...continued from the music theory primer first page.

 

 

Frequency and Intervals

 

We have seen how musical space is organized by octaves from top to bottom. An octave is just the most basic musical interval. An interval is simply the distance between any two notes in musical space. This interval can be described in musical terms (such as "the distance between a certain pitch A and the A above it," or "one octave"), or it can be described in terms of frequency ("the distance between A-220 and A-440," or "220 Hz").

 

In the chart below, every pitch on the standard keyboard is mapped to a musical description (note name) and a frequency (waves per second, or Hertz). In bold we see A-440, the note to which orchestras usually tune, and C-261.6, known as middle C:

 

 

If we look again at the Mixolydian scale, we can see that the distance, or interval, between each adjacent pitch is not always equal.

 

 

C= 130.8 Hz

D= 146.8 Hz

E= 164.8 Hz

F= 174.6 Hz

G= 196 Hz

A= 220 Hz

B-flat= 233 Hz

C= 261.6 Hz

 

Note, for example, that the distance between F and G (about 21.4 Hz) is more than twice as great as the distance between E and F (9.8 Hz).

 

Long ago, musicians realized that a basic scale, such as the Mixolydian, is made up of two different kinds of intervals. The larger one (such as the distance between C-D, D-E, F-G, G-A and Bflat-C) they called a whole tone or whole step. The smaller one (such as between E-F and A-Bflat) they called a semitone, or half-step.

 

[Aside for music theory geeks: because each octave represents a doubling of the frequency of the octave below, the distance between notes measured in Hertz is always getting smaller as you ascend through musical space. For example, The distance between two adjacent whole steps, like C-D and D-E will not be the same in Hertz.]

 

Every whole step can be divided into two equal half-steps. The black notes of the keyboard fill in the half-steps missing in the simple scale forms, which are represented by the white notes of the keyboard. These black notes represent the "accidentals" (sharps and flats) in musical notation: C-sharp (also known as D-flat), D-sharp (also known as E-flat), F-sharp (also known as G-flat), G-sharp (also known as A-flat), and A-sharp (also known as B-flat).

 

The distance between any two most-adjacent keys on a keyboard, regardless of color, is always a half-step. Similarly, any two adjacent half-steps together comprise one whole step, regardless of color. In terms of musical perception, all half-steps are intervallically equivalent, as are all whole steps. (If you're interested in more detail, read about equal temperament.)

 

It's somewhat confusing at first, but the distance between a B and C-sharp is a whole step, no different than the distance between a C and D. The color of the notes on the keyboard (black or white) has no bearing on this basic fact of musical space.

 

The Structure of Scales

 

A scale is a collection of notes that are indigenous to a certain key, ordered in stepwise fashion, usually from bottom to top, and contained within a single octave. The Mixolydian scale (above) is a classic example. The notes in that scale all belong to the musical world of C-Mixolydian. "C" means that C is the first note of the scale, and the most important note in the musical universe being described; and "mixolydian" refers to a particular intervallic structure in the notes that go up from the first one (in this case, C).

 

Take any basic melody, such as Freres Jacques :

 

 

The pitches found in this melody are C, D, E, F, G and A. Of the seven different possible notes of a scale, this melody determines the identity of all but one. Whether this last should be B or B-flat is left undetermined. (In fact, it could easily be either one!) Nevertheless, we can discern which note of the scale is the tonic, which means the most important note of them all. I'll explain more about this later, but for now you'll just have to trust me: the most important note in this melody is C.

 

So, we can say that Freres Jacques is in the key of C. If the missing note is B-flat (which it may have been in the historical context in which the tune first emerged), the collection of pitches would delineate our old friend C Mixolydian (see above). If, however, the missing pitch is B-natural (which would be more typical in our age), then the collection of pitches describes C Major.

 

Check out this C Major scale:

 

 

If we assign a number to each note of the scale, starting with 1 and ascending the scale in stepwise fashion, we get C=1, D=2, E=3, F=4, G=5, A=6, B=7 and the top C=8. These are called scale degrees. G, for example, is scale degree 5 in the C Major scale.

 

If we look at each adjacent interval in the scale, we note that some are whole steps and some are half-steps. In fact, all are whole steps but two. Between E and F is a half-step (between scale degrees 3 and 4), and between B and C is a half-step (between scale degrees 7 and 8). The rest are whole steps.

 

Once we put this in the more general terminology of scale degrees, we get the definition of a major scale: a major scale is one where all the steps are whole, except for half-steps between scale degrees 3-4 and 7-8. Then, we can start on any note and make a scale that sounds equivalent to C Major--that is, one that has the same intervallic structure. In other words, if we keep the half-steps between 3-4 and 7-8, we always end up with a scale that sounds major.