Pitch, Scales, Harmonies & Time

Let's start at absolute square zero, pitch. You know what pitch is, it's how high or low a note or tone sounds. To get to the idea of a scale you can think either in frequency, or string length, i.e. the length of a vibrating string, fastened firmly at both ends. The frequency of a pitch is the number of pulses per second. Think of a tuning fork. When you strike the tines they start oscillating back and forth, beating the air. The beating sends out pulsing compression waves in the air, which reach our ears and we hear a tone. The number of beats per second is the frequency and that determines the pitch. Switch over to the vibrating string, fastened at both ends and pulled tight to a certain tension. You pluck it and it swings back and forth, like the tines of the tuning fork. The frequency of the string is the number of oscillations per second of the vibrating string.

There is no law of nature or physiology that says we should hear any one pitch as special compared to another. We don't have "middle C" wired into our brains. [Actually, some of us do—those with "perfect pitch".] But one particular pitch has become a kind of golden standard, and that is the pitch whose frequency is 440 cycles per second. It is called "A above middle C" and instruments tuned to that standard are said to be in "concert pitch". The choice is arbitrary and it is adopted by convention. Without adherence to such a convention, no two trumpets could play the same pitches! Imagine a duet of two trumpets that disagreed on the frequency of basic pitches.

It is only when one pitch is compared to another, that we have any particular reaction to pitches. The one relative pitch that seems to be universally recognized is the combination of two notes where the frequency of one is double the frequency of the other. Imagine our string together with another, identical string, stretched between two fastening points, under exactly the same tension, but with the distance between its fastening points being half the distance of the first. Physics dictates that the frequency of the second string when it's plucked will be double that of the first. Our ears recognize a special quality in the relative pitches of the notes we hear from the two strings and we call it an octave. This relationship between the frequencies of the octave interval was discovered by the Greek mathematician Pythagoras.

Take a third string, half as long as the second string, and you get a third pitch one octave above the second one.

Now that we have the idea of pitch and octaves, we can discuss scales. A melody is a series of notes, or pitches, and the scale is the sequence of notes available for making melodies. We have a simple scale already—the scale of octave. This won't get you very far, in fact, only far enough to get the first two notes of "Somewhere Over The Rainbow". To make melodies we need more intervals between the octaves. How many, and how far apart the frequencies are is open to choice, sort of. "Sort of" because the choice has been made for us by others who have gone before us. In the West, at least, musicians have followed a notion of dividing the octave into 12 nearly equal intervals, called semitones or half tones. They are equal in a few senses: first, our ears hear them as equal, and, second, the steps in frequency from one tone to the next are adjusted by the roughly same fraction. The word "roughly" looms very large in some musical circles. How the frequencies of the tones are adjusted is the subject of "tuning". There are great volumes written on the subject. One scheme of tuning, called equal tempered tuning makes the intervals exactly equal by computing the frequency of each successive note by 1.059463094 and change, which is the 12th root of the number 2, i.e. 1.059463094 times itself 12 times is 2.0, almost. Another tuning is called just tuning. Here is a paper with a good discussion of it: What Exactly is Singing in Tune? by Brad Needham.

Now we have a way of sub-dividing an octave into discrete pitches. You might ask why 12 sub-intervals? Why not more or less? The answer is that the 12-tone subdivision evolved through tradition and we hear them as though they were facts of nature because of the conditioning of our brains by all the music we hear, right from before birth and through the musical toys we experience as newborns.

Now, what is a scale? A scale is a subset of these 12 tones that are the building blocks for composing melodies. When people speak about a "minor scale", or a "major scale", or "the blues scale", they are just summarizing the idea of preferred notes from which we will make up melodies. Think of a scale as the beaten path of melody making.

The first logical subset of the 12 tones is the entire set itself. It is called the "12-tone scale". Historically, this is the most recent, most modern scale to form the basis of music. The one we are most familiar with is the diatonic scale, containing seven of the twelve notes, eight if you count the octave above. It's the scale we sing as do-re-me-fa-sol-la-ti-do and played on the piano keyboard in the key of C as C-D-E-F-G-A-B-C, i.e. all the white keys between middle C and the C above. Key of C? What's that? That is the starting pitch of the scale, wherever it may be located on the keyboard. Notice that the intervals are not all equal because E and F have no intermediary note (no black key), and likewise with B and C. Remember, we call the interval of two adjacent notes in the 12-tone scale half-tones, making a whole tone two of these intervals, i.e. C to C# is a half-tone, while C to D is a whole tone. Thus the intervals E/F and B/C are half-tone intervals. So, the intervals in the diatonic scale go this way: starting-tone, +whole-tone, +whole-tone, +half-tone, +whole-tone, +whole-tone, +half-tone.

When you think of this sequence without reference to how we hear things, thinking only logically you might wonder, why the half-tones in the middle? Why not a scale comprised of only of whole-tones? Imagine that our scale had six notes or seven counting the octave above the starting tone. That would be C, D, E, F#, G#, A# C, all whole tones. As an exercise if you have an instrument, try playing "Twinkle, Twinkle, Little Star" on such a scale. You'll find it very strange sounding.

There is a subtlety here that is easy to miss, namely that the familiar scale can be seen as two sub-scales strung together where the intervals are whole-tone/whole-tone/half-tone. Thus C D E F are a little scale and G A B C are another little scale with the same tonal relations. So it is not so arbitrary, after all.

How come, then, we should find this diatonic scale so intuitive, so natural, so right it seems like a law of nature? We get this scale with our mother's milk. We hear melodies from infancy. Music is all around in the air and in our culture it is based on the diatonic scale of tones.

Adopting a scale does not mean abandoning the notes that don't fall on the scale. Those other notes can be used as "passing notes", little departures from the familiar notes of the scale. Most classical music and popular music before the advent of rock and roll could be characterized as following the diatonic scale. Consider the classic tune "Somewhere Over the Rainbow". The lyrics (by E.Y. Harburg) will remind you of the melody (by Harold Arlen):

Somewhere over the rainbow 
Way up high, 
There's a land that I heard of 
Once in a lullaby. 

Somewhere over the rainbow 
Skies are blue, 
And the dreams that you dare to dream 
Really do come true. 

Someday I'll wish upon a star 
And wake up where the clouds are far 
Behind me. 
Where troubles melt like lemon drops 
Away above the chimney tops 
That's where you'll find me. 

Somewhere over the rainbow 
Bluebirds fly. 
Birds fly over the rainbow. 
Why then, oh why can't I? 

If happy little bluebirds fly 
Beyond the rainbow 
Why, oh why can't I? 

Can you spot the place where we get a passing note, off the diatonic scale? It comes in the second syllable of the word "away" in the line "Away above the chimney tops". That is a semitone above the 4th degree of the scale. That's an F# in the key of C.

Now we have notes, octaves, the 12-tone notes that are semi-tone intervals, and the diatonic scale. The next consideration is the way we hear intervals, which is called harmony. We are acculturated to find certain intervals pleasing and others not pleasing. Intervals made by combining one note with another note two up the scale are called thirds, and they sound pleasing to our ears: (1,3), (2,4), (3,5), (4,6), (5,7). The fifth interval is pleasing, too, e.g. C with G. Today, people find the third interval more pleasing than the fifth, but a few hundred years ago it was the opposite.

Intervals of adjacent notes on the scale are less pleasing, less harmonious, but not completely harsh, provided they are a whole tone apart. Two adjacent intervals in the diatonic scale are just a half tone apart, namely the 3rd and 4th (E and F in the key of C), and the 7th and 8th (B and C in the key of C).

Certain intervals sound special to our ears, namely the fifth (notes 1 and 5 C/G), the fourth (notes 1 and 4 C/F), and the third (notes 1 and 3 C/E). Why? It's hard to say, but perhaps there is some connection with the ratios of their frequencies:

  • The ratio of the fifth interval is 3:2
  • The ratio of the fourth interval is 4:3
  • The ration of the third interval is 5:4
When we combine two third intervals, 1 and 3, and 3 and 5, we get the harmonic foundation of our scale, called a tonic chord, which is C-E-G in the key of C. The tonic chord is the simplest chord, made with the two sweetest intervals available. Alas, a steady diet of nothing but tonic chords is like trying to make a meal out of dinner mints or cotton candy.

Now we are ready for the idea of a chord progression. A chord, then, is nothing more than a combination of intervals, pleasing to the ear, based on the scale on which the melody is being played. Melodies, in turn, are heard against the harmonic context of the chords played underneath them, by strumming on a guitar, for instance, or with the left hand on the piano. The series of chords played along with the melody is called the chord progression.

A logical question to pose is this: Why should I need more than one tonic chord if we have chosen a particular scale? Take "Yankee Doodle" in the key of C, i.e. using the scale that starts on C. We'll play the melody in the right hand and accompany with the left hand using the tonic C chord (C-E-G). Things go pretty well until we hit the note for the second syllable in the word "pony", when it sounds sour. Likewise when we hit the melody with "in his hat" and "called it macaroni." The reason is that those notes in the scale clash with the notes of the C tonic chord. If we substitute a tonic chord based on the G scale for the pony, and a tonic chord based on the scale in F, and another G tonic chord for the macaroni part, our ears find it ever so much more pleasing. Voila, you have a chord progression.

Our ears have a few very emphatic tastes about how two chords sound when played in succession. In particular, when we have been hearing a tonic chord on the first note of the key of the melody (called the "root" of the melody), and then we hear a chord played on the fifth note of that key (referred to as the "five" chord, or just "V"), we anticipate, quite strongly, a return to the tonic on the root. Since every tonic chord is the V chord of some other chord, we can string together a series of V-to-I resolutions that produce cascading, satisfy resolutions. This series of resolutions is called the cycle of fifths, and it is the harmonic foundation of most non-classical music we hear today—pop, jazz, country, blues.

Now we are in a position to identify the scale and chord progressions most often used in blues. First, the scale. In spite of the fact that blues is often associated with sad laments, the characteristic note of the minor scale is not so common in blues melodies. Two exceptions stand out: The Thrill Is Gone and St. James Infirmary. Most blues melodies are best understood as based on a major, diatonic scale, with one amendment: the seventh note of the scale is flatted (in the key of C):  C  D  E  F  G  A  Bb  C There are always exceptions, like those two I noted. And there are notes you hear very often in blues melodies that are not on the scale, especially the flatted 5th (Gb in the key of C). That note, Gb in the key of C, is loaded with tension because we expect to hear it resolved to G natural, the regular 5th degree of the scale. In fact, so common is the note in blues melodies that sometimes the flatted 5th is listed in the blues scale itself.

Traditionally, blues songs follow a prescribed structure in which the melody is played in 4/4 time over 12 measures, broken into three sections of four measures each. The first four bars (or measures) are played against a tonic chord on the first note of the scale, the I chord. The next two measures are played against a tonic chord played on the 4th note of the scale, the IV chord. On the 7th measure the melody is returned to the I chord. On the 9th measure the chord changes to the V chord for two measures, and returns to the I chord on the 11th measure. It looks like this:

|--1--|--2--|--3--|--4--|--5--|--6--|--7--|--8--|--9--|--10-|--11-|--12-| |------ I Chord --------|--IV Chord-|--I Chord--|--V Chord--|-- I Chord-| 
The lyrics of blues songs often follow a pattern where the first two lines of lyrics are sung over the first four bars and then repeated when the chord changes in the 5th measure to the IV chord (called the "subdominant chord"). Another line or two of lyrics are song on the V chord (the "dominant chord") that resolves the first, repeated lyric. This pattern of lyrics is summarized by the term "A-A-B rhyming".

For information on modal scales see Wikipedia: http://en.wikipedia.org/wiki/Musical_mode