Understanding the Pythagorean Comma

The Pythagorean comma is an intriguing and complex concept in music theory that cleverly intertwines itself with mathematics. This anomaly, at its core, refers to a tiny pitch difference, or musical interval, that arises in the Pythagorean tuning system. Named after the legendary Greek mathematician Pythagoras, this tuning scheme operates based on a stack of perfect fifths, each expressed mathematically as the ratio 3:2.

In Pythagorean tuning, a perfect fifth is derived by multiplying the frequency of a given note by 3/2. If you were to start from a specific note, and climb 12 perfect fifths from there, you might expect to land on the same note you started with, but seven octaves higher. An octave, in musical terms, is a doubling or halving of frequency. Consequently, going up seven octaves would imply multiplying the frequency by 2^7 or 128. However, in reality, when we go up 12 perfect fifths (multiplying the frequency by (3/2)^12), we end up with a frequency that is slightly higher than 128 times the original frequency. The ratio of these two frequencies, (3/2)^12 : 2^7, simplifies to 531441:524288. This slight surplus is the Pythagorean comma.

The exact size of the Pythagorean comma can be calculated using the formula 12*log2(3/2) - 7, which equals approximately 0.23 semitones. In other words, after twelve perfect fifths, we've overshot our target by about a quarter of a half step. It's worth noting that this is a logarithmic measure, reflecting the fact that our perception of pitch is logarithmic in nature – a doubling or halving of frequency always corresponds to the same perceived interval, regardless of the absolute frequencies involved. This is what makes the mathematics of music both fascinating and complex.

Why Does the Pythagorean Comma Exist?

The existence of the Pythagorean comma is deeply rooted in the mathematical foundation of pitch and the intricate characteristics of the harmonic series. In Pythagorean tuning, the frequency ratio of 3:2, representing a perfect fifth, is of paramount importance. The sound waves of two notes with this ratio have a simple, pleasing relationship, with the wave of the higher note fitting neatly into the wave of the lower note three times for every two times the lower note vibrates. This results in a consonant, harmonious sound.

However, there is another fundamental principle in music: the concept of octave equivalence. This principle holds that notes separated by an exact number of octaves are perceived as musically equivalent, i.e., they sound 'the same'. The challenge arises when we attempt to reconcile the 3:2 ratio of the perfect fifth with the 2:1 ratio of the octave.

If we proceed by stacking twelve perfect fifths (each with a frequency ratio of 3:2), we don't return to the exact point where we started in terms of octave equivalence. Specifically, we exceed our initial frequency by a ratio of 531441:524288, which is the Pythagorean comma. This discrepancy emerges because the numbers 2 and 3 are not commensurable – there's no power of 2 that equals a power of 3, and no number of stacked fifths (each with a ratio of 3:2) that can align us back precisely to a note we began from. The incommensurability of the octave and the fifth is what makes the Pythagorean comma an inevitable result of the Pythagorean tuning system.

The Implications of the Pythagorean Comma

The implications of the Pythagorean comma extend significantly into the realm of music, particularly concerning tuning systems and temperament. The existence of this comma signifies that we can't craft a tuning system that aligns perfectly with both the harmonic series (which gives us the 3:2 ratio of the perfect fifth) and octave equivalence (which gives us the 2:1 ratio of the octave). Some form of compromise is imperative, and the nature of this compromise initiates the emergence of different 'temperaments', or tuning systems.

The Impact on Tuning Systems

Pythagorean tuning, which adopts one such approach to tackling the Pythagorean comma, chooses to overlook it for the sake of maintaining perfect fifths. Consequently, this causes some intervals, notably the diminished sixth, to sound noticeably 'out of tune'. This infamous interval, often called the wolf interval, is where the discrepancy caused by the Pythagorean comma is most apparent. As a result, the system works well for music that primarily uses perfect fifths, but less so for music that uses a broader variety of intervals.

Modern Equal Temperament

Equal temperament, the most commonly employed tuning system in Western music today, introduces a different approach. Instead of preserving the perfect fifth at the expense of other intervals, equal temperament 'spreads out' the Pythagorean comma across all twelve half steps in the octave. This results in all intervals (except the octave) being slightly 'out of tune', but not so much that it's generally noticeable to the untrained ear. It allows for free modulation between keys, which was a major development in the history of Western music.

Practical Existance Of Pythagorean Comma

In practice, the Pythagorean comma and its implications are not often consciously utilized in composing music. However, the choice of tuning system has a significant effect on the sound of a piece, and different tuning systems have been preferred at different periods in history. For instance, music from the Renaissance and Baroque periods, when mean-tone temperament was commonly used, can sound quite different when performed on modern instruments tuned to equal temperament.

One might argue that any piece of music that modulates to different keys or uses chromaticism is indirectly making use of the Pythagorean comma. For instance, in Bach's "Well-Tempered Clavier", the composer wrote preludes and fugues in all 24 major and minor keys. In a sense, the entire work can be seen as an exploration of the possibilities afforded by the well-tempered system (a system similar to, but not exactly the same as, equal temperament), and by extension, an exploration of the consequences of the Pythagorean comma.

Perception and The Role of Context

Our perception of music and the tuning systems it employs is heavily influenced by cultural and historical context. In Western music, we are so accustomed to equal temperament that other tuning systems can sound out of tune to us. This can be seen in the reception of some historical performances using period-appropriate tunings – they may sound strange or even dissonant to an untrained ear accustomed to equal temperament.

However, in other musical cultures, different tuning systems are the norm, and equal temperament may not be used at all. For example, many forms of Indian classical music use just intonation, a system that prioritizes consonance and harmonic purity over the ability to modulate to different keys. The scales (ragas) used in this music might sound 'out of tune' to a Western listener, but they are not – they are simply tuned differently.

The Pythagorean Comma in Modern Composition

While the Pythagorean comma is primarily of interest in the context of tuning and temperament, some modern composers have explicitly incorporated it into their work. For example, James Tenney's piece "For 12 Strings (rising)" uses a sequence of pitches that ascend by the interval of the Pythagorean comma, creating a unique spiral of sound.

Other composers, such as Harry Partch and Ben Johnston, have created their own tuning systems that make explicit use of microtonal intervals like the Pythagorean comma. These compositions often require specially built or modified instruments to perform, and they can challenge our expectations of what music can sound like.

The advent of digital audio technology has made it easier than ever for composers and musicians to experiment with different tuning systems. Today, anyone with a MIDI controller and a DAW can create music that uses the Pythagorean comma or other microtonal intervals, opening up a world of sonic possibilities.