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Comment by utunga | Hacker News Reader

Wow - that's fascinating.. sounds like you are saying that 'sometimes even a perfectly tuned piano is not hitting the 'right' pitch for a song' is that right? From a music theory perspective I'd love to know more.. Can you provide any links for an interested amateur to learn more?

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Pianos are tuned to an equal-tempered scale-- essentially (and, of course, over-simply) you "spread out" the "out-of-tuneness" that results from the fact that the Western 12-tone chromatic scale doesn't fit inside a perfect system of frequency ratios. See here for lots of math that I don't understand at all:

http://en.wikipedia.org/wiki/Equal_temperament

As a side (historical) note, one of the earliest champions of Equal Temperament (or Well-Temperament) was Bach, who wrote The Well Tempered Clavier as a kind of advertisement for this tuning system. Prior to systematized equal temperament, it was impossible for an instrument to play in all 24 keys without being re-tuned. On a Well-Tempered instrument, you can play straight through all 24-keys without stopping to be retuned.

Singers (and brass players, and any other instrument that allows for fine, sinlge-cent level tuning) often use a different system called Just Intonation:

http://en.wikipedia.org/wiki/Just_intonation

Just Intonation is "more in tune" than Equal-temperament, but it is not noticeable by most people. I teach music for a living and have a degree in music theory and I have a very difficult time telling the difference.

EDIT: Yes, I did oversimplify by conflating equal temperament and well-temperament, but in the modern debate, they do tend to get lumped together.

As I understand it, "well temperament" is something of a blanket term for many systems that attempted to find a universal tuning. We don't really know how Bach's clavier was tuned for his notion of well temperament.

As noted, pitch is insanely complicated. My knowledge is incomplete, but I'll try and clear up some things.

First off, an A is not always the same as another A. Some time in the last century (wikipedia if you want the exact date, in fact, if you want further reading, just wikipedia some key phrases from this response) it was decided upon that concert A should be 440Hz - this, as I understand it, was largely in search of consistency between manufacturers. Before that, A had been set at a number of points, going down into the 40teens.

That's only the beginning of complications. As it turns out, even if you can get everyone to agree on a base frequency, different methods of calculating the proper frequency yield different results. Pythagorean Tuning uses whole number ratios to calculate the frequencies, starting with 3:2 (a perfect fifth). [Quick note: if you've never seen Donald Duck in Mathmagic Land, youtube it now] But this method has the unfortunate outcome of giving slightly different values for the diminished fifth and the augmented fourth - which are theoretically the same note.

To ancient musicians, this was like getting a wooden train set and putting the curved pieces together in sequence, only to find that they didn't form a circle. The proverbial last piece is known as the wolf interval*.

So people started to fudge it. This is called temperament. There's a bunch of different ways (mathematically and mechanically) to do it, but I don't really have the time or understanding to get into all of them so I'll skip forward to present day. If you walk into your local music shop and play a chromatic scale on a random keyboard, you'll likely hear what's called Equal Temperament. In short, divide your octave into equal parts (in the case of this example, 12) and you have your 12 notes to a chromatic scale.

And it's just that simple. Of course I skipped over tons, but I'm similarly on my way out the door, so deal with it. Or e-mail me if you really care.

If you're not a musician, producer, technician, or the like, [http://en.wikipedia.org/wiki/Music_and_mathematics] should be more than you'll ever need to know. In fact, I wish I had known about that page 10 years ago.

Not enough? This is the textbook I used in school on the topic. It's a little less historical and more sciencey, if that's your bag: http://www.amazon.com/Acoustical-Foundations-Music-John-Back...

Another complication to note is that pitch != frequency. Pitch is how we perceive the thing. The same frequency at different volumes can be perceived as different pitches. So you really can't ever find perfect without knowing some exact context.

See wikipedia entry on pitch perception for more examples:

  http://en.wikipedia.org/wiki/Pitch_(music)#Perception_of_pitch

It turns out that there is really no such thing as a "perfectly tuned piano". Something is always a little dissonant. Different tunings can move this dissonance around, but (on a standard 12-notes-per-octave keyboard) they cannot eliminate it.

Here is a nice article from Slate about all this:

http://www.slate.com/id/2250793/

(EDIT: GMTA, it seems.)

I'm a complete amateur but have come across different scales, different tuning intervals and such - http://www.midicode.com/tunings/temperament.shtml this gives an almost understandable rendition of what's happening.

I found this article quite interesting...

http://www.slate.com/id/2250793/pagenum/all

It covers the history of tuning systems and how many different definitions there are of 'in tune'.

For an in depth and also entertaining history of equal temperament (and some other tuning schemes) you should check out Temperament: How Music Became a Battleground for the Great Minds of Western Civilization by Stuart Isacoff.

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