# How do I name the inversion of an augmented octave?

Say we have D flat going up an augmented octave to D natural. The inversion would be D flat adjacent to D natural, or D natural adjacent to D flat.

We know that when an augmented interval is inverted, it becomes diminished, and vice versa. If we don't know what the original interval was (i.e. it could have been D going up a diminished octave to D flat), how do we know whether these adjacent notes are augmented or diminished?

I have read the posts about the diminished unison being a nonsense, but periodically I have students who ask me about this. Is there an arcane rule, or an exception to the rule?

• Are inversions well-defined for intervals greater than a seventh? Commented Apr 22, 2023 at 15:16
• @Edward indeed. Is an inverted 10th a 13th? A 6th? A 3rd? I think the answer to your question is "no." But an augmented octave is a sort of octave, so it is arguably a bit fuzzy. I've debated with Richard whether diminished unison is a useful concept; inversion of an augmented octave is probably the best argument in favor of it, but practically speaking the diminished unison is more confusing than helpful. It only makes sense in very abstract contexts such as this one. Commented Apr 22, 2023 at 15:54
• @phoog Diminished unison does make some sense. Suppose you’ve got a harmony shifting from major to minor. Then you may get a diminished unison in the third. Of course a diminished union is identical to an augmented union down. And of course the concept of "going up" a diminished unison when you in fact go down is a bit weird. In that sense both diminished unison up and augmented unison down are in fact just different labels for the same interval, that is a unison with a change in alteration.
– Lazy
Commented Apr 22, 2023 at 16:29
• @Lazy When shifting from major to minor, the third moves a descending augmented unison. This is no different from, say, a descending augmented second or any other descending interval like a major sixth. Commented Apr 22, 2023 at 16:41
• @Edward In the sense I think you're asking: correct, the function `invert-interval` is injective, but not surjective, so non-invertible. Commented Apr 22, 2023 at 18:37

## 2 Answers

The perfect unison is the smallest possible interval. There are two ways to consider this:

1. Intervals are always measured from the lowest note.
2. Intervals can be thought of (non-uniquely) the number of half-steps involved, which will always be a non-negative number.

Thus, a "diminished unison" will always be identified as an augmented unison.

Compound intervals invert to simple intervals, so given a simple interval and told it in itself is an inversion, it's impossible to know the "origin" interval.

Simple intervals are defined as the P1 through the P8. An augmented octave is a compound interval.

Compound intervals are inverted by displacing one of the notes by two octaves or both notes by one octave. Thus, given an A8 — say, Db to D — the Db would move up one octave and the D down an octave, forming a diminished octave.

Since a compound interval always inverts to the same interval as its simple correspondent, this is as expected. An augmented octave corresponds to an augmented unison, which inverts to a diminished octave.

Sources:

1. Ultimate Music Theory, "Octave: Simple or Compound".
2. Ultimate Music Theory, "No Diminished First".
3. Wikipedia, "Interval (music): Inversion" (citing Prout, Ebenezer [1903]. Harmony: Its Theory and Practice, 16th edition. London: Augener & Co.).
4. Markville Music Department — Rudiments of Music, "Compound Intervals and Inversions" (PDF).
• I’d like to challenge that assumption. If we take an interval to be a difference in pitch it might make sense to always measure from the lowest note. But if you were to take consecutive notes this would not make sense. It makes more sense to say: Go down a major third than it is to say: Have in reverse order gone up a major third from the next note. But the factorization into an absolute interval and direction is equivalent to a signed interval. In the sense of a distance metric assuming positivity is reasonable, but as a pitch difference having a signed interval does make sense.
– Lazy
Commented Apr 22, 2023 at 16:22
• @Lazy How did we know the interval was a major third, ascending or descending? That was determined according to the lower pitch. Commented Apr 22, 2023 at 16:23
• Why would you say so? A major third is made up of two whole steps. Now take C4-E4. That is C4-D4-E4. Two whole steps. Take E4-C4. That is E4-D4-C4. Two whole steps. From a singers perspective: If we sing a major third down we do not think: What is the lower note that would have the current note as major third, but we think: What is the major third down from this note.
– Lazy
Commented Apr 22, 2023 at 19:21
• @Lazy Major thirds can go up and they can go down, but how do we know it's a major third in the first place? Now I can anticipate you might say, "because it's two whole steps down". But again, whole steps can go up or down, but how do we know it's a whole step to begin with? And reducing it to a half step, how does one define a half step, which is a non-directional concept, as are all intervals (thus the need to specify whether it's ascending or descending). Commented Apr 22, 2023 at 19:35
• @Aaron A halfstep might not be directed, but a directed halfstep is equivalent to a signed halfstep, which is exactly the point.
– Lazy
Commented Apr 22, 2023 at 20:13

A complementary interval can be seen as the interval from the higher note to the octave of the lower note, or as the interval to be taken from the higher note to achieve the octave of the lower note, or the inverted interval from the lower note taken up an octave. Now, as Edward pointed out these concepts generally assume we remain within one octave.

But if we were to ignore this: An augmented octave is up one octave and a half step, so the complementary interval would be a diminished unison. Now, I’d reason it would make sense to define complementary intervals on interval classes modulo the octave. This way two intervals are equivalent iff. they differ by an octave. In this sense the complementary interval class would simply be the negative interval class, and deciding on a representant witin an octave we’d actually see the augmented octave as augmented unison with complementary representant as diminished octave.