Just Third

The problem became much worse in the late middle ages, when the third became regarded as a harmony. Now the Pythagorean third, a ration of 81/64=1.2656 was not harmonious, but very close to that ratio is the ration of 5/4=1.25. Two notes a ratio of 5/4 apart again share quite a few harmonics ( every fifth harmonic of the lower is the same as every fourth harmonic of the higher note), and our theory that the ear hears shared harmonics as harmonious would again predict that this interval would again be harmonious (not as much as the perfect fourth or perfect fifth, but certainly much more so than the tone). Again the music theory bears this out, because around the fourteenth century the major third came be be regarded as harmonious, at the same time as the concern over the deviation from the Pythagorean system became acute. This third, called the just major third, does not fall into the Pythagorean system at all. And it also makes the problems more difficult. The semi-tone between the major third and the perfect Pythagorean fourth is now the ration of 4/3 over 5/4 or 16/15$\approx$ 1.0667 is much larger than the semi-tone we discussed (1.059) and is much much larger than the Pythagorean semi-tone. This drive to bring the third into harmony (instead of disharmony- remember that the major third is one of the principle intervals used for the various men's voices in Barber-shop quartet singing), simply compounded the problem. Semi-tones were proliferating like mad. The fact that the next ratio or 6/5 is very near what was called the minor third brought in still another semi-tone, the ratio between the major and minor third or 25/24$\approx$ 1.041 which is even a smaller ratio than the Pythagorean semi-tone of 1.045.

At the same time as this was happening, music was getting richer. Musicians began to want to do what is called modulation, in which the piece of music would start in one scale, and part way through switch to another scale (ie have some other note, usually the perfect fifth of the first scale, be the base or Do of the new scale. Instead of boringly (as we would now say) just twiddling around with one scale, and one set of pitches and harmonies, one could add excitement by now using another set of notes and pitch relations as well. But things now were becoming intolerable. Two notes which in the first scale bore some relation ( say between the just third and the prefect fifth) now had a new relation ( that between the major sixth and the octave) of the new scale. If one tuned ones keyboard so all the relations in the original octave sounded good, the relations in this new scale didn't. And one could not simply re-tune in the middle of a piece.