Tuning math tools

The following is a list of basic, accessible tools to find the exact differences between frequencies (440hz, 432), ratios (3:2, 7:4), and cents.

Difference between frequencies in cents

To find the difference between frequencies in cents, use this formula (works on Google).

1200log ₂(hz/hz)

Example

The difference between 440hz and 432hz is roughly 32 cents.

1200log ₂(440/432) ≈ 31.77

Frequency ± cents

To add or subtract cents from a frequency, use this formula (works on Google).

frequency(2^(±cents/1200))

Examples

• 440hz + 50 cents is approximately 453hz

440(2^(50/1200)) ≈ 452.89

• 440hz - 50 cents is approximately 427hz

440(2^(-50/1200)) ≈ 427.47

Ratio in cents

To find the amount of cents in a ratio, use this formula (works on Google). Make sure to write the ratio with "/" on a calculator instead of ":".

1200log ₂(ratio)

Example

The amount of cents in a 3:2 ratio is approximately 702 cents.

1200log ₂(3/2) = 701.96

Difference between an interval and ratio in cents

To find the difference between an interval and ratio in cents, use this formula (works on Google). Make sure to write the ratio with "/" on a calculator instead of ":".

1200(semitones/12-log ₂(ratio)

Examples

• The difference between a perfect 5th in 12tet and a justly-tuned fifth (3:2) is approximately 2 cents.

1200(7/12-log ₂(3/2) ≈ -1.96

• The difference between a major third in 17tet and a justly-tuned major third (5:4) is approximately 37 cents. Thank goodness most western music uses 12 notes per octave instead of 17, which would sound like absolute garbage. The mikes wouldn't mind such a reality though.

1200(6/17-log ₂(5/4) ≈ 37.22