A-1

C0

C1

C2

C3

C4

C5

C6

C7

Intervals

Measuring the speed of sounds.

Flashing lights.

So music’s got all these Notes, but what do we do with them? When we start to wrap our head around notes and what they represent, we will quickly want to know how to do more with them. At some point we as artists start thinking about how we can combine these notes together to creeate more interesting sounds. This is where the concept of the interval comes in to play.

Definition

Interval

noun

a space between two things; a gap.

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:

6 Inches
1234567891011

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

3 Hours
12 1 2 3 4 5 6 7 8 9 10 11

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

Pitch Distance

In music we are concerned about the difference between the speed of frequencies. If one frequency vibrates 100 times every second, and another frequency vibrates 200 times every second, then the interval is the diffference between those two speeds.

200Hz - 100Hz = 100Hz

So, in music, an interval is the measurement of the difference in pitch between two notes.

We can also think of these differences as ratios, through reduction, which can help us understand the relative quality of an interval by the simplicity of the ratio.

If the first frequency is 100Hz we can divide it by itself to reduce it to it’s smallest possible mathematical representation, like so:

100 / 100 = 1

And if the second frequency is 200Hz we can divide it by the first frequency, like so:

200 / 100 = 2

This gives us the simplest possible way of expressing the relationship between these two frequencies mathematically:

1 : 2

We could even represent this relationship visually with some sort of graph, maybe like this:

In this graph each wave repesents one of our frequencies. They both start moving in the same direction, but one of the waves moves twice as fast as the other one. In the time it takes the first wave to complete vibration cycle, the second wave has completed two vibration cycles. This means that the ratio of viration between the two waves is:

1 : 2

This is the simplest intervalic ratio that exists in music, other than 1:1, and this will be the first interval that we name, and we are going to refer to it from here on as an octave. Now, you may be wondering why in the world we would call it that, and without any other context you would be very right to question this decision. So, to understand why this choice was made, we first need to talk about note names.

Lines, Spaces, and Letters

In musical notation, we use letters to represent notes on a keyboard. Not all of the letters, I might add. Only a few. We start with the letter A and go through the alphabet to the letter G. If this is new information for you, then I suggest reading all about Notes and how they work in music.

But before we dive into all of the different intervals, I want to draw your attention back to the ruler and clock from earlier and point something out that you might know, but not really realize. Since they are both devices used to measure something in discrete amounts they both have markings on them to help guide us in measuring. If we take a look at a piano keyboard:

One might notice that there are some repeating patterns of keys here as well. In fact, if one were to add a marker in the middle of each note and desaturate the color of the black keys a little bit, it wouldn’t look that much different than a basic ruler:

7 steps

In our Keyboard RulerTM the distance from one marker to the next marker is one step. In this example we are measuring 7 steps from the starting key to the target key.

Well…

to be perfectly honest…

calling it a step is really only half the story…

Half Step

A Half Step is the distance between two adjacent notes. It is the smallest interval in music. C to Db in this example. We can use the letter H to represent a Half-Step.

C

Db

Whole Step

A Whole Step is simply two half steps added together. It is the second smallest interval in music. C to D in this example. We can use the letter W to represent a Whole-Step.

C

D

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:

Part 1

The first part of the name is derived from the distance, measured in the number of letters, which will give us the “number” in the name of the interval. C to D is a 2nd because there are two letters counted, C and D. From C to E is a 3rd because there are three letters counted, C, D and E. The counting includes the notes of the interval as well as the notes in between.

1st

C

2nd

D

3rd

E

4th

F

5th

G

6th

A

7th

B

Part 2

The second part of the naming system is based on the quality of the interval. We use the terms Major, Minor, Augmented, Diminished, and Perfect.

If we want to create a 2nd we must go from one note to the next, for example, C to D. But the problem we run into is that there are two D’s: D and Db. So we have to devise a way to distinguish between the two D’s.

The following example illustrates this concept even further. Although we have not officially discussed this idea yet, there is actually another D that we can conceivable have: D#. If I start on the note C and land on D#, then that interval is STILL considered a second, because the ending note is still some type of D.

C Minor 2nd

1st

C

2nd

Db

C Major 2nd

1st

C

2nd

D

C Augmented 2nd

1st

C

2nd

D#

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.

Color Intervals

Minor 2nd

The interval from C to Db is called a Minor 2nd. It’s a 2nd because there are two letters involved, which are C, and D It is minor because it is the smaller of the two 2nd intervals, which is Db in this example.

Minor 2nd

1st

C

2nd

Db

Major 2nd

The interval from C to D is called a Major 2nd. It’s a 2nd because there are two letters involved, which are C, and D. It is major because it is the larger of the two 2nd intervals, which is D in this example.

Major 2nd

1st

C

2nd

D

Minor 3rd

The interval from C to Eb is called a Minor 3rd. It is a 3rd because there are three letters involved, which are C, D, and E. It is minor because it is the smaller of the two 3rd intervals, which is Eb in this example.

Minor 3rd

1st

C

2nd

D

3rd

Eb

Major 3rd

The interval from C to E will be a Major 3rd. It is a 3rd because there are three letters involved, which are C, D, and E. It is major because it is the larger of the two 3rd intervals, which is E in this example.

Major 3rd

1st

C

2nd

D

3rd

E

Minor 6th

The interval from C to Ab is called a Minor 6th. It is a 2nd because there are six letters involved, which are C, D, E, F, G, and A. It is minor because it is the smaller of the two 6th intervals, which is Ab in this example.

Minor 6th

1st

C

2nd

D

3rd

E

4th

F

5th

G

6th

Ab

Major 6th

The interval from C to A is called a Major 6th. It is a 6th because there are six letters involved, which are C, D, E, F, G, and A. It is major because it is the larger of the two 6th intervals, which is A in this example.

Major 6th

1st

C

2nd

D

3rd

E

4th

F

5th

G

6th

A

Minor 7th

The interval from C to Bb is called a Minor 7th. It is a 7th because there are seven letters involved, which are C, D, E, F, G, A, and Bb. It is minor because it is the smaller of the two 7th intervals, which is Bb in this example.

Minor 7th

1st

C

2nd

D

3rd

E

4th

F

5th

G

6th

A

7th

Bb

Major 7th

The interval from C to B is called a Major 7th. It is a 7th because there are seven letters involved, which are C, D, E, F, G, A, and B. It is major because it is the larger of the two 7th intervals, which is B in this example.

Major 7th

1st

C

2nd

D

3rd

E

4th

F

5th

G

6th

A

7th

B

There are also a few more intervals that fall out of Minor / Major paradigm. We skipped over these intervals earlier on to keep the Minor / Major thing going, but it’s time to revisist them now.

Power Intervals

The perfect invtervals are called “Perfect” because of their place in history. They were once known as the most consonant (pretty) intervals by those who created their names, and those people were quite religious (Monks). They felt that 4ths and 5ths were so perfect, they represented God in music. Hence, Perfect 4ths and 5ths.

Perfect 4th

The interval from C to F will be a Perfect 4th. It is a 4th because there are four letters involved, which are C, D, E, and F. It is perfect because, it’s just so pretty.

Perfect 4th

1st

C

2nd

D

3rd

E

4th

F

Perfect 5th

The interval from C to G is called a Perfect 5th. It is a 5th because there are five letters involved, which are C, D, E, F, and G. It is Perfect because, well, it’s Perfect.

Perfect 5th

1st

C

2nd

D

3rd

E

4th

F

5th

G

The Octave

And finally we get back to the interval that we started with at the very beginning. The interval of an octave is 8 letter steps away from itself, and is also known as a Perfect 8th. It comes from the latin word octo, meaning 8. So, from any note to itself will be an ocatave. In this example, we are demonstrating an octave from C3 to C4.

Perfect 8th

1st

C

2nd

D

3rd

E

4th

F

5th

G

6th

A

7th

B

8th

C

This covers all the intervals within an octave, except for one. There is an interval between the 4th and 5th that is very much not perfect. It is sharp, brash, and dissonant (music theory for not pretty). It is known as the:

The Tri-Tone

The way we name it is either by augmenting a Perfect 4th, or diminishing a Perfect 5th.

Augmented 4th

The interval from C to F# is called a Augmented 4th. It is a 4th because there are four letters involved, which are C, D, E, and F. It is augmented because we are raising a Perfect 4th one more half-step, in this case, to F#.

Tri-Tone

1st

C

2nd

D

3rd

E

4th

F#

Diminished 5th

The interval from C to Gb is called a Diminished 5th. It is a 5th because there are five letters involved, which are C, D, E, F, and G. It is diminished because we are lowering a Perfect 5th one more half-step, down to Gb.

Tri-Tone

1st

C

2nd

D

3rd

E

4th

F

5th

Gb

The interval from C to F# or Gb will be a either an Augmented 4th or Diminished 5th. It is either a 4th or a 5th based on which note name is used. If F is the target note name then it will be called an Augmented 4th. If G is the target note name then it will be called a Diminished 5th.

Tri-Tone

There is another name for this interval as well, and that name is the Tri-Tone. This name is derived from the fact that this interval can be found by moving either up or down on the keyboard three whole steps, or whole tones. If I start on C and move up three whole steps I end up on F# / Gb. If I do tha same thing moving down the keyboard to the left, I will end up on the same note.

Tri-Tone

1st

C

2nd

D

3rd

E

4th

F#

5th

Gb

When all the intervals are combined together in one octave, this is what we get.

All the Intervals

1st

C

2nd

D

3rd

E

4th

F

5th

G

6th

A

7th

B

2nd

Db

3rd

Eb

4th

F#

5th

Gb

6th

Ab

7th

Bb

8th

C

The intervals are color coded based on their quality. Strong, powerful perfect intervals are Red. Warm, rich major intervals are Amber. Cold, somber minor intervals are Blue, and dark, unsettling diminished intervals are Purple. Depending on the context however, an interval may also be considered augmented, which are represented by a bright and sharp yellow.

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

Interval
Letters
Half Steps
Whole Steps
Unison
1
0H
0W
Minor 2nd (Half Step)
2
1H
1/2W
Major 2nd (Whole Step)
2
2H
1W
Minor 3rd
3
3H
1 1/2W
Major 3rd
3
4H
2W
Perfect 4th
4
5H
2 1/2W
Augmented 4th
4
6H
3W
Diminished 5th
5
6H
3W
Perfect 5th
5
7H
3 1/2W
Minor 6th
6
8H
4W
Major 6th
6
9H
4 1/2W
Minor 7th
7
10H
5W
Major 7th
7
11H
6 1/2W
Perfect 8th (Octave)
8
12H
6W

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 :-)

C Minor 2nd

C Minor 2nd

1st

C

2nd

Db

C# Minor 2nd

2nd

D

1st

C#

D Minor 2nd

1st

D

2nd

Eb

All three of these intervals are the same type of interval, but not the same exact interval, as they all start on the next note of the keyboard, but they all move up one note to create the interval. You will also notice that in the first interval the second note is Db, but in the very next interval we switch the name to C#.

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.

C2

7th

D

6th

E

5th

F

4th

G

3rd

A

2nd

B

1st

C

2nd

D

C4

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.

C2

6th

E

5th

F

4th

G

3rd

A

2nd

B

7th

Db

1st

C

2nd

Db

C4

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

C2

6th

E

5th

F

4th

G

3rd

A

2nd

B

1st

C

2nd

D

3rd

E

C4

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

C2

5th

F

4th

G

3rd

A

2nd

B

6th

Eb

1st

C

2nd

D

3rd

Eb

C4

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

C2

5th

F

4th

G

3rd

A

2nd

B

1st

C

2nd

D

3rd

E

4th

F

C4

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

C2

4th

G

3rd

A

2nd

B

5th

F#

1st

C

2nd

D

3rd

E

4th

F

5th

Gb

C4

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.

The Dissonance Dance

So, if we were to look at each interval and explore their ratios and relative dissonances we can re-order from this perspective to gain a new understanding of how they relate to each other, and to our perception of them.

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.

Intervals Tool

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