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A better explanation of the problem:


Also, if you listened to samples in the Slate article and couldn't hear any difference, try this:


terrible example, because he's using pure sine tones. What makes intervals sound in or out of tune are the harmonics, which clash if it's not just tuning. No one can hear the difference between just and tempered thirds on a tone with no harmonics -- our aural circuitry is just not that precise.

What do you mean that no one can hear the difference? If you have any musical training you should know to listen for "beats" which happens because as the sounds come in and out of sync. The closer together the notes are to the proper interval, the easier it is to hear those beats on a pair of sustained notes as they come in and out of sync. Even though the tones are so close together than nobody could hear the difference if they were played separately.

The beats aren't as strong as when two slight variations on the same note are played. But they are still easy to hear in that link.

Really? The aural difference was painfully obvious, especially with the thirds and fifths. With the actual songs it was less noticeable because any given chord didn't play for long before changing.

>What makes intervals sound in or out of tune are the harmonics, which clash if it's not just tuning.

How come people use tuning forks and pipes for tuning?

Skip to the bottom of http://paws.kettering.edu/~drussell/Demos/superposition/supe... and looks what happens when you have 2 notes that are almost, but not quite, the same. That results in very easy to hear "beats", which makes it very easy to tell whether or not you have the same exact note. You don't know which direction you are off, but the slower the beating, the closer you are to being right.

Since most of the energy in a note is at the pitch the note is theoretically at, this is easy to hear when comparing a pure tone to a normal musical instrument with lots of harmonics. Furthermore it is actually somewhat harder to hear it than when you are comparing two musical instruments that both have harmonics, because you get more complications you need to ignore in the harmonics when you're listening for that conflict in the base note.

There are two things at work here. There's harmonics, where you're working with a relationship between two or more frequencies; and then there's the absolute frequency, which is needed to get different instruments to harmonize. Tuning forks give you an absolute frequency.

>Tuning forks give you an absolute frequency.

Well, yes, that's the point of my question. If harmonics were essential to the concept of "in tune", we wouldn't tune by using an instrument that has essentially no harmonics.

Harmonics are how you figure other notes are in tune with your absolute note - they're related to how the frequencies interfere. But that only tells you how to tune one note relative to another. It doesn't help you get a bunch of instruments in tune with one another, even if they're in different locations, etc.

Perhaps if you explained your confusion more, it could be answered better.

Because it is convenient :-)

That is a _much_ better explanation, thanks. In particular, the transposition examples -- 16, 17, and 18 -- really made it clear for me.

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