Recently I came up with interesting discovery how to get equally tempered tones by using a stack of just intervals 3/2 and 5/4. Since they are a part of most harmonic cord, major triad, should be caught very accurate.
The kernel of my theory is using the stack of 7 perfect fifths (3/2) and 1 major third (5/4) to get a single equally tempered tone.
(3/2)^7 * 5/4 * 1/16 = 1.3348388671875
The ratio must be divided by 16 (or 2^4) because the tone I'm looking for went 4 octaves up.
In this example the result is perfect fourth. Mathematical precision is till 5th digit after the decimal point. The error is 0.00128 cents. Repeating same stack 11 times gives a final error of 11 * 0.00128 = 0.01408 cents.
I wrote an article about this discovery but many musicians claim this is not possible in terms of acoustics because lot more events occur, for example inharmonicity and octave stretching. Due to imperfection of the instruments this remains just theoretical.
Although this may never be achieved in practice, I believe the math behind this theory is exceptional. Many roots of 2 could be expressed by small fractions of the first 3 prime numbers (2, 3 and 5). Everything from 2^(1/12) to 2^(11/12) could be calculate very accurate by stacking above expression 11 times. Reversing the ratios is legit too.
Is it possible to tune a piano using this method of stacking 7 P5s and 1 M3? I'm expecting better accuracy up to 246 times but what would be the actual results?
So far the best procedure I can offer for tuning is this one:
Is it efficient enough? Is it going to accumulate errors or it's error free?
Another rough procedure that pushes the tuning process to its maximum is:
Since the stack contains twice less steps but larger ratios, is there any possibility to get it done without accumulating errors?