two instruments. Good opportunity to start the low harp at the piano's A. 49 strings would then cover 4 octaves, and the high and low chromatic harps overlap by 1.5 octave.

Marc Schaefer, aka Enthalpy

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In this discussion on December 16, 2018, I suggested graphite-loaded polymers to replace ebony at fingerboards and other parts. Besides better known POM and PEEK, liquid crystal polymers are available as pellets too, and their properties seem excellent for parts of string instruments. A decently documented LCP is Vectra, from Ticona=Celanese:
hipolymers.com.ar

Compositions page 11: A is the most common base resin, E would have had higher losses. 9nn is pure, 2nn contains graphite choppers.

Vibration damping page 22. Pure Vectra A950 has losses ~6% and E~10GPa while Dalbergia melanoxylon has losses ~0.6% and E~20GPa lengthwise, but E drops a lot in the R and T directions. With 30% graphite choppers, A230 offers ~3% losses and E~30GPa, stiffer than ebony. Lossy materials may improve the parts of string instruments that shall not vibrate - maybe, as theere is some debate about the violin's fingerboard.

Can a musician safely hold this material in his hands many hours a day for 50 years? I'm no expert, but at least the unloaded resins A950, B950 and C950 are compliant with FDA regulations for food contact, page 33. I suppose Graphite choppers don't harm.

Far less nice, the price page 11 is somewhere between PEI and PEEK, ouch. Neither did I see rods for sale, only pellets meant for injection, but Hoechst-Celanese did provide rods of A950 to a research team. Shall the luthiers contract a plastic injection company to make rods for subsequent machining, or rather to inject the net instrument shape? Good to know: Vectra gets anisotropic upon injection.

Datasheets of A950 and A230:
Selection - A950 - A230 at tools.celanese.com

The A950 composition shows better Charpy (break by shock) figures than POM. Resilience drops with graphite choppers.

Both compositions absorb very little moisture.

Marc Schaefer, aka Enthalpy

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13 hours ago, Enthalpy said:

Shall the luthiers contract a plastic injection company to make rods for subsequent machiningï»¿?

Rods would be made by extrusion, not injection, so the unusual resin costs no new tool.

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The horizontal harp proposed here on February 24, 2019 could get a frame of cast magnesium alloy. Passing in the soundbox, it looks easy, lightweight and cheap. Magnesium, optionally the same part, perhaps with integral ribs, might compose the bottom and sides of the soundbox too.

Fibre composites too might make the frame and optionally the faces of the soundbox not meant to vibrate.

Of course, I have nothing against other alloys like aluminium or zinc, nor against assembling parts. And absolutely nothing against wood.

Marc Schaefer, aka Enthalpy

Edited by Enthalpy
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The Japanese koto and other string instruments have their soundboard made of Paulownia tomentosa (kiri) instead of spruce
Koto at Wiki - wood-database.com - Paulownia at Wiki

This broad-leaved tree provides lighter wood than conifers, even spruce: 280kg/m3 against 400kg/m3. Its lengthwise E-modulus is 4.4GPa against 10GPa, so the figure-of-merit E/rho3 is 200 against 156 for flexural wave speed at identical mass. Its fracture stress is 38MPa against 70MPa, so the figure-of-merit sigma/rho2 is 485 against 438 for bending resistance at identical mass.

I have no other data: ER, ET, acoustic losses, ease of working... It grows very quickly, at Australian producers too, but the tree is undesired in the US hence its wood rare and expensive there.

Japanese luthiers use it traditionally, so its other properties are known. At the linked wood-database.com, one luthier tells she prefers it over spruce for guitars.

Worth a try at pianos? Bowed instruments? Others? At identical resonant frequencies, it would make soundboards lighter, hence louder and more responsive hopefully.

Marc Schaefer, aka Enthalpy

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• 3 weeks later...

I suggested here on February 27, 2019 to use liquid crystal polymers (LCP), optionally loaded with graphite choppers, for parts of string instruments. This includes bow parts made of LCP, especially the frog.

Whether LCP improves the manufacture or the play over Diospyros and Dalbergia remains to see, but at least it would preserve rare species and help the musicians to cross borders.

Marc Schaefer, aka Enthalpy

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• 2 weeks later...

Can Japanese string instruments replace ivory by sustainable materials? The koto has big bridges and small plectrums, the shamisen a bridge and a big plectrum, of ivory
bbc.com
as do more instruments, outside Japan too. Ivory is difficult to replace, data from Marie AlbÃ©ric' thesis:
tel.archives-ouvertes.fr page 47

            Ivory    POM-CF       LCP    LCP-CF
------------------------------------------------------------
Density      1700      1470      1400      1500    kg/m3
Young          12      10.4       9.1        26    GPa flex
Resistance    320       170       158       228    MPa flex
Damping         ?         ?      0.06      0.03
Friction        ?         ?         ?         ?    On string
------------------------------------------------------------

I suppose, but haven't computed, that bridges don't load the material to the limit. The replacement should mimic ivory's density and stiffness, damping too. Ceramics would be too stiff and traditional polymers, here POM-CF, not enough. Liquid crystal polymer (LCP) becomes too stiff with 30% graphite choppers, so less choppers would adjust Young's modulus. Some heavy filler can increase the density. Or would glass choppers achieve both at once and keep the bulk colour?

At plectrums, the flexibility of ivory seems impossible to imitate by the material alone. 2.7% flexural strain at break are inaccessible to metals, ceramics, nor the polymers listed here. Other fibres may, especially meta-aramide, with high damping then, and they need a matrix with big flexural strain, but has their composite the necessary stiffness and resistance?

Orienting long fibers can help. Stretching LCP during the extrusion or injection too, as this hardens it much.

Adjusting the shape softens plectrums, bridges too if they have shallow stressed parts. 26GPa against 12GPa allow parts equally stiff 1.47* thinner and broader at same volume and slightly more at same mass, not enough. But I hope plectrums can be marginally heavier.

• If the shamisen's plectrum stays wide towards the handle instead of narrowing linearly, unloaded LCP 1.54* thinner and 4.80* wider has the same fexibility and strength as ivory - flexibility alongside the string if needed to soften the sound would be more difficult, with radial grooves.
• If the koto's plectrum isn't curved like a nail but flat near the finger, and wider or possibly a bit thicker, it gets flexible too if needed.

Maybe.
Marc Schaefer, aka Enthalpy

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• 1 month later...

I suggested to replace spruce or sycamore with yew (Taxus baccata), here on
February 10, 2019 12:33 AM,
to increase the flexibility of the midrib of traditional harps while keeping the resistance.

This applies to much of the harp's soundboard too. Wherever the strings' tension makes the soundboard too stiff, yew will be more flexible and louder. It would apply to any instrument whose strings tension limits the soundboard.

How does yew sound in a harp? This must be experimented. Yew was sought after for mandolins, not only for long bows.

The definitive way to build loud harps should be my soundboards parallel to the strings, which don't suffer the same limitations hence shouldn't benefit from yew.

Marc Schaefer, aka Enthalpy

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• 3 weeks later...

Here are some thoughts about the cimbalom.
en.wiki -Â fr.wiki -Â  cimbalombohak.sk - cimbalom.hu

Big warning: I don't play the cimbalom nor any related instrument, so much here is probably b**ocks. If at the end one detail or an other makes sense, fine.

Many instruments are called cimbalom, the name varies also much, and other instruments can be very similar. I consider the grand cimbalom, of Hungarian style, developed by Schunda around 1870.

==========

If I see properly, the two sets of dampers are pushed down directly by long beams, possibly less stiff than needed. Hoping to make settings easier and more stable, I suggest:

• Individual movements for the dampers hold at a fixed beam;
• Individual springs to push each damper against a string course;
• One common action on each side, moved by the pedal, to pull all dampers from the strings;
• Optionally, the contact between the action and the dampers can be adjusted individually.

The dampers for the central portions of treble strings still need some transmission.

I wish the strings would sound for longer with the dampers. Enabling fine adjustments must help.

==========

The medium alternates long string courses with others split in two by a bridge (bridges have voids for the uninterrupted strings). Angles by the bridges put courses higher at one end or the other to help the musician hit the desired note.

If no bridge shortened these longer courses, an identical ratio between the full length, the longer part and the shorter part would be the golden number, (1+sqrt(5))/2 ~ 1.618. At identical sound speed in the strings, the intervals would be 8 or better 9 semitones (minor or major sixth) between full length, long part and short part, so 2*9 courses would span 27 semitones at uniform sound speed.

The bridge loses about 2 semitones. That's still 4 semitones more than presently, with notes arranged more logically and with uniform sound speed. Whether this is advantageous, and enough so to learn a new string chart?

==========

Why 4 strings per note? For a strong attack but longer sustain at medium and treble notes, 2 strings offer eigenmodes with no net force on the bridge and soundboard, and 3 strings suppress the roll moments too. 4 bring no further advantage here, and the piano has only 3.

String inharmonicity improves with finer strings, and then more strings keep some moving mass. Though, I believe inharmonicity has been hugely overstated; it's not even a drawback with reasonable diameters like here.

Maybe 4 strings cost less than 3. They replace a time-consuming knot on the instrument's left by a turn, plus one cheap tuning peg and its hole at the right. Pianos share some wires before and beyond the turn among adjacent notes, but their tension is very nearly the same, as opposed to the cimbalom.

I suppose there is some design flexibility here.

==========

Existing instruments widen very slowly at the low notes. Did Schunda consider his design already bulky and heavy enough? Consequently, the nearer strings are far too short, see the drawing: the farther medium strings are healthy 1.23*C or even 1.35*C long (as compared to the sound speed in air) but the nearer medium strings drop to 1.03*C or 0.83*C, and the nearer bass strings to meager 0.37*C, usually a receipe for bad sound.

Small cimbaloms widen much more strongly at the nearer strings. Would it hamper playing the big instruments? At least, longer near strings would keep a decent sound speed.

The farther medium strings could keep their length and the nearer be 1.3* as long. With the present notes chart, the nearer medium strings would be 1.08*C and 1.32*C long, perfect for plain steel strings and for the transition to spun bass strings. If keeping straight bridges, the lowest bass string would be 1.63* longer, or more decent 0.60*C. Drawing later and maybe.

The instrument then widens from 1.5m to 2.1m. Many cimbaloms still have a wooden frame. I hope a metal frame would stabilize the tuning and let the instrument weigh less than Schunda's 1870 design, 100kg. Later and maybe.

==========

The angles in the strings hamper the movements of the bridges and put also much pressure on the soundboard, which must be sturdy and is even supported by pillars under the bridges. Efficient reasons for the lack of sonority.

Zig-zags at the bridge, like at the piano, would solve all. They are not possible at the highest strings. Elsewhere, they need an instrument higher at the sides, which a metal frame should enable. Later and maybe.

Marc Schaefer, aka Enthalpy

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Here's a sketch of a big cimbalum that widens enough at the near strings to keep a good tension in the medium register, as suggested here yesterday. Its slope resembles much a small cimbalom.

I fully ignore whether a wider instrument is difficult to play. If it helps to play near the centerline, the strings' tilt can be kept using taller bridges.

For nicer and more uniform timbre, sound is here consistently 1.22* to 1.35* as fast in the medium strings as in the air. It remains faster than in air in half of the bass strings, whose plain steel saves money, then decreases in the spun strings to 0.58* at the lowest note.

I'd keep the mass of the strings, that is, thinner if longer. Less stress is also welcome at the bigger frame.

I've kept the distance from the bass bridges to the outer rails. Bringing less stiffness, strings passing the bridges straight might let shorten this distance. It is very short at a piano. Thanks to its metal frame, the piano also extends its soundboard very far under the agraffes and tuning pegs: to be copied if possible.

Marc Schaefer, aka Enthalpy

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On 6/10/2019 at 1:54 AM, Enthalpy said:

Maybe 4 strings cost less than 3. They replace a time-consuming knot on the instrument's left by a turn

And oops. All pictures show one hook and one knot per string. Bad reason.

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Here I propose a simpler notes chart for the cimbalom. The bass strands keep the usual positions up to B=234Hz, the rest differs:

• Semitones progress smoothly but for three jumps.
• At the jumps, the sections overlap by three semitones, similarly to trill keys at woodwinds.
• The sections have a constant interval. At the violin it helps.

I imagine this chart makes the cimbalum easier to learn and play, but again I don't play it. Example of a usual (but incomplete) chart:
beyondkarpaty.mutiny.net
Hammered dulcimers would resemble more, but the intervals and bass strings differ.

A pair of straight dampers can reach C=1051Hz while most Schunda-like models stop three semitones earlier. They are far from the struck positions.

Schunda had achieved a nearly rectangular instrument shape at the cost of complicated notes chart and very inconsistent string sound speed even among consecutive notes. In my chart, the trapezoidal shape keeps the string sound speed between 1.10*C and 1.16*C at the treble and medium, varying very smoothly even at the section jumps, and decreasing gently to 0.54*C at the bass.

Other string lengths would adjust these example figures, say between 1.20* and 1.27*. All bridges and saddles are straight on the drawing, but curves as at the piano could further equalize the string sound speed or limit the instrument's width. The outer bridges leave 10% non-speaking length in the corresponding strings, less than at a piano, but this can increase if accepting a slower propagation, at all these notes or only the lowest ones.

Schunda's design has 39 strands, my chart has 43 with fewer spun strings. I suppose three strings per strand suffice.

Marc Schaefer, aka Enthalpy

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Catalytic nickel protects against corrosion and is excellent against galling. I used some, with embedded Ptfe particles, at about 600MPa pressure and nearly no speed, against a martensitic stainless steel that galls horribly. The friction was tiny and very smooth without the stick-slip felt with zinc or phosphate layers against tempered steel. Embedded particles of MoS2 or graphite may be good too.

Easy and smooth gliding would improve the bridges and saddles under the strings of some instruments.

• The piano uses steel nails to deflect the strings at the bridge. Wood receives already a gliding surface, the nails not, despite the force is bigger on them. Catalytic nickel with Ptfe should stabilize the tuning earlier.
• At least the cimbalum bends the strings over a metal rod at the many bridges. Easier gliding would equalize the tension among the sections between the bridges to improve the intonation. Especially important as the cimbalum has several strings per note.
• Many instruments have metal saddles or pins near the ends: harp, piano, cimbalom... where the deflection can be big. Better gliding would stabilize the tuning here too, just like violinists put graphite on the wood there.

Tuning pegs would better rub smoothly too. They exist of hard wood or metal presently.

At the violin, stick-slip of ebony pegs in the maple pegbox is a pain. But to replace ebony, nickel should rub strongly (no Ptfe), be black (graphite glides too easily), leave the fingers clean (embedded Ptfe doesn't). My gut feeling is that a hard polymer like LCP, possibly with a filler, has better chances than a metal.

The piano, harp, cimbalom and others have metal tuning pins. Catalytic nickel protects against corrosion, rubs strongly without galling, and hopefully moves smoothly. Steinway pianos have already nickel-plated steel there.

At string hooks, especially where piano strings make a U-turn without a knot, the strong friction of nickel might help tuning.

Marc Schaefer, aka Enthalpy

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