

The centuries-old struggle to play in tune - tkiley
http://www.slate.com/id/2250793/pagenum/all/#p2

======
tkiley
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.

~~~
btilly
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.

~~~
mechanical_fish
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. ;)

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nandemo
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...](http://www.youtube.com/watch?v=BhZpvGSPx6w&feature=related)

~~~
danbmil99
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.

~~~
nandemo
> _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?

~~~
barrkel
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.

~~~
nandemo
> _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.

~~~
barrkel
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.

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tokenadult
Previous submission, submitted with the canonical URL:

<http://news.ycombinator.com/item?id=1283523>

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btilly
Dupe: <http://news.ycombinator.com/item?id=1283523>

