Hi gang, I've finally got you some inharmonicity formulas rounded up and looked over. Here's what I've got so far. Mcferrin Core =(5.3*10^12)*(Dc*0.001)^2/(Freq^2*Lin^4) My revision =(5.3*10^12)*(Dc*0.001)^2/(((T^0.5*20833)/Dc/Lin)^2*Lin^4) (sorry Vern) ---------------------------------------------------------------------------- --------------- Fairchild: from a class handout all measurements in inches Lin = speaking length Dc = plain wire or wrapped string core diameter in 0.001" Dw = wrap diameter in 0.001" D1 = small double wrap step diameter at V bar D2 = small double wrap step diameter at bridge - would just about have to be the same as D1, since it's the same wrap layer L1 = unwrapped wire by V bar L2 = unwrapped wire at bridge L3 = double wrap step length at V bar L4 = double wrap step length at bridge Core =(Dc)^4/(81*Lin^2*T) Single wrap =0.287*((0.89*(Dw^2-Dc^2)/Dc^2)/(1+(0.89*(Dw^2-Dc^2)/Dc^2)))*(((4*SIN((4*L1) /Lin))-SIN((16*L1)/Lin))+((4*SIN((4*L2)/Lin))-SIN((16*L2)/Lin))) Double wrap =0.287*(0.89*(Dc^2-(D1*D2))/(Dc^2))/(1+(0.89*(Dw^2-Dc^2)/(Dc^2)))*(((4*SIN(( 4*(L1+L3))/Lin))-SIN((16*(L1+L3))/Lin))-((4*SIN((4*L1)/Lin))+SIN((16*L1)/Lin ))+((4*SIN((4*(L2+L4))/Lin))-SIN((16*(L2+L4))/Lin))-((4*SIN((4*L2)/Lin))+SIN ((16*L2)/Lin))) If you intend to have the unwrapped ends and step lengths the same throughout the string set, then simply define the L 1-4 variables at whatever you want. Also, decide for yourself what you want the proportion of inner to outer wrap to be for the D1 measurement for scaling purposes. ---------------------------------------------------------------------------- ----------------- Sanderson: from a class handout All measurements in inches Dc = plain wire and wrapped string core diameter in 0.001" Dw = outside wrap diameter in 0.001" D1 = small double wrap step diameter Lin = speaking length L1 = bare end lengths of wrapped strings, assumed both ends the same L2 = step length, where outer wrap goes beyond inner wrap on double wrapped strings, assumed both ends the same. Core =(330*Dc*0.001)^4/(T*Lin^2) Single wrap =0.287*((Dw^2-Dc^2)/(Dw^2+0.12*Dc^2))*(4*SIN((4*PI()*L1)/Lin)-SIN((16*PI()*L 1)/Lin)) Double wrap =0.287*((Dw^2-D1^2)/(Dw^2+0.12*Dc^2))*(4*SIN(4*PI()*(L1+L2)/Lin)-SIN((16*PI( )*(L1+L2))/Lin)-4*SIN((4*PI()*L1)/Lin)+SIN((16*PI()*L1)/Lin)) Again, you must assign your own proportion of inner to outer wrap for scaling purposes. ---------------------------------------------------------------------------- ---------------------- I found that all three of these formulas for plain wire agreed pretty closely. Mcferrin's diverges pretty wildly as you get into the wrapped strings. Being based on frequency derived tension, and the tension at any given frequency being determined by wrap mass, that's understandable. In my revision, I substituted a resultant frequency derived from the core length, tension, and diameter instead of a presumed frequency for the unison position, and it falls in line nicely with the other core computations. No wrap formulas for this one. Fairchild's formulas are the most versatile, but probably give you more options than you really need for scaling work. Also, the values I get for the single wrap formula are roughly 1/12th those of the Sanderson formula, and I don't see why that is. Perhaps I've messed something up that I keep overlooking. If anyone has a correction, please post it. Sanderson's formulas are more condensed and easier to work with in a scaling spreadsheet. All three of these core formulas have a constant that I presume can be adjusted as a scaling factor to calibrate them to measured results in an actual piano. For normal scaling purposes, they're probably close enough, but for computing the perfect tuning, you might want to look into the calibration possibilities. If you find something we should know, or something really annoying, post us your windage adjustments. Ron N
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