The centuries-old struggle to play in tune (opens in new tab)

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Who votes these up? Don't we all know by now that clicking the "Print this article" or "Single page" link on a multi-page article will transport us to that magical single page?

(In case you're wondering, I didn't vote the parent down.)

    http://www.cerlsoundgroup.org/Continuum/

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

(1) Axiom: We hear at a logarithmic scale. What we perceive to be a difference (or interval) between 2 notes is actually a ratio of frequencies. (I think biology may explain this axiom.)

(2) Axiom: Say you hear 2 notes, of frequencies f1 and f2 respectively. When the f1/f2 ratio is a simple rational number (like 2) or (3/2), it sounds good. When the ratio is more complicated (like 19/17), it sounds worse. (Physics can explain that axiom.)

(3) Definition: when the f1/f2 ratio is 2, we call that an octave. The 3/2 ratio is a fifth. The 4/3 ratio is a fourth. As a side note these ratio were basically the only ones that were used. They didn't really used thirds or sixths, probably because of their more complicated ratios, which may have sounded bad to their ears. [1]

(4) Theorem: There is no way in hell you can make an octave out of fifths (they won't perfectly tune together). Informal proof: this is because you can't find any (i,n) ∈ ℤ², such that (3/2)ⁱ = 2ⁿ. As a side note, you can come relatively close: (3/2)¹² = 129,75 which is close to 2⁷.

So, a mathematical impossibility prevents you to perfectly tune the two most basic intervals ever. Ouch. We have to compromise, then. We can sacrifice a few chords (which if played will cause severe ear damage); or cheating a little bit on every ratio, so no chord sounds outright wrong, nor exactly right; or we can try to find a middle ground between these two extremes.

The "sacrifice" strategies was originally favoured. They sounded better, but restrained what you could play. Now, we favour the "cheating a little" strategy (also called "equal temperament"). They give you more liberty, but sounds rather dull on old music meant to be played with an old fashioned tuning (to trained ears, at least :-).

Hope this helped.

[1]: http://pipolitics.com/video-streaming/kaamelott-saison-2-epi... is an excellent joke on the topic. (This is a flash video in French, unfortunately. I hope you understand it, or can find a friend who does).

Equal temperament uses frequencies that are scaled by powers of the twelfth root of 2. Moving up one semitone is the same as multiplying the frequency by 2^(1/12). If you look at this table

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

you can see how the resulting values approximate certain ratios. Ratios matter because two tones that are related by a simple ratio will tend to have greater harmonic relationship and be more 'consonant'.

Other tunings emphasize different ratios and give up the equal spacing of notes -- equal spacing when plotted by frequency on a logarithmic scale.

It's really frustrating to try and figure out the precise notes - but I can improvise fairly well on my own or with another string. I definitely need to train my ear better, but knowing that there's this interval problem now - it'll probably help, because I know that there's something I'm going to have to adjust for.