A little History: Who is Pythagoras?
Pythagoras is a Greek mathematician from the 6th century B.C. (circa 569 - circa 475 B.C.E.). He was born in Samos. Even though he apparently conviced everybody that the world is a sphere, for some reason he is best known for the infamous theorum below regarding triangles:
In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Or:
a2 + b2 = c2
I found an interesting web page which demonstrates proof of this theorum. If you've got your Java, then click here to visit.
The Importance of "12"
Among his ideas/theories about numbers Pythagoras believed that at its deepest level, reality is mathematical in nature. (Now let's keep in mind that this man lived thousands of years before the "relativist" philosophies of Stucturalism -- long before people started questioning the concepts of "reality" and "nature"). Anyways, it seems he set out to discover relationships between numbers and his natural environment.
In regards to music, Pythagoras is credited with developing our understanding of the harmonic series - the overtone series.
As the story goes, Pythagoras noticed that a blacksmith's anvil created tones which were relative to the weight of his hammer. (However, some academics claim that this story is absurd). Regardless of the source of this story or whether or not it's true, it seems true that Pythagoras was perhaps the first person to define the "consonant" acoustic relationships between strings of proportional lengths. Specifically, strings of equal tension (regardless of their material: gut, steel, rope, etc.) of proportional lengths create tones of proportional frequencies when plucked.
For example, a string that is two (2) feet long will vibrate x times per second (Hertz). While a string that is one (1) foot long (x/2) will vibrate twice as fast: 2x. And furthermore, those two frequencies create a perfect octave.
- x Hertz -- Fundamental Pitch
- 2x Hertz -- An Octave Higher
Likewise, dividing the length of that string into halves, thirds, quarters, and fifths will create an octave,
a perfect fifth an octave, and a major third respectively.
- Length = x -- Fundamental Pitch
- x/2 -- An Octave Higher (2/1 the Fundamental Frequency)
- x/3 -- An Octave plus a Perfect Fifth (3/1 the Fundamental Frequency)
- x/4 -- An Octave plus a Perfect Fifth plus a Perfect Fourth (= Two Octaves) (4/1 the Fundamental Frequency)
- x/5 -- Two Octaves plus a Major Third (5/1 the Fundamental Frequency)
In dividing the length in this manner, Pythagoras exposed the first four overtones which create the common intervals which have become the primary building blocks of musical "harmony".
- An octave
- A Perfect 5th
- A Perfect 4th
- A Major 3rd
Pythagoras also acknowledged these intervals, not only as they relate to the fundamental frequency, but to each other and found these ratios:
- 1:1 = Unison
- 2:1 = Octave
- 3:2 = Fifth
- 4:3 = Fourth
- 5:4 = Major Third
Pythagoras went a few steps further however and he acknowledged that twelve (12) was the "most divisible" small number and that these basic ratios can be expressed in regards to the number twelve (12).
- 12:12 (unison)
- 12:6 (octave)
- 12:8 (fifth)
- 12:9 (fourth)
He therefore summized that twelve (12) was the most ideal musical number; thousands of years of musical composition can attest to the notion that he was mostly right.
The Tri-Tone
Pythagoras and his contemporary acousticians/musicians/mathematicians believed that the most consonant interval was a unison (i.e. the same pitch sung or played by two or more voices).
The other intervals, according to the overtone series, were progressively more dissonant. The octave then would be the next "most consonant", then the fifth, the fourth, the major third, the minor third, the major second, then the "most dissonant" minor second.
Pythagoras apparently made one other important observation however about our octave: when the octave is divided exactly in half, an extremely dissonant sound is produced. This sound has become known as our tri-tone. To repeat: I said when the octave is divided in half -- I'm not talking about those strings anymore. This means that if you put your fingers on middle "C" on a piano, and the "C" one octave above, the "F#" which lays in the exact middle of the octave will produce the tri-tone. Moreover, if there are twelve (12) semi-tones in an octave -- the 6th note is the tri-tone.
Overtones Through the Ages
Although very little has changed in respect to these early observations about overtones there have been some drastic changes in musical composition since the days of Pythagoras and his buddies.
The Reign of the Perfect Intervals
Some of the earliest written/recorded music from the Gregorian period (around 600 A.D.) shows that the composers/musicians of that time used the intervals of unisons and octaves almost exclusively. Following some time later the intervals of fifths and fourths were commonly used. The music I'm referring to here is commonly known as "Gregorian Chant" or "Organum".
Then sometime prior to or during the Renaissance composers began to focus more strongly on the major third and used that interval as a basis for building melody and chords.
The Major Third
Even into the Baroque period while Johann Sebastian Bach was composing music, the major third was of utmost importance. Bach's music, despite his use of relatively complex chords and being very "chromatic" at times, always shows a great respect for the major third. For example, all of his compositions end with a major chord, even if the rest of the piece is written in a minor key. Furthermore, Bach and others of the Baroque period treated the intervals of fourths and fifths differently than their predecessors. While the monks of 600 A.D. would have sung entire melodies harmonized in fifths, Bach would have scoffed at them and claimed that those fifths (which recur from each note to the next as in parallel harmony) cause dissonant sonorities. Thus, it seems somewhere between 600 A.D. and 1600 A.D. musicians modified their use of the fifth and fourth evidently to accomodate their newfound favourite: the major third.
It wasn't until much later, perhaps mid-Romanticism (but still very rare), that a piece of music was able to end on a minor chord.
Our Tolerance for Small Intervals and the Inevitable 12-Tone "Pan-Tonality"
In the late 19th century it seems composers (perhaps starting with Debussy, Ravel, Shostakovish) began to disregard the major third in favour of other harmonic constructs. Debussy, despite the common practices of his contemporaries, wrote music using parallel octave and fifth harmonies. This of course seems like a return to the days of "Gregorian Chant", but in terms of 19th century musical taste it was very radical. Composers by this time were familiar with using extended chord structures including multiple layers of thirds: while "C E G" create a major chord, "C E G B D" create a "major 9th chord"), and had all but completely disregarded the diatonic scales to generate new sounds and dramatic harmony.
As Henry Cowell observes in his book, "New Musical Resources": "the history of harmony has followed the series of natural overtones" (Cowell, 1930). It seems that our tolerance for dissonant harmony, our acceptance of smaller overtones and therefore more complex sonorities, has grown/changed according to those early observations of Pythagoras. As time passes, musicians and composers continue to explore the higher overtones and audiences become more accepting of the common intervals. What was once considered extremely dissonant is now relatively consonant.
In the early half of the 20th century, Arnold Schoenberg developed a new method of composing music which did not favour any of the intervals, while considering all of them. He called it the "12-tone method". In this method of composing music, all intervals are treated equal (for the most part) and all notes are treated with equal import. Some folks call this music atonal, while others call it pan-tonal...and a host of other names have followed in this vein: serialism, dodecophonic, weird, etc.
Regardless of its name, 12-tone music seems to be a culmination so-to-speak of 2600 hundred years of acoustic/harmonic/musical evolution. Beginning of course with Pythagoras' observations about intervals and the number 12, continuing through a slow but logical acceptance of the lower partials of the overtone series, and ending with 12-tone music which uses all intervals equally and is not restricted harmonically to diatonic/tonal implication.
So then, what's next? As Henry Cowell suggests in his book, "New Musical Resources", composers will continue to explore the higher partials of the overtone series. Those intervals divide the minor second into smaller and smaller bits leaving only microtones. While some composers of the 20th century have created microtonal music, audiences seem reluctant to accept it. (I'm speaking of course of the Western-Euro-North-American musical tradition. The musical traditions of Indonesia, Africa, Asia, India, etc. have been very accepting of microtones for many hundreds of years.)