So in the remaining lessons in this module, we're gonna be talking about the explanation of consonance. What is it that makes tone combinations either pleasing, consonant, or relatively displeasing? And I emphasize relatively, because they're all relatively pleasing, it's just a matter of degree. So we're gonna be considering three explanations. A mathematical explanation, a physical explanation, and a biological explanation. So in this lesson let's begin and discuss mathematical explanation of consonance. This goes back to Pythagoras. And you'll remember that Pythagoras was the ancient Greek philosopher whose major contributions I think people would say were in mathematics and geometry. But he was very interested in music as well, as was his school of Pythagorean philosophy. And he had a very simple idea of what made music consonant. And his idea was that this was dependent on the ratios of strings, the lengths of strings. And he was half right and half wrong. His framework for thinking about this was in terms of a philosophy that, as those of you who know anything about ancient Greek philosophy will be aware, had a very spiritual aesthetic context to it. And he thought that the ratios that were musically significant were these ratios of unison of the octave of the perfect fifth and the perfect fourth. And he limited his concept to these ratios because, as we'll see and as all modern musicians know, it gets very complicated after these simple whole number ratios. Things get out of hand pretty quickly. So Pythagoras was interested in the philosophy and the spirituality of these ratios, and didn't want to go beyond what he thought were these simple perfect ratios. That he limited his concept of music too in fact he forbid ratios that quickly in the post-Grecian era and he, even in the era of ancient Greece, were being played all the time. So this is a lyre. The instrument of ancient Greece, or one of the instruments of ancient Greece. And you can see that is has seven strings. I mean some lyres had four strings, some had ten, this one has seven, but you can appreciate that these Pythagorean ratios were not what the popular music of the day was limited to. They were playing music much as we hear it today in the folk music and popular music of the time. But Pythagoras has been sort of the dominant historical figure in thinking about music in ancient Greece because of his philosophical and spiritual concept of what musical tones were all about. So the bottom line of Pythagorean ratios was that they were based on string lengths. And Pythagoras was aware that strings that were of the same length were played together sounded good on two lyres for example. That string lengths that were in the ratio of two to one, an octave, sounded consonant, attractive, pleasing. That ratios of three to two, the perfect fifth, were also pleasing. And the perfect fourth, four to three, were pleasing as well. So he limited His concept of music and it's held over in Pythagorean tuning to these four ratios that were expressed as the so-called tetrachord in ancient Greek music. But be aware that the popular music of the day wasn't limited to Pythagoras' rather austere concept of what music could and should be based on mathematics.