Pythagorean Major Third and Semi-tone

However there is another interval. If we go up by two tones, (a major third) we get $\left( {9\over 8}\right)^2= 81/64$. We note that this interval shares almost no harmonics ( the 81st harmonic or the 64th harmonic of a note take it outside the range of hearing for most notes). This interval, the major third, was considered a dissonance, not a harmony in the Pythagorean system used in the west until about the 14th century.

We note that again, two tones do not add up to a fourth. There is again something left over. This amount left over, this difference between a Pythagorean major third and the perfect fourth, is called a semi-tone. Under the Pythagorean system, this semi-tone is the ratio between the perfect fourth, 4/3, and the Pythagorean major third, 81/64, and is thus a frequency ratio of 256/243$\approx$ 1.045. Note that this is significantly smaller than the semi-tone we used of about 1.059. Note also that two of these Pythagorean semi-tones do not make a Pythagorean tone.

In the Pythagorean system, there are different semi-tones, some smaller, some larger, depending on exactly where they occur in the music.

While the Pythagorean system is a rational system, it has problems. We note that since semi-tones differ in size, that two semi-tones do not make a tone, that six tones, or twelve semi-tones do not make an octave, one could easily run into trouble. How does one tune one's instrument, especially if the instrument has fixed tones like the a lute, a viola da Gamba (the family of bowed string instruments which has frets like a guitar) a harpsichord or an organ. In particular what happens if one wants to play starting on a different note than the one the instrument was designed for (lets say because the singer's range of singling does not encompass the same range as the octave the instrument is tuned for)? does one re-tune the whole the instrument each time one changes key? (how does one re-tune the organ, which would require shortening or lengthening the organ pipes?).