We will use the mathematical ratio to compare two major thirds. For the first third we will use Pythagorean tuning and for the other we will use just intonation (see this article for information on the different tuning systems). In the following table we see the frequencies of the notes and calculate the mathematical ratio of each interval:

Temperament | C4 | E4 | Mathematical ratio | |
---|---|---|---|---|

Major third Pitagorean tuning |
260 | 329.06 | 329.06 / 260 = 1.2656153846 | |

Major third Just intonation |
260 | 325 | 325 / 260 = 1.25 |

The major third of the just intonation is smaller than that of the Pythagorean tuning (1.25 is smaller than 1.2656153846). But the numbers do not help us to understand the difference in size. How big or small is the difference? How noticeable is the difference? Using cents makes the comparison easier.

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