Intervals are the key to understanding everything you will ever learn in music or play on your instrument. Think about it: scales are made up of intervals, chords are made up of intervals, melodies are made up of intervals, the music you play, write, listen to ..... breaking it down ... intervals are the primary building blocks.
I have received many harmony related questions, e.g. modes and scales and the answer to these questions always lie in the concept of intervals and the knowledge surrounding them. So here's my advice - study them to the degree of perfection. Know them in your sleep!
I hope with this article I will provide you with all necessary theory, knowledge and exercises to master this topic for good :-) If you have any questions regarding this topic please use the Music Theory Forum to post them.
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Western Music, its melodic and harmonic concept, is based on a very mathematical system to describe the relationship between two single notes. These relationships are called intervals. The term "interval" comes from Latin and means something like ‘the valley inbetween'.
An interval describes the distance between two notes
That's it!?! I had to wait for this page to load just to get this wise description!?! Well, let's get serious. The sentence above is simplistic of course, but in essence it stands for what intervals are: A mathematical concept, a strict meassurement.... In the same way you would explain to somebody the way from point A (your favorite pub) to point B (your favorite blues bar), you describe the distance between two notes. The difference being that with music, it's intervals that are your miles and minutes.
Before we go into further detail let's just have a quick look at how we come across intervals with two examples. Note that these can be seen vertically, meaning single notes stacked upon each other, ie a chordal approach, and horizontally, meaning single notes played one after the other, ie a melodic approach.
2 single notes:
Note 1 Note 2
|____________|
1 interval
triad (3 single notes):
Note 1 Note 2 Note 3
|____________|__________|
1st interval 2nd interval
|_______________________|
3rd interval
Feel free to examine 4 notes or apply the above to a scale. So how about one single note you might ask. Well, as long it's just one note- no real interval is involved. But let's say we play the same note twice (either on the same instrument or on two different instruments) then an interval is involved. In our metaphorical example this would mean that your favorite pub is also your favorit Blues bar (ya a lucky guy:-).
Ok let's dig a bit deeper.
You are all familiar with the concept of time as we measure it. If I say "1 hour", you have a clear understanding of what this means, the same with minutes and seconds. In the same way intervals are a concept with units and a corresponding terminology.
The Terminology
In order to measure the distance between two notes we have to define two main families of intervals.
The first family is called Perfect Intervals. The second family is called Major or Minor Intervals. Whereas our Perfect Interval family solely exists of one option, (ie Perfect), the Major or Minor Intervals family can take 2 forms: either Major or Minor, whereas Minor means “not as far away as Major", (ie Minor is always one half step below major).
Below is a graphical visualisation of the above said. Remember, we are talking about a distance here.
It wouldn't be in the nature of science (and music theory is a vast science) if there weren't some options/complications. No worries though - it's all structured and logical.
To both families of intervals there are 'alterations', which help us in certain situations to be more accurate in describing the distance between two notes.
This is how it goes: altering from our two main families we can distinguish between Augmented and Diminished, whereas Augmented expands upon the original family (by a half step) and Diminished lowers it (by a half step).
Let's look at our visual examples again:
A Quick Review
So far we have learned some important terminology which is used to describe the distance between notes. We know that there are Perfect Intervals and that there are intervals that can be either 'Major or Minor'.
Note: Intervals can only belong to one of these families - it can't be Perfect and Major or Major and Minor or Perfect and Minor. An interval can only be Perfect or Major or Minor.
Then we learned that there are options in the 2 main families called Diminished and Augmented. Augmented expands upon the original interval family and Diminished lowers it.
Now that the terminology is out of the way we can talk about the ‘units' that are used in describing music and thus intervals and their names.
The Units: Interval Names
Intervals are always named in relation to a Root note and then by their order of appearance. It's best to take a look at all of the natural note names (all white keys on the piano) and thus the C major scale.
As you can see you start counting from the Root (1). Apart from the Root (1) and the Octave (8) all notes are named according to their order of appearance. The word 'Root' became more accepted than previously 'the prime' or 'the first' as it better describes the starting position. If we play the Root twice at the same time it is called a 'Unison' (Latin for together). The Octave is a relic from the classical naming of intervals.
Now I'm sorry to tell you that saying Second, Fifth or Sixth is not enough. This would be too easy and would result in us authors being unemployment :-). This is where our terminology comes in that we learned in the beginning.
Perfect Intervals
Root, Fourth, Fifth and Octave are Perfect
In our C Major example this would be the notes c, f, g and again c. 'Perfect' describes those intervals that are the framework for building scales and chords. Perfect Intervals stay the same, whether you build a major or a minor scale. An exact translation of Perfect Intervals from its German origin would be "Clean Intervals" (reine Intervalle). It describes that they are consonant Intervals and therefore sound clean and perfect.
Let's take another look at our C Major Scale
Let me just point out some observations (some more obvious than others) to the image above as we can look at it in different ways:
- Perfect Root and Perfect Octave are the same note, one octave apart
- It takes 12 half steps (6 whole steps) to get from the root to the Perfect Octave
- There are 6 different other note names between Root and Perfect Octave
- it takes 2 whole steps and a half step to get from the Root to the Perfect Fourth
- it takes 5 half steps to get from the Root to the Perfect Fourth
- there are 2 different other note names between Root and Perfect Fourth
- I'll leave the rest up to you..... make up your own to better visualise the above.
Example: Root is d
Question: What is the Perfect Fourth of d?
To figure this one out we have a few different options, based on our observations above:
1) it takes 2 whole steps and a half step to get from the Root to the Perfect Fourth:
root = d, 1st whole step = e, 2nd whole step = f#, half step = g
Perfect Fourth from d is g.
2) it takes 5 half steps to get from the Root to the Perfect Fourth:
root = d, 1st half step = d#, 2nd hs = e, 3rd hs = f, 4th hs = f#, 5th hs = g
Perfect Fourth from d is g.
3) It's really important and the whole reason for this article, that you learn this by heart! : Perfect Fourth from d is g.
Memorising Intervals
So how do you approach memorising all the intervals of all possible roots. Here's my tip. Learn the intervals from notes without sharps and flats first and then add flats or sharps when needed.
We just figured out that the Perfect Fourth from d is g. If I'd ask you for the Perfect Fourth from db, which is just a half step below d, we do not need to start counting again - we just lower the result from d by a half step, thus gb. The Perfect Fourth from db is gb. The Perfect Fourth from d# is g#. See how this works?
So if ya know all Perfect Intervals starting on a natural note name you can easily transform this to all other notes.
And here they are - memorise them (I excluded the octave as this is pretty obvious):
Root Perfect Fourth Perfect Fifth
c f g
d g a
e a b
f bb c
g c d
a d e
b e f#
A common Pitfall
With the above list of intervals starting on natural note names, you may already have some queries. Sorry for me being so cautious but I wanna make sure that this is all clear and logical to you (this is our main goal here
The example starting on the note f might bother you. The Perfect Fourth has a flat in it. Why? and why isn't it an option to say that a# (which is the same pitch as bb) is the perfect fourth of f?
The answer, again, lies in the observations we made earlier, which are basically our rules to work out an interval. Let me take those observations apart with the f to a# example:
a# is 2 whole steps and a half step way from f = it could be the perfect fourth
a# is 5 half steps away from f = it could be the perfect fourth
BUT
there are no other 2 note names between f and a#. The only note name inbetween is g. Basically a# includes the note name a, which is the third note you come across when starting to count from f. Thus a# cannot be the fourth and must be some kind of third.
Exercises: Perfect Intervals
I encourage you to print out these exercises and fill in the blanks. Repeat them a few times and try to memorize the intervals until you feel comfortable with them. If you have any troubles, try to apply the rules we setup earlier to get to the specific interval. If you want to check whether you are right, refer to the page "Solutions of Exercises" at the end of this article. Good luck!
Exercise 1: Name the Note Names
Root P. Fourth P. Fifth Octave
c f g c
g
d
a
e
b
f#
gb
db
ab
eb
bb
f
Exercise 2: Name the Note Name
Perfect Fifth from g = ____ Perfect Fourth from db = ____
Perfect Fifth from g# = ____ Perfect Fourth from a# = ____
Perfect Fifth from eb = ____ Perfect Fourth from cb = ____
Perfect Fifth from d# = ____ Perfect Fourth from c = ____
Perfect Octave from bb = ____ Perfect Fifth from b = ____
Exercise 3: Name the Interval
d to g = _____________ ab to db = _____________
f# to c# = _____________ e to b = _____________
cb to fb = _____________ a to e = _____________
c to g = _____________ f to bb = _____________
f to c = _____________ bb to eb = _____________
Major and Minor Intervals
Second, Third, Sixth and Seventh can be Major or Minor
To repeat myself, the second family of intervals is divided into two possibilities: Major or Minor. The difference is in a half step. A Minor Interval is always a half step below the Major Interval. So to find a Minor Interval we have to know the Major one. We again take our handy C Major scale for example. As the name tells you in a Major scale everything is Major if it is not Perfect! Still with me? Confused? Please take the time and read this paragraph again.
First we repeat our Perfect Intervals.
Now we apply the rule from above and name the other Intervals.
In order to find the Minor Intervals we lower the Major Intervals by a half step, ie we are adding a flat to the note.
example: c to d is a Major Second, c to db is a Minor Second
So if we change all Major Intervals to Minor we get
Again. It is very important for you to understand that Minor Intervals are an alteration of Major Intervals. That does not mean they are less important or less used. It is just very helpful to think about them in this way to avoid naming the notes wrongly.
Exercises: Major and Minor Intervals
Exercise 1: Name the Note Name
Major Major Major Major Minor Minor Minor Minor
Root Second Third Sixth Seventh Second Third Sixth Seventh
c d e a b db eb ab bb
f
bb
eb
ab
db
gb
f#
b
e
a
d
g
Exercise 2: Name the Note Name
Major Seventh from g = ____ Minor Seventh from a = ____
Minor Second from g# = ____ Major Sixth from d = ____
Minor Sixth from eb = ____ Minor Sixth from c = ____
Major Third from a = ____ Major Second from f = ____
Major Second from bb = ____ Minor Third from ab = ____
Minor Seventh from c = ____ Minor Sixth from g = ____
Major Second from f# = ____ Major Third from d = ____
Minor Second from eb = ____ Major Seventh from bb = ____
Major Third from g = ____ Major Second from db = ____
Minor Second from bb = ____ Minor Third from c = ____
Exercise 3: Name the Interval
d to f = _____________ ab to g = _____________
f# to g# = _____________ e to f = _____________
cb to eb = _____________ a to f = _____________
c to bb = _____________ f to d = _____________
f to e = _____________ bb to c = _____________
d to c# = _____________ ab to f = _____________
f# to g = _____________ e to g = _____________
cb to bb = _____________ a to c# = _____________
c to a = _____________ e to f# = _____________
bb to a = _____________ gb to bb = _____________
Diminished and Augmented Intervals
As we learned earlier on, Diminished and Augmented Intervals are alterations of the two major families. Let's have a closer look.
Alteration of a Perfect Interval
Diminished lowers a Perfect Interval by a half note. Augmented raises it by a half note.
For example c to g is the Interval of a Perfect Fifth. c to gb is a Diminished Fifth and c to g# is a Augmented Fifth.
NOTE: In naming the interval it is very important to include the basic interval note name. We derive our two altered intervals by changing the perfect one. We took the g and added a flat or a sharp.
Again, you might be tempted to say that gb could also be named f# , which is an alteration of f. Learning music theory is like learning a language. There are logical grammar rules that you have to stick to. Sure, you could survive in making grammar mistakes but it could be embarrassing if you work in a professional environment. f is the Perfect Fourth of c, so f# is the Augmented Fourth of c and therefore not an alteration of g.
Alteration of Major/Minor Intervals
For example c to d is the interval of a Major Second. c to db is a Minor Second. C to dbb (D double flat) is a Diminished Second and c to d# is a Augmented Second.
Note again that dbb is the same pitch as c, but is referred to as Diminished Second, as our starting note is d.
To be honest, I cannot remember having to deal with a diminished second since I took classes in traditional harmony. One use of this would be if you are writing for strings like Violin or Cello. Because they are not fretted instruments, there is a slight intonation difference in playing a dbb and a c and thus beeing musically distinctive would be important.
Nevertheless, trust me. If you understand this weird stuff everything else in the future will be a piece of cake.
Exercises: Augmented and Diminished
Exercise 1: Name the Note Name
Dim. Seventh from g = ____ Dim. Fifth from f# = ____
Aug. Second from a = ____ Dim. Seventh from d = ____
Dim. Fifth from e = ____ Aug. Fourth from f = ____
Aug. Fourth from c = ____ Aug. Second from c = ____
Exercise 2: Name the Interval
d to g# = _____________ ab to b = _____________
f to g# = _____________ e to db = _____________
cb to d = _____________ c to bbb = _____________
f to bbb = _____________ d to ab = _____________
f# to gb = _____________ cb to d = _____________
c to gb = _____________ bb to e = _____________
The Tritone
After discovering that interval names are based on their order of appearance from the root, we mentioned two exceptions: the first note is called the Root and the eighth note is called the Octave. Another Interval that has its own name is the Tritone. The Tritone describes exactly the middle of an Octave and is equal to a Diminished Fifth (well, or Augmented Fourth). For centuries this Interval was avoided and called the Devil's Interval or "devil in music" (diabolus in musica) because of its strong dissonance, but for today's music it plays an important role in creating musical tension.
More than One Octave
Up to now we have dealt with intervals occurring within the range of one octave. Now we extend our range to two octaves. In general there is no big difference in naming the intervals. We take the same approach as with one octave and 'count' the notes.
The underlined Intervals are very important. Ninth, Eleventh and Thirteenth are called Tensions and play an important role for constructing chords and chord scales.
For Tenth (3rd up an octave), Twelfth (5th up an octave), Fourteenth (7th up an octave) and Fifteenth (two octaves) it is more common to name them by their interval quality and add one Octave. e.g. play me the minor third up an octave, or play me the Fifth up an octave. This will make more sense when you know about Chords and Chord scales (Modes). Just remember that Thirds, Fifths, Sevenths and Octaves keep their original names even when they are one Octave higher, out of practical reasons.
Note that all previously explained rules about interval naming stay the same. The d (Second) up an octave (Ninth) still belongs to the Major or Minor family and the g (Fifth) up an octave still to the Perfect family.
The Number System
Once you understand what we have looked at so far, it will be a small step up to using the commonly used abbreviations for intervals. Actually, it's just a matter of replacing the interval names with an equivalent number and adding abbreviations to show the quality of the interval. Thus a Third becomes a 3 and a Ninth a 9. To specify the quality we use abbreviations like P for Perfect and M for Major.
Example for Perfect Intervals Example for Major or Minor IntervalsAt this point I'd like to mention that there are a lot of different abbreviations in use. There is no worldwide standard that regulates exactly how to abbreviate. Sometimes you will see a “dim" for diminished or a triangle for major (especially in Europe). A Minor Interval can also be written with a dash (-7) or with a “m" (m7) in front of the number. The abbreviations I use is a result of my studies at Berklee College of Music and my believe that they are the ones least likely to cause confusion.
Perfect Fifth = P5 Major Seventh = M7
Diminished Fifth = b5 Minor Seventh = b7
Augmented Fifth = #5 Diminished Seventh = o7
Augmented Seventh = #7
The 'Useful' Interval Chart
If you count all intervals and their alterations you would end up with 50 different intervals within two octaves. This fact could be very confusing if you read about them for the first time assuming that every interval is important. Yes, they are, but the important thing is that you understand the theory and are able to construct every interval that is possible, on the fly.
Below I give you a table of intervals that you deal with every day. They are the building blocks of scales and chords and you should be familiar with them. I included the full interval name, an example based on the note c as the root, and the corresponding abbreviation with the number system.
Interval Name Note Name Index
Root c 1
Minor Second db b2
Major Second d M2
Minor Third eb b3
Major Third e M3
Perfect Fourth f P4
Diminished Fifth gb b5
Perfect Fifth g P5
Augmented Fifth g# #5
Minor Sixth ab b6
Major Sixth a M6
Diminished Seventh bbb o7
Minor Seventh bb b7
Major Seventh b M7
Perfect Octave c P8
Minor Ninth db b9
Major Ninth d M9
Augmented Ninth d# #9
Perfect Eleventh f P11
Augmented Eleventh f# #11
Minor Thirteenth ab b13
Major Thirteenth a M13
Complementary Intervals
Up to now we always talked about intervals that are going 'up', ie the second note is always a higer note than the root.
The system of intervals also works into the other direction, ie. downwards.
Let's say we take a P4, which is 5 half steps away from the root. These 5 steps can be seen starting on the root in either direction, up or down. The difference is the note that we land on. If we go up a P4 from c we already know that the answer is f. Now, if we go down a P4 from c we end up on the note g, which as we should know is the P5 of c in an ascending direction. This is where the term 'complementary intervals' comes in.
Let's have a closer look:
Expanding upon what we mentioned above, we can state that each interval has a complementary interval. Here are the rules for generating complementary intervals:
1) Complementary intervals always add up to nine. If you go down a 4 you end up on a 5 (4+5=9). If you go down a 5 you end up on a 4. If you go down a 6 you end up on a 3 (6+3=9), etc ...
2) Perfect Intervals stay Perfect. Major turns to Minor and vice versa.
Examples:
P5 down = P4 M6 down = b3
b2 down = M7 b7 down = M2
Conclusion
Phew, quite a lot of information I've thrown at you but I hope in a fairly logical and helpful way. Again, please feel free to post any queries or suggestins in the learn forum. I'll be glad to answer them.
One final note for guitarists who have read the first issue of this article at guitar4u: You might wonder where all the guitar related material and exercises have gone. Well, I have decided that this topic has to be written in such a way that it is useful to all musicians. Don't dispair, I'm going to write a separate, more in depth article just for applying intervals on the guitar.
Thanks and cu all soon,
Guni
Solutions of Exercises
Perfect Intervals
Exercise 1: Name the Note Names
Root P. Fourth P. Fifth Octave
c f g c
g c d g
d g a d
a d e a
e a b e
b e f# b
f# b c# f#
gb cb db gb
db gb ab db
ab db eb ab
eb ab bb eb
bb eb f bb
f bb c f
Exercise 2: Name the Note Name
Perfect Fifth from g = d Perfect Fourth from db = gb
Perfect Fifth from g# = d# Perfect Fourth from a# = d#
Perfect Fifth from eb = bb Perfect Fourth from cb = fb
Perfect Fifth from d# = a# Perfect Fourth from c = f
Perfect Octave from bb = bb Perfect Fifth from b = f#
Exercise 3: Name the Interval
d to g = perfect fourth ab to db = perfect fourth
f# to c# = perfect fifth e to b = perfect fifth
cb to fb = perfect fourth a to e = perfect fifth
c to g = perfect fifth f to bb = perfect fourth
f to c = perfect fifth bb to eb = perfect fourth
Major and Minor Intervals
Exercise 1: Name the Note Name
Major Major Major Major Minor Minor Minor Minor
Root Second Third Sixth Seventh Second Third Sixth Seventh
c d e a b db eb ab bb
f g a d e gb ab db eb
bb c d g a cb db gb ab
eb f g c d fb gb cb db
ab bb c f g bbb cb fb gb
db eb f bb c ebb fb bbb cb
gb ab bb eb f abb bbb ebb fb
f# g# a# d# e# g a d e
b c# d# g# a# c d g a
e f# g# c# d# f g c d
a b c# f# g# bb c f g
d e f# b c# eb f bb c
g a b e f# ab bb eb f
Exercise 2: Name the Note Name
Major Seventh from g = f# Minor Seventh from a = g
Minor Second from g# = a Major Sixth from d = b
Minor Sixth from eb = cb Minor Sixth from c = ab
Major Third from a = c# Major Second from f = g
Major Second from bb = c Minor Third from ab = cb
Minor Seventh from c = bb Minor Sixth from g = eb
Major Second from f# = g# Major Third from d = f#
Minor Second from eb = fb Major Seventh from bb = a
Major Third from g = b Major Second from db = eb
Minor Second from bb = cb Minor Third from c = eb
Exercise 3: Name the Interval
d to f = minor third ab to g = major seventh
f# to g# = major second e to f = minor second
cb to eb = major third a to f = minor sixth
c to bb = minor seventh f to d = major sixth
f to e = major seventh bb to c = major second
d to c# = major seventh ab to f = major sixth
f# to g = minor second e to g = minor third
cb to bb = major seventh a to c# = major third
c to a = major sixth e to f# = major second
bb to a = major seventh gb to bb = major third
Augmented and Diminished
Exercise 1: Name the Note Name
Dim. Seventh from g = fb Dim. Fifth from f# = c
Aug. Second from a = b# Dim. Seventh from d = cb
Dim. Fifth from e = bb Aug. Fourth from f = b
Aug. Fourth from c = f# Aug. Second from c = d#
Exercise 2: Name the Interval
d to g# = augmented fourth ab to b = augmented second
f to g# = augmented second e to db = diminished seventh
cb to d = augmented second c to bbb = diminished seventh
f to bbb = diminished fourth d to ab = diminished fifth
f# to gb = diminished second cb to d = augmented second
c to gb = diminished fifth bb to e = augmented fourth