Why call a 4th or 6th interval an 11th or 13th?

Okay, gonna get the new theory subtopic area going here - plus, I’ve been wondering this:

With intervals, why do they call the 4th and 6th interval an 11th or 13th, respectively? Or, a 2nd, is called a 9th? I can understand the note itself is the same, just one octave higher. But what if I’m playing the 6th interval from a root note I’m using? It seems awkward to think of it as an unwieldy 13 intervals up the scale.

And, speaking of intervals, are the 4th and 5th interval called “perfect”, because they are the same in both a major and a minor key, or is there some other reason?

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Great question Vik! A few things:

First of all: Intervals describe literal musical distances between two notes.
So the distance between your open E string and your open A string is a (perfect) 4th. But the distance between your open E string and the A on the 2nd fret of the G string (one octave higher than the open A string) is a (perfect) 11th.

So the reason you would say 11th instead of 4th, is if that’s the distance you’re trying to describe specifically.

Here’s the confusing part: Interval terminology is very similar to scale degree terminology, and musicians often use them in a very mixed up way that confuses the hell out of those who aren’t “in the know.” I won’t get into all the details here, but if you hear someone refer to an 11th, they may be using scale degree terminology, not interval terminology, meaning they’re talking about it in relation to the current chord/key. Sometimes it makes sense to talk about notes in the next octave if they’re a chord extension, which happens a lot in jazz and other music that uses extended harmony (meaning beyond the 7th chord, which is as complex as a lot of rock/pop/blues gets).

So in that case, it has to do with how the chords are voiced on guitar/piano/etc., they tend to be up far away from the root note down in the bass.

I don’t know if there’s a good “reason” for how things are, but this is how things are:
2nds, 3rds, 6ths, and 7ths get modified like this
Diminished << Minor << Major << Augmented

Unisons, 4ths, 5ths, and Octaves get modified like this:
Diminished << Perfect << Augmented

Moving to the left makes an interval smaller, moving to the left makes it bigger. Examples:
Major third = 4 half steps (of distance between the two notes)
Minor third = 3 half steps
Diminished third (uncommon) = 2 half steps

Perfect 4th = 5 half steps
Diminished 4th = 4 half steps
Augmented 4th = 6 half steps

Great walkthroughs on this stuff for free on musictheory.net:
Generic Intervals
Specific Intervals
Writing Intervals
Inverting Intervals (good lesson but least important of the four, IMO)

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Perhaps to add a bit to what @JoshFossgreen already explained: if you keep piling thirds on top of each other, you get chords, right!? Three of them make up a triad, which is the root, the third and the fifth. If you keep going in the same manner, you add the 7th, the 9th, the 11th, and the 13th - that is what jazz musicians often do to “create” chords with lots of “color”. (And there is even more varieties, since the intervals involved can be diminished, minor, perfect, major, augmented).

And, indeed, using the 11th instead of the 4th (and similarly for the others) creates a different “color” for what is essentially the same chord - this is what is called “voicings”, in essence different arrangements of the same notes on, e.g., a keyboard (probably easiest to visualize this on a keyboard).

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As usual, your concise way of explaining things is way better than those websites - thanks Josh!

This actually makes a ton of sense now - thirds getting piled on top of one another - cool.

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