Deduction of the equation of frequencies of a piano

[Bruno Tolentino]

I'd like of know from where originated the funcion f(n) presented in this page (https://en.wikipedia.org/wiki/Piano_key_frequencies)!?

Probably, f(n), is the solution of some differential equation and I'd like of understand how this diff equation was architected.

If someone can answer this doubt I will be so much happy!

 

[Vanadium 50]

There's no differential equation. This is just equal tempement - you have twelve equal steps every octave, and each octave doubles the frequency.

 

[Rob Benham]

Unless it's on a piano. The region known as the temperament is tuned in a logical manner, but from then on, the tuner applies a bizarrely difficult series of modifications to obviate discordant resonances. This is known as 'Keyboard Stretch'. For example, A 440 could still be near to 440, but by the time one had reached 1760, it could be as high as 1765! I have no idea how this seems to gel with the rest of the orchestra.

I once spent a year or so developing an electronic tuner - one which resonated the string continuously while tuning or analysis of case resonance was undertaken. The head of electronics at Essex University UK thought the chip was made by Fairchild and had gone the same route while a graduate student. The chip wasn't totally accurate, but human ability to sense frequency it seems is limited to about 6 cents so it was better than a lot of tuners.

Despite months of trying, I was never able to master the skill of a classical tuning.

I later caused the main oscillator to be allowed to swing 100 cents either side of the main frequency. (100 cents is a half tone) This allowed sustained investigation into unwanted resonances. There was a well known case of a plastic chocolate box in a cupboard causing tuners much grief.

The whole idea of my machine was to tune concert grands in a noisy environment but I concluded it was not a viable investment. A while later a tuner turned up at my home in Texas and the subject of electronic aids came up. He then demonstrated a tuner which he used 'When tired.' After tuning the temperament, he let the device run and it then produced the stretch for that particular piano. My device would have been totally out-classed.

An App for my phone now produces a frequency counter and generator that's more accurate. And it was free.