Thread: just like the other from one to the other.

Originally Posted by grapes

Look at where those fractions in your table come from. In your table, CEG is obviously 4:5:6, as I said. Taking GBD, G:B is clearly 4:5, but the D would have to be one octave higher, or 9/4, so G : D is 4:6.

FAC goes back down, one octave lower.

I agree that the numbers in my "traditional" scale are totally consistent with your method, I had just never heard this method of constructing them before.

What I have heard is that the modern twelth-root-of-two system was developed as an approximation to the traditional scale, that doesn't require retuning instruments whenever there is a change of key. Is this accurate?

Originally Posted by grapes

The musical notes can be derived from harmonics a little more simply, maybe equivalently. A major chord is just the 4th, 5th, and 6th harmonics. In other words, three notes whose frequencies have the ratios 4:5:6. The notes CEG form a major chord, and you can go down from there, as FAC is a major chord, or up as GBD is a major chord. Those are the seven notes of our western music.

Yes, this does give the 7 notes of the scale correctly and a great way to remember the relative frequencies.

However it does not explain why. Why 4:5:6? And why repeat 3 times? These things are arbitrary. Harmonics theory follows logically from extremely simple principles - that there are standing waves and that nature must be non-linear.

Originally Posted by Coelacanth

What I have heard is that the modern twelth-root-of-two system was developed as an approximation to the traditional scale, that doesn't require retuning instruments whenever there is a change of key. Is this accurate?

I believe it is!

Originally Posted by RayTomes

Yes, this does give the 7 notes of the scale correctly and a great way to remember the relative frequencies.

However it does not explain why. Why 4:5:6? And why repeat 3 times? These things are arbitrary. Harmonics theory follows logically from extremely simple principles - that there are standing waves and that nature must be non-linear.

Clearly the 1:2 ratio is what we call an octave. So, the second harmonic is just the octave. The third harmonic is 1:3, but the ratio between the two is 2:3, which is the 4:6 of the major chord. To add a third note, we have to go to 1:5, since 1:4 is just two "octaves". Then, to get a chord of three notes, we're pretty much forced to go with 4:5:6. As you take this process out further and further (not just three times), the identities repeat--they fall back on themselves almost. That's the approximation that Coelacanth is talking about.

Of course, there are other musical systems. But I didn't see them mentioned in your work.

Originally Posted by grapes

Just tempering, sure. I mean, "just tempering". If you continue that little game, going up or down octaves (doubling a frequency gives the same "note"), there is a discrepancy. The (6/4)^12 should be 2^7 (starting at C, CEG, GBD, DgA, AdE, EaB, Beg, gbd, dFa, aCe, eGb, bDF, FAC, back to C) but it's not, though close.

Quite possibly, the first time this was done, the names were not attached to the notes. But the note in the middle of the 9th chord would've been (6/4)^8(5/4) times the frequency of the original starting note, the first note of the first trio. That's 1.5^8 times 1.25, or 32.036, very close to five octaves, 2^5. It probably seemed like a reasonable thing to call it the same note.

Its a funny thing about the 12th root of two. If you want to try and approximate the most useful ratios for music using some root of 2 then the best roots are 5, 7, 12, 19, 31, ... a kind of Fibonacci series. I don't think that it keeps doing this though.