Harmony/Some Basic Concepts, The Major Scale, & Intervals

Some Basic Concepts
Pieces of music are said to be in a certain key. Beethoven's fifth symphony is in C minor, for example. A key implies a set of notes which relate to a central note, the note after which the key is named. All notes in this set are related to a central tone in a hierarchical fashion. This central tone is called the tonic or key center. Thus, the tonic of Beethoven's fifth symphony is C. What's more, it is said to be in C minor. This is because the composition mostly (but not exclusively) uses notes from the C minor scale and closely related keys. Scales are collections of notes, usually represented in ascending order from tonic to tonic. The word scale comes from the Latin scala meaning ladder or staircase. Scales are the foundation of our study of harmony.

The Major Scale
The first scale we will examine is the major scale. The major scale is one of the most common scales to hear and most listeners will be quite familiar with its sound: An interval is a measurement of the distance between tones. The major scale is made of our two smallest sizes of intervals: half-steps and whole-steps. A half-step moves from one note to the next immediate note. Half-steps are represented by playing any two keys which are directly next to each other on the piano (whether black or white) or one fret to the next on a fret board, e.g. B to C, or C to Db. Whole-steps are made of two half-steps. Whole-steps are represented on the piano by playing two notes that have one key directly between them; on fretted instruments, whole-steps would be the notes 2 frets apart, e.g. C to D or E to F#.

The major scale is a seven note scale, usually represented as a full octave, tonic to tonic, thus eight notes. The arrangement of whole-steps and half-steps that constitute a major scale is whole-whole-half-whole-whole-whole-half. Starting on C, the major scale would be C D E F G A B C. Starting on G, the major scale would be G A B C D E F# G. Each note is assigned a scale degree, usually represented by capped Arabic numerals ($$\hat 1, \hat 2, \hat 3...\hat 8$$), but are also sometimes indicated by plain Arabic numerals or Roman numerals. Each scale degree also has a traditional name, here listed with the notes of a C major scale:
 * C ($$\hat 1$$) is the Tonic meaning tone.
 * D ($$\hat 2$$) is the Super-tonic meaning above the tonic.
 * E ($$\hat 3$$) is the Mediant meaning half-way between the tonic and dominant.
 * F ($$\hat 4$$) is the Sub-Dominant meaning a dominant below the tonic.
 * G ($$\hat 5$$) is the Dominant, traditionally the second most important note in the scale, after the tonic.
 * A ($$\hat 6$$) is the Sub-mediant
 * B ($$\hat 7$$) is the Leading-tone, so-called because it leads back to the tonic, and it indeed has among the strongest pulls back to the tonic.

Intervals
As previously mentioned, intervals measure the distance between two tones. We distinguish between intervals in two different contexts: a melodic interval is the distance between two notes sounded consecutively or horizontally; a harmonic interval is the distance between two notes sounded simultaneously or vertically. Intervals can go up or down; be above or below. Intervals consist of two parts: size & quality. An interval's size is represented by a number indicating the number of letter names the interval spans, including the first and last tone. So, any kind of a A (A-flat, A-natural, A-sharp) up to the closest C of whatever inflection (C-flat, C-natural, C-sharp) is a 3rd of some sort. Any kind of C up to the next G is a kind of fifth. An interval's quality depends on the relation of the top note to the lower note's major scale and belongs to one of two classes: The first class of interval qualities applies to 2nds, 3rds, 6ths, and 7ths. The options for this class's quality of interval are: The second class of interval qualities applies to 4ths, 5ths and octaves. The intervals one might term 1st or 8th are called a unison or an octave respectively. This class contains the intervals dubbed perfect consonances. The options for this class of interval are: Intervals larger than an octave are called compound intervals. They are either named by the number of tones they span, like intervals smaller than an octave, e.g. 9th, 13th, (intervals are only commonly named by number up to the 13th by convention) or they are named with smaller 'equivalent' intervals plus the number of octaves the interval spans, e.g. a 2nd plus an octave, a 6th plus 2 octaves. This reflects a concept called octave equivalence. Different notes with the same exact name (notes related by one or more octaves) are treated as mostly (although not totally) equivalent. Notes with different names, but are the same frequency are also somewhat equivalent, but can differ in function. This means that intervals larger than an octave are thought of as somewhat equivalent to or related to the interval which would be created by changing the octaves of one of the tones in order to bring the interval within the span of an octave. In order to determine an interval's quality, first calculate its size, which determines to which of the two classes it belongs. If the upper note is in the lower note's major scale, then it is major or perfect. An interval can be enlarged (augmented) by either raising the upper note, lowering the lower note, or both. An interval can be made smaller (or diminished) by raising the lower note, lowering the upper note, or both: Remember to consider any key signatures when calculating intervals. Ordered by size, the qualities are: Intervals are usually abbreviated with a: Other than the unison and the octave, a few intervals have special names. Minor and major 2nds are also called half-steps and whole-steps, respectively; an augmented 4th or a diminished 5th is also called a tritone. A tritone is so named because it is made of three whole-steps and is half the size of an octave. Let's calculate a couple intervals together:
 * augmented
 * major
 * minor
 * diminished
 * augmented
 * perfect
 * diminished
 * When a perfect or major interval is enlarged, it becomes augmented.
 * When a major interval is diminished (made smaller), it becomes minor.
 * When a minor or perfect interval is diminished it becomes diminished.
 * diminished < minor < major < augmented
 * diminished < perfect < augmented
 * Capital A for augmented, A2
 * Capital P for perfect, P5
 * Capital M for major, M3
 * Lowercase m for minor, m7, often drawn with an line over it to distinguish it from major, m7
 * Lowercase d for diminished, d6

Here we have an ascending melodic interval, C4 to A-flat4. First, we need to determine its size, so we count up from C, with C starting as one (not zero). This tells us it is some kind of 6th, meaning it can be augmented, major, minor, or diminished. To determine its quality we will see how the top note relates to the bottom note's major scale. In this case, a major scale produced on C contains an A-natural, which would be a major 6th. In order to arrive at the interval given, we have to lower the top note which diminishes (or shrinks) the interval, thus the interval shown is an ascending minor 6th. Shown here is a harmonic interval spanning G3 to B3. To find its size we count the number of letter names the interval spans: G, A, B; a 3rd. To determine its quality will build a major scale on G: G, A, B, C, D, E, F#. We find that the upper note is in the lower notes major scale; thus it is a harmonic major 3rd. This is a descending melodic interval, Bb3 to F3. The notes span from an inflection of B through A & G to F, a 4th. The F major scale contains Bb as its 4th member, so this is a descending perfect 4th. Here we have a larger looking harmonic interval. Tip: It is a good idea to practice identifying the size of intervals visually by calculating the number of lines and spaces that constitute its size. With direct practice, a student can and perhaps ought to be able to identify intervals up to a 10th, more or less instantaneously. This interval spans 9 tones: A, B, C, D, E, F, G, A, B; a 9th. B is the second member of A's major scale so it is a major 9th or an octave plus a major 2nd. This is an ascending melodic interval, Eb4 to F#4. This interval is a second, but play it on a keyboard and it might seem a little large. A major scale built on Eb has an F-natural. The given note is F#, so it has been augmented once from the major 2nd: so it's an augmented 2nd. Which is enharmonically equivalent to a minor 3rd. Here we have a melodic interval, D#4 to Ab4. If we count the note names, we arrive at an interval size of a 5th, easy enough. But when we go to calculate the quality, we run into a problem. What does the D# major scale look like? Although there is a major scale built on D# theoretically (D#, E#, F-double sharp, G#, A#, B#, C-double sharp), in practice we just use D#'s enharmonic equivalent, Eb. Most musicians aren't quick at calculating un-used theoretical major scales, so we need to determine the quality in a different way. The easiest way is to temporarily ignore the sharp on the D and calculate the interval from D4 to Ab4, which results in a diminished 5th. This interval is then further diminished by raising the lower note, thus we have a doubly diminished 5th. While this might seem esoteric, this interval could easily occur in the keys of G or e-minor as two notes which pull to notes to two notes of an e-minor chord for example. Theorists recognize interval qualities being doubly and sometimes triply diminished or augmented. Not to be confused with the verbs diminish and augment which mean to shrink and expand an interval, respectively. These actions can also be said to have been done 'doubly' or 'triply'. Very often these challenging intervals are represented with their enharmonic equivalents. Two or more note names can apply to a single note depending on their context. These notes are enharmonically equivalent. These note names can differ in their function, the role they play in a particular scale and key, but the tone is the same. Since individual notes can be represented by different names on the staff, so too can an interval have different names. Very often composers will write enharmonically equivalent notes in place of notes which make a passage difficult to read, even if another option might be more 'theoretically accurate'. Lastly, intervals can be inverted. An interval is said to be inverted when one of the notes is displaced by an octave (or more) changing the vertical order of the notes. The size of an interval after inversion is easily determined by subtracting the size of the interval from 9. Thus 2nds invert to 7ths, 3rds invert to 6ths, 4ths invert to 5ths, and '''tritones invert to tritones. And vice-versa.''' The quality of an interval after inversion follows simple rules as well:
 * Augmented inverts to diminished
 * Major inverts to minor


 * Perfect inverts to perfect
 * Vice-versa for all the qualities

Further Work
It is very important for students to be able to quickly identify intervals and other theoretical concepts. To this end, you should practice these exercises until the calculations become fluid.