An Introduction to Intervals, Scale Degrees, and Chord Construction

-Drew Peterson

	Any discussion of music theory presupposes a basic working knowledge of the 
language in which it functions. Therefore, before getting into anything more complex, 
having some sort of an idea what phrases such as "interval" and "degree" mean in 
reference to musical notes is absolutely crucial. So, if you're a complete stranger to
this whole music theory bit, this lesson's for you. And if you have this stuff down 
cold, hopefully as it goes on you'll still find things of use to you, when we get into 
application. so, without further ado...
	An "interval" is simply the distance between two pitches. Intervals are 
measured in "steps" and "half-steps," with the "half-step" being the smallest possible
interval in the western musical system (microtonal scales, with intervals smaller than 
a half step, make up a large part of eastern music, in particular indian music, and 
while it's possible to play them on a guitar with careful bending, this is not widely 
done.) a half-step is the distance from one fret to the next on a guitar. For example,
if you fret the G on the 3rd fret of your low E string, and move up one fret to G#, 
that's a half-step. Two half-steps, not suprisingly, add up to a whole step. If 
instead of going to G# from G, you went to A, that would be a change of a whole step. 

  half-step	  step
|-----------|-----------|
|-----------|-----------|
|-----------|-----------|
|-----------|-----------|
|-----------|-----------|
|---3---4---|---3---5---|


	Larger intervals exist, but they're defined in terms of these two- a step and 
a half interval, for instance, or a three-and-a-half step interval. Think of it as 
similar to the American system of inches and feet (or the Metric system of centimeters 
and meters).
	Intervals go from being an abstract concept that's nice to know to something 
that's actually useful in making music when you apply them to scales. To really 
explore this, you have to understand the idea of scale "degrees." A scale degree gives
the relationship between a given pitch and the "tonic" or starting pitch, of the 
scale. As there are 7 pitches in any given diatonic scale (8 if you include the 
octave), they are named 1 through 7 (or 8, counting the octave). for example, here's a
C major scale with all the scale degrees labeled.

    1   2   3   4   5   6   7   8
|-----------------------------------|
|-----------------------------------|
|-----------------------2---4---5---|
|-----------2---3---5---------------|
|---3---5---------------------------|
|-----------------------------------|

	The first scale degree is usually called the "root," and the 8th is (if we're 
talking about 7-note diatonic scales; exceptions apply to different scale forms) 
called the "octave." 
	There are a few things that you should pay attention to here. First is the 
step/half-step pattern in the major scale. If you'll notice, there's a whole step 
between the first and second degrees of the scale, a whole step between the second and
third degrees, a half-step between the third and fourth degrees, a whole step between 
the 4th and 5th degrees, a whole step between the 5th and 6th degrees, a whole step 
between the 6th and 7th degrees, and a half step between the 7th and octave. This 
doesn't have any huge relevance to anything important in guitar playing, but if you're
playing lines linearly on the neck, it's useful to remember this pattern- it makes 
moving up and down the neck a bit easier;

W = Whole
H = Half
      W   W   H   W    W    W    H
|---------------------------------------|
|---------------------------------------|
|---------------------------------------|
|---------------------------------------|
|---3---5---7---8---10---12---14---15---|
|---------------------------------------|

	Either you could memorize this for all 12 keys, or you could just learn the 
pattern of half-steps and steps, and shift around on the fly without having to think 
about it too much. Also, it helps you come up with your own "patterns" of scales to 
play- if you know why each note is in the scale, then it's a lot easier to find them 
in different positions, and easier equals faster. And fast is good.
	From scale degrees, the next step up is back to intervals. A scale degree 
tells you the relationship of a pitch to a given scale, whereas an interval is 
slightly more generalized- it simply gives the relationship between two pitches, 
regardless of where they fall in the scale. Most of the time, if nothing else is 
specified it'll be assumed that we're starting from the first degree of the scale 
(this is true for chord construction, etc), but this isn't always the case, which is 
why this is such a cool concept. An interval in relation to two scale degrees is sort 
of like a bar chord shape in relation to an open E chord- an open E is always an open 
E, but a bar chord can be played everywhere and isn't tied to a specific pitch. 
Likewise, a scale degree is a highly specialized interval where the "starting pitch" 
is specified, while a regular, run-of-the-mill interval is merely the relationship 
between two unspecified pitches.

	Intervals draw their names from the scale degrees, too. for example, the 
smallest type of interval is a "second." A "Major second" (abbreviated M2) is a 
distance of two half steps, or in the key of C starting from the root, C to D. A 
"third" is the next largest interval. A "Major third" (M3) is two whole steps (4 half 
steps), and in the key of C starting from the root, would be C to E. Next is a 
"fourth." a "perfect fourth" (either p4 or just 4, i usually just use the number) is, 
in the key of C major, starting from the root, (assume all this "in the key... from 
the root" stuff from now on) C to F. then comes the "fifth," or a "perfect 5th" of C 
to G. Next is the the 6th- in C, that's a major 6th from C to A. The is the 7th, in C 
a major 7th from C to B. and then an octave would just be C to C. oh, and C to C when 
the C's are the same pitch is called a "unison."

  unison  M2  M3    4   5    M6    M7   octave
|--------------------------------------------|
|--------------------------------------------|
|-----------------------------2----4----5----|
|---------0----2----3----5-------------------|
|----3----3----3----3----3----3----3----3----|
|----8---------------------------------------|

	Ok, so i'm sure you're asking, "why all the "majors" and "perfects?" Well, it 
simply means that they match the intervals of a major scale (not a suprise, as this IS
a major scale). So, the logical next question here would be, "what happens when they 
DON'T match an interval from the major scale?" Well, there's a few different ways they
can be re-named, and here, the fact that some are called "major" and some are called 
"perfect" makes a difference. First, we'll discuss the major intervals.
	If you lower the upper note of a "major" interval a half-step down, you get a 
"minor" interval (abbreviated with a lowercase "m"). This doesn't necessarily mean 
that it's the type of interval you'd find in a "minor" scale- this is true in most 
instances, but a minor scale still has a major second, for example. If you apply this 
to the intervals mentioned previously, a minor second is now a half step, a minor 
third is now three half steps, a minor 6th is now 4 steps above, and a minor 7th is 
now 5 steps above. 

     m2   m3   m6   m7
|------------------------|
|------------------------|
|-------------------3----|
|---------1----6---------|
|----4----3----3----3----|
|----8-------------------|

	If you lower it another half step, you get a diminished interval (usually 
abbreviated with a degree sign, but that's a bit tough on a computer, so "dim" is used
most often on the internet). The only diminished intervals you really see commonly is 
the diminished 7th- a diminished 2nd is basically a unison by a different name, a 
diminishhed third is essentially a major 2nd, and a diminished 6th is essentially a 
perfect 5th. True, a diminished 7th is basically a major 6th, but they have different 
harmonic functions. The other intervals show up far too rarely to be woth worrying 
about (I don't think I've ever seen them, outside of drill sheets for theory classes I 
took).
	If, instead of lowering a major interval by a half step, you raise it, you get
an "augmented" interval (either "aug" or +) This is another interval you don't see too
much in action in the rock world, but it's worth keeping in mind. If you wish to 
experiment with these, simply raise the major interval a half-step and play them 
together- in the interest of brevity I'm going to skip on the notations.
	
	The perfect intervals skip the "major" and "minor" tag and go straight to 
diminished and augmented. (there's a reason for this difference in nomenclature- if 
you'll take it on blind faith, skip this next bit, but if you care... Basically, it 
has to do with what happens when you invert a given interval. A major third, for 
example, becomes a minor 6th when you invert it over the 3rd. Try it- play an A on the 
5th fret of the high E, and a C# on the 6th fret of the G; an inverted major 3rd. Now, 
measure the interval from the G# up to the A, relative to G#: you have a minor 6th. 
Perfect intervals, on the other hand, do not change from major to minor when you 
invert them. A perfect 5th inverted becomes a perfect 4th. Thus, they were dubbed 
"perfect," a term that initially seems a little odd but becomes considerably less so 
when you consider that formal musical study was well within the domain of the clergy 
until the advent of the Rennaissance. Speaking of faith, lol... But anyway.) Unisons 
can be diminished or augmented, but it happens quite rarely; basically, your 4ths and 
5ths are the only ones that you'll see with any great frequency being diminished or 
augmented. Also, a diminished 5th and an augmented 4th are essentially the same pitch,
but they serve different purposes musically, so it's worth keeping it in mind that 
although the "notes" are the same, the musical roles are different. 

     dim4 dim5 +4   +5
|------------------------|
|------------------------|
|------------------------|
|----2----4----4----6----|
|----3----3----3----3----|
|------------------------|


So, in summation...

-an interval is the difference between two pitches.
-a scale degree is a way of naming a certian pitch by the interval between itself and 
the root of its parent scale.
-a major interval (M) or a perfect interval (P) is an interval that, when treated as a
scale degree, is the same as the distance between the root of that scale and the note 
of the major scale that is the same distance from the root as the interval in 
question.
-a minor interval (m) is one that is one half step lower than its major equivalent.
-a diminished interval (dim) is one that is one whole step less than its major 
equivalent or one half step less than its perfect equivalent.
-an augmented interval (aug or +) is one that is one half step above its major or 
perfect equivalent in the major scale.

	Ok, that about covers naming intervals. You can do all kinds of cool things 
with these, but the most fundamental use of intervals is in building chords. Take a 
typical open E chord;

   E major  5th   M3
|-----0-----------------| E
|-----0-----------------| B
|-----1-----------1-----| G#
|-----2-----------2-----| E
|-----2-----2-----------| B
|-----0-----0-----------| E

	Now, if you'll notice, the difference between the E and B is an interval of a 
perfect 5th. And the interval between the E and the G# is a major third. Or, a E major
chord is composed of the intervals of a perfect 5th and a major third. Now, do the 
same with an E minor chord, and you'll get a perfect fifth and a minor third. Major 
3rd= major chord, minor 3rd= minor chord. See a pattern? 
	There's a reason for this. The smallest number of pitches you need for a 
"chord", according to the western musical system, is three, some sort of a third, some 
sort of a 5th, and a root. This type of chord, composed of three pitches, is called a 
triad. There are four primary triads in the western musical system; major, minor, 
augmented, and diminished. (this is true of the tertiary, or third-based, harmonic 
system, at least- there are other options out there, such as quartal, or fourth-based,
harmony, but this is the most fundamental starting point)
	Major and minor triads are by far the two most common. A major triad is a 
root, a major third, and a perfect 5th. Or, alternately, if you look at the example 
above, you can see that a B is a minor third above a G#- you can think of a major 
triad as a major third with a minor third stuck on top. This is particularly useful 
for tapping- it's a lot easier to picture a triad in terms of thirds than in 3rds and 
5ths:

|-12h16t19p12h16t19-| etc.
|-------------------|
|-------------------|
|-------------------|
|-------------------|
|-------------------|


    <----------M3----------><--------m3-------->
 |--X--|-----|-----|-----|--X--|-----|-----|--X--|

OR
    <--------------------5-------------------->
    <--------------M3------->
 |--X--|-----|-----|-----|--X--|-----|-----|--X--|


	I think you'll agree that the top one is easier to visualize quickly- it's a 
four fret gap between the root and the third, and another three between the third and 
the 5th	, as upposed to trying to picture 4 frets between the root and the third, and 
7 between the root and the 5th. 
	A minor triad, meanwhile, is composed of a minor 3rd and a perfect 5th, or a 
minor third with a major third on top. 
	Meanwhile, a diminished triad is a minor third and a diminished 5th, or can be
treated as two consecutive minor thirds, while an augmented triad is a major third and
an augmented 5th, or two consecutive major thirds.

       Emaj  Emin  Edim  Eaug
|------------------------------|
|------------------------------|
|------4-----4-----3-----5-----|
|------6-----5-----5-----6-----|
|------7-----7-----7-----7-----|
|------------------------------|

	Additionally, intyervals can help you figure out more complicated chords. For 
example, take suspended chords; Esus4 means that the third is dropped and replaced 
with a 4th, giving a chord with a ambigious quality to it that is neither major nor 
minor (hence the "suspended" name). (keep in mind that the names of the intervals here
are in reference to the root, not to the note immediately below it. Also the 4th and 
2nd 5th are an octave above their normal pitches, but that's not important)

|-----0-----| E  octave
|-----0-----| B  5th 
|-----2-----| A  4th  
|-----2-----| E  octave 
|-----2-----| B  5th 
|-----0-----| E  root 

	Or a Bmadd9, the chord from the minor section of Satch's "Always With Me, 
Always With You." Well, a 9th is basically a 2nd up an octave (do the math- if your 
octave is 8, your second is a 9th). In reality, you could just call this an add2 
chord, but for some reason guitarsits tend to think of it as a 9th. whatever, either 
is essentially correct. This would simply mean to take a Bm chord and add an 
additional 9th to it:

|-7------------------|  octave
|-7------------------|  5th 
|-7-----------7------|  m3  
|-11-------11---11---|  9th (2nd)
|-9------9---------9-|  5th 
|-7----7-------------|  root

	And then there's 7th chords. A straight "7th chord" is simply a triad with an 
interval of a m7 added in. A "major 7th chord" is simply a 7th chord with a major 7th, 
as opposed to a minor. And when you see chords like E9, D13, then that's assumed to 
have, in addition to a major triad (unless the chord is notated as being minor, like 
Em13), it has a 7th, and all the odd intervals up to the one given (i guess it has 
something to do with stacked thirds, similar to triads, but i've never really sat down 
and worked that out). For example, a E9 chord contains the root, M3, 5th, m7, and M9 
(or 2nd, basically) intervals, while a thirteenth would add the 11th and 13th degrees 
to this. Obviously, you run out of fingers REAL quick doing this, so  it's acceptable 
to "drop" intervals. The 5th, while essential to rock guitar, really has very little 
effect on the "sound" of the chord in more complex voicings, and is usually one of the
first to go. Somewhat suprisngly, the root is also commonly dropped; the idea is, the 
bass guitar can suggest the fundamental harmony while the guitar suggests upper 
extentions. ASs you get into more complicated chords, it's common to drop some of the 
intermediary upper voices: in the case of a 13th chord, a lot of the time the 11th 
will go, as well as sometimes the 9th, as it's more important to get the "13th" sound 
in there than to be technically correct by representing all the pitches in the chord. 
For example, here's a few common voicings, spelled out in intervals on the side:

  A7
|---5---| octave
|---5---| 5th
|---6---| M3
|---5---| m7 
|---7---| 5th 
|---5---| root     1, M3, 5, m7

 Cmaj7
|-------|
|---5---| M3
|---4---| M7
|---5---| 5th 
|---3---| root    1, M3, 5, M7
|-------|

 A9
|---7---| 9th
|---5---| 5th
|---6---| M3
|---5---| 7th 
|---7---| 5th 
|---5---| root    1, M3, 5, m7, M9

  D9
|-------|
|---5---| M9
|---5---| m7 
|---4---| M3
|---5---| root   1, M3, 5, m7, M9
|-------|

  E13
|---14---|   M9
|---14---|   M13 (M6, but up an octave- count it)
|---13---|   M3
|---12---|   m7
|---11---|   M3
|---(0)--|  (root)  M3, M7, M9, M13 (over imaginary root of E)


and a little Hendrix...

 E7#9
|--------|
|----8---|  aug9 (#9)
|----7---|  m7
|----6---|  M3
|----7---|  octave
|----0---|  root      1, M3, m7, #9 (aug9)

	In dominant chords (chords containing a major third and a minor 7th), the 
upper extensions are typically "modified" with sharp or flat signs, rather than 
interval names- thus, here we have a E7#9, not an E7aug9, although either is 
technically correct. It's a musical convention, and is slightly more conveniant.

	So, memorize this stuff. It's incredibly important to your udnerstanding of 
harmony, and will help you grow musically by leaps and bounds. Enjoy!