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 interval and ratio in cents

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

1200(semitones/edo-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