Using the traditional approach, if you want to be able to quickly name any interval from any note, you basically have to memorize them all; needless to say, this is a huge amount of data. If you have some time for thinking, you can memorize the intervals between the notes A, B, …, G (without accidentals) and then derive the others by memorizing how accidentals change the intervals (for example, if you put a sharp in front of the bottom note of an interval, it ‘shrinks’, i.e. the major third becomes the minor third, the minor third becomes the diminished third etc.). Both these approaches are highly ineffective (in terms of time/naming ability ratio). In this article, I am going to teach you my own approach using which you will be able to name any interval quickly (and also analyze any interval quickly) by memorizing only a tiny fraction of the amount of data you would have to memorize using the traditional approach. (You can also read my article about The numeral notation, which is a complete theory based on this approach.)
Above, you have a table associating to every musical note a number (notes after Ebb (E double flat) and Cx (C double sharp) are not included, but you can draw the rest of the table yourself if you want—the letters F C G D A E B are repeated in the same order over and over again; just the accidentals are added.
Well, here’s what you’ll have to do first: memorize the number associated with every note. The easiest way to do so is to memorize the notes F, C, …, B, and remember that
|Minor second up||Major seventh down||–5|
|Major second up||Minor seventh down||+2|
|Minor third up||Major sixth down||–3|
|Major third up||Minor sixth down||+4|
|Perfect fourth up||Perfect fifth down||–1|
|Perfect fifth up||Major seventh down||+1|
|Minor sixth up||Major third down||–4|
|Major sixth up||Minor third down||+3|
|Minor seventh up||Major second down||–2|
|Major seventh up||Minor second down||+5|
The second step is to memorize the table of intervals you see above. The first and the second column show two different ways to look at an interval, because when you go, for example, one major third up from C, you end up on E, which is the same as the minor sixth down from C (just one octave higher). However, it is not necessary to memorize the whole table; it suffices to memorize only a half of it, because of the following rule:
For example, the perfect fifth down is –1, so the perfect fifth up is +1. Although the table doesn’t contain augmented and diminished intervals, it is not necessary to memorize them, because they are created just by adding # or b to the note (and of course, # and b cancel); for example, the augmented fifth (i.e. +1) above C = –2 is just –1#, i.e. G#.
How to use this all in practice?
It might seem at first that this approach is not in any way better than the traditional one; it is necessary to memorize a lot of things, and one must even perform certain arithmetical operations. However, the point is to think about the symbols A, B, C#, Db etc. as about different names for the numbers. If you learn German, for example, you just learn to call the numbers 1, 2, 3, … “eins”, “zwei”, “drei”, … The concept of number is still the same, only the name changes. In a similar fashion, one should think about “C sharp” as being a name for the number 5.
Once you start thinking about the numbers using the names given above, the rest comes automatically; when someone says E, what you really hear is the number 2. When you want to say what +4 above it is (which should internally mean the same as “major third up” for you), you immediately see that it is 6 (because you are used to work with small numbers since you were a kid). Then, all you have to do is to speak out loud the name of the number 6, i.e. “G sharp”.