one of the few piano tuners who knows that the right "inharmonicity" is bliss

Albert Ketenzian must be one of the few piano tuners on the planet who understands that the overtones of a single string must be slightly greater than x2, x3, x4 ect in frequency compared to its fundamental frequency so that the note itself sounds correct to human ear brain.


Then the octaves will be tuned slightly higher than x2 (in order to eliminate beats between the second overtone of the low string and the fundamental of the high string). The art is a compromise really but great pianos have the right scalling (choice of diameter, length for each note) so that the right inharmonicity takes place at all strings. So that all notes, overtones, combinations of notes sound correct to ear brain. A piano evolution of hundreds of years.


In short taking a given length and diameter of piano string...


Assuming we do not know at what note it will sound best. We might for example attach one string to a wooden board for experiments. Wearing protective eye googles, protective gloves etc is important to avoid self injury as at some point the string will break.


Starting at low tension the string is very inharmonic. Sounds like bell.


Increasing the tension it sounds like some sad oriental instrument.


Further increasing tension we might come to a point where its second overtone is double in frequency to the fundamental. The sound is OK like a cheap synth.


Further tiny increase of tension creates a magical sound as now the seconds overtone and other overtones are higher than exact x2, x3, x4 etc. This is the right inharmonicity to ear brain for that particular sounding string.


Further increasing tension it still sound nice but the sound may be tense. This is perhaps overshooting the inharmonicity.


Further increasing tension the string might break.


X2 frequency relation stuff is for kids. Well in fact kids may know better.


Bibliography:



Octaves and Pitch - Gauffin, Sundberg
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.431.8104&rep=rep1&type=pdf


http://www.mmk.ei.tum.de/fileadmin/w00bqn/www/Personen/Terhardt/ter/top/octstretch.html


Ear brain octave vs engineering octave - euroelectron


The Pleiades tuning for Yamaha DX7 II, Korg Triton


Musical Acoustics - Donald Hall


Conceptual Physics - Hewitt


Piano servicing, tuning and rebuilding - Reblitz