Pythagoras saw ratios and numerical connections everywhere, and it was a connection between simple ratios and harmonic sounds that particularly excited him. This was possibly the primary reason that from his time the ancient Greeks considered music as a mathematical science. Pythagoreans believed that only through numbers may we achieve comprehension of things that would otherwise remain unknown; and that it is not only in aspects of nature that is seen the manifestation of number but also in the creations of art and music. They believed that it was in music the power of the number showed itself most clearly, and the discovery of concord in music and number strengthened their belief in the divinity of the number. It is likely that the Egyptians and Babylonians had already conceived of much of this; documents show that both of these societies knew about Pythagoras’ ideas including his famous theorem well before Pythagoras was born, but he still gets the credit. In any case, originality in philosophy often consists not in having new thoughts, but in making clear what was not clear before. Pythagoras certainly learned about arithmetic and music during his time in Egypt, Babylon and India, but nevertheless such music/math ratios are emphasised in the experiments attributed to Pythagoras and his followers.
So what is this concord linking sound and number? The story goes that one day Pythagoras passed a blacksmith shop and heard the sound of hammers beating on various size anvils. As previously explained, in a musical context, two notes are consonant if they sound pleasing to the ear (the antonym for this is dissonant). Further investigations revealed that the masses of the blacksmith hammers were in simple number ratios to each other. It appeared that there was a simple number connection between the two sounds that produced concords, that is, consonant sounds. Pythagoras continued his investigation away from the blacksmith’s forge, taking his investigation to the medium of a stretched string – probably on a monochord (as in the slide) or lyre. The ancient Greeks knew that pitch was identified with frequency, and they could already determine the rate of vibration of a musical note, and therefore determine the ratio of frequencies of two notes that construct an interval.
Starting with a fixed length of string which we can call the fundamental, halving this produces a sound an octave above. Hence when we double a frequency –say 440 to 880, the pitch will raise an octave. From the fundamental, the new pitch is of the ratio 2:1. Playing the string at one third produced a 5th higher again. At one-quarter length, the string vibrates in 4-parts giving 2 octaves above the fundamental, and at one-fifth of the length of a string, we get a major 3rd, higher still. These first 4 notes of the overtone series form a major chord, the foundation of Western harmony. They are related to mathematically pure ratios.
1:1 Fundamental (Unison)
3:1 5th 3:2 5th within octave range
4:1 2 octaves
5:1 Major 3rd 5:4 3rd within octave range (not in Pythagoras’ time, he didn’t get this far)
The notes that sound harmonious with the fundamental correspond with exact divisions of the string by whole numbers. This discovery had a mystic force.
Hence a fundamental connection between music and mathematics had been established. Consonant musical sounds were related to simpler number ratios. The simpler the ratio, the more consonant was the musical interval. The Pythagoreans knew that by multiplying these fundamental ratios they could determine other pitches, and hence constructed a musical scale as the basis for music composition. This was done primarily through a cycle of perfect 5ths, and then reducing the octave of the note to fit it into the octave range. For example, to find the ratio that gives us a whole tone (M2nd) we can add P5 + P5 (3:2 x 3:2 = 9:4). To bring it down an octave, divide by 2 = 9:8.
And in this way we can calculate the diatonic intervals of a scale.
45:32 A4 (4th + m2 = 4/3 x 16/15 = 64/45) (1.40) tritone
64:45 D5 (5th – m2 = 3/2 x 15/16 = 45/32) (1.42) tritone
The intervals of a perfect 4th and 5th are so called because they sound particularly pleasing to many people, and since the time of the ancient Greeks, this particular feature of the scale, along with the unison and octave is at the heart of all music. These four intervals – the perfect unison, 4th, 5th and 8ve are the basis of all musical consonance. The significance of the musical scale is evident in the fact that every culture we know of has the octave as the basis for its music. Although it has been claimed that some music for example Indian and Arab –Persian music use “micro-tuning” –scales with intervals much smaller than a semitone, close analysis reveals that their scales also rely on 12 or fewer tones and the others are simply expressive variations, similar to the American blues tradition, where a blues note is somewhere between 2 diatonic notes.
In the next post I’ll discuss Music in Ancient China.
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Also by Michael Griffin
Music and Keyboard in the Classroom: Fundamentals of Notation is a unit of work for general music middle school classes. Designed around the mastering of practical skills, it integrates theory, aural and history, and allows students to progress at their own rate. View Table of Contents. “This has been a great buy; the books are just superb! Interesting topics with a wide range of pieces. Great content with clear progression of learning. Fascinating teaching philosophy! BRAVO!” -The Grieg Academy, London. Available at Amazon.com
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View Table of Contents. “We have been using your keyboard course and the results have been amazing!” – St George College, Australia
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Modern Harmony Method: Fundamentals of Jazz and Popular Harmony (Third Edition, 2013) is a clear and well organised text suitable for students of arranging and composition, and for classically trained musicians wishing to grasp the beautiful logic of jazz harmony. Essential understandings include chord selection, voicing, symbols, circle of 4th progressions, extensions, suspensions and alterations. Included in the 107 pages are explanations, examples, exercises and solutions. The course can be started with students in year 9 and worked through to year 12 musicianship, composing and arranging. Available at Amazon.com