equal temperament....explaination in simple terms

[iconic]

Please no large words.....i think its an applied system to an instrument inherently out of tune intonation wise to what are considered perfect intervals.....guessing though to be honest.

Edited July 26, 2013 by iconic

[Guest bassman7755]

OK try this: in order to be able to play in any key and have the exact same relative tuning of the musical intervals (e.g. to have a "major 3rd" that sounds the same in the key of C as it does in the key in of C#) its necessary to have perfectly uniformly spaced semitones.

This system of uniformly spaced semitones is called "equal temperament".

The problem is that natural overtones do not exactly line up with these uniformly spaced semitones.

On guitars the major 3rd is the problem interval - for example if you tune the G and B strings to that they "ring" nicely when playing them on the same fret then you will be quite out of tune with respect to equal tempered tuning.

Edited July 26, 2013 by bassman7755

[paul_5]

pre Bach (the chap who devised our 'tempered scale' system notes were worked out mathematically (based on Pythagoras - the triangle guy), so that (for example) C# and Db were slightly different in pitch from each other which meant that changing key could be a bit of a bugger, as the keyboard instrument (spinnet, virginal, harpsicord etc...) had to be tuned to the new key. What Bach did was to work out a pitch that sat equally in between the two, and just use that for both pitches.

Bach then went on to publish the 'Well Tempered Clavier' to demonstrate his system. The books featured keyboard pieces across all of the keys that could be played without retuning the instrument.

Bingo!

Do you know Bach's organ works?








of course it does - he had 23 children!

[dlloyd]

It can be quite a complicated subject, but basically "consonance", when two notes sound good together, comes from the harmonic overtone series.

Rest your finger at the 12th fret of the A string (exactly half the length of the string), pluck the string and remove your finger... you get a note that is higher (a harmonic) than the note you would normally get on an open string... without going too far into the mechanics of it, that higher note is twice the frequency of the open note... or what we call an octave.

Do it again at the 7th fret (exactly third the length of the string)... same story, but even higher... you get a note that is 3 times the frequency of the open note... or what we call a perfect 12th... or a perfect fifth above the original harmonic. It's also a bit quieter than the original octave harmonic.

And so it goes on, getting higher and higher and quieter and quieter.

If you keep doing this, you find that the harmonics that end up being the loudest form the intervals that sound the best when played together because they complement each others' overtones.

The most widely used scale in western music is the major scale... it's formed from combinations of low order harmonics and produces nice, consonant melodies and harmonies. Without explaining the maths behind it (which aren't complicated), you get a scale comprised of the following intervals:

1: 1/1
2: 9/8
3: 5/4
4: 4/3
5: 3/2
6: 5/3
7: 15/8
8: 2/1

So if you wanted to create a major scale from A (55Hz), you'd get:

A = 55.00 Hz
B = 61.88 Hz (= 9/8 x 55)
C# = 68.75 Hz
D = 73.33 Hz
E = 82.50 Hz
F# = 91.67 Hz
G# = 103.13 Hz
A = 110.00 Hz

Sounds lovely, but what if I want to play in B major? Let's assume we're filling out a piano keyboard... we're starting from the B we got from the A=55Hz scale we've already built:

B = 61.88 Hz
C# = 69.61 Hz
D# = 77.34 Hz
E = 82.50 Hz
F# = 92.81 Hz
G# = 103.13 Hz
A# = 116.02 Hz
B = 123.75 Hz

Some of the notes we don't have, so we can add extra notes (black keys or whatever), and some of the existing notes are okay, like the B, E and G#, but look:

C#, 5/4 x A 55Hz = 68.75 Hz
C#, 9/8 x B 61.88 Hz = 69.61 Hz

F#, 5/3 x A 55 Hz = 91.67 Hz
F#, 3/2 x B 61.88 Hz = 92.81 Hz

If we use the existing C# and F# from the A major scale, B major will sound terrible!

In fact, every major scale except for A major will sound terrible.

We get past this by reducing the consonance of the A major scale to make it sound just "quite good" instead of lovely so that every other key sounds "quite good" instead of terrible. We do this by making all semitones equal. It's a log2 scale, so we use a formula reflecting that

1: 2^(0/12)
2: 2^(2/12)
3: 2^(4/12)
4: 2^(5/12)
5: 2^(7/12)
6: 2^(9/12)
7: 2^(11/12)
8: 2^(12/12)

The 2 comes from the doubling of frequency per octave. The denominator, 12 comes from the number of subdivisions we want (like frets, semitones) and the numerator is the number of frets/semitones above the reference note. All semitone intervals sound equal now so we call it 12 tone equal temperament.

[iconic]

Wow....thanks guys...excellent stuff

[essexbasscat]

[quote name='dlloyd' timestamp='1374842214' post='2153927']
It can be quite a complicated subject, but basically "consonance", when two notes sound good together, comes from the harmonic overtone series.

Rest your finger at the 12th fret of the A string (exactly half the length of the string), pluck the string and remove your finger... you get a note that is higher (a harmonic) than the note you would normally get on an open string... without going too far into the mechanics of it, that higher note is twice the frequency of the open note... or what we call an octave.

Do it again at the 7th fret (exactly third the length of the string)... same story, but even higher... you get a note that is 3 times the frequency of the open note... or what we call a perfect 12th... or a perfect fifth above the original harmonic. It's also a bit quieter than the original octave harmonic.

And so it goes on, getting higher and higher and quieter and quieter.

If you keep doing this, you find that the harmonics that end up being the loudest form the intervals that sound the best when played together because they complement each others' overtones.

The most widely used scale in western music is the major scale... it's formed from combinations of low order harmonics and produces nice, consonant melodies and harmonies. Without explaining the maths behind it (which aren't complicated), you get a scale comprised of the following intervals:

1: 1/1
2: 9/8
3: 5/4
4: 4/3
5: 3/2
6: 5/3
7: 15/8
8: 2/1

So if you wanted to create a major scale from A (55Hz), you'd get:

A = 55.00 Hz
B = 61.88 Hz (= 9/8 x 55)
C# = 68.75 Hz
D = 73.33 Hz
E = 82.50 Hz
F# = 91.67 Hz
G# = 103.13 Hz
A = 110.00 Hz

Sounds lovely, but what if I want to play in B major? Let's assume we're filling out a piano keyboard... we're starting from the B we got from the A=55Hz scale we've already built:

B = 61.88 Hz
C# = 69.61 Hz
D# = 77.34 Hz
E = 82.50 Hz
F# = 92.81 Hz
G# = 103.13 Hz
A# = 116.02 Hz
B = 123.75 Hz

Some of the notes we don't have, so we can add extra notes (black keys or whatever), and some of the existing notes are okay, like the B, E and G#, but look:

C#, 5/4 x A 55Hz = 68.75 Hz
C#, 9/8 x B 61.88 Hz = 69.61 Hz

F#, 5/3 x A 55 Hz = 91.67 Hz
F#, 3/2 x B 61.88 Hz = 92.81 Hz

If we use the existing C# and F# from the A major scale, B major will sound terrible!

In fact, every major scale except for A major will sound terrible.

We get past this by reducing the consonance of the A major scale to make it sound just "quite good" instead of lovely so that every other key sounds "quite good" instead of terrible. We do this by making all semitones equal. It's a log2 scale, so we use a formula reflecting that

1: 2^(0/12)
2: 2^(2/12)
3: 2^(4/12)
4: 2^(5/12)
5: 2^(7/12)
6: 2^(9/12)
7: 2^(11/12)
8: 2^(12/12)

The 2 comes from the doubling of frequency per octave. The denominator, 12 comes from the number of subdivisions we want (like frets, semitones) and the numerator is the number of frets/semitones above the reference note. All semitone intervals sound equal now so we call it 12 tone equal temperament.
[/quote]

A big Thank You for taking the time to write that post great stuff

[benthos]

Thanks, paul_5 and dlloyd for expanding my understanding of this. Fascinating stuff

[skej21]

[quote name='iconic' timestamp='1374821275' post='2153533']
Please no large words.....i think its an applied system to an instrument inherently out of tune intonation wise to what are considered perfect intervals.....guessing though to be honest.
[/quote]

I'd try an explain but this is so much easier to understand.

WARNING! Watching this video WILL make everything you listen to immediately afterwards sound dreadfully out of tune lol.

http://youtu.be/BhZpvGSPx6w

Edited July 27, 2013 by skej21

[paul_5]

[quote name='skej21' timestamp='1374925949' post='2154841']
I'd try an explain but this is so much easier to understand.

WARNING! Watching this video WILL make everything you listen to immediately afterwards sound dreadfully out of tune lol.

[media]http://youtu.be/BhZpvGSPx6w[/media]
[/quote]


AAARRRGGGHHHHH! EVERYTHING'S WRONG!

[iconic]

Years of kart racing, moto !, enduro, trials and over a thousand miles of track days maybe why i cant hear a difference.....i will try again tomorrow.