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I was just listening to a rock song that is in a minor key, and the phrase ends with a v moving to a ♭VI chord.

Since the dominant chord is taken from the natural minor and doesn't contain the leading tone, is this still a deceptive cadence?

My theory textbook says that a deceptive cadence needs a chord with the leading tone followed by any chord but the tonic. It also says a chord without the leading tone followed by any chord but the tonic is a half cadence. Because the roots follow that of a deceptive cadence, does it make more sense to analyze it as that rather than a half cadence?

From 28-32 seconds. Song is in A minor, chord progression is Am-G-Em-F.

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Since in rock, you can put together any 4 diatonic triads (the diminished triad usually excluded), loop them, and call it a chord progression, I'm hesitant to use functional labels here. To call the cadence "deceptive" is to imply that the alternative progression Am-G-Em-Am was expected, or at least would work. What do you think? The classical trick of first writing a deceptive and then resolving to the corresponding authentic cadence: it is quite foreign to this style, which instead chooses a chord progression and sticks to it.

That said, there are popular songs that use deceptive cadences where the dominant chord lacks a leading tone, see e.g. In Every Age (youtu.be/nR3I-McnF5c), 0:15: VII-VI is deceptive, VII-i is authentic.

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Are you asking about "classical" cadences?

If so, then you really need two things:

  • the leading tone
  • the actual end of a phrase of music

This excludes the "plagal" cadence as either: not a cadence, but a coda-like gesture after a perfect cadence, or modal not tonal music.

My theory textbook says that a deceptive cadence needs a chord with the leading tone followed by any chord but the tonic.

Yes, that's a common way to state it. Although, by far, the second chord is the sub-mediant chord (rather than the generic non-tonic chord criterion.)

I didn't listen to the example, but I agree with @Mirlan. Rock music often loops a set of chords, and those changes shouldn't necessarily be conceived in terms of cadences. Rock style harmony very often deliberately eschews the V I progression fundamental in classical, functional harmony. Not always, but very frequently. When that is the case, I think it best to just skip functional harmony analysis and use other analysis methods.

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