: Use cases for modulating keys? So I'm just learning how to modulate keys and first of all I want to make sure I understand the concept correctly.... the way I've learned it is basically using a chord function in one scale

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You seem a little mixed up about the relationships diatonic chords have in major compared to minor keys.

Your example hints at this. I - ii - V - i. The key has moved (in a good way - more later) into the parallel minor. So far so good. That's where it starts to fall apart.

I > ii, let's say in key C, is C > Dm. i > ii in key Cm is Cm > Do. It's not necessarily a direct swap. And the same will occur for most of the rest of the chords, too. True, I is equivalent to i, and IV is equivalent to iv. V can, and probably will, stay as V.

Your 'V' of a new key works really well. It just happens that in C and Cm, that V is G.
It also works with other modulations. In key C, modulating to key D, use A (V of D), or use G♯ to get to C*sharp;. Or E (V of Am) to get to Am. Major to major is straightforward, and happens lots. Major to minor won't work easily unless there are only three chord involved - tonic, sub-dominant and dominant.

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Perfect cadence: (I IV V I or iv iv V i)

Starting from any key you can choose a chord of any degree and continue from there with a perfect cadence.

I IV V I VI => i vi V i

major -> minor -> dim7

You can also go across the parallel minor chord of a major chord and making a vii dim7 of the new tonic:

C -> cm -> cdim7 -> Db

Try this modulation with arpeggios the 12 chromatic keys.

direct modulation:

Sounds always good when passing to the mediant chord.

(Without a cadence you can join directly a song played in C with a transposition or another song in the mediant key Ab or Eb (introducing by the dominant7 also other mediants are common: C , Bb7 Eb etc.)

a very common modulation is the key change by introducing the new tonic by the secondary dim7 the 2nd inversion of its vii7/I46 - V7 - I

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You seem to be describing two difference scenarios, but I can try to explain both and then it may become clearer to you.

I-ii-V-i... Once we land on the i chord, we find what scale uses the notes in the i chord as a ii chord and then play the same functions in the progression in the new scale.

First, it's better to refer to a key or mode rather than a scale. Also, you really should indicate key or mode when giving Roman numerals, especially when asking about modulation.

So, assuming we are talking about major/minor keys let's start on C. C: I ii V i shows the tonic chord changes from major I (C major) to minor i (C minor.) That's a change in mode rather than key.

The simplest way to play the same chord functions in the minor mode are: Cm: i iio V i. In addition to the change of the tonic chord to minor, the ii chord becomes diminished iio and the V chord stays the same (a major triad.)

It might help to name the chords, like C Dm D C changes to Cm Ddim G Cm.

The second scenario is changing from C to A#. I assume you mean both keys are major. It's much more sensible to call the second key Bb (A# major or minor are odd key signatures, you normally use Bb major or Bb minor.)

So, that's C: I ii V I to Bb: I ii V I with chord names C Dm G C to Bb Cm F Bb.

How do you use these ideas?

For the first scenario, the change of mode, I think you understand. A common use is to darken the mood. You could repeat a phrase and switch from major to minor for the second iteration. Or you could start up a totally new section in the minor. You can play with the endings a bit to emphasize the surprise of the change. Instead of I ii V I try I ii V V hold the V then make the mode change i iio V i.

The second scenario, the modulation, presents a problem using your example I ii V I progression. I may have misunderstood you, but I think you mean to repeat the progression transposed to another key. That process is more like a harmonic sequence. It's tricky to describe the distinction briefly, because there are similarities between the two, and a harmonic sequence can be used as the means to modulate.

A harmonic sequence is when a chord progression is repeated but transposed up or down. It's common to use only a two chord progression in sequence. Sequences can be diatonic C: [ I6 IV ][ viio6 iii ] or chromatic C: [ I6 IV ][ V6/V V ]. I used brackets to show the sequenced (transposed) pairs of chords. I think you can see why just transposing a progression to another key is like a harmonic sequence.

So, transposing C: I ii V I to Bb: I ii V I is like a sequence. But, you might ask if a modulation is a change of key, and that transposition goes to a new key, why isn't it a modulation?

I suppose you could call it some kind of direct modulation, but it's more common to have some common chord between the two keys. More importantly the I chord doesn't normally start the modulating phrase. Think of it like modulating to a new tonic.

For an example let's start in C and modulate to Bb. A common chord between the two keys is F major. We will have two phrases. The second one will end on a Bb chord, but not start with it. | C: I ii V I | Bb: V6/V V6/IV I64 V7 I | with names | C Dm G C | C/E F Bb/D Eb Bb/F F7 Bb |

It's easier with notation.

The first 2 bar of the modulating sequence could be interpreted in C or Bb. That's typical of the common chord area of a modulation.

Modulating down a whole step from major isn't really common in classical style, but I wanted to keep with your example.

Jazz does something like A: ii V I I | G: ii V I I | F: ii V I I (Bm7 E7 Amaj7 | Am7 D7 Gmaj7 | Gm7 C7 Fmaj7...) but it's just another application of the modulating by sequence principle: don't start with I, repeat the pattern sequentially, end with a cadence in the new key.