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T-rev's Guitar Hint of the Biennium


Past Hints

9/7/1998   -    Partial Capoing
1/11/2001   -    Anchoring
3/1/2001   -    The harmonic
10/22/2002   -    More about capos
3/24/2007   -    Intervals and consonance

Intervals and Consonance - Definitions and Derivations

This is a beginners' music theory lesson that covers the definitions and origins of some basic "intervals", such as "perfect fifth" and "major third". It should be on your level if you know the names of notes, what a scale is, and simple arithmetic. First, here are couple models that are helpful and/or interesting when thinking about some of these music terms:

1) The piano keyboard. It's the easiest way to picture note names, especially if you focus on the keys of C major and its "relative minor" key A minor. Both the keyboard and the naming of notes are designed around those keys. You can see there's no black key between B and C or between E and F. If you play only white keys everything will sound like it "fits" in the key of C or A minor. You can see the patterns of chords and scales as they fall on the keyboard.

2) A guitar string and frets. A different layout of the notes good for visualizing different qualities of those note (and relationships between notes) than a keyboard. For one thing, the black and white keys all become equal, as each note--whether natural (white piano key) or sharp/flat (black) gets its own fret. Also, you can see that the frets, although they are the same pitch interval ("half-step" aka "minor second") apart, are not the same distance apart, which leads to some interesting connections...

Octaves

An "interval" is just the difference in pitch between two musical notes. The most basic interval is not a second, third, fourth or fifth, but an octave. Both of the above models reveal important things about the octave.

First, it's called an OCTave because it is an interval of EIGHT. Eight what? Eight notes of a major or minor scale to get from a starting note back to another of itself. And if the key is C major or A minor, those notes will fall on white keys (if the minor scale is a "natural minor" scale). An octave is the 8th scale step, so really it is 7 scale steps up from the first scale note. Counting the black keys, it is 12 "half-steps" up from the first note.

(Note that for counting notes over multiple octaves, there are only 7 scale notes for each octave, because the eighth (and 15th and 22nd, etc.) Is really the first note of the next octave. There are 12 half-steps per octave, and the 13th note counting by half step is the first note of the second octave.)

On a guitar, half-steps are frets, so the 12th fret is an important one because it plays the note that is an octave above the string with no fret applied. Notice that the 12th fret is at the exact midpoint of the guitar string. Or at least, of the part of the string that can vibrate, between the "nut" and "saddle". (The nut is essentially the zeroth fret. The saddle is at the other end, on or part of the "bridge".) Playing the 12th fret divides the length of vibrating string exactly by 2, and it multiplies the frequency at which the string vibrates by 2. In other words if the whole string with no fret applied was vibrating at 110 times per second, then it will vibrate 220 times per second when the played at the 12th fret. (This important principle can be generalized: the frequency of the note produced is inversely related to the length of the string vibrating.)

Consonance

So an octave means a change by a factor of 2 in the frequency of the pitch. If on one string of a guitar you played the 110 Hz note (Hz stands for the unit Hertz, which means "times per second". For example, computer processors speeds are given in megahertz, MHz, or gigahertz, GHz.) and simultaneously on another string played the 220 Hz note, those notes would resonate and match together very closely. Why? Think about all those individual little vibrations in slow-mo. Each one of the 110 Hz vibrations will match up with one of the 220 Hz vibrations, and every other 220 Hz vibration will match up with 110 Hz one. The two frequencies mesh together. This is called "consonance"; consonance is the opposite of dissonance. For any two notes, the more of their little vibrations that match up, the more consonant they are; and the fewer that match up, the more dissonant those notes are together.

Notes that are an octave apart are as consonant as any two different notes can possibly be, and they sound very similar to each other. So much so that they have the same name. For example the eight B flats on an 88-key piano keyboard, octaves apart from each other, are all different notes but yet all B flats.

The perfect fifth

The next most consonant interval is the "perfect fifth". Why? Remember, dividing length of a guitar string is multiplying frequency. A string can be divided up an infinite number of ways (if you aren't limited to frets), but most of them would not give a frequency that's consonant with the open unfretted string. In order to have lots of those little vibrations matching up, you need to multiply the starting frequency by a whole number. (Or you could multiply a base frequency by two different whole numbers to come up with two new frequencies that would be consonant with each other.)

The next whole number after 2 is 3, so instead of dividing the string in half, you divide it in thirds. This happens at the 7th and 19th frets. The 7th fret cuts the strength to 2/3 of original length, so the frequency is multiplied by 3/2 (the reciprocal of 2/3). So if the open string vibrates at 110 Hz, then played at the 7th fret it vibrates at 165 Hz. For every other 165 Hz vibration, there's a 110 Hz one; and for every third 110 Hz vibration, there's a 165 Hz one. That's a perfect fifth interval: from the open string to the 7th fret. So a perfect fifth is seven half-steps.

(The 19th fret cuts the string to 1/3 original length, which means tripling the frequency to 330 Hz. That is 165 Hz doubled, so it's an octave higher than 165 Hz, as it should be since it is 12 frets above the 7th fret. So the 19th fret is an octave plus a perfect fifth above the open string. Also note that 1/3 the original length is half as much as 2/3 the original length, so again, the frequency should be doubled, and the math all works out whichever way you figure it.)

Why is it called a "fifth"? Because it is the interval between the first and fifth notes of a major scale.

The perfect fourth

Next in our search for most consonant intervals comes dividing the strings into fourths. This occurs at the 5th (3/4), 12th (2/4), and 24th (1/4) frets. Of course, 2/4 is the same as 1/2. We already know the 12th fret is 1/2 the length and doubles the frequency. And 1/4 is really just one half of one half, so the 24th fret (if you have that many frets) is 440 Hz, which is one octave above the 12th fret and two octaves above the open string.

But 3/4 the length (at the 5th fret) is more interesting. That's 4/3 of the original frequency of 110 Hz, so it would be 146 2/3 Hz. And it so happens that this is the interval called a "perfect fourth". So perfect fourth is five frets, i.e. Five half-steps.

The math works out cleaner if we start at 165 Hz, which multiplied by 4/3 is 220 Hz. So that's another example of a perfect fourth: from 165 Hz to 220 Hz, or from the 7th fret to the 12th fret on our example string. Again, five frets.

Why is it called a "fourth"? Because it is the interval between the first and fourth notes of a major scale.

Inversions

An important consideration here is that, in a way, the perfect fifth is the SAME as the perfect fourth! That's because the fourth is an "inversion" of the fifth. To invert generally means to flip something, and in music you do this by shifting a note by an octave. For example, if we start with our 110 Hz and 165 Hz notes, and then we raise the 110 Hz note an octave to 220 Hz, then the interval is inverted; the low note has flipped up to become the high note. And the interval between 165 Hz and 220 Hz is a perfect fourth.

You could say that a fifth plus a forth equals an octave. Or equivalently that a fifth minus an octave equals a fourth ("negative", meaning down instead of up).

Remember that on our guitar string, unfretted is 110 Hz, 7th fret is 165 Hz, and 12th fret is 220 Hz. So seven half-steps (frets) is a perfect fifth, and five more half-steps is a perfect fourth. Seven frets plus five frets equals 12 frets. A fifth plus a fourth equals an octave.

Another example: start with 220 Hz (12th fret) and 330 Hz (19th fret). That's a perfect fifth. Then drop the 330 Hz an octave to 165 Hz (7th fret). That's a perfect fourth. A fifth minus an octave equal a fourth.

One more way to do the math: remember that a perfect fifth is a frequency factor of 3/2 (e.g. 110 to 165). And a perfect fourth is a frequency factor of 4/3. So fifth plus a fourth is 3/2 multiplied by 4/3, which equals 12/6 or just 2. That means doubling the frequency, which of course is an octave. So again, a fifth plus a fourth equals an octave!

Thirds

So, what's next? Divide that string into five parts. 4/5 of the original length is at the fourth fret. (The other divisions are near the 9th, 16th, and imaginary 28th fret.) That's 5/4 of the original 110 Hz, or 137.5 Hz. This is a "major third" interval: four frets.

Divide the string into six parts. 5/6 of the original length is at the third fret. (The other divisions are at frets 7, 12, 19, and imaginary 31.) The frequency is 6/5 times 110 Hz, or 132 Hz. It is a "minor third" interval: three frets.

Note that a major third plus a minor third equals a perfect fifth. The math works either way. Adding the half-steps: 4 + 3 = 7. Or multiplying the frequency factors: 5/4 x 6/5 = 6/4 = 3/2.

Why a major third called that? Because it is equal to the first and third notes of a major scale.

Why a minor third called that? Because it is equal to the first and third notes of a minor scale.

Temperament

Now throw all that out the window. Or rather, back up a bit in the history of western music. Remember that the piano keyboard is designed for the key of C major (and A minor)? It used to be that it was also tuned for C major, so that the intervals you'd commonly play in C major would be as consonant as possible. In other words, they kept the intervals true to those ratios of whole numbers. This is called "just intonation".

With just intonation, you could still play in other keys, but they wouldn't sound the same, because the intervals would be different. The major third above "tonic" would be a different interval in one key than in another. ("Tonic" means the same note as whatever key you are in; in C major a C note is the tonic.) Aside from the pitch being a little higher or lower, it gave each key a slightly different feel.

But eventually people figured out that by splitting the differences you could have a happy medium that broke every octave into 12 equal half-steps. This is called "equal temperament". It isn't perfectly optimum for every chord, but it's really close, and it simplifies things and lets you play equally well in any key (e.g. E flat major) without any retuning in between. Very handy!

(Of course, some instruments, like the human voice, are not limited to discreet half-step intervals. A good barbershop singer will not necessarily sing the exact same pitch for a given note when harmonizing in two different keys. The C# he would sing as a major third in the key of A would be different from the Db he would sing as a perfect fourth in the key of Ab.)

Pitch deviations

That perfect fifth that should be 165 Hz with just intonation is actually 164.814 Hz at the 7th fret with equal temperament, about 0.1% flat. (Practically speaking for a guitarist, that's so close that the error due to temperament is not significant, especially alongside other factors such as guitar construction and setup and even how hard you push on the string.) For comparison, each equal tempered half-step is about a 5.95% increase in frequency.

The perfect fourth that should be 146.667 Hz is actually 146.832 Hz at the 5th fret, about 0.1% sharp, still really close.

But the major third that should be 137.5 Hz is actually 138.591 Hz, or 0.8% sharp. That really is a perceptible deviation, especially when harmonizing with other notes if you play more than one string simultaneously.

The minor third that should be 132 Hz is actually 130.813 Hz, or 0.9% flat. Even though the deviation is even bigger than for the major third, in practice this is less noticeable since playing a fret will tend to sharp the note a bit, which in this case works to compensate for the deviation rather than exacerbate it.

Other intervals

There are a few other intervals possible within the 12-half-steps-per-octave framework. There are whole number ratios for them too, but maybe a simpler way to derive the other intervals is by the differences or sums of third, fourth, and fifth intervals:

A "minor second" is one half step, which is equivalent to a perfect fourth minus a major third.

A "major second" is two half steps, which is equivalent to a perfect fifth minus a perfect fourth.

An "augmented fourth" is six half steps, which is equivalent to two minor thirds. The same size interval can also be called a "diminished fifth", depending on context. (Note that this interval is a perfect fifth "diminished", or decreased, by a half step.)

A "minor sixth" or "augmented fifth" (again multiple names for the same size interval) is eight half steps, which is equivalent to two major thirds. (Note that this interval is a perfect fifth "augmented" or increased by a half step.)

A "major sixth" is nine half steps, which is equivalent to a perfect fourth plus a major third. The same size interval can also be called a "diminished seventh".

A "minor seventh" is ten half steps, which is equivalent to two perfect fourths.

A "major seventh" is eleven half steps, which is equivalent to a perfect fifth plus a major third.

Intervals larger than an octave are numbered according to the same pattern. For example, an octave plus a second is a ninth. (The terms ninth, eleventh, and thirteenth are used primarily to refer to the corresponding notes in 9, 11, and 13 chords.) But most larger intervals are just broken down as one or more octaves plus one of the smaller intervals listed above.

Enharmonic equivalents

Notes that are the same pitch but have different names, such as D# and Eb, are called "enharmonic" equivalents. That term also applies to intervals that are the same size but have different names depending on context, such as augmented fourth and diminished fifth. For enharmonically equivalent notes, the appropriate name generally depends on what key the music is in. In the key of Bb you would call the perfect fourth Eb, but in the key of E you would call the same note D#.

Why? The first four notes of the Bb major scale are Bb,C,D,Eb. Since the third note is a D, it makes sense to call the next note an "E-something" rather than a "D-something". (Put another way, the key signature for the key of Bb has a flat on E so that each note of the scale can have its own line or space on the staff.) Likewise in the key of E the D# note is the major seventh and should not be called Eb. As a loose rule of thumb, use flats in keys with the word "flat" in their name (e.g. Bb, Eb, Ab, etc.) and use sharps in the other keys. That "rule", however, doesn't necessarily hold once you stray from a plain major scale. In the key of C you could use a Bb chord, and you wouldn't call it A#, because Bb is a minor seventh, not a sixth.

For enharmonic intervals, the designation is determined by the number of scale notes spanned by the interval. For example, within a C major scale, the interval from B to F spans 5 notes--B, C, D, E, and F--so it is some kind of fifth. It's size is 6 half-steps, so it is a diminished fifth. Still within a C major scale, the same size interval occurs from F to B but spans only four notes: F, G, A, and B. So in that case it is an augmented fourth. (Note that according to this rule the augmented fifth and diminished seventh do not ever actually occur in a major scale, and are really more for chord construction. But chords are important to guitarists, so those names are good to know.)

Final notes

You can see that all the major and perfect intervals form a major scale (if the same note is the first note of each interval). The minor intervals mark some of the notes on certain minor scales. Note that all the minor intervals are a half-step less than the corresponding major intervals. An augmented chord contains an augmented fifth. A diminished chord contains a diminished fifth, and a diminished 7 chord contains a diminished fifth and seventh.

Note the subtle difference in meaning between the name of an interval as opposed to chord components. Component notes of a chord are generally referred to by the interval from the root of the chord to the note, neglecting any extra octaves. For example, take a D7 chord on guitar: xx0212:
 • Any 7 chord has four notes: root, third, fifth, and seventh. This particular D7 is in root position, meaning that the bass note on the D string is the root of the chord; it is not inverted. (If a chord is inverted then the bass note is not the root of the chord, e.g. a D/F# chord.)
 • The next note, on the G string, is the fifth of the chord, which as you'd expect is a perfect fifth interval above the bass note.
 • After that is the seventh of the chord, on the B string, a minor seventh interval above the bass note, but only a minor third interval above the previous note in the chord.
 • The last note in the chord is the major third of the chord, but it is not a major third above the bass note. Instead, it is major third plus an octave above the bass note. It is also an augmented fourth interval above the note on the B string and major sixth interval above the note on the G string.

Summary

Summing up:

interval namefraction of
original
string
length
frequency
multiplier
# of
half-steps
equal
tempered
flat or sharp
vs. just
intonation
common
enharmonic
equivalent
inverts to
octave1/2212exactunison
major seventh8/1515/8110.68% sharpminor second
minor seventh9/1616/9100.23% sharpmajor second
major sixth3/55/390.91% sharpdiminished seventhminor third
minor sixth5/88/580.79% flataugmented fifthmajor third
perfect fifth2/33/270.11% flatperfect fourth
diminished fifth5/77/561.02% sharpaugmented fourthaugmented fourth
perfect fourth3/44/350.11% sharpperfect fifth
major third4/55/440.79% sharpminor sixth
minor third5/66/530.90% flatmajor sixth
major second8/99/820.23% flatminor seventh
minor second15/1616/1510.68% flatmajor seventh

Here's an Excel spreadsheet I made that easily calculates pitches, notes, frets, harmonics, fraction string lengths, and pitch accuracies.

By the way, 440 Hz is the standard pitch for the note "A", specifically, the next A above "middle C" (which is the C closest to the middle of a piano). Middle C on a guitar is the second smallest string played at the first fret. Our example string of 110 Hz played open is actually the A string on a standard-tuned guitar, which is two octaves below 440 Hz "concert pitch".

T-rev 3/24/2007


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