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In major and minor keys, the perfect cadence is V to I

Will this structure remain the same in a different mode?

I.e. E Phrygian Mode in the key of C (E F G A B C D), the tonic chord is the iii of the key of C major. So, the scale would be iii IV V vi viii(dim) I ii iii (expressed relative to C major). So would viii(dim) to iii be a perfect cadence?

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A couple comments may help.

A perfect cadence is distinguished from an imperfect cadence by its dominant->tonic motion in the melody, not necessarily in the harmony.

When describing the harmonic properties of the mode, it's simpler to consider the example to be in the key of the mode itself, not the relative ionian scale. So E-phrygian is harmonized i ♭II ♭III iv vø ♭VI ♭VII. So, you really can't form a perfect cadence in E-phrygian because there's no D♯ to put in the melody. vø->i might be the closest you can get without altering the scale to create a secondary dominant. Diminished->tonic, even a minor tonic, gives a strong resolution. Or ♭II->i is quite dramatic, too.

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With respect to your second question, I would say in this case yes, it would be a perfect cadence. B --> E is V --> I (key of Emin in this example). This is the same as representing B as the viii(dim) and E as the iii in the key of Cmaj.

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Perfect cadences are all about functional harmony. Harmony with dominant seventh chords that include nice sharp leading notes itching to resolve to the tonic.

Modes need a different language and a different approach.

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