Tuning and temperament.

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Tuning and temperament.

Postby SysExJohn » 04 Apr 2015 17:08

Ever wondered why we tune our keyboard instruments the way we do?
What exactly does a piano tuner do?

Eh? Why would I do that? Hasn't it always been this way?
No, not by a long chalk.

To explore this we need to go back many, many centuries.
To Pythagoras to be precise who, we understand, discovered musical ratios after some experimentation.

He invented a device called a monochord. A long box with a single string streched from end to end.
He also had a moveable 'bridge'. He could position the bridge under the string anywhere he wanted.

What he discovered was that certain ratios produced pleasing sounds.
Precisely in the middle produced a sound exactly in harmony. We call it an octave.
It has a ratio of 2:1, or twice the frequency of the original note.

It's easy arithmetic. 2:1 means multiply by two divide by one.
So if the A below middle C is 440Hz then the next A is 440 x 2 / 1 = 880.
Going the other way we invert the ratio, so 1:2.
440 x 1 / 2 = 220Hz.
Not too difficult was it?

Okay. what about the notes in between?
Well Pythagoras discovered one of the most pleasing other harmonic ratios was 3:2.
We call that ratio a fifth. (Many of you know this already.)
Counting from C as 1, using just the white notes, 2 is D and so on until we reach G. G is the fifth note of the Major scale.
G should be in a ratio of 3:2 to C.
If C were 500Hz (it isn't) then the G above would be 500 x 3 / 2 = 750Hz.
Played with C they would sound very harmonious.

Now if we were to work our way all the way around the circle of fifths,
C to G to D to A and so on,
tuning each new note as the fifth above the previous one, we'd eventually arrive back at C again ... wouldn't we?
And everything would be precisely in tune wouldn't it? Would it? Surely it would, wouldn't it?

Actually, NO!
Why not?

Because seven octaves (the ratio 2:1) is not the same as 12 fifths (ratio 3:2).

Let's use 100Hz as the starting frequency to make the arithmetic easier.
Octaves. 100 x 2 /1 = 200... 200 x 2 /1 =400. then 800, 1600, 3200, 6400, 12800.
Fifths. 100 x 3 /2 = 150... 150 x 3 / 2 = 225... Then 337.5, 506.25, 759.375, 1139.0625, 1708.59375, 2562.890625, 3844.3359375, 5766.50390625, 8649.755859375, 12974.6337890625.
And there it is, we have arrived at C = 12,974.634Hz tuning by fifths, whereas we know it should 12,800.
That's a blow isn't it?
The difference in ratio is known as a Pythagorean Comma in tuning circles.

So what to do?
Well, when tuning e.g. a piano, what we do is we make each fifth slightly out or tune, flatter than it should be.
By a 12th of a Pythagorean comma. We temper each fifth.
If we make each fifth exactly the same proportion flat we call it Equal Temperament or ET.

In the history of tuning many didn't like ET, they described it as bland.
They were messing around a lot with temperaments, tuning each fifth flat by a different amount, around Bach's time, 300 years or so ago.
They came up with lots and lots ... and lots of different ways to tune.
They were used to hearing what was called tone colour or in German, 'Affekt'.
We now believe Bach had his own way which we surmise he described on the title page of his seminal Das Wohltemperirte Clavier (The correctly tuned keyboard).

So, I hate to tell you this, but your keyboard is completely out of tune except for the octaves.

Any use?
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