# Why is C-F# a diminished fifth instead of an augmented fourth?

It fits the strict definition of both, so it seems a bit weird to me that we choose to call it a diminished fifth. Seems very arbitrary as they are the same.

• Do we call it a diminished fifth? I wouldn't. – badjohn Apr 29 '17 at 22:28
• Are you maybe getting confused by the fact that, nowadays, we use the term "tritone" for both augmented fourths and diminished fifths? In the Middle Ages and renaissance, "tritone"—meaning three tones or whole steps—only referred to an augmented fourth because that's the interval that results when you go up by three tones. Now—at least in equal-tempered contexts—we tend to use "tritone" for all intervals that are separated by six half steps. However, we can and do distinguish between diminished 5ths and augmented 4ths in most common-practice situations. – Pat Muchmore Apr 29 '17 at 23:56
• Possible duplicate of Why do notes have multiple names? – 11684 Apr 30 '17 at 1:12

C to F is a perfect fourth. C to G is a perfect fifth. Found as those same notes in both major and minor. Making the interval one semitone greater than a perfect is called augmenting, making it smaller, diminishing. Thus C to F# is an augmented fourth; C to Gb is the diminished fifth. So it doesn't fit the definition of +4 and o5 - C>F# is only +4. Going the other way, F#>C is o5, while Gb>C is +4.

Yes, in equal tuning it's the same note, and asking to hear the two, the same thing would be heard. However, writing them down, they would look different, and that would give reasons why they have different names.

The interval is, as Pat says, a tritone - the two notes are as far away from each other as possible, looking at a circle with all notes around it, and the sound is quite dissonant - thus being christened the 'Devil's interval', and banned for many years in the early days of music.

• Not sure about "Devil's inrerval", but I am pretty sure it has been referred to as "diabolus in musica" by a number of writers. – Old John May 5 '17 at 17:07
• Also, the other way of describing the distance is that all intervals have an inversion over the octave, in the harmonic sense. The "tritone" doesn't, or rather, it does, but it is the same interval. Which means that it is the same upside down, and can't be reduced to a tetrachord, which was the original building block for scales in ancient Greek culture. Although, the enharmonicity of the tritone came with modern tuning, the diabolus in musica is mostly due to it's unavoidability in tuning systems, although I think this also might refer to the Pythagorean third. – Hatebit May 5 '17 at 19:31
• Interestingly, the requirement to avoid "diabolus in musica" in early sacred music led to two different "B" notes in the scale, B flat and B natural. They were notated "round b" and "square b." We get our flat and natural signs from those, and it's why the Germans call B flat B and B natural H. – BobRodes May 6 '17 at 2:17

If you are moving upward, C to F# is NOT a diminished 5th. It's an augmented 4th. The names of intervals are determined by how many letters are involved in the notation of the interval. For example, from C to Gb, moving upward, IS a diminished fifth, because counting C-D-E-F-G involves five letter names. But if you spell the same sounding interval C-D-E-F#, then it is an augmented fourth. (Four letters involved equals a fourth.) On the other hand, if you are counting downward (which in general you shouldn't do), from C to F# IS a dimished fifth. C-B-A-G-F#, five letters, hence a fifth.

To reiterate, the name of any interval is determined by how many letters are involved in getting there.

Not actually relevant to the question, but an interesting sidelight in classical theory, is that a diminished fifth needs to resolve inward, to a major or minor third, while an augmented fourth needs to resolve outward, to a minor or major sixth. In ear training tests, when the teacher plays a tritone, you need to wait for the resolution before answering the question!

• Resolution depends on context. If enharmonically equivalent, the difference is only notational, and the argument about resolution is unnecessary. True about ear-training tests, I remember those. Also, why not count downwards? In many cases, you have to. For example, transposing instruments may transpose an interval above or below. – Hatebit May 5 '17 at 19:26

The trivial answer is that by choosing to call it C - F# you've defined it as an augmented 4th. 4 letters, C,D,E,F - make a 4th. If you want some sort of 5th, call the top note some sort of G. End of.

But there's a solid question here too. We often write a sharpened 4th rather than a flattened 5th. You're more likely to see a melody and chord written as A in my example than as B. But harmony theory prefers to call the interval a b5. There's a simple reason. Common Practice (and Jazz) theory likes everything to be a triad, an extension of a triad, or a modification of one of these. The 'pile of 3rds' thing. It will just about tolerate an added 6th, and goes all shifty when the 11th comes up. But, mostly, it likes triads. Therefore conventional chord naming prefers a modified 5th - which is a triad note - to a modified 4th.

You use inclusive counting and you start on C (C being 1) C --> D is a second, C --> E is a third and C --> F a fourth.

Now, in this case, C with whatever accidental going to F with whatever accidental makes a fourth. C# to F gives you a diminished fourth, C to Fb also gives you a diminished fourth. C to F# gives you an Augmented fourth and Cb to F also gives you the same interval.

In the other case you mention, C with any accidental in front of it, going to G with any accidental in front of it gives you a fifth. C - G# gives you an augmented fifth, Cb to G also gives you that interval.

You don't get to make a fourth a fifth just because you are working with enharmonic equivalents.

Because it's not C-Gb. Oooops.

I've made the same mistake as the OP when asking the question.

If C-F# is not augmented 4th, C-Gb is not diminished fifth. NOT. As pointed out in other answers, there is no way this could be, since C-F is always a fourth.

• This does not provide an answer to the question. To critique or request clarification from an author, leave a comment below their post. - From Review – Richard May 5 '17 at 18:58
• This was originally a veritable lapsus lingua, but I left it as it addresses some of the points of the question. I've edited the answer. No sarcasm intended. – Hatebit May 5 '17 at 19:14