String Instruments

[Enthalpy]

What lets the tuning of a metal string drift?

Humidity has no expected quick effect on steel. Creep acts very slowly at a piano, where margin below the proof strength might matter, knots quality too.

1. Cold-drawn high-carbon steel expands by 10.4ppm/K. This compares with the string's stretch: 1111MPa/210GPa = 0.53% for 1.1*C. The sqrt drifts the frequency by -980ppm/K.
2. Young's modulus drops by 300ppm/K or less: 210GPa to 205GPa from +20°C to +100°C, accelerates above. At constant length, the sqrt drifts the frequency by -150ppm/K.
3. The change of the string's speaking length, for instance 12ppm/K, is negligible.
4. The frame's expansion and deformation matters much.

Stretching the string by 0.74% for 1.3*C reduces to -704ppm/K the thermal expansion effect. But if a string used at 0.8*C isn't overspun, the thermal expansion effect climbs to -1860ppm/K. The relative importance of Young's modulus drift goes the opposite way.

Strings of cold-drawn titanium alloy, if practical, would expand less: 9.3ppm/K for Ti-Al6V4 vs 0.57% stretch at 1.1*C, while Young's modulus drops by 450ppm/K. Cold-drawn austenitic stainless steel seems to reduce its Young's modulus as quickly as carbon steel, but the X2CrNiMo17-12-2 expands by 16.2ppm/K (at least when annealed!) and the PH 15-7 Mo by 9ppm/K in condition RH950. Prior to cold-working, duplex X2CrNiMoN22-5-3 reduces its Young's modulus as quickly, but expands by 12.5ppm/K.

Gut, polyamide and fluorocarbon polymers behave differently.

==========

A perfect steel or cast iron frame expands by 10.4ppm/K too, leaving -150ppm/K due to the drift of Young's modulus.

So if you tune at +20°C a cimbalom with hypothetic perfect steel frame and play it outdoors at +10°C, it goes sharp by 0.15%. Inaudible to most people, more so if all strings drift equally. In contrast, a wooden frame does drift, over temperature with some woods, and by humidity always.

If the temperature changes by 2K in your room or concert hall, the piano with iron frame drifts by 0.03%, inaudible.

==========

The strings' tension deforms the frame, whose Young's modulus drops with temperature too. So should its metal expand faster as a compensation?

I don't believe so. The frame must deform far less than the strings to make the tunings independent. Also, the frame's deformation varies among the strings, so thermal expansion couldn't compensate it everywhere.

Better a stiff frame whose expansion compensates only the strings. The frame is naturally bulkier than the strings anyway, and it must vibrate less, but its shape too must be stiff.

==========

The frame can compensate the strings' Young's modulus drift too. At cold-drawn high-carbon steel stretched for 1.1*C, it acts as 0.15* the thermal expansion, so 12.0ppm/K at the frame would let play from 0°C to +40°C without the 0.3% frequency drift.

For instance the stainless duplex X2CrNiMoN22-5-3 offers 12.5ppm/K, is strong and has nice fabrication capabilities. The martensitic X20Cr13 would be less perfect with 10.2ppm/K and the usual austenitic alloys less good with 15.8ppm/K.

Aluminium expands more: AA2014 22.7ppm/K, AA5083 23.8ppm/K. 10K variation would detune steel strings by bad 1.1% and 2K by not good 0.2%.

For Ti-Al6V4 strings too (harder alloys exist), a frame expansion of 11.9ppm/K would be good. That is, titanium strings could coexist with steel ones, overspun or not.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Steel strings rust, slowly at a piano, faster at mallet instruments that play sometimes outside and are closer to the musician's hands, and more so at plucked instruments. While six strings are easily replaced, zithers and dulcimers can have 30 strands of 3 strings.

But how stable can stainless strings be? Experiments shall decide. Stainless steel is abandoned at the piano; I suppose only austenitic steel was tried. Creep and losses may be worse than with carbon steel. All must be hardened by deep cold-work, but I don't have good data for this condition, so the following is unreliable.

Martensitic stainless behaves much like carbon steel. Tempered below 300°C, the X20Cr13 stays tough and offers YTS>=1400MPa before cold-work improves it, as is expected but not documented. More alloying elements and less C, like Cr17Ni2Mo, resist corrosion better but offer less hardness and toughness. Variants of X20Cr13 with more carbon exist in some countries, others contain V and similar to form hardening carbides.

Among them, X11CrNiMo12 (Böhler T552 and elsewhere) is a turbine alloy with known behaviour at 500-600°C, including creep. Cold-work and under-tempering aren't documented, logically. Tempered at 570°C and without cold work, it offers YTS>900MPa. Expansion 10.3ppm/K and E-modulus drift -174ppm/K (acting -87ppm/K on the frequency) suit a cast iron frame better than high-carbon steel does.

Precipitation-hardening martensitic steels harden by ageing after easier cold-working and they resist corrosion far better than high carbon steel does. I have no modulus drift data about the Maraging Ni18Co12Mo5Ti (tough YTS~2360MPa without cold work) but expansion like 9.9ppm/K could fit cast iron. The stainless X3Cr13Ni8Mo2Al aka PH13-8 (Böhler N709 and elsewhere) offers tough YTS~1400MPa which cold work supposedly improves, expansion is 10.3ppm/K like carbon steel. The stainless PH15-7Mo becomes martensite by cold-work prior to ageing (YTS~1800MPa, can improve?), it expands by <9ppm/K to suit a cast iron frame better.

Ledeburitic stainless resembles high-carbon steel. Hrc=57 to 60 as tempered nearly suffices, but can it be drawn, can wires be bent? Expansion 10.1ppm/K and E-modulus drift -174ppm/K for X90CrMoV18 suit a cast iron frame better than high-carbon steel does.

Austenitic stainless harden by cold work and stay tougher than carbon steel. X12Cr17Ni7 achieves quickly 2000MPa and more, X2Cr17Ni12Mo2 needs deeper area reduction but resists finger corrosion better. Undrawn 15.6ppm/K would fit a frame of austenitic stainless steel or copper alloy.

Precipitation-hardening austenitic steels expand even more: 16.2ppm/K for X5Ni26Cr15Ti, whose response to cold drawing isn't documented.

Duplex stainless strings would excel against corrosion. YTS and toughness respond to cold reduction similarly to X12Cr17Ni7. 12.5ppm/K for X2CrNiMoN22-5-3 suggests a frame of duplex or austenitic stainless steel.

CoCr20Ni16Mo7 resists corrosion better than all steels. It's known to exceed 2050MPa by cold-work plus ageing. 12.3ppm/K would match a duplex or austenitic frame.

Nickel alloys for turbines are optimized against creep and known to harden by deformation. Expansion of 12.3ppm/K and E drift of -313ppm/K would let the alloy 718 fit an austenitic frame.

Marc Schaefer, aka Enthalpy

[Enthalpy]

At instruments with many strings, the tuning pins and wrestplank (or pinblock) are uneasy to design, due to the big force, the many tuning pins fitting in limited space and budget, the quest for easy and stable tuning. At hitch pins (or hanger pins), the design without tuning is easier.

Usually as depicted below, cylindrical accurate steel tuning pins hold by force in tight holes in a wooden wrest plank. Wood provides strong friction and, thanks to its elasticity, it demands a lesser accuracy in the hole tightness.

Pins are of hardened steel at most zithers, dulcimers and cimbaloms but a nickel layer would prevent corrosion and bring smooth strong friction as at many pianos; maybe nickel adheres better on medium-carbon steel like 30CrMoV9 tempered at 500°C for hardness.

At an example cimbalom, a D=0.8mm string that propagates at 1.3*C pulls almost 800N, and if it leaves the pin just 10mm above a stiff wrestplank, the bending moment is 8N*m. This induces 380MPa in a D=6mm pin, and wood's elasticity worsens the lever length.

==========

If wood shall provide the pin 1600N friction from 0.4 coefficient on 25mm height, it needs 8MPa contact pressure at many close locations, too strong for the cross direction, demanding a plywood construction. Banal plywood may fail at hammered dulcimers; Steinway pianos have 7 plies of rock maple at 45° directions.

Humidity and also heat let wood expand. Wood creeps also, letting the tuning drift within weeks and months at a new or restringed dulcimer. Here are suggestions for a steel pinblock that shall stabilize the tuning of steel strings. The steel isn't critical, it could be cast iron too, or match a frame steel for easier welding, or be duplex or austenitic stainless steel for best temperature stability of steel strings. Prior experience tells me that electroless nickel doesn't gall against stainless steel, but the pinblock may get electroless nickel too.

On figure A1, the hole guides the pin on most height but a ring of metal holds it firmly. The stronger deformation is more easily adjusted. The coefficient of friction must be experimented. Taking 0.3 with nickel, 1600N friction result from elastic 300MPa in a h=2mm dR=1.4mm ring whose D=6mm is expanded by 9µm. Or plastic 550MPa in a h=1.5mm dR=1mm ring expanded by 0.1 to 0.3mm, but the initial force may drift due to creep.

Reamers and grinding machines achieve the 9µm difference accurately. Reamers can be customized, here to several diameters. Without ribs, a customized milling tool guided by the hole can make the outer shape. A Cnc milling machine does it too and can leave ribs in the pinblock, and so does casting.

The tight parts could have slits or other shapes that ease the deformations. I wouldn't use polymer nor elastomer rings as they creep badly, but separate metal parts could bring more elasticity, like coiled spring wire. They must hold without play at the pinblock.

A second ring at the top would exclude dirt and guide even better the tuning pin, figure A2.

On figure B1, the toppling moment creates 2*F+F on a shallow gliding fitting, so µ=0.3 would provide only 0.9*F, but the shaft widens. D=6mm to 9mm lets rub 1.35*F at the string, supposed to suffice even if µ=0.23. Maybe.

On figure B2, the shaft and the pinblock have a thread whose 30° slope multiplies the rub force by 2 and diameter by 1.23, so 2*F with the deeper fitting and µ=0.3 bring 1.48*F at the string.

On figure C, the tuning pin and the pinblock have fitting cones pressed in an other to rub enough. The user turns the pin as he presses, so little force suffices. Violinists do it with two fingers while holding the pegbox with the others, so the tuning hammer (=wrench) will suffice at a cimbalom or piano as it does at a harp. Reamers exist for the euronorm with dD/dh=0.02, but more slope might define the pin's height better.

Rubbing 1600N at D=6mm H=25mm means some 11MPa compression or 57ppm deformation, so 12.6-10.7=1.9ppm/K mismatch between X2CrNiMoN22-5-3 and 30CrMoV9 change little over 10K variation. Or make the tuning pins of the same steel, cold-drawn if needed.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Manufacturer sites suggest that cimbalists rarely tune their 133 strings. A street musician can't often enough. But humidity and temperature let the wooden frame and soundboard detune the instrument. A metal frame shall stabilize the cimbalum's tuning, once its temperature matches the strings. Some manufacturers have steel or graphite between the wooden pinblocks, but metal everywhere would improve. Someone wanting the typical detuned cimbalom sound can still achieve it. So here it goes, and much applies to other instruments.

==========

When humidity and temperature change the soundboard's stiffness, string passing the bridge straight keep their tension at the piano. Avoiding the push enables also a thinner, louder soundboard. The same at a cimbalom:

Every second strand of strings arrives higher at the saddles and optionally the tuning pins and hitch (hanger) pins. The dampers, saddles, optionally pinblock must adapt to this. I happily leave this design to someone else.

At most pianos' bridge, the strings make a zigzag around two inclined nails

agraffes exist too, by phoenix
phoenixpianos.co.uk
and since the strands are spaced at the bridge of a cimbalom, dulcimer or zither (and this could apply to more instruments), we might pinch the strings delicately so they glide more easily:

Spring manufactures bend spring wire to any shape, nonrecurring costs are reasonable. Many designs are possible, including one spring for many notes. The top shape must push on all strings. Here the ends centre the top on the bridge. One end hard to detach would prevent losing the spring. Springs give a force more predictable than screws or others. A layer of catalytic nickel loaded with Ptfe would ease the friction where possible, including at the saddles, but on the springs it's uneasy.

When redesigning a cimbalom bridge, one might try strings of equal length within a note. At high notes, when the fundamentals of the short steel strings are in unison, different lengths prevent it for the partials, and possibly we perceive it. Pianos make this effort.

==========

To match steel strings, I propose a frame of duplex stainless steel, but normal ferritic or martensitic steel, or cast iron, or austenitic stainless, would be decent to:
scienceforums

A grand cimbalom has 20 strands of 4 plain steel strings. I take D=0.7mm and D=0.8mm stretched to 1.1*C, or 1111MPa, so they cumulate 40kN roughly, spread over 0.48m. To simplify, I take uniform 0.8m from left to right pins, and the frame must preserve accurately the 4200µm string stretch.

I can't estimate decently the bass strings. They will need some more frame, extrapolated from the plain steel strings section.

Beams 80mm and 300mm below the strings receive 54.6kN and 14.6kN. 10.9cm² and 2.9cm² duplex there cumulate 8.8kg (more for the bass strings) and deform by 200µm, acting together 346µm on the strings. So tuning needs 3 passes after re-stringing, but in normal use, even if all strings had been a quartertone wrong, the first string drifts by 0.24% as all others get tuned, and this is inaudible to most people.

The 50MPa stress doesn't determine the metal amount. If spread at two locations, 1/4 and 3/4 of the 0.48m, it leaves 5.5cm² at the beams compressed by 27.3kN each. Even a 15mm*37mm plain section buckles at 32kN, and much more if supported laterally by skewed beams. Tubes reach more quickly the air temperature, but welded struts, cast parts, or a plate milled or laser-cut, are resistant enough.

The pinblock of (duplex stainless) steel
scienceforums
can be welded on the beams and deforms little between the beams. If it's 80mm wide and keeps 2*140mm² steel at the rims, EI=83000 SI, and over 4*0.12m with two support, 4*10kN bend it by 9µm only. 20mm thick plates would weigh 2*6.0kg but they can be cast or milled to leave steel just at the top, around the pins and at ribs between the pins, for maybe 2*2kg. Intuitively, limited torsion needs a closed section, obtained by welding plates or a truss.

==========

The drawing shows strings at one single height, hiding complexity around the pinblocks.

The soundboard holds at the metal frame but must be free enough to expand. Some flexible metal parts could contribute its movements, as on some Camac harps.

This first attempt claims a metal frame is feasible, stable and light. Simpler and cheaper must be possible.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Between the pedal and the dampers of a cimbalom, dulcimer and similar, I wanted to propose a steel cable in a housing, like for bicycle brakes, but this exists already. One example:
saitenart.ch
At bicycles they rub, but at cameras they don't.

Most cimbaloms and hammered dulcimers have 4 strings for most notes. Maybe they have an excellent reason I didn't grasp. Standard acousticians will answer "inharmonicity" but I'm not convinced.

[Enthalpy]

Users and luthiers confirm that strings rubbing at bridges hamper the tuning of cimbaloms and similar instruments.

A string deflected by 25mm over 250mm pushes 0.1* its tension on the bridge. If µ=0.3 at the narrow metal contact, 3% tension mismatch let the string move, so both sides of the string can have 1.5% detuning, or 1/4 of a semitone. High strings are deflected 5 to 7 times, adding the rubbing force. If one section of a choir is well tuned, the others may be off.

Sites recommend to pull the strings from the bridges the equalize the tension, or to strike the strings forte, or to tune first the section farthest from the tuning pin, then nearer and nearer. Dutchak cimbalom puts tuning pins at both ends of the high strings.

==========

My strings chart of June 16, 2019
scienceforums
has at most 2 bridges per string, easing the problem.

Violinists pencil their ebony nut (saddle) so the strings glide more easily. Improves metal pairs too?

I suggested here on June 17, 2019
scienceforums
to cover the bridges' metal and saddles with catalytic nickel that embeds Ptfe particles. At high pressure, it cuts µ by nearly 10.

The piano's pins let the string enter and leave the bridge parallel to inject no force, but their strong zigzag and small bias lets the strings rub much and press little at the bridge. It's the best existing design, but maybe a smaller zigzag and more pin bias is acoustically as good and eases the tuning. Nickel with Ptfe should be tried too. Adopt the design at the cimbalom?

My springs at the bridge thrive to push hence rub as little as the vibration needs
scienceforums
their force is repeatable and adjusted by design, or even by choosing a fastening position.

Let's imagine (beware I didn't try!) that a 10mm deflection over 300mm, or rather the equivalent force, suffices to apply the vibrating string on a bridge. µ~0.04 from nickel+Ptfe limit the tension mismatch to 0.1% and the frequency to 0.07%, imperceptible even with two bridges.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Instruments smaller than the grand cimbalom are more common: hammered dulcimer, small cimbalom, tsambal, Hackbrett in the Alps, and many more names. Their range starts rarely below G below the treble clef and exceeds sometimes the E above. These small cimbaloms could get a subset of my strings chart described on June 16, 2019 here. This example ranges from G to A, exceeding most small cimbaloms, with only 12+11 choirs, including two notes overlap between the string sections, and offers the logical arrangement of the notes.

G strings propagating the sound 1.20* faster than the air have 1.042m speaking length, 1.146m between the saddles, approximately 1.26m between the outer pins, so the instrument could be about 1.3m wide. This design option without wound strings will sound better and consistently, as all strings propagate between 1.20 and 1.26*C with smooth transitions between the sections.

24mm spacing between note pairs let the strings occupy 0.3m only, for an instrument <0.4m. Then 1.3m aren't so bulky, similar to a bass guitar, and the instrument with case is easily carried on one's back.

==========

Other intervals are possible between the sections, for the big cimbalom too. One example:

• The lowest section has 11 choirs, say from G to F
• The three others share 14 choirs starting just above
• They range from E to F at octave intervals.

The instrument spans then 3 octaves and a seventh, more with optional bass strings. It needs 25 choirs, making it 60mm longer, versus 84mm if the usual bass string layout adds that range. The string's propagation speed is almost as consistent. Learning seems easier.

Marc Schaefer, aka Enthalpy

[Enthalpy]

At hammered string instruments, the straight strings I proposed arrive with alternating heights at the hitch and tuning pins, as noted here on July 08, 2019. Here's a design idea after all, with metal pinblocks.

For minimum deflection at the saddles, I propose to bring the lower ends below the pinblock and the higher ones above. The strings being straight, the ones ending high at right are low at left and reciprocally. So the low ends can hang at hitch pins below the pinblock while the other end have tuning pins above the pinblock.

For more thickness at reasonable weight, the pinblock can be cast or milled to leave height at the pins, and if possible at the saddles to improve the stiffness.

The sketch isn't exact to the pixel, but 4 or 5 strings per choir look easy. Here the tuning pins limit to 12mm choir period, and a wider pinblock would accommodate them behind the hitch pins for compacity. Conical tuning pins are simpler; the string exits threaded pins at constant height, as is known.

The soundboard can extend below the pinblocks to the box, which is said to improve pianos. Pianos have wide holes in the iron frame above the soundboard.

Some felt at the hitch pins can reduce noise. Hanging the strings is less easy below the pinblock. I suggest hollow hitch pins to peep through, optionally holes in the pinblock. Springs of durable material and proper strength shall hold the strings while not stretched. They can be offset, so the string passes at the right of the spring. While operating there, one would wisely protect the soundboard, for instance with a mirror.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Could manganese steel make musical strings? Also called Hadfield steel or Mangalloy,
wikipedia
this steel containing 11 to 15% Mn and other condiments hardens quickly by cold-work to UTS>2000MPa while staying tough, good start for a string.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Here's a string chart for a hammered dulcimer, hackbrett or small cimbalom with octave intervals between the sections, as suggested on July 13, 2019 11:50 PM here.

14+14 choirs give it three octaves and a seventh, reaching higher notes than the grand cimbalom. The sections overlap by two semitones, more at the bass bridge because room is available. Additional bass strings are possible.

The chosen shape and sizes on the diagram give the plain steel strings a consistent fast propagation, from 1.17 to 1.28* the velocity in air, and the sections join smoothly. The instrument is 1.12m wide between the saddles at G, and 0.34m long at the strings, so including the soundbox and transport box, it fits on one's back.

Marc Schaefer, aka Enthalpy

[Enthalpy]

If an instruments has many strings, they pull strongly on the frame, which must be stiff for stable tuning, hence massive. Besides the shape, the frame's material determines the trade-off.

What matters is E/rho, for which low-alloyed Fe, Ti, Al, Mg are essentially equal and outperform Picea abies (Norway spruce) only against humidity and temperature. Cu alloys are worse, and the few better (quasi-) metals are brittle or unaffordable: Si, B, Be, Mo, Cr...

Some ceramics would outperform Fe, for instance alumina (380GPa for 3920kg/m³). The thermal expansion wouldn't be bad (7 to 8.4ppm/K for alumina), but most ceramics are brittle and expensive.

Frames of graphite fibres composite exist, at least at Cimbalom Vsiansky
cimbaly.cz
and while this is the preferred usable material to improve E/rho, the thermal expansion is 3ppm/K or less.

==========

Lighter materials would make parts thicker hence simpler than steel if not lighter, but they expand faster than the strings at heat.

10 to 12% Si in Al makes usual cheap cast alloys and reduces aluminium's 24ppm/K to 20ppm/K. That's much for steel strings but not bad for cold-drawn austenitic steel, nice combination. I've seen no alloy beyond the eutectic 12% Si.

At a pinblock, thermal expansion detunes the strings little. The tuning pins must hold firm despite the pinblock's expansion: added parts, thread rather than tight cones...

Metal Matrix Composites mix ceramic particles or fibres in a metal. The composite is stiffer and it expands less, so aluminium would be the matrix to match the strings. In 2019, these composites are still expensive, uncommon, and often brittle. 25vol% of SiC particles in AA2024 bring 115GPa and 15.5ppm/K, not bad for steel strings, and potentially excellent for austenitic steel. AlSi12Fe gets too brittle if adding particles.

==========

Titanium aluminide gamma-TiAl is lighter and stiffer than Ti: rho~3900kg/m³ E~165GPa for Ti-48Al2Cr2Ni, ratio 1.6* better. The break elongation is only 1 to 3% at room temperature, but brass, cast aluminium and zinc alloys, cast iron are no better. 10.9ppm/K would match steel strings directly.

Making parts of gas turbines, this alloyed intermetallics must be presently unaffordable for string instrument frames. Whether this can evolve, say after sports items generalize it? Or could a banal foundry produce it?

1ppm/K more would stabilize steel strings even better, other strings would need more. I've not found what more Al does.

The other intermetallics Ti3Al and TiAl3 seem abandoned for staying very brittle.

==========

Weight matters for many instruments more than for the piano: harps, hammered strings, many plucked strings.

Marc Schaefer, aka Enthalpy

[John Cuthber]

Could a combination of materials do for a piano frame what this does for the pendulum of a clock?
https://en.wikipedia.org/wiki/Gridiron_pendulum

Even better, could it change the length of the frame in such a way as to compensate for changes in both the length and the tension of the strings?

That way, temperature wouldn't affect the tuning.

The old joke about "it was in tune when I bought it" might become a thing of the past.

 

[Enthalpy]

Here's a metal frame for the Hackbrett of July 17, 2019. Call it hammered dulcimer, small cimbalom, santur etc as you like, and adapt it to plucked strings too.

The strings occupy 0.51m to 1.12m width between the saddles and only 0.35m length. To simplify the soundboard, I take one beam before the strings and one beyond, spaced 2*0.21m.

Strings shall have D=0.5 to 0.6mm. It's rather 0.4 to 0.5mm at hammered dulcimers and 0.7 to 0.8mm at the grand cimbalom. 1.23*C mean propagation speed needs 1389MPa stress and 273 to 393N tension, so 4*(14+14) strings pull 2*18.6kN.

========== Struts

A D=42.40mm e=3.2mm tube of 22-05 duplex holds 22.9kN compression 30mm below the mean string height, and a D=17.20mm e=2.30mm one 4.3kN traction 130mm below the strings. They deform by 290ppm and 200ppm to cumulate 403ppm on the strings whose normal stretch is 6710ppm, so these parts let the first string drift by 1 semitone after re-stringing the instrument. If retuning the whole instrument by 1/3 semitone, the first tuned string drifts only 0.1% due to these components. The four tubes weigh 8.1kg.

A pair of thin diagonal struts avoid other deformations.

Riba Edelstahl claims to deliver these tube sizes and Thyssen Krupp to manufacture starting at D=21.34mm e=2.77mm. Metric sizes seem absent. When air temperature changes, the strings reach it quickly, and tubes follow faster than plain material does. Plain duplex would need 25mm*23mm to buckle at 40kN, and be as stiff and heavy.

If milling, cutting by laser or water jet the upper, lower and some skewed struts from 25mm*120mm*1.12m and 0.51m plates, 38kg raw material costs a bit. Casting instead should be cheap, it also enables I-shaped sections. Stiffness can be traded for mass and cost. Implanting the wide tube higher would improve.

Cast iron, banal steel, martensitic or ferritic stainless would stabilize steel strings less perfectly, austenitic steel worse.

========== Pinblocks

On the cited design, the pinblocks are skewed by 42.4°, so only two F=13.7kN bends and twist them, and their deformations act *0.738 on the strings, but they are twice X=0.28m long between the struts.

Let's take duplex plates 10mm thick and 100mm wide, where narrower ends change little. The distributed load bends EI=167*10³ by 5FX³/(24EI)=0.38mm at the centre. Including the angle, the pinblock pair shortens by 0.55mm the middle strings, whose 0.8m stretch by 5.4mm. Bending lowers the first string by 1.7 semitone after re-stringing, and by 0.2% if retuning the whole instrument by 1/3 semitone.

A plates pair weighs 8.8kg, which is the paid raw material, unless holes are drilled or cut near the plates' centreline. This wastes no stiffness, gains weight, lets peep through, and is said to make a piano louder.

The torsion inertia moment of this flat section is far from the sum of r²*dS (many books are wrong) but, according to
Dubbel, Taschenbuch für der Maschinenbau, C2.5 Torsionbeanspruchung, Tabelle 7
it's J=0.31e³W for W/e=10, so G=76GPa give GJ=2.4*10³.

The F=13.7kN can act less than 5mm from the plate's midheight to create 69N*m spread over X=0.28m, so the plates twist 5.4mrad at the centre. If a string passes 15mm above midheight, and including the angle, the pinblock pair alters the length by 0.12mm by torsion, less than bending does.

Thin plates bent along their stronger axis can buckle by twisting to expose their weaker axis. For a similar case of a beam held at one end and loaded at the other, taken from
P. Dellus, Résistance des matériaux
F=4,013*sqrt(EI*GJ)/L², with GJ as previously and EI=1667 for the weaker axis, to buckle at 100kN, so 13.7kN look safe.

========== Comments

The pins near the outer edge of the plates and the saddles pushed outwards reduce the instrument's width for the same string length. Extending the soundboard laterally makes a louder instrument and lets put the bridges farther out for a smaller instrument.

This design has nearly converged with the piano, but for the positions of the struts.

Marc Schaefer, aka Enthalpy

==========

Hi JC, I come back soon!

[John Cuthber]

On a tangentially related note.  (and I accept, it's a serious "tangent"), if I was minded to repeat Cavendish's experiment on weighing the Earth, What would be the best material from which to make the suspension wire?

[Enthalpy]

On 7/21/2019 at 4:31 PM, John Cuthber said:

Could a combination of materials do for a piano frame what this does for the pendulum of a clock?
Even better, could it change the length of the frame in such a way as to compensate for changes in both the length and the tension of the strings?

Fun: I thought the same about a day ago.

The pendulum needs a constant length. This is more difficult than stabilizing the tune of a piano, where the causes of drift are

1. The mismatch of thermal expansion between the strings and the frame
2. The drop of the strings'  E modulus at heat, small contribution
3. The change of the speaking length is negligible.

For steel strings, the drop of E modulus acts as some 2ppm/K additional expansion. Figures in this thread, there:
scienceforums
so stretching steel strings over a cast iron frame is already good, prior to any compensation. I propose duplex stainless steel just to compensate these last 2ppm/K. If a Ukrainian cimbalist tunes his instrument at +20°C and goes to the street to play at 0°C, steel+iron would detune by 0.67% = 0.11 semitone, which is perceptible. But with a wooden frame, the same scenario is disastrous.

This here holds for metal strings. For fluorocarbon, polyamide and gut, I have no hope. They creep and can react to humidity. So the target instruments are the cimbalom and zither families.

14 hours ago, John Cuthber said:

[...] repeat Cavendish's experiment [...] What would be the best material from which to make the suspension wire?

Huge preference for steel. Polymers are only lighter, not stronger at identical cross-section, and they are twisted.

The "Extra-extra" sort from Roeslau guarantees 1720MPa for d=5mm (wooooow!)
roeslau-draht.com
so if you're caring and live with SF=1.7, you can suspend 2t under d=5mm.

At this wide diameter, cold-stretch isn't very efficient. Maraging steel is stronger, something like >2200MPa, it may creep a bit more:
dynamicmetals.net
looks like Allegheny has changed its name, address and activity meanwhile.

Gravitational interferometers plan to use wires of single-crystal sapphire, but that's more a question of mechanical thermal noise. Wires from the same crystal as the mirrors have smaller mechanical losses hence put less vibration density outside the mechanical resonance. Cavendish didn't care about this.

The one easy improvement over Cavendish's setup is the readout, with optics and electronics. You can measure precisely the period with and without attracting masses, subtract the wire's stiffness. And depleted uranium is available too, cheaper than tungsten, nonmagnetic.

[Enthalpy]

Here's a top view of the Hackbrett with metal frame and octave tuning.

The upper tubes can be narrower for a compact instrument. Accept stronger interactions in string tuning, or take thicker tubes and wait longer until they reach a changed air temperature. The extreme case is the mentioned laser- or jet-cut 25mm plate. Casting could make advantageous I beam profiles. For comfort and aspect, an adhesive strong film can cover the upper tubes, of polyimide, colored polyester etc, or they can be painted.

Putting the metal framework outside the soundbox might ease assembling but I prefer the aspect of wood. The upper struts should be as high as possible for stiffness.

The bass bridge and material cost limit the width of the pinblocks, about 116mm on the sketch. Holes near the centreline can be laser- or jet-cut for weight and loudness - or cast or mill the part. Width matters less at the ends, which could be straight if the aspect is better or the soundboard louder or simpler.

A breaking string is badly dangerous, so looking under the pinblocks was a bad idea. Keep a mechanical access there, or add a window?

Duplex pinblocks can be wide, so aluminium wouldn't be stiffer against bending, but it's cheaper, and thicker is stiffer against torsion. For instance Al-Si12Fe can be cast. Steel, cast iron, ferritic stainless would stabilize the tuning nearly as duplex does, and austenitic stainless but worse. If Aluminum-Matrix-Composite or gamma-TiAl are or become affordable, fantastic, see July 21, 2019 here.

The pinblocks bend much with struts before and beyond them. A longer instrument like a grand cimbalom would worsen that. Struts around 1/4 and 3/4 the length (or more struts) would improve this bending a lot, but the upper struts are then lower so they deform much more. The lower struts must then be much lower, which needs a deeper soundbox like the grand cimbalom has. Assembling has different issues then.

Marc Schaefer, aka Enthalpy

[Enthalpy]

It's a market study, and this abstract doesn't tell whether their titanium aluminide gamma-TiAl is ingot, wrought semi-products or whatever. There
marketwatch.com
7500usd/t, and a piano manufacturer would buy amounts similar to aeroplane engine manufacturers, while a Hackbrett or harp manufacturer wouldn't.

Replacing 180kg cast iron, a 100kg frame would then cost 750usd, but processing must be much more expensive.

1996 report about designing with titanium aluminide by Pratt&Whitney (hence aeroplane engines):
pdfs.semanticscholar.org
they cast with oversize, then machine.

About difficulties with casting, and IHI's attempt to cast at net size:
ihi.co.jp
TiAl reacts with the mould and the liquid is viscous, but string instruments frames can accept sturdy shapes.

TiAl compositions are given in atomic percent, so Ti48Al2Cr2Ni means nearly 1 mole Al for 1 mole Ti. Absolutely meaningful for intermetallic alloys, but not the most frequent convention.

[Enthalpy]

Thoughts about a soundbox for the Hackbrett suggested here on July 23, 2019.

At its lowest note, G=197Hz, lambda/2=0.87m in air resembles the mean dimensions 0.8m*0.4m, so the radiation resistance would be roughly 5.6mohm*F² = 217ohm if the soundboard moved uniformly. 1m/s rms (arbitrary for comparisons) would displace 0.30m³/s rms and radiate 20W.

A mean 0.82m*0.37m soundbox 0.1m deep would provide 31dm³ and 0.22µF or j0.27mS at 197Hz. The same 0.30m³/s rms create 1.1kPa rms in the box whose 0.84m² dissipate 6*10^-9*SP²F^0.5 = 89mW by conduction. Radiation is stronger even if multipolar and inefficient, so conduction losses need no deep box at this big soprano.

The unstressed soundboard of Picea abies (Norway spruce) weighs some 1.7kg with bracings, bridges and strings: the 0.30m² piston has 19H inductance. Take a Helmholtz resonance at B=248Hz with holes. The soundbox 0.1m deep for the frame suffices for Helmholtz: its 0.22µF need 1.9H of which the holes provide 2.1H and pass most throughput, good.

If located between the soundboard and the box' walls, the metal frame would leave 2mm*(0.6m+1.0m) = 32cm²clearance around the soundboard. 6mm thickness make 2.3H leak, achieving the Helmholtz resonance without added hole. For 1m³/s rms throughput (arbitrary for comparisons), 313m/s at the slit dissipate 2.2*10^-3*SV²F^0,5 = 65W, so Q=55 at the resonance. This is decent as more losses add. The slit's width and height adjust L and Q, probably too variable and unstable.

Alternately, the metal frame with some façade can surround the contiguous soundbox at front and aft for easier assembly. The soundboard can still have slits there, or better reach all walls. Then 770 holes, D=2mm h=3mm, achieve reliably the 2.0H with Q=28. That's one or several rosaces, where the hole or slit width adjusts Q. A CNC laser or milling machine does it better. The instrument's front panel can accommodate them.

To make an idea of the table's data, I tuned resonant frequencies with a spreadsheet
HackbrettBracings.xls
based on unjustified and bizarre assumptions:

• Only the table vibrates.
• Lowest resonance at F#=372Hz, a fifth above Helmholtz. Next resonances rather close.
• Picea abies for the plate and the bracings in both directions.
• Rectangular table in the computations, 0.8m*0.4m.

This resulted in a table hold at all edges and bracings stopping before the edges. The four first modes have a single wavelength in the 0.4m table length, and one to four antinodes in the 0.8m width, which are wider apart than in air, so such a table should radiate well all frequencies. The table has slightly more flexural stiffness in the 0.4m direction, which is the plate's L direction.

Alternatives:

• Vibrate the bottom too! It improves violins, and cimbaloms do it. Increase then the stiffness in the 0.8m direction, interleave the modes.
• But a thicker plate and ribs in the R direction only would be much heavier. Bidirectional ribs seem better.
• Paulownia tomentosa makes lighter tables than Picea abies at identical resonances. At least one guitar luthier prefers it.
• Cross-laminate spruce on a balsa core. Example:
  Spruce - spruce X - balsa - balsa X - balsa - spruce X - spruce, or end-grain balsa.
  The sandwich is much lighter than a ribbed plate and can be anisotropic, tapered, crowned.

Some may apply to the piano and others too.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Harpists have 47 strings to retune often, cimbalists sometimes neglect to tune their 137, and zithers don't help their musician neither. Metal frames will stabilize the tuning, but only in the future and for metal strings. So could something help tuning these instruments?

Maybe a motorised tuning hammer (which is a wrench) would. It would listen to the played string, determine the correction, and apply it up to perfect tune, taking the chosen pitch into account. I've seen tuning forks that indicate the necessary correction, but applying it autonomously seems uncommon or inexistent.

Here an example of shape, others are possible, like a T more common at cimbaloms.

Achieving perfect tune within one decay time would save much time. For high notes, the tool must be smart enough to work on repeated pluckings.

==========

Safety demands error-free operation. Many strings are a tone or semitone short of breaking, and their debris are badly dangerous for the user and the instrument. Tuning a harp a bit too high explodes its soundboard.

Perhaps the safest way is when each tuning pin of the instrument tells the hammer what note is correct. If the hammer can read, it suffices to engrave near the tuning pin a barcode or number, like Yamaha's note number. Or like did old low-tech reliable access control cards, each tuning pin could have a unique set of passive resonating circuits whose frequency the hammer senses by contact to obtain for instance a note name and octave number. One componentless printed circuit board can bear pairs of resonating circuits, say around 40.68MHz, for a dozen tuning pins or more. A current return path isn't mandatory. Multiple strings in a choir could share the resonators. May I please express the wish that manufacturers follow a single standard, and don't reproduce the present mess of octave numbering?

Without additions to the instrument, the hammer could listen to the played string and emit a sound at the height it supposes correct. Only if the user confirms, would the hammer turn the tuning pin.

I emphasize that identifying the height of a played string is difficult. A harp or grand cimbalom have a wide height range. To perceive the played string longer, the hammer must accept a big dynamic range. Though, many electronic tuners don't identify properly notes played by a piano.

The hammer must make more checks, especially whether the undertaken action has the expected effect on the heard note.

==========

The instrument could incorporate an actuator per string. Useless at a piano, useful to few people at a guitar, too expensive for a zither or hammered dulcimer, but for a harp it's conceivable, and extremely profitable for professionals. Share the electronics, put one geared motor per string. At 47 strings *20€ it's affordable, and one microphone or accelerometer per string is cheap. At least, each motor corresponds to one note without confusion.

A harp could even tune itself while played, when a pedal is in flat position. Don't forget to make jokes at other instruments.

Flip a switch, strike a proooooiiing, the harp is tuned, life is nicer.

Marc Schaefer, aka Enthalpy

[Enthalpy]

This spreadsheet claims that a soundboard of kiri is 0.82* as heavy as spruce, and balsa covered with spruce 1.04*. This holds at identical propagation speeds, here the nearly isotropic ones I chose for the Hackbrett on July 28, 2019.

PiceaPaulowniaSandwich.xls

The comparison makes many subjective assumptions:

• Thick narrow bidimensional ribs.
• Is balsa any good as a sounding material? It sounded "pk pk" last time I tried.
• 0.5mm thin veneer of spruce. Maple would be even thinner.
• Available kiri has controlled fibre orientation.
• Spruce fibres parallel to next balsa layer.
• The core's bad G modulus doesn't spoil the sandwich.

Harps cover a core with veneer, but it's harder than spruce, and they keep bars, a better combination that I didn't model here. A sandwich could also be more durable than a thin plate with ribs and demand less labour.

A hollow core makes a lighter sandwich but may need too much work. Drill many holes, assemble from ribs, possibly along several directions...

A guitar luthier prefers kiri (Paulownia tomentosa) over spruce (Picea abies or Picea sitchensis) when available, and the present model with ribs converges with her observations, as do figures-of-merit for plain material. Worth trying at all instruments with a soundboard?

I've read that fibres have badly controlled orientation in commercial kiri. For the koto, luthiers apparently don't care, despite the wood is very anisotropic. Is a better control desirable? Can assembled smaller planks achieve it from a tree less tall than a spruce?

Marc Schaefer, aka Enthalpy

[Enthalpy]

Could Liquid Crystal Polymers make varnishes for soundboards?

Known as Vectra, Zylin, Vectran and more, they are decently stiff as bulk material: E~10GPa for 1400kg/m³, much worse ratio than spruce along the fibres, but far better across. A deformation that orients the fibres stiffens them to >200GPa: would thin layers brushed in cross directions improve? And they would replace a varnish that lacks stiffness.

LCP are very impervious to moisture, nice. Skin contact should also be innocuous, with some LCP allowed even for food contact.

Drawback: they dampen the vibrations. More than most polymers do, but what do existing varnishes?

Other drawback: they have very few solvents, and even fewer one acceptable on the skin and on wood.

==========

Some guitars, and maybe more instruments, assemble wood with banal lime. I wouldn't be surprised if hide glue dampens less the vibrations. Violin luthiers use it exclusively despite its brittleness.

Marc Schaefer, aka Enthalpy

[Enthalpy]

I described on July 23, 2019 a metal frame running outside the soundbox for a Hackbrett: before, behind, and at the sides above the box. Here's an attempt for the grand cimbalom with sixth tuning as on June 16, 2019. The instrument is heavier and less stiff than with metal beams through the box as on July 08, 2019, but it's easier to assemble. Octave tuning as on July 17, 2019 with added bass strings would further complicate the present attempt.
July 23, 2019 - June 16, 2019 - July 08, 2019 - July 17, 2019 on scienceforums

 

The bridges and saddles could spread less at the bass strings for an instrument smaller and maybe easier to play. Here as previously, the bass strings drop to 0.54*C, the others vary smoothly between 1.10*C and 1.16*C.

I take round 60kN total string tension. The pinblocks are skewed by cos=0.783 so F=21kN pull perpendicularly to each X=0.4m half-pinblock. 1.1*C stretch the strings by 0.54% so a semitone detuning is 320ppm, or for middle strings 169µm per side, 244µm perpendicularly to the pinblocks.

Using 22-05 duplex everywhere, E=200GPa G=76GPa rho=7820kg/m³, so tubes of D=48,26 e=5,08 and D=17,20 e=2,30 at 30mm and 180mm below the mean string plane feel 36kN and 6kN and deform by 261ppm and 279ppm, cumulating 360ppm = 1.1 semitone at the strings. They buckle at safe 111kN and weigh 15kg together.

The e=15mm W=180mm pinblocks can be narrower at the ends. Holes leave metal at the outer 34mm; round holes aren't best. EI=1,1MN*m² in 5FX³/(24EI) bend by 253µm = 1.0 semitone at middle strings.

21kN spread on 0.4m shear by 54µm = 0.2 semitone at middle strings the pinblocks' 2*15mm*34mm.

Twisting inertia is 0.24e³W for each 15mm*34mm part of the pinblocks as from Dubbel, together GJ=4.2kN*m². If 21kn spread over 0.4m pull 5mm outside the middle plane, 5.0mrad let middle strings 15mm away from the plane stretch by 75µm = 0.3 semitone by torsion.

At a narrow cantilever beam according to Dellus, here EI=4.8kN*m² at the thin direction, buckling would occur for 4.1*(EI*GJ)^0.5/L² = 115kN, a first hint for the present case.

Cumulated 2.6 semitone at the centre strings (but 1.1 at the extreme) mean 4 tuning passes after restringing and 1 pass to shift by 0.2 semitone the whole instrument, not bad. Temperature and moisture shall have no effect.

Two pinblocks cost 34kg material and weigh some 19kg. Width at the centre brings much stiffness, costs money but adds little mass. Casting seems interesting, including the struts optionally. Thicker edges at T shaped pinblocks would catch broken strings and improve the stiffness versus mass tradeoff.

Marc Schaefer, aka Enthalpy

[Enthalpy]

Positioning the big tube higher at the metal frame suggested for cimbaloms, Hackbrett and similar is advantageous. Both tubes experience a force less amplified, and their deformations act less amplified on the strings.

At least the distal tube can be level with the mean string height. Then the small tube feels no force and contributes no string deformation - it can be lighter, maybe suppressed. As compared with tubes 30mm and 180mm below the strings, the big tube's elasticity acts (5/6)²* as much on the strings, so it detunes by 0.7 semitone instead of 1.0, or it can be lighter, and the small tube doesn't add 0.1 semitone.

How high can the proximal big tube be, ideally level? Alas, I don't play these instruments. The mallets strike the strings head down, but I ignore by how much. The strings are also higher than the average where striked. So a D=48.26mm tube doesn't have to be 30mm below the strings.

[Enthalpy]

All piano soundboards I've seen have the plate's fibres in one direction and bracings in the cross direction, where the plate alone would be too flexible.

The spreadsheet HackbrettBracings.xls from July 28, 2019 in this thread claims that a thinner plate, and bracings in both directions, would be much lighter at identical resonant frequencies. The instrument would then be louder.

This needs tall narrow ribs that keep stiff where they cross, not trivial. Cimbalom Vsiansky do it, with shallow wide ribs
cimbaly.cz
and some guitar luthiers too. Maybe one direction can be above the plate and the other below.

This should combine with the use of kiri, Paulownia tomentosa.

Unless the balsa sandwich of August 03, 2019 03:45 PM is better. It's not the lightest, and I doubt about its damping, but it looks easier to build.

Marc Schaefer, aka Enthalpy

[Enthalpy]

After all, here's a grand cimbalom with my octave tuning and a metal frame outside the soundbox. It may be easier to learn and adds a high fifth of dubious quality to E=2649Hz, as high as some hammered dulcimers reach, within the piano's bad last octave.

46 choirs spaced by 12mm give the usual length, and the sections overlap by 2 semitones; if playable, less spacing would stiffen the frame. String velocity between 1.10*C and 1.18*C, dropping to 0.52*C at low notes, keep the frame 1.6m wide, but if playable, longer strings sound better.

The string chart shares the low notes with existing big cimbaloms. The Hackbrett of July 23, 2019 here could make a tiny change and share exactly this grand cimbalom's string chart, but for low strings.

The pinblocks are but longer than on August 12, 2019 here, but they keep the 1.0 semitone flexural stiffness thanks to their arch design, here of maximum width. The skewed struts are 10mm wide, the arc 38mm and the chord 30mm, with 166mm midwidth height, all 15mm thick. The tubes stay as before. Torsion is less bad as I evaluated here on August 12, 2019 as the skewed struts improve that. Strings reduce the mutual detuning by pulling at the arcs, but I did NOT check all buckling risks.

This construction enables a bigger soundboard that breathes better. The soundboard and box can stop before the pinblocks for easier use, at my previous designs too. Something at the pinblocks' edge should catch broken strings.

The pair of arched constructions weigh 16.8kg of duplex, gained 2kg, but cost 40kg if cut from a 15mm sheet. Carbon steel would be cheaper, and casting even cheaper, of duplex or cast iron. TiAl would save 1/3 mass, metal matrix aluminium about as much. Making the arch of tube seems complicated.

Marc Schaefer, aka Enthalpy