The centuries-old struggle to play in tune | Hacker News Reader

The centuries-old struggle to play in tune

slate.com

26 pointstkiley17 comments

Quick mathematical summary of the problem:

An octave is a 2:1 ratio between frequencies, so 880 hz is one octave above 440 hz. A perfect fifth is a 3:2 ratio between frequencies, so 660 hz is a perfect fifth above 440.

In the modern western system of music, twelve perfect fifths is harmonically equal to seven octaves. In other words,

(2/1)7 == (3/2)12

Unfortunately, we know this is mathematically untrue.

Furthermore, three major thirds is harmonically equal to one octave:

(2/1) == (5/4)3

This also is mathematically untrue.

Hilarity ensues.

I'll also summarize the advantage of equal temperament:

Regardless of what key the song is in, a certain interval is always the same exact ratio. A major third in the key of F is the same as a major third in the key of Bb. This is good for instruments like the guitar and piano, which aren't made or tuned for a single immovable key. Contrast that with harmonicas, for examples, each of which is made for only a certain key.

You are using multiplication where you should be using exponentiation. If one fifth is (3/2) then 2 fifths is (3/2) times (3/2) which is (9/4). Your math looked semi-plausible when comparing fifths and octaves, but comparing 3 thirds and an octave it was way off.

Therefore the first comparison should be 7 octaves which is (2/1)^7 = 128, versus 12 perfect fifths which is (3/2)^12 = 531441/4096 = 129+3057/4096 = 129.746337890625.

Similarly for thirds, you're comparing one octave (2/1) = 1 with 3 thirds (5/4)^3 = 125/128 = 1.953125.

As you can see, the ratios are close, but not quite right. Hence the problem.

I think we just lost some carats in the mix. The original comment meant to type exponentials but they didn't come out right on the page.

Someday HTML will support TeX and we'll never have this problem again. ;)

A better explanation of the problem:

http://www.yuvalnov.org/temperament/

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

http://www.youtube.com/watch?v=BhZpvGSPx6w&feature=relat...

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.

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?

Because it is convenient :-)

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.

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.

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.

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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