Which are all the musical intervals that are valid?

Question

I have been reading around many sources, but it seems there is not a full standardized way in naming intervals, some sources say that such a thing as a diminished first exist.

So I started making an interval chart and realized something, intervals have two variables, the type (1st, 2nd, 3rd, ...) and the quality (diminished, minor, perfect, major, or augmented). But not all the qualities are to be used by any type.

Types 1,4,5 can use diminished, perfect, or augmented.

Types 2,3,6,7 can use diminished, minor, major, or augmented.

Intervals that are more than an octave apart are called compound intervals, but I don't know if the same rules apply??

So I ended up with something like this:

Now substituting for the Note C I got:

With this chart I think it is easy too see all intervals, but I am not convinced of the validity of all of them, for example, I have never heard someone talk about a diminished tenth interval.

So, do you think this is a valid way to get to any single interval for a note?, expanding this more would be valid?, or is there a limit on interval naming?

Edit: Corrected last chart.

Answer 1

Your chart looks correct, with one exception: in your bottom chart, the diminished 11th should be 15 semitones from C, not 12.

In theory, every interval is possible and valid, with the exception of flagrantly wrong intervals like a "major fourth." With that said, there is a point of diminishing returns: it's almost always pointless to go through the trouble of identifying something as an augmented 27th, when we may as well just call it an augmented 6th. If we really want to get specific, we can just call it a compound augmented sixth.

Which brings me to something that may save you some time: The Rule of 7. This is just a convenient way to translate between simple intervals (those within an octave) and compound intervals. If a third is expanded by an octave, it becomes a (3 + 7 =) 10th. When I mentioned a 27th earlier, that's just a compound version of a (27 - 7 - 7 - 7 =) 6th. Knowing this rule prevents your table from continuing on indefinitely.

And if it's helpful, we sometimes call the number of the interval (6th, 3rd, etc.) the "generic" interval. When we add on the quality (minor 6th, major 3rd, etc.) we call that a "specific" interval.

Lastly, you will occasionally encounter a doubly augmented or double diminished interval. C up to G♭♭, for instance, is a doubly diminished fifth. It's rare, but you will occasionally encounter it.

Answer 2

The number of semi-tones between notes is not sufficient for describing an interval. A C to an F#, for example, has 6 semi-tones and makes an augmented fourth, but the same number of semitones exists from C to Gb, which is a diminished fifth. The change in the type of interval has to do with the relationship between the two notes according to the key.

All intervals exist in sound, but the notation is meant to describe those relationships. Some intervals are impossible or ridiculous to notate.

Say we have Cb to Ab. That is a Major 6th. A minor 6th would be Cb to Abb, which we can still notate. But a diminished 6th, which is quite possible in some keys, is not possible to write with normal notation because it would require an Abbb. Triple flats are not standard, if they are used at all. I have never seen one despite reading some very chromatic music.

Answer 3

In addition to the interval types you've enumerated, there are also doubly augmented and doubly diminished intervals. For example, the C# - Gb interval is a doubly diminished fifth. They have limited practical use but they do exist.

In theory, intervals can be triply, quadruply, quintuply or, in general, multiply diminished/augmented. Such intervals have even fewer practical uses.

Answer 4

You're absolutely right that intervals are naturally represented by a pair of numbers. But I think it's easier to take the numbers to be "number of scale steps" and "number of semitones," rather than "number of scale steps" and "augmentedness." One advantage of the former representation is that you can add/combine intervals by simply adding the two numbers in the pair.

If you plot semitones (horizontal) vs scale steps (vertical), including only major, minor and perfect intervals, it looks like this:

         -2 -1  0  1  2  3  4  5  6  7  8  9
     ...
  -1     M2 m2
   0           P1
   1              m2 M2
   2                    m3 M3
   3                          P4
   4                                P5
   5                                   m6 M6
                                             ...

which makes it easy to see that every number of semitones has exactly one representation as a major, minor or perfect interval, except the tritone (or a tritone plus any number of octaves) which has none.

(Note that I've written 0, 1, 2, ... as the number of scale steps for a unison, second, third, ..., because only if you do that can you combine intervals by adding the numbers of steps. It's unfortunate that the names are off by one from the most convenient mathematical representation.)

If you add (singly) diminished and augmented intervals, you get

         -2 -1  0  1  2  3  4  5  6  7  8  9

  -1  A2 M2 m2 d2
   0        d1 P1 A1
   1           d2 m2 M2 A2
   2                 d3 m3 M3 A3
   3                       d4 P4 A4
   4                             d5 P5 A5
   5                                d6 m6 M6 A6

Now every number of semitones has exactly two names, except for the unison (or any number of octaves), which has three.

You can fill in the entire chart with n-times-diminished and -augmented intervals, all of which are technically valid, but they're almost never used.

Another theoretically nice, but less intuitive, pair representation of intervals is as a number of octaves and a number of fifths. This is completely interchangeable with the scale-steps-and-semitones representation: for every pair of integers in one representation, there's a unique pair of integers that represents the same interval in the other.

You can even represent intervals as a single integer, and preserve the property that you can combine intervals by adding the integers, if you are willing to give up the ability to represent highly augmented or diminished intervals. For example, if singly augmented and diminished is enough, then (2 × number of scale steps + number of semitones) looks like this:

  ... -5 -4 -3 -2 -1  0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 ...

  ... A2 M2 m2 d2 d1 P1 A1 d2 m2 M2 A2 d3 m3 M3 A3 d4 P4 A4    d5 P5 A5 d6 m6 M6 A6 ...

which represents all of those intervals uniquely and pretty compactly, with only one unused value per octave. One octave is 2×7+12 = 26, so this may be called a "base-26" system.

If you want doubly augmented and diminished intervals, then you have to use (4 × number of scale steps + number of semitones) to avoid ambiguity. This gives you a "base-40" system which has actually been used in a few published papers. (And which I think may actually be patented, so be careful.)

Answer 5

The categorysation of intervals as you did is one point. Your list is almost perfect.

Another is the notation respectively the signing. The manual of Band in a Box contains a vaste list of chords. (Pdf will follow)

The validation is another question. All may be valid but not all are used. Someone will be the first one but this would not be a new creation.

Many composer use enharmonic change or interchange and ignore the correct writing ( as use in popmusic or the sheets of a single instrument part. Bartok was a purist in correct writing.