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Enthalpy

String Instruments

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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.

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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

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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

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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.

TuningPinBlockRef.png.13e50eb51c0e9437d3e7719a35db8e7c.png

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.

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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.

TuningPinBlockA.png.351cdea43334b26a76ab2b55cdce8949.png

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.

TuningPinBlockB.png.2ed50f759e5c77984a23566e9ea9458f.png

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.

TuningPinBlockC.png.5be93709ff5f6b13f4b83b3252b6dab3.png

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

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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.

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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:

CimbalomStringsStraight.png.58d8045cfe2137cb056c378bc1f893be.png

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
PianoBridgeWiki.jpg.e35e1048dcd831ec30b3bced948a6901.jpg
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:

PinchStringsBridge.png.12290ab9d1a0627416ee68c1474a9101.png

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.

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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.

CimbalomMetalFrame.png.3f73bce395f47907f8cb13ca84bba0e4.png

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.

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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

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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.

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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.

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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

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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.

HackbrettSimplerChart.png.b2b84f20360d48f8f2a2552467fb5953.png

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.

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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

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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.

CimbalomAlternateHitchTuning.png.a052ee7dc84480f8209a9e10d41cf283.png

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

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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

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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.

HackbrettOctaveChart.png.9424f35b7a2d318a8289e042788ce136.png

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

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