I'm new to music theory, and reading through the basics I learned about interval classifications. But, along with that many questions came to my head. The one I want to ask today is the following:

Say I play a C to Eb interval. That would be a minor third interval, but that's just because I'm reading the note C before Eb. If I read it as Eb to C, that would make it a major sixth interval. So, that means every interval can have more than one name and, are those different names what is known as "enharmonic intervals"?

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    If the C is lower than the Eb (but in the same octave) then it's a minor third. If the Eb is lower than the C (but less than an octave) it's a major sixth. There's no ambiguity there. Those two intervals are complementary, i.e., they add up to one octave. – Matt L. Nov 16 at 20:50
up vote 10 down vote accepted

What you described is not an enharmonic relationship, but rather an inversion. Where one pitch is re-positioned an octave above or below the other pitch.

The inversion of a minor third is a major sixth.

These are the basic interval inversions:

  • Seconds invert to sevenths.
  • Thirds invert to sixths.
  • Fourths invert to fifths.
  • Octaves invert to unisons.

Your original question could be re-worded as...

How to determine if an interval is [an inversion] or not?

There is nothing inherent in an interval to say it is an inversion. So, a minor third is an inversion of a major sixth, but that should not be misunderstood to mean minor thirds are inversions or minor thirds are created by inverting major sixths.

Usually something will be labelled an inversion from some initial reference point. An example that comes to mind is invertible counterpoint. This where we have two melodies in counterpoint and the lower one is raised an octave to place it above the other melody. When that is done all of the intervallic relationships become inverted. So intervals that were thirds in the original counterpoint will become sixths in the inverted counterpoint. In that case we could speak of the sixths as inversions of the thirds. There are other inversion relationship, invertible counterpoint is only example.

Enharmonic relationships - a completely different matter - are when different "spellings" are used for the same thing. Intervals and pitches can both exhibit enharmonic relationships.

For an interval let's consider 3 half-steps. We could spell that as C to E flat (a minor third) or we could spell it as C to D sharp (an augmented second.) Different names and spellings for what are enharmonically the same interval distance of 3 half-steps.

For a pitch example let's look at the E flat and D sharp of the previous example. They are both the same pitch (same key on the piano is another way to look at it) but different spellings are used.

  • I reckon an inverted octave is still an octave! Unison is the same note, but these are still an octave apart. – Tim Nov 17 at 15:18
  • @Tim The maths says that an inverted octave is a unison... but it doesn't rigorously hold up for unisons so I'd say it's down to your judgement. – wizzwizz4 Nov 17 at 18:32
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    @wizzwizz4 - let's take C3 and C4. An octave. C3 then gets shifted to C4... actually, it is unison! You're right, I'm not. – Tim Nov 17 at 18:35

No, enharmonic intervals are intervals that sound the same, but are notated differently. So a minor third (C to Eb) would be enharmonic with an augmented second (C to D# in this instance).

The intervals you give in your example are called "complementary intervals." These are intervals that create an octave when added together.

  • Thanks for the answer Peter. It's more clear to me now. I marked Michael Curtis' answer as the accepted one though because he explains more about it. Thanks! – Bruno Hiram Alva García Nov 17 at 0:47

You are describing an Inversion.

Intervals are always named from the bottom note up. This can be confusing. If you think it's odd that E, the 'third' of a C major triad is both a Major 3rd and a Minor 6th away from the root depending on how the chord's voiced, I'd find it hard to argue with you! But this is how it's done.

'Enharmonic' is something else. It's when a note can have two different harmonic functions, resulting in it being spelt two different ways. F# or Gb. F or E#. (E# isn't 'just being silly'. It's the necessary spelling for the 3rd in a C# major triad. And even pop music is full of those.)

If you want a rule-of-thumb for working out what an interval becomes when inverted, here goes. Subtract the number from 9. Hence a 4th inverts to a 5th, a 3rd to a 6th. Swap Major and Minor, swap diminished and augmented, leave Perfect alone.

I'd say that a better description of an enharmonic interval would be comparing the C-Eb to C-D#. One's a minor third, one's an augmented second.

To determine an interval, always count up from the bottom (lower) note. Imagine that's the root of a key, it doesn't matter what key it's actually in. It's simply the relationship between two specific notes.

You are talking inversions here, and there's a simple formula to calculate that. The rule of nine. As in, a third upside down becomes a sixth, a fourth becomes a fifth, a seventh becomes a second.

Then there's the enharmonic issue, which actually can't be seen as an issue. Only when listened to, generally out of context. As in hear C>D#. It could easily be a minor third, but here it's an augmented second. Rules for this (yes, some theory does have rules!) is that the letter names make the number, but then you have to work out if it's maj., min., dim. or aug.

Perfect intervals are 4th, 5th and octave. If they grow bigger by a semitone, they become augmented; shrink by a semitone, diminished. All others are maj,min, dim, or aug.

An example or two. C>E = maj3. C>Eb = m3. C>E# = aug3. C>Ebb = dim3. Note all are 3rds because all are C>E.

To help with your inversions, as in the question - E>C = m6. Eb>C = maj6. E#>C = dim6. Ebb>C = aug6. Be well aware that Ebb in this situation is not D, and E# is not F. C>F = P4, so F>C = P5. They stay perfect when inverted.

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