What is the difference between the mode and the modal interval?

MATH:

The mode is calculated for individual numbers while the modal interval is calculated for ranges (intervals) of numbers.

Let's assume the following data set: 1, 2, 3, 4, 5, 6, 6, 6, 6, 13, 13, 13, 18, 22, 22.

Now we organize that set into intervals of five:

1-5: 5, 6-10: 4, 11-15: 3, 16-20: 1, 21-25: 2.

The mode is the number 6 (it occurs four times) while the modal interval is the 1-5 interval (which has five numbers, more than any other interval)

MUSIC:

In music, a "modal interval" exists only in the context of voice imitation.

If, for instance, there is a 3- or 4-voice fugue being played or sung, and the first interval of the first voice sung is a fifth, then traditionally the second voice will answer with a fourth.

An example of it being the other way (a fourth answered with a fifth) can be found in Bach's fugue in C minor from the Well Tempered Keyboard. This is called the "modal answer" or the "tonal answer".

In music of the various ecclesiastical modes the "modal interval" is distinguished from the non-modal interval in that it is seen as a good example of voice leading (minor seconds and thirds are seen as good modal intervals in the Phrygian mode, for instance; tritones are never good modal intervals).