Listen to the Piano - Not the salesperson


Pythagoras (6th C. B.C.) observed that when the blacksmith struck his anvil, different notes were produced according to the weight of the hammer. Number (in this case "amount of weight") seemed to govern musical tone. . . . .If you will listen to a piano as you play a sound, strike a key so that the hammer strikes the strings . . .the sound may take time to reach you but each is under 22K....See if you can hear the sound in your imagination before it comes, by judging from the proportions of the string lengths (the shortest string is the farthest to the right.....) and the amount of force with which you strike the key. You can also distinguish what sounds good to you.
Further, he observed that if you take two strings in the same degree of tension, and then divide one of them exactly in half, when they are plucked the pitch of the shorter string is exactly one octave higher than the longer:

Again, number (in this case "amount of space") seemed to govern musical tone. Or does musical tone govern number?
He also discovered that if the length of the two strings are in relation to each other 2:3, the difference in pitch is called a fifth:


...and if the length of the strings are in relation to each other 3:4, then the difference is called a fourth.


Thus the musical notation of the Greeks, which we have inherited can be expressed mathematically as 1:2:3:4

All this above can be summarized in the following.


(Another consonance which the Greeks recognized was the octave plus a fifth, where 9:18 = 1:2, an octave, and 18:27 = 2:3, a fifth;)


This triangular figure of numbers in the shape of the Greek letter Lamda is the Tetrad of the Pythagorians.