Difference between revisions of "1st and 2nd Strings - Intervals"

Playing on a single string is a complete pain. You are constantly changing positions, restricted to simple melodies, and cannot create any harmony. Let's add another string: the 2nd string or B string. Below you can see both the notes of the 1st and 2nd strings together.

E|F|-|G|-|A|-|B|C|-|D|-|E|
B|C|-|D|-|E|F|-|G|-|A|-|B|
0     3   5   7   9     12

E--0--1--3--5--7--8--10-12--
B--0--1--3--5--6--8--10-12--
   E  F  G  A  B  C  D  E
   B  C  D  E  F  G  A  B

If you look at the 2nd string frets that are used for the C major scale, you'll notice they're the same ones as the 1st string, except here we use the 6th fret instead of 7th. On the 1st string, that note is a B - the leading tone, which is a semitone below the tonic note. Here, that note is an F, the 4th, a whole tone below the 5th.

Remember that the 1st string is tuned a 4th above the 2nd string. E is a fourth above B. As we go up the fretboard, we see each note on the 2nd string has its relative 4th directly above it on the 1st string. Starting with open we have:

Now that we have two strings to play simultaneously, we can start to explore the concept of intervals and how they drive all harmony.

• Tuning and Harmonics
• 1st String

Two notes form an interval. This can describe either the distance between them, in terms of pitch/semitones, or the sound produced by playing them together (simultaneously or in sequence). Understanding intervals IS understanding harmony. Scales and chords are composed of a fixed set of intervals that give them their unique sound. Thus, the key of C Major sounds quite like G Major but quite different from C Minor.

You can determine the interval between any two notes by counting the number of notes between them in the common 7-note major or minor scales (or other modes of these scales). For example, C-D is a 2nd interval, while C-E is a 3rd, and C-F is a 4th:

C D E F
1 2 3 4

The quality of the interval will be different depending on which scale you are using and which notes of the scale are being measured. 2nds, 3rds, 6ths, and 7ths will have minor or major qualities - minor is closer/smaller and major is further/larger. 4ths and 5ths on the other hand are usually the same distance in most scales, called perfect; but occasionally, a 5th will be flatted by one semitone, making it a diminished 5th. Likewise, sometimes the 4th is sharped a semitone, called an augmented 4th.

Let's look at all the possible intervals inside the 12-tone chromatic scale (which correspond to each set of 12 frets for each string on the guitar).

All Intervals

Note: the colors used for the intervals are simply used to more easily visualize the size of the interval and its proximity to the root or its octave. They should not be misinterpreted to indicate harmony - color theory and music theory are very different, particularly as used here!

Numbering

The intervals are numbered 2-7 corresponding to the intervals formed within the common 7-note (heptatonic) scales. These scales use one of each of the numbers, counting up with each next note in the scale. For example, the common major scale uses M2, M3, P4, P5, M6, and M7. A note played against itself with the same pitch (for instance on two different instruments) is called a Unison interval. The same notes played as different pitches are referred to as an Octave.

Where the names have ()'s around a word in the description, that word is optional, and often omitted. Major 2nd, perfect 4th, and perfect 5th are all implied by 2nd, 4th, and 5th, as they are present in the common major as well as minor scales.

Qualities/Types

There are 5 different types of intervals:

• Augmented - One half-step larger than the corresponding major or perfect interval
• Major - The larger of the pair of 2nd 3rd, 6th, or 7th intervals
• Perfect - The harmonious 4th and 5th intervals near halfway between root and octave
• Minor - The smaller of the pair of 2nd 3rd, 6th, or 7th intervals
• Diminished - One half-step smaller than the corresponding minor or perfect interval

Counting Up/Down and Octaves

Given any two notes forming an interval, the lower-pitched of the two notes in an interval is typically referred to as the root note. Whether you measure the interval from the higher note to the lower or lower to higher is immaterial - it's the same distance either way. However, this can get confusing if which note is actually lower is ambiguous. For example, if you have a C and a D and the C is the root note, you have a 2nd interval (C D). If the D is lower, you have a 7th interval (D E F G A B C).

 U 2
 C D

 U 2 3 4 5 6 7
 D E F G A B C

When an interval spans 8 or more notes, the intervals could be considered to get larger and larger. For example, the interval from C to the D over one octave higher is technically a 9th (C D E F G A B C D). In terms of harmony, however, these intervals are often transposed to be within an octave. Just as octaves are considered harmonically equivalent, intervals are considered harmonically equivalent to intervals one or more octaves larger. This keeps understanding harmony simpler and let's us focus on note names more than pitch - we have 12 notes to worry about instead of the 49 fretted notes of a 24-fret, 6-string guitar and 88 keys of a full size piano.

 U 2 3 4 5 6 7 8 9      U 2
 C D E F G A B C D  ~=  C D

Qualities in Relation to Common Scales

You'll notice the 4th and 5th intervals have the most harmonious ratios and reside near halfway between the root note and its octave. The tritone in the direct center is actually less harmonious than the neighboring 4th/5th. In the common major/minor scales, it appears in place of a 4th or 5th, so it is often regarded as a4/d5 rather than the tritone.

The other numbered intervals come in pairs of major and minor, with the minor nearer the root and major further. Think minor = smaller and major = larger in terms of the distance between the component pitches.

Alternative Interval Names

Sometimes you may see interval names not listed above, like A6 (augmented 6th) or d7 (diminished 7th). Augmented simply means raised a half-step above a major or perfect interval and diminished means lowered a half-step below a minor or perfect interval. So an A6 is really the same interval as a m7, and a d7 is the same as a M6. The reason the alternate names are occasionally used is that the interval names are sometimes required to fulfill the practice of naming the intervals in a scale while using 2-7 once each, for more exotic scales. Also, interval names imply harmonic function - the alternate names suggest a different function from how such intervals usually act.

Playing Them All

Let's play all of the different intervals - fret the 2nd string using your index or middle finger, and make sure it's curled enough so the the 1st string is left open and can resonate at the same time as the 2nd string. Pluck both strings at the same time, sliding up one fret on the 2nd string each time:

E-0--0--0--0--0--0--0--0--0--0--0--0--0--
B-5--6--7--8--9--10-11-12-13-14-15-16-17-
  U  m2 M2 m3 M3 P4 tt P5 m6 M6 m7 M7 Oct

We start on E on both the 1st and 2nd strings. The 2nd string moves up one semitone each time: F, F♯, G, G♯, A, etc. This allows us to hear all the intervals named in the tables above; however, we are hearing them all against the root note of E, not C.

That we are using E instead of C is immaterial. Remember: Intervals have their own unique sound. A major 3rd between E and G♯ will sound the same compared to a major 3rd between C and E. The pitches themselves are different, but the ratio between them is the same; the relative distance and harmonic interaction between each pair of notes is the same.

The line below the tab indicates the interval. Notice how nasty the m2 and tt intervals are. the M7 can also be a bit nasty, but because there's more space between the notes, it sounds less harsh. On the other hand, the P4 and P5 sound very harmonious. The other notes sound less and less so going from the major 3rd/6th, minor 3rd/6th, and major 2nd/7th.

Don't Get Ahead of Yourself

Notice in the table of all the intervals I've included columns for Just Intonation Ratio, Just Intonation Cents, and 12-Tone Equal Temperament Cents. If you don't know what the hell I'm talking about, don't feel bad. The J.I. Ratio merely indicates roughly how harmonic each interval sounds, with the smaller ratios more harmonic than the larger - this follow the experience of playing the tab above.

Cents are a strict measure of relative pitch that are covered way later on. Each semitone covers 100 cents - each octave spans 1,200 cents. This measure is often used for tuning strings, but applies well for understanding intervals and harmony. I merely want to pique your interest now - guitar is a fun instrument in allowing us to play with pitch and harmony in ways other instruments cannot. Mastering the instrument goes beyond simply knowing the strict intervals listed above but knowing the math and core concepts of harmony as well as how to achieve this in your playing.

For now you should be mainly concerned with the sound and names of the intervals, not the pairs of notes involved in the tab. For example, you should know what a perfect 4th interval sounds like in comparison to a minor 3rd, but not worry about knowing that E-A is a perfect 4th and E-G is a minor 3rd. That will come with time. Don't stress!

Eventually, you'll want to be able to know all the intervals in terms of every note. Like this:

Diatonic Intervals

When playing inside a certain scale or key, we may occasionally omit the major/minor descriptor and just say the interval number. For instance, I might ask you to play the 6th above C without qualifying whether it is a major 6th or minor 6th. Or I might say we're going to play a harmony line a 3rd above any given melody, which will more than likely include both major and minor 3rds.

As mentioned above, common scales only contain 1 interval for each of the numbers 2-7. This applies not only to the root note of the scale, but to each additional diatonic (member) note in the scale. Each scale has its own set of diatonic intervals, just as it has diatonic notes.

As mentioned earlier, intervals can describe the combined sound of two notes as well as the distance between two notes. In a scale, the root note forms a basis point from which we can describe other notes by their distance from it. For instance, I could say to play the note a M3 above root then the note a P5 above the root. Ironically, we can use intervals to determine other intervals. For instance, the interval between the M3 and P5 above the root note is a m3.

--0----4----7--
  |-M3-|        root +4 semitones is a M3
  |----P5---|   root +7 semitones is a P5
       |-m3-|   the M3 to the P5 is 3 semitones - a m3

Let's look at the intervals from two different perspectives inside the key of C Major:

Inside any particular major scale, every note has a different set of intervals for all the other notes in the same key. In C Major, C and A both have major 2nd, perfect 4th, perfect 5th intervals but have different 3rd, 6th, and 7ths. If we added a harmony line a 3rd above the melody, we'd have a major 3rd when the melody played C (C-E), but a minor third when the melody played A (A-C).

When we get to diatonic chords, we will explore the common diatonic intervals for each note of the major and minor scales. For now, let's hear what the different sets of intervals sound like when played in sequence:

E-0--0--0--0--0--0--0--0--
B-5--7--9--10-12-14-16-17-
  U  M2 M3 P4 P5 M6 M7 Oct

E-0--0--0--0--0--0--0--0--
B-5--7--8--10-12-13-15-17-
  U  M2 m3 P4 P5 m6 m7 Oct
 

Notice the difference in the 3rds, 6ths, and 7ths.

Harmony in 4ths and 5ths

Time for an exercise. Let's play the C major scale with a constant interval above it. First we'll use 4ths. Chord charts for how to finger the 4th intervals in open and other positions are provided below.

o|-|
o|-|

|i|-|
|i|-| 

|m|-|
|i|-|

|r|-|
|m|-|

||i|
o|-|

|-|m|
|i|-|

|-|r|
|m|-|

|-|r|
|i|-|

These are pretty much the only ways to do it comfortably - you have a lot of options. Try them all and determine which one is the most comfortable for you. For the Perfect 4th that uses the index finger on both strings, you just flatten the finger, using your fingerprint to fret both strings. This breaks from the desired technique of using your finger tip, but it actually happens quite a lot and shouldn't be viewed with any kind of taboo.

  C4 D4 E4 F4 G4 A4 B4 C4
E-1--3--5--7--8--10-12-13
B-1--3--5--6--8--10-12-13-
  P  P  P  A4 P  P  P  P

Notice the major scale only contains one tritone or augmented 4th. We refer to it as A4 instead of tt here, because we are using 4ths - in that context it is the A4. I find it's easiest to flatten the index finger and shift positions for each interval, but add the middle or ring finger for the A4.

Let's try the same thing but using 5ths:

||-|m|
o|-|-|

|-|-|r|
|i|-|-|

|-|-|p|
|i|-|-|

  C5 D5 E5 F5 G5 A5 Bd5 C5
E-3--5--7--8--10-12-13--15-
B-1--3--5--6--8--10-12--13-
  P  P  P  P  P  P  d5  P

Very similar - only one tritone (or diminished 5th here). Although the interval is identical (including fingering) to the A4, we call it a d5 based on the context. Again, it's easiest to use the index and ring fingers to make the perfect 5th shape and shift positions as you move up the fretboard, switching out the ring finger for the middle finger (or moving the ring finger back a fret) when you reach the d5.

You may notice the 5ths sound rather similar to the 4ths. Remember that 4ths and 5ths are the most harmonious intervals, with ratios of 4:3 and 3:2, respectively.

Those Perfect 5ths are also called power chords, which are a staple of rock and roll music. They might sound a little weird on the 1st and 2nd strings, but try playing around with that shape on the 5th and 6th strings!

  E5 G5 A5 B5 C5 D5 E5 
A-2--5--7--9--10-12-14-
E-0--3--5--7--8--10-12-
  P  P  P  P  P  P  P  

Harmony in 3rds

Playing the 4th or 5th intervals on a melody line isn't very interesting - you almost always have a perfect 4th or 5th. Let's try 3rds:

o|-|
||i|

|i|-|
|-|m|

o|-|-|
||-|m|

|i|-|-|
|-|-|r|

E-0--1--3--5--7--8--10-12-
B-1--3--5--6--8--10-12-13-
  M  m  m  M  M  m  m  M

M = Major, m = Minor

In some cases, we're using major 3rds, and in some cases minor 3rds. Now that is a little more interesting! Again, shift positions for each interval, leaving the index finger on the fretboard but shift between using the middle and ring fingers to make the major and minor shapes.

Intervals Above vs. Intervals Below and Inversions

One last thing about intervals - intervals refer to distance between two pitches. However, we regard the octave of notes as harmonically equivalent. If you increase or reduce both the notes of an interval by one octave, it does not change distance at all. However, if you increase the lower note by one octave such that it now the higher note, or lower the higher note one octave, you still have the same notes but a different amount of distance and thus a different interval. This is called an inversion.

For example, G is 7 semitones above C, making it a Perfect 5th above C. However, it is only 5 semitones below C, making it a perfect 4th below C. Similarly, a G is a perfect 5th above C, but a C is only a perfect 4th above G.

  C    G    C
B-1----8----13
  | +7 | +5 |
  |-P5-|-P4-|

  G    C    G
G-0----5----12
  | +5 | +7 |
  |-P4-|-P5-|

The 7ths/2nds, 3rds/6ths, and 4ths/5ths swap. This makes sense - the further a note is from some root note, the closer that note will be to the root note's octave. Yet notice the major intervals' inversions are minor intervals, the perfect intervals' inversions are also perfect intervals, and the tritone's inversion is also a tritone.

The major/minor swap may seem a little counter-intuitive at first. "Major" is often associated with harmonious and happy, while "minor" is associated with dissonant and dismal. That's more true of the major and minor scales than the intervals; however, I would even contest such a general statement. Songs in major keys can be sad and songs in minor keys happy given how they are written.

Intervals are more basic, however. The major/minor label has nothing to do with their consonance and dissonance, but the size of the interval. It so happens that all the major/minor intervals in a major key against the root note are major, while 3/4 of the major/minor intervals are minor in a minor key.

Chords can be inverted without changing their harmonic function (although it does affect their percieved qualities), but intervals are a bit more sensitive. See the example below.

D----2--------2--
A-2--2-----3--3--
E-0--------0-----
  P5 P4    m6 M3

In the tab above, the low E string and higher D string are always playing an E, and the A string plays B, then C. This demonstrates an E-B P5 interval and its inversion, a B-E P4 interval, then an E-C m6 interval and its inversion, a C-E M3 interval. Inverting the perfect intervals only create a slight difference in perception, while inverting 3rds or 6ths has a larger impact. The composer cannot simply swap them willy-nilly - they clearly have different qualities in terms of overall harmony although they do sound related.

B---0--1--3--5--6--8--10-12-13-
    B  C  D  E  F  G  A  B  C

Here's all the notes from the C Major scale on the 2nd string. Remember from the 1st String lesson that the notes can be grouped into sets of 3 for each position, with either a w-w, w-h, or h-w pattern.

h-w 0--1--3
w-w    1--3--5
w-h       3--5--6
h-w          5--6--8
w-w             6--8--10
w-w                8--10-12
w-h                   10-12-13

Armed with this knowledge, let's play a song...

  1   2   3   4  |1   2   3   4  |1 n 2 n 3 n 4 n|1   2   3   4  |
  q   q   h      |q   q   h      |e e e e e e e e|q   q   h      |
E-0--------------|0--------------|---------------|0-------0------|
B-----3---1------|----3---1------|1-1-1-1-3-3-3-3|----3---1------|

This is the same melody as before, but now we can play it in the key of C major. It's a bit more difficult because you have to make sure you are playing the 2nd string most of the time, but occasionally play the 1st string. And notice we end the song here with the C - E major 3rd interval to spice it up just a tad.

Let's add some more intervals:

  1   2   3   4  |1   2   3   4  |1 n 2 n 3 n 4 n|1   2   3   4  |
  q   q   h      |q   q   h      |e e e e e e e e|q   q   h      |
E-4---2---0------|4---2---0------|0-0-0-0-2-2-2-2|4---2---0------|
B-4---4---0------|4---4---0------|0-0-0-0-0-0-0-0|4---4---5------|

We are back in the key of E major and have the same melody on the E string, but the B string is doing something a little different. There's a little more depth to the piece now. Is that a good or bad thing? It's not necessarily either. Some melodies are suitable for rich harmony, but simple ones like this are usually fundamentally changed if such were added. For instance, try the following:

  1   2   3   4  |1   2   3   4  |1 n 2 n 3 n 4 n|1   2   3   4  |
  q   q   h      |q   q   h      |e e e e e e e e|q   q   h      |
E-4---2---0------|4---2---0------|0-0-0-0-2-2-2-2|4---2---0------|
B-4---4---4------|2---2---2------|4-4-2-0-0-0-2-2|4---0---1------|

Ok, that's weird, right? Sounds almost like a different song. But again the melody is exactly the same.

• Know what an interval is - the "distance" between two notes.
• Memorize all 11 intervals, by name and the number of half-steps that define them.
• Know that the intervals against the root in a major scale are M2, M3, P4, P5, M6, M7 - all major and perfect.
• Play all the interval exercises above without a metronome - go slow and try to hear the sound of the intervals.
• Know how to finger the basic 3rd, 4th, and 5th intervals and play them ascending and descending the C Major scale.
• Learn all the C Major notes on the B string. Play all the notes ascending 1 position at a time, like in the 1st String lesson, then try them descending.
• Play Hot Cross Buns C.E. with a metronome, and try to improve timing and speed, except for the last example - play that one slow and just try to hear the song - don't worry about playing it in-time just yet.

• Equal Temperament and Alternate Tunings
• Beginner Songs