# 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 1.9µW, conduction would waste 0.1µW and viscosity >0.1µW, 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. +-45° 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 0.15µW while conduction to 0.6m2 box wastes 0.03µW, 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 16µW. 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.wikipedia (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 45° 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  to the bridge  so that it the tops 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 détaché 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  to the bridge  so that it the tops 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
------------------------

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 180°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 50€/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

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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:
Biomaterials Science: An Introduction to Materials in Medicine
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