Intonation and tuning of the Classical Guitar
Guitars use what is called the "equal tempered scale" which is also used for most instruments with fixed pitch notes. This tempered scale leads to equally spaced intervals regardless of the tonality used. This is a mathematical solution that divides the scale into 12 equal parts. These divisions do not give us frequencies that provide perfect or beatless 5ths for example, as with some other scales.
This next table shows the frequencies of two of the scales used for tuning instruments, Equal temperament and Just temperament. The examples have been calculated starting on A (220Hz).
| Frequencies variation between Equal and Just temperament | |||||
| Equal Temperament | Just Temperament | Variation | |||
| A | 220.00 Hz | Unison | 1:1 | 220.00 Hz | 0 Hz |
| A# | 233.09 Hz | 25:24 | 229.17 Hz | -3.92 Hz | |
| B | 246.96 Hz | 9:8 | 247.50 Hz | .54 Hz | |
| C | 261.65 Hz | Minor third | 6:5 | 264.00 Hz | 2.34 Hz |
| C# | 277.22 Hz | Major third | 5:4 | 275.00 Hz | -2.22 Hz |
| D | 293.72 Hz | Perfect fourth | 4:3 | 293.33 Hz | -.38 Hz |
| D# | 311.19 Hz | 45:32 | 309.38 Hz | -1.81 Hz | |
| E | 329.71 Hz | Perfect fifth | 3:2 | 330.00 Hz | .29 Hz |
| F | 349.33 Hz | 25:16 / 8:5 | 343.75 Hz | -5.57 Hz | |
| F# | 370.11 Hz | Perfect 6th | 5:3 | 366.67 Hz | -3.44 Hz |
| G | 392.13 Hz | 9:5 | 396.00 Hz | 3.86 Hz | |
| G# | 415.46 Hz | 15:8 | 412.50 Hz | -2.96 Hz | |
| A | 440.18 Hz | Octave | 2:1 | 440.00 Hz | -.18 Hz |
When a player uses harmonics to tune the guitar, the frequencies of the harmonics are not the same as the equivalent fretted notes, except in the case of the octave. How to tune a guitar is a personal thing, but it would help if the player understood that string harmonics are not the same as the fretted note in an equal tempered scale.
The equal tempered scale, because of its equal spacing of notes allows the player to modulate between keys without having to retune the instrument, but it has it problems in that the perfect 4ths and 5ths are not perfect.
If the intonation is 'out' on a guitar, this will be even worse and is most noticeable. Fret placement on the guitar using the equal temperament scale, is determined by what is commonly called the rule of 17. The fret position calculation is simple and is best described using an example.
