Twelve-Tone Equal Temerament (12-TET).

[Correlli]

I posted this stuff over a year ago. Thought I might post it again as it's very interesting stuff (I think anyway).

Source: Wikipedia



Western music is based on the Twelve-Tone Equal Temerament (12-TET). This is a schema by which an Octave is divided into series of equal frequency ratios. The calculation for the below values is a bit complex, so I havn't gone into much detail about that.


12-TET Decimal Values
--------------------------------
Unison - 1
Minor second - 1.059463
Major second - 1.122462
Minor third - 1.189207
Major third - 1.259921
Perfect fourth - 1.334840
Diminished - 1.414214
Perfect fifth - 1.498307
Minor sixth - 1.587401
Major sixth - 1.681793
Minor seventh - 1.781797
Major seventh - 1.887749
Octave - 2.000000


Examples:

(Open A)55hz * 1.498307 = (D) 82.04hz
440hz * 2.0 = 880hz


Bass guitar open string frequencies
-----------------------------------
B = 31hz

E = 41hz

A = 55hz

D = 73hz

G = 98hz

C = 131hz


There is also another shema called Just Intonation Tuning, which is not as flexible as 12-TET.

(0) 1:1 - unison
(1) 135:128 - major chroma
(2) 9:8 - major second
(3) 6:5 - minor third
(4) 5:4 - major third
(5) 4:3 - perfect fourth
(6) 45:32 - diatonic tritone
(7) 3:2 - perfect fifth
(8) 8:5 - minor sixth
(9) 27:16 - major sixth
(10) 9:5 - minor seventh
(11) 15:8 - major seventh
(12) 2:1 - octave

Example:

55hz * (3/2) = 82.5Hz

[dlloyd]

Western music is based on the Twelve-Tone Equal Temerament (12-TET). This is a schema by which an Octave is divided into series of equal frequency ratios. The calculation for the below values is a bit complex, so I havn't gone into much detail about that.

It's not that difficult.

If you plot perceived pitch against frequency, you find it follows a logarithmic curve to the base 2.

That sounds a bit tricky, but all it means is that each time you go up an octave, pitch is doubled.

A1 (open A on the bass) = 55 Hz
A2 (second fret on G string) = 110 Hz
A3 (14th fret, G string) = 220 Hz
A4 = 440 Hz
A5 = 880 Hz, etc.

A2 = A1*2
A3 = A1*4 (or A1*2*2 = A*2^2)

(that "^2" means "squared" in the absence of superscripts)

A4 = A1*8 (or A1*2*2*2 = A*2^3)
A5 = A1*16 (or A1*2*2*2*2 = A*2^4)

And (because this becomes important below)...

A2 = A1*2 ( = A1*2^1)

But what about steps smaller than an octave? In equal temperament, which is how most guitars are tuned, the octave is divided into 12 equal steps.

If we take the octave as being 2^1 times greater than the starting pitch, as shown above, the semitone is 2^(1/12) times greater than the starting pitch.

This 2^(1/12) (also described as the twelfth root of 2) is kind of a magic number when it comes to equal temperament instruments as it also defines string lengths... the length of the string from the first fret to the bridge is equal to that of the nut to the bridge divided by 2^(1/12)

[dlloyd]

Originally Posted by Kiwi Kid

Examples:

(Open A)55hz * 1.498307 = (D) 82.04hz

There's a couple of things wrong with your calculation there

The first thing is that 55 Hz * 1.498307 = 82.41 Hz

The second thing is that a perfect fifth of A is E, not D.

Bass guitar open string frequencies
-----------------------------------
B = 31hz

E = 41hz

A = 55hz

D = 73hz

G = 98hz

C = 131hz

Well, those are approximate values. But you're better to calculate them from A1=55 Hz.

For instance, 73 Hz is 10 cents flat from D's actual tempered value.

[gsys]

One of the many reasons I wish I owned a fretless - so I can have the same pitch control I have on my upright on an electric bass. All I can do with frets is bend things upward. If I want to play a consonant major third above some note or other, I can't very easily now, can I.

[dlloyd]

That's right, a tempered major third is 14 cents sharp of the pure major third.

[lemur821]

There is also another shema called Just Intonation Tuning, which is not as flexible as 12-TET.

(0) 1:1 - unison
(1) 135:128 - major chroma
(2) 9:8 - major second
(3) 6:5 - minor third
(4) 5:4 - major third
(5) 4:3 - perfect fourth
(6) 45:32 - diatonic tritone
(7) 3:2 - perfect fifth
(8) 8:5 - minor sixth
(9) 27:16 - major sixth
(10) 9:5 - minor seventh
(11) 15:8 - major seventh
(12) 2:1 - octave

This could use a little more explanation. There are lots of schema, usually called temperaments or tunings, all dividing the octave in different ways. 12-TET (or 12EDO (Equal Divisions of the Octave), as I prefer to call it) is one. We can also do 19EDO, one of my personal favorites, or 31, or 7 -- any number you like, in fact. The great thing about tunings that divide the octave into equal steps is that you can transpose to your heart's content and the song will sound pretty much the same.

The octave can also be divided unequally. Just temperaments, which, as Kiwi said, derive each note of the scale from an exact frequency ratio (a low-numbered fraction) are one set of tunings which do that. The advantage to those exact ratios is that intervals will be more in tune, and sound better to most listeners. Unfortunately, the unequal steps of the scale mean that you can't transpose and have the music sound the same. It would be like playing a tune in a different mode, rather than in the same mode but higher. That's what Kiwi means when he says just intonation is less flexible.

EDIT: Oh, and another great thing about equal temperaments is that fretted instruments (like the bass) can simply place long frets across the fretboard: the way all our basses are fretted. Tunings with unequal steps have to have funny constellations of frets. Harder to make, and trouble if you like to bend notes.

This is one of my favorite subjects. Taken as a whole, music in tunings other than 12EDO is called "microtonal." It's not really a good name, since it implies that 12EDO is normal and other tunings aren't, but it's currently the most widely known name for it.