Ranking the Intervals

(Still More About Series)


(entry for 12/27/2024)



You may have already noticed, but whether or not you have, there’s a strange feature about the idea of Partials or the Overtone Series that we’ve talked about for the last two posts: namely, that as you go up the ‘steps’ of the series, each interval involved is smaller than the one before.

Go back to the Partials of the note A110 that we already covered extensively. (An illustration of the intervals involved is at the head of this post.) The distance from the first partial (the fundamental) to the second one (the first overtone) is exactly one octave. To be more precise, the next partial up from the fundamental A110 is A220, an octave higher. The next ‘step’ is from A220 to E330, which is a perfect fifth (in ‘pure’ tuning). The next one up is from E330 to A440, which is a perfect fourth. Then from A440 to C-sharp 550 is a major third. And from C-sharp 550 to E660 is a minor third.

In other words, each subsequent step is smaller than the one before.

Now remember that each subsequent step is also half as loud as the one before. That’s just another way of saying that an octave is more important than a fifth, and a fifth is more important than a fourth, etc.

Each smaller interval makes up a smaller percentage of the total sound than the interval below it in the series.

So we can now make a definitive statement. As we go up a ‘notch’ in the Series of Partials, each interval is less important than the one below it, and each interval is mort important than the one above it.

So what’s the most important interval of all? Obviously, it’s the octave. It makes up more than half of the total tone that we hear (when we’re playing a string instrument). The next most important is the perfect fifth, taking up more than 25% of the total. (Because the ‘pure’ fifth is so close to the ‘tempered’ one [see the post on Temperament], the fifth is just as important in the tempered scale as it is in the pure one.)

This brings up one of the most important points in the subject of harmony: the different ‘keys’ that you can play or sing in are related to each other in fifths.

Let’s take an example. Let’s say you’re playing in the key of ‘D,’ which is two sharps. If you go to the fifth note of that scale, which is the pitch A, and play or sing in that key, the new key signature is three sharps. In other words, by going up a fifth, you’ve added a sharp. This goes on and on (theoretically) forever: If you go up another fifth, to ‘E’ major, you’ve added a fourth sharp, and another fifth, to ‘B’ major, adds a fifth sharp. Another fifth, to ‘F-sharp’ major, has a key signature of six sharps, and another fifth, to ‘C-sharp,’ has seven.

Whoops! We’ve run out of room, because there are only seven different letters to put sharps on, so we’ve gone as far as we can go in that direction. (Which is why I said 'theoretically' forever, earlier.)

Let’s go back to that key of D major (two sharps) where we started originally. Instead of going up a fifth, let’s go down a fifth instead. If we count five letters down from D, including the D (music is not math, remember!), we end up on G. And to get the key signature for G major, we subtract a sharp, instead of adding one. So since the key signature for D was two sharps, for G it must be one sharp! Which it is. (Don’t forget their relative minors. The key of ‘b’ minor for two sharps, and of ‘e’ minor for one.)

Go down another fifth, from G down to C. Subtracting another sharp, we get NO sharps (of flats either). This gives us the so-called empty key signature, the one for ‘C’ major and ‘a’ minor.

Now we run into a problem. We’ve run out of sharps. There are none left to subtract. So what do we do if we go down another fifth. A fifth down from C is F. So what’s the key signature for ‘F’ major? Answer: since we can’t subtract a sharp, we add a flat. The key signature for ‘F’ major is one flat. Go down another fifth, from F down to B-flat. The key signature for ‘B-flat’ major is two flats. (As is the signature for ‘g’ minor.

Etc. Every time you go down another fifth to get a new starting note for your scale, you add another flat to the key signature. You keep doing this till you run out of letters. The final key in the flat direction is the key of ‘C-flat’ major (‘a-flat’ minor).

Going back to the notion of the ‘hierarchy’ of intervals, we said that the octave is the most important one, and that the fifth comes next. If you look at the notation at the head of this post, you’ll see that the fourth is the next most important.

The fourth is an important interval in these key relationships also. If you go up a fifth, it’s exactly the same as going down a fourth. For example, a fifth up from F is C. And a fourth down from F is also C. They’re not the same C, but they are both C’s of some kind. So, our ears expect the ‘roots’ of chords (the bottom note of the triad in root position) to move up or down by fifths or fourths. There are certainly exceptions, such as moving up or down by a major second, especially up, but the fact is that our ears ‘want’ root movement to be by fifths and/or fourths, most of the time.

Another relationship between keys to notice is that the third partial of any given key is the same note as the second partial of the key that has one more sharp (or one less flat) than the first key. For example, the third partial in the key is A, but that same A pitch is also the second partial of the key of A, which has one more sharp than D has. (Three sharps vs. two.)

An illustration of this relationship is shown at the head of the post called More About Series, but we'll reprint it here to make things easier.  (In this case it's showing the two keys in reverse order, so that the fourth partial in A is also equal to the third partial in D. Remember, roots can move in either fourths or in fifths, and still create the same relationship.)



This identity of two pitches in two different series is called a Pivot Tone, and is another reason our ear expects key roots to progress in 5ths or in 4ths. (We'll talk more about pivot tones in a later post.)

Which brings us to the concept of the Circle of 5ths, which explains the whole idea of harmony in a very well-organized way. That’s the spot we’ll begin in, in the very next post.


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