Nice writeup, Nick... and the animation at your linked page really clarifies what's happening. I thought I understood what the deal was with partials, and I did, but that animation makes it much more clear and concise. Paul Bruesch Stillwater, MN On Fri, Mar 13, 2009 at 8:01 PM, Nick Gravagne <gravagnegang at att.net> wrote: > William et al, > > > > I remember a tuning class held at a large chapter meeting. Intervals were > played and the beats were obvious to both newbies and veterans. Adjustments > were made and we could all hear the beats speeding up and slowing down. A > fine temperament was set by adjusting the beat rates for even thirds and > sixths, and “quiet” fourths and fifths. A young man asked about coincident > partials: “where exactly do they line up?” > > > > The instructor said he used to know but wasn’t sure; there was some > head-scratching in the room of 35 attendees, but a few had the answers. > “You’ve been reading Braid White’s book, haven’t you?” Virtually all the > veteran tuners adamantly opined that it is best to listen to the “obvious” > beats, those we had been listening to during the demonstration. These > obvious beats “sounding” at the fundamentals are what this list is now > calling “whole tone” or “whole sound” listening or tuning. > > > > That chapter meeting was held in New Jersey in 1973 and I was among the > newbies. I learned to tune by hearing the whole package, although later on I > was pleased to isolate the partials. Tuning then became a balancing act of > checking the whole sound with the partials of choice. > > > > Virgil Smith is not a mathematician, but he had latched onto the concept of > resultant forces. Ten forces of different magnitudes pulling an object in > many opposing directions can all be reduced to one significant force --- the > resultant force. And the object will move steadily in one direction and at > one speed. The energy force in a vibrating string divides itself up among > the multitude of partials; many sine waves superimpose themselves. The > famous French mathematician J. Fourier (1768 – 1830) analyzed this > phenomenon and gave us the famous Fourier curve, the single resultant > curve/force that essentially represented the integral (the whole) of the > many constituent superimposing partials, including the fundamental. The > single curve does not look like a simple sine wave; rather it is bumpy and > strange yet periodic. > > > > For fun, go to > http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html<http://id.mind.net/%7Ezona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html>and see a violin string animation of the Fourier curve as the resultant wave > (the white wave) of partials. You have to build the Fourier pulse by > clicking on the partial selections. > > > > These curves do not simply exist for the convenience of study, they point > to the reality of our physical universe. The simple act of standing up > amounts to the resultant force of a multitude of smaller forces, > equilibriums and gravity. Fortunately, we do not need to analyze these to > simply stand up. What is true of physical mechanics is true of sound. > > > > Now if the temperament note F exists as a single resultant curve, and A > above it the same, then the superimposing of these two single waves running > along a time plot will indicate an interference of 7 bps, and all this will > be experienced by the ear at the fundamental level. Even more fascinating, > the F and A will coalesce into its own single resultant curve, also periodic > in nature. The relatively small energies that exist at the higher coincident > partials could not possibly affect the intensity of the beating effect we > have at the pitch frequencies unless the whole tone resultants are > interacting. > > > > And yet more mind boggling is that a single resultant curve exists for a > sustaining chord played in different positions up the keyboard. There comes > a whole brilliant swirling and shimmering sound, but shot through with tiny > laser beams. Only piano tuners and certain musicians can surgically dissect > these. It seems to me there must be a study or lab experiment that > demonstrates this reality. > > > > RicB: it is not a stretch to borrow from the world of higher mathematics > and refer to partials as “derivatives” and to the combining of all these > derivatives as the “integral”. Math purists might balk due to the implied > functions, but relative to our discussion, we would then have Derivative > tuning as partial-focused, and Integral tuning as whole tone, Fourier > tuning. These sterile terms lack warmth, but they point theoretically in the > right direction. > > > > Regards, > > > > *Nick Gravagne, RPT* > > *Piano Technicians Guild* > > *Member Society Manufacturing Engineers* > > *Voice Mail 928-476-4143* > > > -------------- next part -------------- An HTML attachment was scrubbed... URL: <http://ptg.org/pipermail/pianotech_ptg.org/attachments/20090313/3b1339f7/attachment-0001.html>
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