Logic Pro X Tuning overview
The 12-tone scale used in Western music is a development that took centuries. Hidden in between those 12 notes are a number of other microtones—different frequency intervals between tones.
To explain, look at the harmonic series: Imagine that you have a starting (or fundamental) frequency of 100 Hz (100 vibrations per second). The first harmonic is double that, or 200 Hz. The second harmonic is found at 300 Hz, the third at 400 Hz, and so on. Musically speaking, when the frequency doubles, pitch increases by exactly one octave (in the 12-tone system). The second harmonic (300 Hz) is exactly one octave—and a pure fifth—higher than the fundamental frequency (100 Hz).
From this, you could assume that tuning an instrument so that each fifth is pure would be the way to go. In doing so, you would expect a perfectly tuned scale, as you worked your way from C through to the C above or below.
The following table provides a summary of the various calculations.
Note | Frequency (Hz) | Notes |
|---|---|---|
C | 100 | x 1.5 divided by 2. |
C# | 106.7871 | Divide by 2 to stay in octave. |
D | 112.5 | Divide by 2 to stay in octave. |
D# | 120.1355 | Divide by 2 to stay in octave. |
E | 126.5625 | Divide by 2 to stay in octave. |
F (E#) | 135.1524 | |
F# | 142.3828 | Divide by 2 to stay in octave. |
G | 150 | x 1.5 divided by 2. |
G# | 160.1807 | |
A | 168.75 | |
A# | 180.2032 | |
B | 189.8438 | |
C | 202.7287 |
As you can see from the table, although the laws of physics dictate that the octave above C (100 Hz) is C (at 200 Hz), the practical exercise of a (C to C) circle of perfectly tuned fifths results in a C at 202.7287 Hz. This is not a mathematical error. If this were a real instrument, the results would be clear. As a workaround, choose between the following options:
Each fifth is perfectly tuned, with octaves out of tune.
Each octave is perfectly tuned, with the final fifth (F to C) out of tune.
Detuned octaves are more noticeable to the ears, so your choice should be obvious.