Pythagorean tuning
Introduction
Pythagorean tuning is a system of musical tuning based on the mathematical ratios of the Pythagorean scale, which is derived from the harmonic series. This tuning system is named after the ancient Greek philosopher and mathematician Pythagoras, who is credited with discovering the mathematical relationships between the frequencies of musical notes. Pythagorean tuning is characterized by its use of pure perfect fifths, which are intervals with a frequency ratio of 3:2. This system was widely used in the music of ancient Greece and the medieval period and has had a significant influence on the development of Western music theory.
Historical Background
The origins of Pythagorean tuning can be traced back to the ancient Greeks, who were among the first to systematically study the mathematical properties of musical intervals. Pythagoras and his followers, known as the Pythagoreans, believed that the universe was governed by mathematical principles and that music was a reflection of these principles. They discovered that the most consonant musical intervals could be expressed as simple ratios of whole numbers. The perfect fifth, with its 3:2 ratio, was considered the most consonant interval after the octave (2:1).
Pythagorean tuning was the dominant tuning system in Western music until the late Middle Ages when it began to be replaced by other systems such as just intonation and equal temperament. Despite this, Pythagorean tuning remains an important part of music theory and is still used in some musical traditions today.
Mathematical Basis
Pythagorean tuning is based on the construction of a scale using a series of perfect fifths. Starting from a given note, each subsequent note is derived by multiplying the frequency of the previous note by the ratio 3:2. This process is known as "stacking" fifths. For example, starting from the note C, the next note in the series would be G (a perfect fifth above C), followed by D (a perfect fifth above G), and so on.
The Pythagorean scale is typically constructed within a single octave, which requires the use of octave equivalence. This means that notes that are an octave apart are considered equivalent, allowing the scale to be "folded" back into a single octave. The resulting scale consists of seven notes, corresponding to the white keys on a piano.
One of the key features of Pythagorean tuning is the presence of the Pythagorean comma, a small interval that arises due to the mathematical properties of the tuning system. The Pythagorean comma is the difference between twelve perfect fifths and seven octaves, and it is equal to approximately 23.46 cents. This discrepancy is a result of the fact that the ratio (3/2)^12 is not exactly equal to 2^7. The presence of the Pythagorean comma means that it is not possible to construct a perfectly consistent scale using only pure fifths, which is one of the reasons why Pythagorean tuning was eventually supplanted by other systems.
Construction of the Pythagorean Scale
The construction of the Pythagorean scale begins with the selection of a starting note, known as the "tonic." From this note, a series of perfect fifths is generated by multiplying the frequency of each note by the ratio 3:2. This process continues until a complete scale is formed.
For example, starting with the note C, the Pythagorean scale can be constructed as follows:
1. C (tonic) 2. G (perfect fifth above C) 3. D (perfect fifth above G) 4. A (perfect fifth above D) 5. E (perfect fifth above A) 6. B (perfect fifth above E) 7. F♯ (perfect fifth above B)
To bring the scale back into a single octave, the notes are adjusted by octave equivalence:
1. C 2. D 3. E 4. F♯ (lowered by an octave to F) 5. G 6. A 7. B
The resulting scale is a diatonic scale, similar to the modern major scale, but with slightly different intervals due to the use of pure fifths.
Characteristics and Limitations
Pythagorean tuning is characterized by its use of pure perfect fifths, which give the scale a distinct sound. The intervals between the notes of the Pythagorean scale are not all equal, resulting in a scale that is slightly "stretched" compared to modern equal temperament. This gives Pythagorean tuning a unique tonal quality that is often described as "bright" or "open."
One of the main limitations of Pythagorean tuning is its inability to accommodate all musical intervals with the same level of consonance. While the perfect fifths are pure, other intervals, such as the major third and minor third, are slightly out of tune compared to just intonation. This can result in a phenomenon known as "wolf intervals," which are particularly dissonant intervals that occur in certain keys.
The presence of the Pythagorean comma also creates challenges for modulation, as the tuning system does not allow for seamless transitions between keys. This limitation was one of the factors that led to the development of other tuning systems, such as equal temperament, which allows for more flexible modulation.
Applications and Influence
Despite its limitations, Pythagorean tuning has had a lasting influence on the development of Western music theory. The emphasis on pure intervals and mathematical ratios laid the groundwork for the study of harmony and the development of other tuning systems. Pythagorean tuning is still used in some musical traditions, particularly those that emphasize the use of pure intervals, such as Gregorian chant and certain forms of folk music.
In addition to its musical applications, Pythagorean tuning has also been studied in the context of acoustics and the physics of sound. The mathematical principles underlying the tuning system provide insights into the nature of sound waves and the perception of musical intervals.