On equal temperament

In this article, I use Stengers’ (2010) concepts of ‘factish’, ‘requirements’ and ‘obligations’, as well as Latour’s (1993) critique of modernity, to interrogate the rise of Equal Temperament as the dominant system of tuning for western music. I argue that Equal Temperament is founded on an unacknowledged compromise which undermines its claims to rationality and universality. This compromise rests on the standardization which is the hallmark of the tuning system of Equal Temperament, and, in this way, it is emblematic of Latour’s definition of modernity. I further argue that the problem of the tuning of musical instruments is one which epitomizes the modern distinction between the natural and the social. In turn, this bears witness to what Whitehead calls the ‘bifurcation of nature’. Throughout this article, using the work of Stengers and Latour, I seek to use tuning as a case study which allows social research to talk both of the natural and of the social aspects of music and tuning, without recourse to essentialism or simple social construction. In this way, my argument seeks to avoid bifurcating nature

The temperament of keyboard instruments in England during the late sixteenth and early seventeenth centuries

Meantone is reputedly the temperament for early keyboard instruments. General references are made to it as one of the pre-cursors of equal temperament, and "Das Wohltemperierte Clavier" is often cited as one of the main turning points in the history of keyboard temperament, Bach being held as a champion of equal temperament at a time when meantone was the accepted temperament. Looking further back into the history of keyboard music we would expect to find meantone firmly established as the standard temperament, but the issue is complicated by a Fantasia by John Bull, the difficulties of which can only be explained by reference to equal temperament. If equal temperament is accepted as necessary for the Bull Fantasia, certain questions immediately spring to mind. How strongly was meantone the established temperament for keyboard instruments if equal temperament was known to one of the earliest schools of keyboard music? How can "Das Wohltemperierte Clavier" be considered as such an important milestone in the history of keyboard temperament if equal temperament was conceivable at the beginning of the seventeenth century or even earlier? Finally, if equal temperament was known at the beginning of the seventeenth century, and firmly established in the first half of the eighteenth, why did a firm such as John Broadwood and Sons only make equal temperament its standard keyboard temperament as late as l846? Dr. J. Murray Barbour has cast serious doubts on the assumption that Bach intended "Das Wohltemperierte Clavier" for an equally tempered instrument, whereas a glance at mid-nineteenth century keyboard music, with its enharmonic changes, makes anything other than equal temperament impossible, but the question of the chromatic notes in the works of the English virginalist composers has yet to be fully investigated. This study is an attempt to bring together information about the music of the late sixteenth and early seventeenth centuries and the temperaments and instruments available from which to draw conclusions about the most likely temperaments to have been in use. The available material strongly suggests quarter comma meantone as the standard temperament, and there is sufficient evidence in the music to support the notion that the dissonance caused by the occasional substitution of an E flat for a D sharp was tolerated, while at the same time it seems clear that the Bull Fantasia was written for a clavicymbalum universale, an instrument which offered exactly the nineteen different notes which Bull required for this piece

Tuning Renaissance and Baroque Instruments: Some Guidelines

Provides detailed guidelines for tuning keyboard instruments, including the harpsichord, piano, and organ. Musical examples for tuning unisons between overtones in consonant intervals and octaves are included. Other musical examples illustrate the procedures for setting a quasi-Pythagorean temperament, a meantone temperament, a temperament ordinaire, a Bach-style irregular temperament, and an equal temperament

The Grail of Harmony: Just Intonation Vs. Equal Temperament

Western art music is founded upon the system of tuning known as equal temperament. The European continental based harmonic science which defines classical music was effectively established upon the science of this system. The opposite system is that of just intonation, of which the method of tuning yields scales with partials, tones smaller than those in existent in common practice, commonly referred to as microtones. Eastern music is founded upon this system. There has been debate on both systems sparsely throughout history since earliest recorded antiquity. Numerous performers, scholars, theorists, and scientists have observed problems with common practice equal temperament. Advocates for just intonation address limitations equal temperament causes on the ear in addition to composition. The following annotated bibliography brings to light the benefits of music founded upon just intonation in comparison to the limitations of equal temperament through the work of musicians, composers, scholars, philosophers, and scientists

The Establishment of Equal Temperament

Equal temperament is the foundation for modern music, yet most musicians have no concept of its meaning. In a modern culture that treats music solely as a means of entertainment, there is no need to understand the complex mathematical foundation on which music was created. However, hundreds of years ago, there existed an entire culture that integrated these numbers into their philosophies and way of living. Unfortunately, there were flaws in the relationships between the numbers that were kept secret for several centuries. These imperfections could not be hidden forever, and as the years wore on, people began to fight the old traditions. Many intellectual and religious leaders created solutions to compensate for those flaws and thus old systems of tuning were challenged by these new methods. Humans have always struggled with change and implementation of modern tuning and temperament methods were no exception. Thus, the road to the establishment of equal temperament tuning became a long, and difficult battle against the laws of the universe

Flexible Tuning Software: Beyond Equal Temperament

The premise of this creative Capstone project was to develop a computer instrument that is capable of tuning itself flexibly in such a way as to not require a tempering of the Western scale, as is necessary for fixedly tuned instruments. The difficulty in creating such a system of tuning arises from the mathematical paradox of the musical harmonic series, which is the sequence of frequencies that sound naturally as overtones over a fundamental pitch. They follow a proportional pattern of 1:2, 2:3, 3:4, etc. These small-integer ratios of frequencies represent the consonant (in-tune) harmonic intervals. When an interval does not have a small-integer ratio between its notes, it is perceived as “out-of-tune,” and the two frequencies will compete with each other. The problem becomes apparent when the ratios are used to create the Western scale on a fixed instrument. When tuning such an instrument (i.e. a piano), the intervals inherently cannot all be tuned justly, or according to the appropriate proportions. The nature of the ratios does not allow larger intervals to be explained exactly by the smaller intervals, though they are expected to coexist and be used simultaneously in Western music. For example, an octave in music is the equivalent of three major thirds; however, the justly tuned octave (2:1) is not equal to the sum of three major thirds (5:4); 5:43=125:64, not 2:1. The difference, then, between notes tuned in terms of different intervals is known as a comma, and the attempts to distribute this comma are called temperaments. To overcome the impossibility of perfectly consonant temperaments, we have created a computer program that can function as a self-tuning instrument by mathematically calculating the frequency of each pitch played in relation to the pitch preceding it. Though this allows every interval between sequentially played notes to be in tune, it does mean that pitches are not fixed, and rather are flexible and changing as a piece progresses. The program possesses the capability to play several historical fixed temperaments, namely Pythagorean, Quarter-Comma Meantone, Werckmeister III, and Equal Temperament. However, it also is capable of playing in two non-fixed tuning systems, distinguished as Sequential Tuning and Flexible Tuning. The Sequential Tuning system uses the ratios in the harmonic series to tune each note proportionally to the most recent note pressed. It is ideal for educational purposes, clearly demonstrating the flexibility of the program, but is less practical for tuning purposes, as it disregards held notes and tunes solely based on the notes pressed while the note is held, creating obtrusive dissonances between any held note and the notes tuned sequentially before it is released. The Flexible Tuning system remedies this issue by first tuning notes based on a held note, and if there is no note being held, then on the most recent note pressed. This eliminates the noticeable dissonances of the Sequential Tuning system, while still tuning each note flexibly and in real time based on its interval relationships to other notes played before and at the same time as it. In this way, every interval sounding is the appropriate small integer ratio for that interval in the harmonic series

How Equal Temperament Ruined Harmony (and Why You Should Care) by Ross Duffin

Rasch discusses and critiques Duffin\u27s book on equal temperament

The Battle Between Impeccable Intonation and Complete Chromaticism

Equal temperament represents a way of completing the musical circle, and systematically compensating for the Pythagorean comma. Pythagoras discovered this acoustical problem around 550 B.C., and since that time music theorists have debated how to deal with it. The problem is that no perfect solution exists—something must be compromised. As musical styles developed, specific factors and harmonic tendencies led to the gradual adoption of equal temperament. Early in music history, theorists preferred systems which kept acoustical purity relatively intact. Pythagorean intonation and just intonation serve as two examples. However, the move from modality to tonality decentralized the melody as the dominating feature of a composition. Correspondingly, this raised the importance of harmonic structure, and introduced the idea of modulation. Not all tuning systems allow a performer to easily change keys; most systems contain some type of wolf fifth. This interval sounds exceedingly dissonant, due to its distance from an ideal frequency ratio. Thus, composers had to avoid certain keys, like F# Major or Bb Minor. During this time, hundreds of different meantone temperaments arose, all of which deal with the Pythagorean comma in slightly different ways. These temperaments attempt to balance pure acoustics and freedom of modulation. Eventually, as chromaticism became increasingly common, so did equal temperament. Musicians traded true intonation for the ability to play in any key at any time. While equal temperament is now universally hailed as the standard tuning system, it is not perfect. Rather, it represents a compromise designed to best accommodate the needs of tonal music since the Baroque Era. I will mathematically show the problems encountered when creating a tuning system, and discuss the various known solutions. I will then use historical documentation to show how musicians eventually landed on equal temperament as the most complete solution