I've read about this (temperament) many times, and I still have no idea what they are talking about, and I assume it's because I just can't hear the difference.
I'm not tone deaf, but I guess I'm not very good at tones.
Virtually no untrained ear can hear the difference. It's perfectly normal. Now, if you really have no clue, I can give you a mathematical intuition, so your brain understand what your ear can't hear.
(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 :-).
The actual note doesn't matter at all. Just the ratio. We do have a reference note, but this is only for convenience. (Indeed, the reference note in Europe progressively changed, from 415Hz to 440Hz).
I'm not sure the ratio of two notes played one after the other really counts by itself. However in many instruments, two consecutive notes will tend to overlap (piano, for instance). Also, many instruments (especially those with strings, like claviers and violins), resonate better when the note you play has a "good" ratio with the natural notes of the instruments, even if you don't play them! Some instruments were even designed around this principle. So you have to maintain a good ratio with respect to these "base" note at all times.
In practice, the ratio of two notes played one after another is indeed important.
I'm not sure I understand your last question. Actually, you can change the tune of a piano. So, the optimization you speak of is possible even on a modern piano. Just re-tune it.
A final note about why "simple ratios" sound better: When you make a string resonate, it doesn't do only one note. It does its base frequency (say f), and many others (every multiple of f). So, in a piano, when you play a G at 100Hz, it also plays at 200Hz (a G), 300Hz (a D), 400Hz (a G) etc. Note that there exist actual keys whose main frequencies are 200Hz, 300Hz, 400Hz and so on. They will resonate, which will make the sound richer, louder.
The problem with equal temperament is that the only ratio which is really respected is the octave (2 to 1). In such a case, the D I mentioned above won't be quite at 300Hz, and won't resonate well. So equal temperament became practical only when instruments became loud enough to make up for the loss in resonance.
If you are mathematically inclined, this might help. It's how I, as a non-musician, came to understand tuning a little.
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
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.
Comments
I feel sad that I simply can not hear this.
I've read about this (temperament) many times, and I still have no idea what they are talking about, and I assume it's because I just can't hear the difference.
I'm not tone deaf, but I guess I'm not very good at tones.
Virtually no untrained ear can hear the difference. It's perfectly normal. Now, if you really have no clue, I can give you a mathematical intuition, so your brain understand what your ear can't hear.
(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).
Thanks it does help. So the actual note doesn't matter? Just the ratio of the notes when played together?
What about the ratio of two notes played on after the other? Is that very important too?
Could you make a piano with multiple keys each tuned to match a particular ratio better?
The actual note doesn't matter at all. Just the ratio. We do have a reference note, but this is only for convenience. (Indeed, the reference note in Europe progressively changed, from 415Hz to 440Hz).
I'm not sure the ratio of two notes played one after the other really counts by itself. However in many instruments, two consecutive notes will tend to overlap (piano, for instance). Also, many instruments (especially those with strings, like claviers and violins), resonate better when the note you play has a "good" ratio with the natural notes of the instruments, even if you don't play them! Some instruments were even designed around this principle. So you have to maintain a good ratio with respect to these "base" note at all times.
In practice, the ratio of two notes played one after another is indeed important.
I'm not sure I understand your last question. Actually, you can change the tune of a piano. So, the optimization you speak of is possible even on a modern piano. Just re-tune it.
A final note about why "simple ratios" sound better: When you make a string resonate, it doesn't do only one note. It does its base frequency (say f), and many others (every multiple of f). So, in a piano, when you play a G at 100Hz, it also plays at 200Hz (a G), 300Hz (a D), 400Hz (a G) etc. Note that there exist actual keys whose main frequencies are 200Hz, 300Hz, 400Hz and so on. They will resonate, which will make the sound richer, louder.
The problem with equal temperament is that the only ratio which is really respected is the octave (2 to 1). In such a case, the D I mentioned above won't be quite at 300Hz, and won't resonate well. So equal temperament became practical only when instruments became loud enough to make up for the loss in resonance.
If you are mathematically inclined, this might help. It's how I, as a non-musician, came to understand tuning a little.
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.