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An interval is usually defined as the distance between two notes or pitches. The term distance refers to the space between two notes or pitches, which can be measured in three primary ways described later. There is a distinction between the terms note and pitch, which is important from a theoretical standpoint. Notes usually represent a specific pitch, while the pitch is the frequency that a person hears. A pitch's medium is the air, while a note's medium is usually the staff. This distinction is important because the most common way of naming an interval depends upon notation, and strictly speaking is not transitive to pitches. There are two basic ways by which an interval may be classified, and they can usually be analyzed the same way:
Occasionally there is a distinction between the "general" name of an interval and the "specific" name of an interval. The general name for an interval is derived from counting the number of lines and spaces between two notes including the line or space on which both notes exist. The specific name for an interval is the general name plus one of five potential qualifiers: major, minor, diminished, augmented, perfect. This "general vs. specific" distinction, however, only accounts for the "letter-based" system for naming an interval, which is delineated below. Although it tries to make things more specific by creating more distinctions, the distinctions are useless.
The letter-based name, which is the most common way of identifying an interval, can be found by counting the number of staff positions the two notes encompass. The pitch class {C-E} is a major third because the number of staff positions the two notes encompass is three. While the letter-based name of an interval (major third, minor second, diminished unison, etc) is by far the most common, there are two other ways to name an interval: numerical pitch labels, and scalar distances. A numerical pitch label is found by counting the number of semitones between two notes. For example, C4 has a numerical pitch value of 60 and E4 has a numerical pitch value of 64; therefore, the numeric pitch label is 4. This is also sometimes known as the chromatic distance. In many ways, this identification system makes more sense than the more well-known letter-based system: why is a major third actually only 2 whole tones apart?
The other method of identifying intervals is based on the distance between two notes in a given scale. For example, in D major, the distance between D and B has a scalar distance of “5” - the first note is not counted. The three different systems can be very confusing since the pitch classes {C-E} can be called a major third, chromatic 4th or scalar second! Consonance and Dissonance
It is essentially a postulate that some intervals sound better than others. This cannot be proven since it is inherently subjective; however, a generalization can be made based on the accounts of millions of people worldwide who cover their ears in horror when they hear the all-so-evil diabolus in musica (the tritone). The basic Pythagorean theory of numerical relationships in music, is that more constant intervals have closer whole number ratios that can be found by dividing the frequency (measured in hertz) of the upper note by the frequency of the lower note. For example, the generally accepted tuning standard for the pitch A4, with slight deviations across countries, is 440 hertz. A3 has a relative value of 220 hertz. Therefore, the interval has a 2:1 ratio. The perfect fifth has a ratio of 3:2, the perfect fourth has a ratio of 4:3, the major third has a ratio of 5:4, etc. On the other hand, dissonant intervals have ratios that are inherently less farther whole number ratios. For example, the augmented fourth has the ratio 25:18 in just intonation.
Consonant intervals are those which inherently sound “good”, while dissonant intervals inherently sound “bad”. While the following further sub-classification is largely irrelevant to music today, it did have great significance to composers during the middle ages. This most detailed breakdown of intervals, theorized by Johannes de Garlandia (ca.1205-1255), is as follows:
It can essentially be said that all constant intervals in major-minor tonality are the unison, octave, and those that create major and minor triads. In root position, the superimposed tones of a triad from the root are a third and a fifth. These are the most consonant intervals. The other consonant intervals (fourth and sixth) can be created by inverting the triad so that the root is no longer the bottom note. Dissonant intervals, on the other hand, cannot be created by the triad; rather they are created by means of melodic activity. Dissonant intervals are the result of step-wise melodic motion. Therefore, it can be said that melodic activity and motion is really the result of dissonance.
The perfect fourth is a special case: it is sometimes consonant and sometimes dissonant. In early polyphony, the perfect fourth served as a stable interval. Over the many centuries, however, composers experimented with adding color to their compositions by using thirds and sixths. The most important thing that resulted, besides new musical structures, was the basic triad, which served as the foundation of later music. When the triad was commonly used, it effectively changed musical structure, texture, and the status of the consonant fourth. When the third was used in conjunction with the fourth, it became apparent that the fourth sounded less like a consonant inversion of the fifth, and more like an “active” interval which had melodic tendency towards the third. Therefore, when the fourth is found in a melodic passage, and the note following the fourth is a third, the fourth is said to be dissonant. If the interval functions as an inverted fifth, it is consonant.
There are two classifications of intervals as they relate to structure: simple intervals and compound intervals. A simple interval is any two notes whose distance is less than an octave. A compound intervals is two notes whose distance more than an octave. For example, a major tenth is really just a compound major third (an perfect octave + a major third). A major seventeenth – even though it spans two octaves – is still a compound major third. Generally speaking, any interval larger than a tenth is impractical to talk about in common practice harmony. For that reason, a person may simply call a major nineteenth a compound fifth. Inversions
The word inversion literally means "the reverse of the normal order of something". In music, intervals, chords, melodies, and voices can be inverted. In this chapter we will discuss the inversion of the simple interval, which can be achieved either by moving the top note down one octave, or the bottom note up one octave. This is also known as a registral inversion. For example, if someone wanted to invert a major third (C-E), he or she would would move the C up one octave or the E down one octave. The inverted chord is a complement of the original term. In both cases, it would produce a minor sixth. If someone wanted to invert a perfect fourth, the interval would become a perfect fifth. If one wanted to invert a diminished fourth, it would become an augmented fifth. There are two facts that a person may notice when inverting an interval:
