Now that we have formalized our notation at the fundamental level of music notation–a single pitch–we need a method to measure the distance between two pitches.
Any two-note combination is called a dyad, and the distance between the two pitches of a dyad is an interval. Intervals are the fundamental building blocks of melody and harmony. At their simplest, intervals need only measure the distance between two pitches, but there are many variables in music for which we must account.
In the example below, each interval represents the concept stated at the beginning of its staff, but each measure also has an important intervallic relationship to the measures above and below it. Using these, develop explanations for how we find each of the following:
Our goal when measuring intervals is intrinsically tied to the tonal system that we use, diatonic harmony. The simplest way to measure the distance between two intervals would be to measure the distance by the shortest possible interval–in this case, a half-step (minor second). While easily understandable, this method does not relate to our concept of tonality. Instead, counting half-steps creates interval-classes in which intervals are considered equal regardless of the pitches. For example, the interval of G to D-flat has six half-steps which is identical to the interval from G to C-sharp. Both even use the same pitch-classes, however, any person familiar with diatonic harmony will immediately associate these two intervals with different key centers. (G to D-flat is strongly associated with the key of A-flat major/minor, whereas C-sharp to G likely implies D major/minor.) The context of these two intervals is critical in determining their function in tonal harmony, so we must use a system that differentiates between the two.
In a diatonic labeling system, every interval has a size and a quality.
For example, in a minor second, labeled m2, the m indicates the quality of the interval and the 2 indicates the size of the interval.
Interval size can be determined by considering either:
Both these methods will correctly identify interval size, although counting letter names yields results without requiring the presence (or visualization) of a staff. Do not forget that you must include both letters when counting. (e.g. The ascending interval from A to C is a third, because you must count A (1), B (2), and C (3).)
This means that any interval that has the same two letters, regardless of accidentals or key signatures, will always have the same size. Using our previous example, the size of the interval between G and D-flat is a fifth: G (1), A (2), B (3), C (4), and D (5). We can change the bottom note to any other G (G-sharp, G-flat, etc.) and the top note to any other D (D-sharp, D-natural, etc.), but the size of that interval will always be a fifth. Yet if we exchange the D-flat for its enharmonic equivalent, C-sharp, we alter the letters and turn the size of the interval into a fourth.
Interval quality is difficult to examine without beginning to think about our concept of tonality and keys, because it is designed to describe tonal intervals. One of the most common and straightforward methods for finding interval quality requires a strong familiarity with the twelve major scales:
From this, our interval hierarchies can be grouped into two distinct hierarchies:
Note that even though perfect intervals use a different hierarchy than major/minor intervals, both hierarchies share the terms diminished and augmented.
Let’s practice some examples using this method:
C to E:
C, as the tonic of a major scale, we can use the C-major scale to find that the naturally occurring E in the key of C major is E-natural.E-natural would be a major third *(M3) above C.D to G-sharp
D, as a tonic, we can use the D-major scale to find that the naturally occurring G in the key of D major is G-natural. Therefore, G-natural is a perfect 4th above D, because all naturally occurring intervals in a major scale are either perfect or major depending on their size.
G-sharp is one half-step above the perfect interval, we go one step up the “perfect” hierarchy to find that this interval is an augmented fourth (A4).F to E-double-flat
F, as do, we can use the F-major scale to find that the naturally occurring E in the key of F major is E-natural.E-natural would be a M7 above F.E-double-flat is two half-steps below the major interval, we go two steps down the major hierarchy to find that this interval is an diminished seventh (d7).Melodic and harmonic intervals are one of the simplest concepts to understand, although the describing them can be problematic. At first, students often describe harmonic intervals as occurring “at the same time” while melodic intervals occur “at different times”. While this is a simple explanation, we must be careful in how we apply the terms interval and pitches. The interval is the space between the two pitches, therefore, the interval cannot occur “at the same time” or “at different times”. The pitches can occur simultaneously or consecutively, but the interval always exists as a fixed measurement. This is how we can label harmonic and melodic intervals using the same system.
I recommend that we think of intervals as existing on either a horizontal or vertical axis, because we can visualize axes (as in the plural of axis) easily on musical staff notation. If two pitches occur “at the same time”, they will be aligned vertically on a staff. If two pitches occur consecutively, they will represent unique points on a horizontal line that runs parallel to the lines of the staff. This may seem overly technical, but it is an important distinction.
A final note: melodic intervals can have two modifiers attached to them, ascending or descending. Ascending intervals start with the lower of the two pitches, whereas descending intervals start on the the higher pitch. Harmonic intervals cannot be ascending or descending.
Simple intervals include any interval that is equal to or smaller than an octave. Compound intervals are any interval larger than an octave.
To label compound intervals, we count letter names as we do for simple intervals. We can find a compound interval by adding 7 to any simple interval. For example, a 2nd becomes a 9th. A 4th becomes an 11th. An 8ve (octave) becomes a 15th.
Conversely, if we see a compound interval, we can find its simple equivalent by subtracting 7. A 12th is a 5th plus an octave. A 10th is a 3rd plus an octave.
You may ask why we don’t add eight considering that we are adding an octave. The answer lies in how we find the size of intervals. When we find interval size, we count the letter names and include the starting pitch. When we add an octave, we have already used the top note so we are missing one letter. For example, a fifth from A to E includes the letters A B C D E. If we add an octave, the first E was already included in the first interval of the 5th, so we are only adding seven letters F G A B C D E.
The difference between chromatic and diatonic is probably the most straightforward of interval classifications in usage, but it has a level of nuance that students often miss and causes confusion later. Simply put, diatonic intervals contain only pitches that belong to the current tonality, whereas chromatic intervals contain at least one pitch that does not belong to the current tonality.
Of course, this relies on you knowing what the current tonality is. In many situations, this means that you can find chromatic pitches, and therefore chromatic interals, wherever you find a note with an accidental. So if the key signature matches the current tonality–for example, the key signature has two sharps and the tonality is D major–any pitch that does not have an accidental is a chromatic pitch and all intervals formed with that pitch would be chromatic intervals.
An easy way to think of inverted intervals is to consider an inversion to be an interval in which one pitch is fixed and the other is transposed by one octave toward the fixed pitch.
To determine the size of an inverted interval, it is easiest to simply memorize the interval pairs, so:
Each of these pairs adds up to nine, so if you ever forget or doubt your memorization, you can find an inversion by simply subtracting the interval size from 9. For example, if there is a written 3rd, subtract 3 from 9 to find that the inversion of a 3rd is a 6th. Note that for compound intervals, you must use subtract from 16 or use negative numbers and absolute values. Because of this, it is easier to reduce compound intervals to a simple interval before inverting.
To find the qualities of inverted intervals, you simply need to memorize three pairs:
We will explore the mathematical underpinnings of why inversions form these pairs in Unit 23.