Tuning System

Musical Term
Dividing the octave into 12 equal intervals

"Equal Temperament" is a musical tuning system that divides the octave into equal intervals, ensuring that all half steps between adjacent notes have the same frequency ratio. This tuning system allows for consistent and flexible modulation between different keys, but it comes at the cost of sacrificing the purity of certain intervals.

Equal Temperament is a widely used tuning system, particularly in keyboard instruments like the piano and many modern instruments. In this system, the octave is divided into 12 equal parts, each representing a half step or a semitone. The frequency ratio between adjacent notes in Equal Temperament is the twelfth root of 2 (approximately 1.05946), which ensures that the distance between each half step is the same.

However, this uniform division of the octave comes with a trade-off. In traditional tuning systems like "Just Intonation", intervals are tuned based on pure integer ratios, resulting in perfectly harmonious intervals. In Equal Temperament, intervals are adjusted slightly to achieve equal spacing, which means that certain intervals are no longer pure. For example, the perfect fifth and major third intervals are slightly narrower in Equal Temperament compared to pure tuning.

Despite this compromise, Equal Temperament offers significant advantages. It allows for seamless modulation between different keys, as all keys have the same pattern of intervals. Musicians can play in any key without encountering jarring dissonances when transitioning between different tonalities.

Tuning System
A tuning system having intervals that are acoustically pure

"Just Intonation" is a musical tuning system that uses pure whole number ratios to define the relationships between musical intervals. In this system, intervals are tuned to express the most harmonious and simple frequency ratios, resulting in pure and resonant harmonies. However, due to the nature of these ratios, it can lead to challenges when modulating between different keys.

In Just Intonation, musical intervals are tuned according to ratios of whole numbers. This tuning approach aims to achieve the most consonant and harmonious intervals by using simple ratios of frequencies. For example, the perfect fifth interval has a frequency ratio of 3:2, which means that the higher note vibrates three times for every two vibrations of the lower note.

Unlike "Equal Temperament", Just Intonation focuses on creating pure and naturally resonant harmonies. However, this system has its limitations when it comes to playing in different keys. Since each key has a unique set of intervals, modulating between keys can result in intervals that are no longer pure, causing dissonance. This makes Just Intonation more suitable for certain types of music, such as choral and acapella singing, where complex modulations are less common.

Despite its challenges, Just Intonation remains a valuable tuning system for musicians and composers seeking to achieve specific harmonic qualities and resonance. It is particularly prominent in music traditions that prioritize rich harmonies and use instruments that allow for adjustments to individual pitches, such as the human voice and certain string instruments.

Tuning System
A tuning system that aimed to balance the purity of certain intervals while allowing modulation between different keys

"Meantone Temperament" is a historical musical tuning system that aimed to balance the purity of certain intervals while allowing modulation between different keys. It achieves this by slightly adjusting the sizes of various intervals, resulting in more consonant harmonies in specific keys but causing dissonance in others.

During the Renaissance and Baroque periods, musicians sought to balance the drawbacks of Pythagorean Tuning and the constraints of Just Intonation. Meantone Temperament emerged as a compromise solution. In this tuning system, specific intervals, such as the major third and minor third, are tuned to be more pure and consonant, while other intervals are slightly adjusted.

The most common form of Meantone Temperament is the "quarter-comma meantone", which divides the octave into equal parts, resulting in slightly narrower fifths and slightly wider major thirds compared to Pythagorean Tuning. This adjustment improves the purity of thirds in certain keys, such as those closely related to the tonic, resulting in more harmonious and expressive music.

However, Meantone Temperament has limitations when modulating to distant keys. Due to the adjusted intervals, certain keys become more dissonant, and some intervals sound out of tune. This led to the development of more flexible tuning systems like Well-Temperament and eventually Equal Temperament.

While Meantone Temperament is historically significant and provides insights into the challenges of tuning, it is less practical in modern contexts due to its limitations in accommodating modulation and key changes.

Tuning System
The system of tuning

"Musical temperament" refers to the system or method used to tune the intervals between the notes of a musical scale, ensuring that the instrument's pitches are in harmony across various keys and tonalities. Different temperaments address the inherent mathematical challenges of tuning in an attempt to strike a balance between pure harmonic ratios and practicality in playing various musical compositions.

The concept of musical temperament arises from the need to accommodate the mathematical relationships between musical intervals within the limitations of musical instruments. Because of the way frequencies interact, it's impossible to create a tuning system where every interval is perfectly in tune with one another across all keys. Therefore, temperaments involve a compromise between ideal harmonic purity and the practicality of playing in different keys.

Historically, various temperaments have been developed, each with its own set of rules for tuning. Some temperaments prioritize certain intervals, such as the pure fifth, while slightly altering other intervals. One common temperament is "equal temperament", where the octave is divided into twelve equal parts, resulting in each half-step being equidistant. This system enables modulation to any key, making it suitable for instruments like the piano that require versatility.

Other temperaments, like "just intonation", strive for pure harmonic ratios but can result in certain keys sounding more in tune than others due to the intricate relationships between intervals. Some historical compositions were tailored to specific temperaments, influencing the character of the music when played using historically accurate tuning.

Different historical periods, musical styles, and cultural preferences have led to the exploration and development of various temperaments. As technology advanced, especially in electronic instruments, musicians gained more flexibility in adopting and experimenting with different temperaments beyond the constraints of traditional acoustic instruments.

Tuning System
A musical tuning system that based on using simple integer ratios to define the intervals between notes

"Pythagorean Tuning" is a musical tuning method based on using whole number ratios to define the relationships between musical intervals. In this system, intervals are adjusted using simple integer ratios, but switching between different keys may result in dissonance.

Pythagorean Tuning originates from the viewpoints of the ancient Greek philosopher Pythagoras (570-490 BC), who believed that musical intervals should be defined based on whole number ratios to achieve the purest harmonies. In this tuning method, intervals are adjusted based on simple integer ratios such as 1:1, 2:1, and 3:2. For example, the perfect fifth interval has a ratio of 3:2, meaning the higher note's frequency is 1.5 times that of the lower note.

However, Pythagorean Tuning presents challenges, especially when modulating between different keys. Since different keys have distinct interval ratios, modulation may result in intervals no longer being pure, leading to dissonance. This restricts the practical use of Pythagorean Tuning in performance and playing.

Although Pythagorean Tuning is less common in modern music, it still finds applications in certain specialized music genres and experimental music. It emphasizes the purity and harmony of intervals but is limited by its inability to accommodate flexible modulation.

Tuning System
Perfectly in-tune intervals complement significantly out-of-tune intervals

"Well Temperament" refers to a class of historical musical tuning systems that seek a compromise between the pure intervals of Just Intonation and the flexibility of Equal Temperament. In Well Temperament, the sizes of certain intervals are adjusted to enhance the consonance of common keys while allowing modulation to other keys without excessive dissonance.

Unlike Equal Temperament, which evenly divides the octave into 12 equal semitones, Well Temperament systems adjust specific intervals to achieve better consonance in certain keys. These adjustments create a variety of different Well Temperament tunings, each with its own unique set of interval modifications.

The goal of Well Temperament is to strike a balance between the purity of intervals in Just Intonation and the ability to play in a variety of keys without extreme dissonance. While the intervals in Well Temperament are not as pure as those in Just Intonation, they are more usable across different keys than the intervals in Equal Temperament.

Various composers and theorists from different historical periods have proposed their own versions of Well Temperament, resulting in a range of tunings with varying interval adjustments. Some famous examples include Kirnberger Temperament and Werckmeister Temperament.

Although Well Temperament has historical significance, it has largely been replaced by Equal Temperament in modern music due to the latter's flexibility in modulating between keys. However, Well Temperament remains an important part of understanding the evolution of tuning systems and their impact on the performance and interpretation of historical music.

Tuning System