Part 2: Genesis of a Theory

(The history of temperaments is long and complex. Anyone interested should consult the sources listed for a more complete explanation.)

Just as Pythagoras was to provide a starting point for Partch’s theories, Pythagoras, ironically, must also be credited with generating the idea that led ultimately to Equal Temperament: the cycle of 5ths.

Dividing a string into 3 parts––a number that held critical significance for Pythagoras in its representation of the tetractys, (the pyramid of dots; an equilateral triangle with 10 equally spaced dots inside, which according to Theon of Smyrna revealed to the initiate the mystery of universal nature),––Pythagoras generated the interval of a perfect 5th. By continuing the process, generating a fifth above the 5th, he was able to construct the heptatonic scale. But this process did not yield a cycle. The generating of ascending perfect 5ths produces a never-ending spiral in that the 12th 5th is close in pitch to the 1st, but is sharp by 23.5 cents. This remainder is known as the Pythagorean Comma. Pythagoras didn’t accept this irrational spiral, and adjusted all of the 5ths flat by 23.5/12 cents so that they would fit into 1 octave, and yield his divine circle. His actions are manifest of changes that occurred to the human psyche around 600 BC, referred to by the composer Dane Rudhyar as “ The Great Mutation”: the gradual replacement of Mysticism with Rationalism.

Partch’s innovations came through the rejection of the Pythagorean idea of ascending 5ths, instead basing his scale instead on a fundamental law of acoustics, the Overtone Series.

Equal temperament (the intonation system that most western music is based on) was a later development of Pythagoreanism, and is a system of intonation which ultimately resulted from the development of fixed pitch keyboard instruments in the mid 14th century. Europe’s musicians were determined to have keyboards because of the ease in which they could produce major chords as never before, a sound that the people of Europe loved. But it was soon realized that the freedom of intonation they were so used to in the voice would be a tremendous task to apply to keyboards.

“Theorists and builders wanted to apply the Just Intonation of the Ptolemaic Sequence [the modern diatonic major scale in Just Intonation], but they were prevented from doing so by the limitations of (1) comprehension; (2) the mechanics of instrument construction (3) the hand; and (4) the familiar notation.”6

In other words the technology and education at the time would not allow for such a complex system to be realized, and it was because of this that the compromise of Equal temperament was adopted.

Brief Explanation of Temperaments

When two pitches of differing frequency are sounded together or in succession there is a relationship between them, and it is this relationship that the ear senses and is known as an interval in Western music theory. In physical terms, the two notes are related by a comparison of their wavelengths, and the resulting combination represented by their ratio.

“By plucking a string of a given tension and then while plucking it while it is stopped at the exact midpoint, we produce the interval two parts to one part, or one vibration to two vibrations, the ratio 2/1.”⁷

Since the ear hears two simultaneous tones in a small ratio as sounding consonant, Just Intonation uses pure intervals and refers to pitches not by note name but by ratios, all relating to an arbitrary 1/1 ratio, analogous to the open string referred to above. In Equal Temperament, the pitches become more important than the intervals as the demands of harmony and modulation necessitate that pitch identities are fixed throughout the 12 diatonic keys. If we attempt to tune the intervals Just in one key, they will not sound Just in a remote key, in other words, a Just ‘E’ in the key of ‘C’ would be different than a Just ‘E’ in the key of ‘C#’. The compromise is made so that the “E” is equally in (or out) of tune in both keys. This compromise was brought about to necessitate the advancement of musical composition, around the time of Bach, enabling composers more freedom to modulate and use all 12 keys equally, and to use more complex harmonies.

Today, Equal Temperament’s uniformity makes it the perfect tool for creating music in a commodity based culture/economy. Although we have the technology to make viable complex systems of temperament, these possibilities have only been explored on the fringe. Equal Temperament possibly represents a convenience demanded of today’s global economy. But the widespread domination of equal temperament as an unfortunate by-product of imperialism has impoverished world music culture. The following quote from a letter sent to Partch, which he published in his book Genesis of a Music, illustrates this:

“ To go back from Equal Temperament on keyed instruments is to scrap the music of two centuries. We may have entered on an evil course-it has ruined singing for instance-but we shall have to go on with it… The Indians are up against it too: they have imported the harmonium, the issue of which is inevitably European harmony, though they don’t know it. A 25-note harmonium has been invented for them but they won’t use it-too difficult-they are settling down complacently on a 12 note scale, and contenting themselves with the dozen or so ragas it will play, and scrapping the many scores of them they used to sing…it’s all very sad.” –A.H. Fox-Strangways ⁸

Partch’s answer to Equal Temperament

Harry Partch’s theory was a lifelong endeavor and work in progress. His book Genesis of a Music was written as an explanation of his theory and is too extensive to discuss here. What follows are Partch’s summaries of “The Four Concepts”.

“1. The scale of musical intervals begins with absolute consonance (1 to 1), and gradually progresses into an infinitude of dissonance, the consonance of the intervals decreasing as the odd numbers of their ratios increase.” [This is a property of the natural overtone series.]

POSTULATE: Every ratio of a monophonic system is at least a dual identity.

2. Over-number tonality, or Otonality (“major”) is an immutable faculty of the human ear.

3.Under number Tonality, or Utonality (“minor”), is the immutable faculty of ratios, which in turn represent an immutable faculty of the human ear.

4. In terms of consonance man’s use of musical materials has followed the scale of musical intervals expressed as Concept One; from the earliest times it has progressed from the unison in the direction of the great infinitude of dissonance.” ⁹

With these 4 concepts, Partch established that major and minor tonalities could be explained as natural products of the overtone series, and consequently developed the 43 tone scale in which all of his works are written. Partch’s new system of tonality required him to abandon almost all currently used musical instruments, and to create an array of instruments that are no less magnificent than the works written for them.

Next: Part 3: The Instruments »

1. Harry Partch: America's first Microtonal Composer
2. Part 2: Genesis of a Theory
3. Part 3: The Instruments
4. Endnotes, Discography & Bibliography