On Math and Music: Intervals

Some believe that the most fundamental aspects of reality -- everything that we are and everything that we perceive -- ultimately come down to mathematics. It is the language of science and perhaps the only surviving bastion of irrefutable truths in the aftermath of the Age of Enlightenment. It is this irrefutable quality, this perfection of sorts, that also lends mathematics a certain beauty; in fact, the origin of all beauty may come down to simple mathematical relationships.

Our brains perceive mathematical relationships in countless ways, but perhaps none are so direct as the way in which we process sound. Suppose I were given two tuning forks, one designed for a frequency of 440 Hz and the other for 880 Hz. When I strike the first, the metal begins to vibrate, moving back and forth at a rate of 440 times per second. This vibration, in turn, causes the surrounding air molecules to oscillate at the same frequency, an oscillation that travels outwards from the tuning fork and reaches my ear. Assuming that the tuning fork continues to vibrate at this frequency, my brain will interpret the oscillation of air molecules as a steady and constant "pitch."

Now suppose I strike the second tuning fork, which vibrates 880 times per second. This new pitch corresponds to a frequency twice that of the first, and if I strike the second tuning fork while the first is sounding, they together produce a sound something like this. Notice how smoothly the pitches blend together. The interval heard here is called an "octave," a musical term reserved for any pair of pitches with a frequency ratio of 2:1. Our ear easily identifies the relationship between these two pitches because their frequencies are in a small, natural-number ratio to one another. By contrast, listen to the major seventh, an interval that corresponds to a frequency ratio of about 15:8. The blend is not nearly so pleasing to the ear.

When two pitches blend together well, like the octave, they are referred to as "stable," or a "consonance." Those that don't blend so well, such as the major seventh, are referred to as a "dissonance." In medieval music, the consonances were the octave (2:1), the perfect fifth (3:2, listen), and the perfect fourth (4:3, listen). In the early 15th century, starting with the Burgundian School, intervals of a major third (5:4, listen) and a major sixth (5:3, listen) began to be treated as consonances, allowing for developments such as triadic harmony.

John Dunstaple and Triadic Harmony: The Burgundian Three, Part III

Album: Dunstable: Sweet Harmony Masses and Motets
Track: "Quam pulchra es" (Track #1)
Composer: John Dunstaple
Instruments: 3 vocals
Musical Form: Motet
Year: ~1410-1453
Composition for Comparison: "Twist and Shout" by the Beatles and "Surfer Girl" by the Beach Boys

The elegant melodies of Binchois inspired a generation of musicians and Dufay's epic compositions elicit awe and respect even from modern listeners, but the real revolution of Renaissance music began with John Dunstaple. This great English composer is credited with introducing a technique that is often taken for granted by the 21st-century ear: triadic harmony.

In short, a musical triad consists of the root, third, and fifth of a major scale and contains intervals of a third between successive notes. For you music-theory beginners out there, I've selected some modern, well-known songs to demonstrate what this sounds like. The first is "Twist and Shout" as performed by the Beatles, a rendition that should be familiar to anyone who has seen Ferris Bueller's Day Off. The majority of the song uses a single lead vocal with a pair of backing harmonies, but if you play to 1:25 into the recording, you'll hear the Beatles build into a triadic harmony, one note at a time. It begins with John Lennon singing the root note ("Ahhh...") over the twanging of George Harrison's lead guitar. Next George chimes in with the third, followed soon after by Paul with the fifth. At 1:28, they're in a full triadic harmony. It only holds a complete triad for the few seconds after 1:28 -- the sequence continues after that to the seventh, breaking the triad. The notes of a triad can also be struck simultaneously: C Triad.

It's not uncommon to hear full triadic harmony in modern music, particularly in pop music of the '50s and '60s. In "Surfer Girl," a song by the Beach Boys from 1963, they utilize it throughout much of the song, crafting the melody over top of the moving triads. In fact, the arrangement is in many ways similar to that of "Quam Pulchra Es," one of John Dunstaple's motets from the mid-15th century. This motet is one of the more blatant examples of how the interval of a third was being gradually promoted to the status of a consonance (that is, a stable musical interval) in the early Renaissance period. For those that have been following my Journey, this sudden addition of musical triads should be a very noticeable (and very welcome) change to early polyphony. Compare it to this early mass movement by Guillaume de Machaut, where fourths, fifths, and octaves dominate the harmonic structure. Although such intervals achieve a more "pure" sound, they are almost too easy for the human brain to process and we may be left feeling like something is missing.

Triadic harmonies were commonplace through most of the 15th century in England, but it won't be until the late 15th and early 16th centuries that the musical changes initiated by John Dunstaple and his English contemporaries begin to permeate the music coming from continental Europe. Unfortunately, because of Henry the VIII's dissolution of the monasteries in the 1530's, very little music survives from England in the preceding century and the process by which triadic harmony was introduced into polyphony may forever remain a mystery. Dunstaple was famous enough that some of his music has come down to us from continental sources, but the catalog of surviving music doesn't even approach that of contemporaries like Dufay and Binchois. This is indeed a shame, as many argue that John Dunstaple was the most influential English composer of all time.

Related Links: Allmusic, YouTube, Triads in Medieval Music, Intervals