Alas, while Renaissance theorists considered just intervals the foundation of melody and harmony, there was also a fly in the proverbial ointment, in the form of the growth of independent instrumental music featuring fixed-pitch fretted and keyboard instruments. The polyphonic music of the middle ages and the Renaissance was predominantly vocal music, and the human voice, when properly trained and coupled to a sensitive ear, is readily capable of the subtle intonational adjustments required to perform sophisticated music in Just Intonation. The same can hardly be said for fretted strings or keyboard instruments. A player of a lute, guitar, or viol can make some expressive adjustment of pitch, it is true, but certainly has not the same degree of flexibility as a singer. An organ or harpsichord can produce only those tones that its pipes or strings have been tuned to. For reasons that will not be explained here, but which will be made plain in subsequent chapters, a fixed-pitch instrument intended to play in perfect Just Intonation in more than a few closely related keys requires far more than twelve tones per octave, an arrangement that had already become standard by the fifteenth century. In fact, some experimental keyboard instruments with far more than twelve keys per octave were built during the sixteenth and seventeenth centuries, but, presumably because of their added cost and complexity, these instruments did not become popular and the mainstream of musical thought and activity adopted a different solution to the problem of intonation on fretted strings and keyboard instruments: that of temperament.
The basic premise of temperament is that the number of pitches required to play in different keys can be reduced by compromising the tuning of certain tones so that they can perform different functions in different keys, whereas in Just Intonation a slightly different pitch would be required to perform each function. In other words, temperament compromises the quality of intervals and chords in the interest of simplifying instrument design and construction and playing technique. Many different schemes of temperament were proposed in the Renaissance and Baroque eras, but, at least where keyboard instruments were concerned, they eventually coalesced into a type of tuning known as meantone temperament. (According to many writers, equal temperament was always the preferred system for lutes and viols, because it greatly simplified the placement and spacing of the frets.) Meantone temperament aims to achieve perfect major thirds and acceptable major and minor triads in a group of central keys, at the expense of slightly flatted fifths in those same central keys and some bad thirds and triads and one very bad fifth in more remote keys. The exemplary variety of meantone temperament, called quarter-comma meantone, produced, in a twelve-tone realization, eight good major triads and eight good minor triads, with the remaining four triads of each type being badly mistuned. Meantone tunings satisfied the needs of composers for a time, but as instrumental music became more complex and the desire to modulate to more remote keys increased, the bad triads became a barrier to progress. As a result, musicians gradually adopted another system, twelve-tone equal temperament.
There is some uncertainty as to who deserves the credit or blame for the invention of equal temperament. It seems to have been the product of many minds working along convergent lines over a number of decades, if not centuries. The French monk and mathematician Marin Mersenne gave an accurate description of equal temperament and instructions for tuning it on a variety of instruments in his most important work, the Harmonie Universelle (1639), thereby contributing substantially to its popularization, but the practical adoption of equal temperament, like its invention, was a gradual process, occurring at different rates in different countries. Equal temperament seems to have first found favor for keyboard instruments in Germany, where some organs were so tuned as early as the last quarter of the seventeenth century, although it was still a subject of debate there 75 years later. Meantone seems still to have been the predominant system in France in the mid-eighteenth century, and in England meantone continued to be the predominant tuning, at least for organs, until the middle of the nineteenth century. The commonly held assumption that J. S. Bach was an advocate of equal temperament and that he wrote the twenty-four preludes and fugues of The Well-Tempered Clavier to demonstrate its virtues is at least debatable. The term "well temperament" was used in the eighteenth century to describe a species of temperament in which all keys were usable and in which the principal consonances of the most central keys often retained their just forms. In well temperaments, different keys had different characters, depending on their closeness to or remoteness from the key on which the tuning was centered. This latter characteristic was considered desirable by many Baroque composers and theorists, who believed that different keys had characteristic colors and emotional effects.
Twelve-tone equal temperament, unlike meantone, mistunes all consonant intervals except the octave. Also unlike meantone, twelve-tone equal temperament favors perfect fifths over thirds. The equally tempered perfect fifth is approximately two cents narrower than the just perfect fifth (one cent = 1/100 tempered semitone or 1/12 octave), whereas the equally tempered major third is approximately fourteen cents wider than the just major third, and the equally tempered minor third is approximately sixteen cents narrower than the just minor third. In a sense, the rise of equal temperament can be seen as a partial resurgence of the old Pythagorean doctrine, since the Pythagorean tuning also produced good perfect fifths (and fourths), wide major thirds, and narrow minor thirds. The major advantage of equal temperament over meantone is that every key in equal temperament is equally good (or equally bad). There is no contrast in consonance between keys, so all twelve tones can serve equally well as keynotes of major or minor scales or as the roots of major or minor triads.
Equal temperament was not adopted because it sounded better (it didn't then, and it still doesn't, despite 200 years of cultural conditioning) or because composers and theorists were unaware of the possibility of Just Intonation. The adoption of twelve-tone equal temperament was strictly a matter of expediency. Equal temperament allowed composers to explore increasingly complex chromatic harmonies and remote modulations without increasing the complexity of instrument design or the difficulty of playing techniques. These benefits, as we shall see, were not without costs.
Throughout the Baroque and Classical eras, while music, at least on keyboard instruments, was dominated first by meantone temperament, then by equal temperament, theorists continued to explain musical consonance as the product of simple, whole-number ratios. Considerable advances were made in the scientific understanding of sound production by musical instruments and of the human auditory mechanism during this period. Ironically, Mersenne, who played such a significant role in the popularization of twelve-tone equal temperament, also first detected and described the presence of the harmonic series in the composite tone of a vibrating string and in the natural tones of the trumpet. Mersenne was also the first theorist to attribute consonance to ratios involving seven, the next step up the harmonic series from Zarlino's senario. Later theorists, most notably Jean Philippe Rameau (1683–1784), appropriated the harmonic series as further support from "nature" for the primacy of whole number ratios as the source of consonance. It apparently did not strike most of the theorists of the seventeenth and eighteenth centuries as problematic that, although they formed the theoretical basis for the whole of contemporary harmonic practice, simple-ratio intervals were gradually being purged from musical practice in favor of tempered approximations.
In the nineteenth century, a vigorous attack on equal temperament was mounted by Hermann von Helmholtz (1821–1894), surgeon, physicist, and physiologist, and father of modern scientific acoustics and psychoacoustics. Helmholtz considerably advanced scientific understanding of the production and perception of musical sound, and proposed the first truly scientific theory of consonance and dissonance. He was a strong advocate of Just Intonation, and deplored the effect that equal temperament had on musical practice, particularly with regard to singing. Contemporary with Helmholtz's studies there was a good deal of interest in the invention of experimental keyboards for Just Intonation (primarily organs or harmoniums), particularly in Great Britain. Among those engaged in this activity, the most notable were General Perronet Thompson, Colin Brown, and R. H. M. Bosanquet. Unfortunately, the proposals of Helmholtz and the other intonational reformers of the nineteenth century appear to have had no detectable effect on contemporary musical practice, although Helmholtz's work, in particular, was to have a significant influence on musicians of subsequent generations. Nineteenth century composers were still enamored of the facility for modulation and for the building of increasingly complex harmonies that equal temperament provided, and it was not until these resources were exhausted that any alternative was seriously considered.