Magic number 1.0595472

At your risk.


This is on letting a good quality piano and one's brain decide what the optimum A4 frequency be on that particular piano so that overtones of strings sound correct to listener's brain rather than taking A4=440Hz as a tuning reference?. We know for example jazz men women tune A4 at 444Hz making the piano sounding better even at so complex chord intervals.


This experiment has been going on since a bit after Olympic Games in Athens ie 2004.


A few thousand euros miraculously found had been spent on tools and top quality materials in order to rebuild a great quality Pleyel model 9 upright piano made in around 1921. The original Firminy factory in France strings had been repolished and reinstalled, one each day.. The piano was ready on about 2008. Everything possible was done so that it could withstand higher string forces than when tunning at A4=440Hz.


While increasing above 440Hz a sweet spot had been found where all intervals including major 3rds sounded amazing while testing playing Prelude in C - Bach from well tempered clavier.


Then further increase (psychological inertia overshoot) had been done past 450Hz in an attempt to make all 5ths perfect (beatless). The circle of 5ths of Pythagoras had been closed as 3rd overtones were higher than 3:2 by the increased tension of strings.
But the piano sounded bad especially on high major 3rds which had a lot of beat and the piano although still sounding very fresh and nice it had an element of tension.


Over the last years tuning had been once by hand decreased to A=451Hz which sounded very nice but with still a bit of tension .


During the last 3 or more years the piano had been left untuned but played a lot. Natural detuning.


Now it is out of tune on quite a few notes, intervals etc.


But today it had been observed that A flat major (which is little used anyway) sounds amazing. And this included playing octave or double octave plus major third ie high C which had been problematic as already mentioned.


Using an online signal generator (www.onlinetonegenerator.com) and by beat comparison it was found today that bA3 is 209.8Hz and bA4 is 420Hz while both sounding great in tune (beatless) and at the same time the octave interval sounding correct to ear brain which only happens on state of the art instruments of anytime.


Now 420 : 209.8 = 2.002


This agrees with the well known fact that more than times 2 is needed for a subjective octave or the octave sounds a poor one.


Then the 12th root of the exact 2.0019066 had been taken which gives 1.0595472 which can be called the Pleiades equal temperament half tone ratio number as opposed to 1.05946 which is the 12th root of 2, the well known poor engineering? equal temperament.


Then using 1.0595472 by successive multiplication, from bA3=209.8Hz, A4 had been found to be 445Hz.(


Similarly C4 (major 3rd wrt bA3 is calculated 264.4Hz. Playing this frequency on the tone generator agrees with Pleyel.


Similarly B3 (minor 3rd wrt bA3) calculated with the magic number as 249.555Hz agrees with Pleyel as it currently plays very well these particular above intervals.


After all this concentration the overtones of bA3 could be distinguished by ear brain. Even the very high C (5th overtone), C6 that could be heard sounded a very satisfactory double octave major 3rd and agreeing with the note itself played on Pleyel which plays very well this interval.


These results are promising so an attempt will be made to tune the piano at A4=445Hz which should be near to an optimum for the particular piano.