equal temperament tuning system

equal temperament tuning system

Here are the frequencies of all the strings on a standard 6-string guitar in standard tuning (EADGBE), showing the open string (fret 0) and notes up to the 12th fret (one octave higher):


6th String (E2, 82.41 Hz)

FretNoteFrequency (Hz)
0E282.41
1F87.31
2F#92.50
3G98.00
4G#103.83
5A110.00
6A#116.54
7B123.47
8C130.81
9C#138.59
10D146.83
11D#155.56
12E3164.81

5th String (A2, 110.00 Hz)

FretNoteFrequency (Hz)
0A2110.00
1A#116.54
2B123.47
3C130.81
4C#138.59
5D146.83
6D#155.56
7E164.81
8F174.61
9F#185.00
10G196.00
11G#207.65
12A3220.00

4th String (D3, 146.83 Hz)

FretNoteFrequency (Hz)
0D3146.83
1D#155.56
2E164.81
3F174.61
4F#185.00
5G196.00
6G#207.65
7A220.00
8A#233.08
9B246.94
10C261.63
11C#277.18
12D4293.66

3rd String (G3, 196.00 Hz)

FretNoteFrequency (Hz)
0G3196.00
1G#207.65
2A220.00
3A#233.08
4B246.94
5C261.63
6C#277.18
7D293.66
8D#311.13
9E329.63
10F349.23
11F#369.99
12G4392.00

2nd String (B3, 246.94 Hz)

FretNoteFrequency (Hz)
0B3246.94
1C261.63
2C#277.18
3D293.66
4D#311.13
5E329.63
6F349.23
7F#369.99
8G392.00
9G#415.30
10A440.00
11A#466.16
12B4493.88

1st String (E4, 329.63 Hz)

FretNoteFrequency (Hz)
0E4329.63
1F349.23
2F#369.99
3G392.00
4G#415.30
5A440.00
6A#466.16
7B493.88
8C523.25
9C#554.37
10D587.33
11D#622.25
12E5659.26

🎸

The formula for calculating the frequencies of musical notes on a guitar or any other musical instrument is based on the equal temperament tuning system, which divides an octave into 12 equal parts (semitones).

Formula:

Components:

  1. fn – The frequency of the note at the n-th fret.
  2. f0 – The frequency of the open string (fret 0). For the 6th string, f0 = 82.4 Hz.
  3. n – The number of frets from the open string (e.g., n = 1 for the 1st fret, n = 12 for the 12th fret).
  4. 2n/12 – The factor by which the frequency increases as you move n frets up.
    • In equal temperament tuning, moving up 12 frets doubles the frequency (an octave).
    • Therefore, for each fret, the frequency increases by the 12th root of 2 ≈ 1.059463.

Step-by-Step Explanation:

  1. Octave Relationship:
    • Doubling the frequency raises the note by one octave.
    • Example: Open E2 is 82.41 Hz, and at the 12th fret (one octave higher), the note E3 is 82.41 × 2 = 164.8282
  2. Dividing the Octave:
    • In the equal temperament system, the octave is divided into 12 semitones.
    • Each semitone increases the frequency by the 12th root of 2 ≈ 1.059463
  3. General Formula for Any Fret:
    • For nn frets, the frequency is scaled by 2n/12
    • Multiply this factor by the frequency of the open string (f0) to get the frequency at the n-th fret.

Example Calculation:

For the 6th string (E2, f0 = 82.41 Hz):

  • 1st fret (F): f1 = 82.41 × 21/12 = 82.41 × 1.059463 = 87.31 Hz
  • 5th fret (A): f5 = 82.41 × 25/12 = 82.41 × 1.33484 = 110.00 Hz
  • 12th fret (E3): f12 = 82.41 × 212/12 = 82.41 × 2 = 164.82  Hz

This formula works for any string or instrument tuned to equal temperament.