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What is a mediant?

What is a mediant? Here are some definitions.

Noun
  1. (music) The third degree of the diatonic scale.
  2. (mathematics) A rational number whose numerator is the sum of the numerators of two other given rational numbers and whose denominator is the sum of the denominators of those same two other rational numbers.
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Examples
Further along it requires the reader to understand the cycle of fifths and key relationships such as the mediant.
In an interrupted cadence, the lower mediant is substituted for what would have been the new tonic.
The Mass in C has a tonal plan based on mediant and submediant.
A minor, therefore, is related to a major key with its tonic on C, the mediant or third degree of the scale of A minor.
The true augmented sixth and the true cadence gain in significance as a contradiction to the false cadencing around the mediant.

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