Definition

An interval is simply the measurment of the space, or displacement, between two things. What is important is what type of thing the interval is measauring.

For example, a ruler is a physical device that is used to measure physical distance:

And a clock is a physical device that is used to measure the passage of time:

In music the concept of an interval is used to measure something a little bit different, however…

Nomenclature

The name of each interval has two parts. This is because there are more notes on a keyboard then there are letters in the musical language.

This is because the “number” in name the interval name does not come from the number of steps.

no no no no no no noooooo.

It actually comes from counting the letters on the keyboard.

The problem is that there are more notes on a keyboard then there are letters in the musical language. We deal with this by introducing a second concept into the naming system

So, naming intervals involves two steps:

Intervals

The following is description of each interval in tonality. We start by learning about 2nds, 3rds, 6ths, and 7ths, as they form a logical group.

Steps

Here is a chart listing all of the intervals within an octave. You will notice that Half-Steps and minor 2nds are the same. This is true of Whole Steps and major 2nds as well. We will use Half steps and whole steps to measure the distances of each interval.

• H will stand for Half Step

• W will stand for Whole Step

Qualities per Interval

Not all intervals use all the avaiable qualities

2nds, 3rds, 6ths, 7ths - Diminished, Minor, Major, Augmented Unisons, 4ths, 5ths, 8th - Diminished, Perfect, Augmented

…soooo…that’s it…right?

…well…not quite.

You see, there are 12 notes on the piano keyboard. If you are a little heazy about that then read this article about Notes and then come back here. If we think about intervals for a minute we will probably come to the realization that every interval could probably be repeated on every note of the keyboard. For example, the interval that is one note to the immediate next note on the keyboard is called a minor 2nd, or a half-step…but couldn’t we create the same interval starting from every note on the keyboard, rather than just C?

But, of course :-)

Reciprocals

All intervals have a reciprocal interval as well. This means that:

• All 2nds are also 7ths

• All 3rds are also 6ths

• All 5ths are also 4ths

…depending on which direction you move on the keyboard, either right (up in pitch), or left (down in pitch).

Here are a few examples:

In this first example the starting, or origin, note is C, and the ending, or target note, is D. If we move up the keyboard in pitch to the right the nthe interval that is created between C3 and D3 is a Major 2nd.

But if we move down the keyboard to the left to the first D, we end up creating the interval of a Minor 7th. No matter which direction we go the ending note is still D, but the interval changes from a smaller interval to a larger interval. The quality will alos be reversed much of the time. In the previous example the interval went from major to minor. This will happen with 3rds and 6ths as well.

We can do the same thing with Db and it will give us a simialr outcome when transposed down on octave.

A Major 3rd will become a Minor 6th when inverted.

A Minor 3rd will become a Major 6th when inverted.

A Perfect 4th will become a Perfect 5th when inverted.

A Tri-tone will stay the same when inverted, as it splits the octave in half evenly.

So…all good…everything is done…nothing more to discuss…on we go…toooo…

WAIT!!!!!

There is…ummmm…actually one more thing that we should probabaly talk about…

Consance to Dissonance

When we listen to intervals we hear them as being somewhere on a spectrum from completely consonant (sounds really nice) to completely dissonant (sounds completely awful). And it turns that we can kind of understand why this is the case…the way do this is by examining the relationship between each frequency in an interval using the concept of a ratio.

Check out this graphic of two wavy lines:

Now, you may be looking this graph and you might be thining “Da heck? I only see one wavy line…what’s going on?“…and you would be both right and wrong…you see, there is actually two sine waves (the wavy lines) being drawn in the graph, but the ratio between them is 1:1, effectively meaning that they are the exact same thing they are layered on top of each other.

Let’s try a different set of sine waves, shall we?

I hope the difference between them is a bit more clear in this example. The ratio between these two sine waves is 1:2. What this means is that the first wave starts at a position of rest, which is the middle red line, then it moves all the way up, then all the way down past the middle red line, then back to the middle red line to start the process all over again. In audio we call this journy a cycle, or period, which essentially mean the same thing.

If we look at the second wave, we will notice that in the same time that the first wave completes one cycle, the second wave actually completes two cycles. They actually look like they meet right in the middle, and they kind of do, except that each wave is moving in the opposite direction, and that is not what we are looking for. This is what is known as being completely out of phase with each other.

What we are looking for here when we describe the ratio between the two waves is when do they both meet together at 0 again and moving in the same direction. In this particular example, they both end up doing that after one and two cycles, respectively. This is why the ratio between them is 1:2, or 2:1…doesn’t really matter which number is first. I generally put the lower number, as we generally think of frequency in this way, from lowest to highest. So long as you understand what each number means and what it represents then it’s up to you how you want to think of it.

Interval Ratios

Unison (Same wave twice) - 1:1

Octave - 1:2

Perfect 5th - 2:3

Perfect 4th - 3:4

Major 6th - 3:5

Major 3rd - 4:5

Minor 3rd - 5:6

Minor 6th - 5:8

Minor 7th 5:9

Major 2nd - 8:9

Major 7th - 8:15

Minor 2nd - 24:25

Tri-Tone (Augmented 4th / Diminished 5th) - 32:45

If we examine the order of the intervals we will notice that the perfect intervals, like octaves, perfect 5th, etc…are generally considered to be the most consonant to the human ear. They also exhibit the simplest ratios. Next up we find the intervals of 3rds and 6ths, major, then minor, which also tracks with our understanding. 2nd’s and 7th’s are up next and finally the tri-tone finishes the list up with the most complex ratio and is gnerally considered the most dissonant interval, although one could make an argument that the major 7th or minor 2nd are equally as unsettling.

This is not to say, however, that we should avoid using any of these intervals, as they all serve a purpose in helping us compose interesting music. If we only used octaves our music would not be alkl that interesting, as their would be a lack of tension and drama that could then give way to release. Our music needs all of these intervals in order to really be compelling, and so learning them all and how they can be used to create complelling music should be the ultimate goal.