T
he growing complexity of music from the Renaissance motets and hymns of John Dunstable and Guillaume Dufay, to the Italian operas of Monteverdi, culminating in the brilliance of J.S. Bach, coupled with the growing popularity of keyboards, seemed to make some sort of temperament inevitable. The 15th through the 17th centuries witnessed a long running debate on the relative merits of pure harmony and tempered harmony, and what form such temperament should take. All of this, as we have seen in Chapter 1, led to the tuning showdown of 1706 between Bach’s cousin and the young man credited with the co-discovery of the equal tempered solution, Johann Georg Neidhardt.We know from Franchino Gafori’s Practica musica in 1496 that organists commonly tempered the fifths and thirds. The organ at St. Martin’s cathedral in Lucca had split keys with separate digitals for D# and Eb, and for G# and Ab. We learn from Bartolomeus Ramis de Pareja’s Musica practica in 1482 that the break had been made with Pythagorean thirds that Ramis claims were “tiresome for
singers.”38 And just as the singers were bringing the thirds into harmonic tune, the keyboards were forcing them back out.
Then, in Brescia, Italy, Giovanni Maria Lanfranco gave tuning rules that approached equal temperament in which the fifths are tuned so flat “that the ear is not well pleased with them,” and the thirds “as sharp as can be endured.”39 This description can hardly be interpreted as an endorsement for the system, but it does give us some idea of how sensitive ears first reacted to the tempered intervals.
In the meantime, the performing musicians were charging ahead with stylistic advances, variations, ornamentations, diminution, augmentation, modulation, inversions and retrograde motion, all made possible by the relative simplicity of keyboards. It was about this time, mid-16th century, that Henricus Glareanus misnamed all the Greek modes when assigning names to the 12 ecclesiastic modes. It was also the age when violins, previously considered crude folk music instruments, were becoming accepted among serious composers and performers.
In 1555 Don Nicola Vincentino built a harpsichord-like instrument that he called an Archicembalo with 31 notes per octave arranged on six ranks of keys.
The first rank consisted of the 7 white keys of the diatonic major scale, on the second rank were the five black keys, and above those seven alternative black keys, forming the 19-tone octave:
C, C#, Db, D, D#, Eb, E, E#, F, F#, Gb, G, G#, Ab, A, A#, Bb, B, B#, C The first certain voice of meantone temperament was that of Arnolt Schlick (1455-1525), the blind organist for Count Palatine at Heidelberg. In his Spiegel der Orgelmacher und Organisten in 1511 he gave a tuning formula that was an early form of meantone. He also mentioned the practicality of split-key organs as a solution. Twelve years later Pietro Aaron in Venice described his meantone temperament with “sonorous and just” thirds, and fifths “a little flat.”
Ranks four and five were “enharmonic” tones with the purpose of reviving ancient Greek scales employing quartertones. The final rank repeated the diatonic scale of the first. Vincentino failed to interest his contemporaries in bringing back the Greek enharmonic genera, but the 19-tone scale gained some support, and was revived a generation later by Michael Praetorius in Germany.
The theoretical battles of this era are best witnessed, however, in the great clash between Gioseffo Zarlino, choir master at the Basilica San Marco in Venice, and Vincenzo Galilei of Florence, a former student of Zarlino, a lute player, and the father of the more famous Galileo Galilei. Zarlino maintained that the natural voices of the singers demonstrate the true intervals, and that those intervals for the diatonic major scale are represented by the Ptolemaic Sequence (1/1, 9/8, 5/4, 4/3, 3/2, 5/3, 15/8, 2/1). He reconfirms that singers did not sing Pythagorean thirds. He pointed out that arithmetic and harmonic proportions could coexist in the same musical schema, and he showed how the number series 1-2-3-4-5-6 created both the major and minor tonalities. Zarlino designed keyboards of 17 and 19 tones per octave, but also accepted temperament for keyboards and lutes, with the
understanding that such temperament compromised the consonance of intervals and triads. He described meantone temperament, and the Euclidean geometric construction used to calculate the mean proportional.
Franco de Salinas (1513-1590), another blind organist, thoroughly
examined the growing tuning problem, and investigated, by ear obviously, all the known or hypothesized solutions for 12-fixed-tone instruments which, it is interesting to note, he calls “artificial instruments.”40 He set out the rules for meantone which had already been formulated by Schlick and Aaron before him.
He noted that for the thirds and sixths to be “made sweeter,” the fifths had to be flattened. For fretted instruments he proposed equal temperament, at least in theory, but not by giving precise formulation. He proposed for the placing of frets on viols that “the octave must be divided into twelve parts equally proportional.”
And finally, he proposed a 24-tone enharmonic matrix based on just intonation which allowed some, but not all, pure major and minor triads.41
Zarlino pointed out that “every composition, counterpoint, or harmony is composed principally of consonances. Nevertheless, for greater beauty and charm dissonances are used,” and that “the ear not only endures them but derives great pleasure and delight from them ...”42 Here Zarlino was referring to pure harmonic
The lute, with its fixed frets, limited the flexibility of the player, and demanded some form of temperament.
(From Marin Mersenne's Book of Instruments, 1636).
intervals, not tempered intervals. He was a choir master with a keen ear, and although he investigated all the options, he clearly favored the pure harmonies that the voices intuitively and naturally found.
Vincenzo must have been overwhelmed with excitement at his discovery.
Unfortunately, his math did not quite add up, and his “midpoint” was flat of the true midpoint or octave of the string. His semitone is the same 18/17 semitone used by Arabic lutists, but it is slightly flat of a truly equal semitone, and this error is compounded over the twelve divisions. Vincenzo’s octave is flat by about one-eighth of a semitone. Zarlino, clearly the better mathematician in addition to his keen ear, wasted no time in publicly replying to Vincenzo and pointing out his error. He published a “Musical Supplement” to his earlier Dialogo, setting the record straight, and reasserting his position that the natural voice and the true harmonic ratios are in agreement.
Commenting on this controversy in the nineteenth century, German scientist, musician, and authority on acoustics Hermann Helmholtz gave Zarlino
Vincenzo Galilei attacked the theories of his former teacher in his 1580 Dialogo della musica antica et della moderna. He claimed that the voice does not teach the true intervals, but rather the voice is taught correct intervals by singing to instruments properly tuned by correct theory. Vincenzo Galilei’s version of correct theory was that of Aristoxenus, a series of tempered fifths that would fit into an octave. He maintained that art does not simply follow nature, but improves upon nature, and he devised a system for tuning the lute that he claimed created twelve equal semitones. To begin, he divides the string length into 18 parts, and the first part from the nut he marks as the first fret. (This is the arithmetic proportion, and gives an 18/17 semitone, flat of the Ptolemaic 16/15.) The length from this first fret to the bridge he again divides into 18 equal parts and marks the first part as the second fret. This procedure he continues until he has marked off
12 semitones of proportionally equal size (18/17). “This brings me to the midpoint of the whole string; the first and lower octave thereof I find I have divided into twelve equal semitones and six tones, as said by Aristoxenus.”43
Their argument over art, nature, and the perfect tuning system raged on until Zarlino’s death in 1590, and thereafter was taken up by others. A generation later, Johannes Kepler, who inadvertently discovered the laws of planetary motion while trying to show that the distances of the planets conformed to the harmonic series, corrected Vincenzo’s semitone and calculated the correct string lengths to make 12 equal semitones equal one octave. His calculations were published in his Harmonices mundi of 1619.
Knowledge of the overtone series, or harmonic series, would change everything. It gave 18th century Harmonists a physical foundation for the harmonic intervals.
credit for reintroducing "the correct intonation," and added that "singers were then practiced with a degree of care of which we have at present no conception. We can even now see from the Italian music of the fifteenth and sixteenth centuries that they were calculated for most perfect intonation of the chords, and that their whole effect is destroyed as soon as this intonation is executed with insufficient
precision."44
Everyone was getting into the act. Galileo, son of Vincenzo, who had already seen the moons of Jupiter through his telescope, began investigating the laws governing the vibration of strings. And French mathematician and
philosopher René Descartes gets credit for being the first voice in history to acknowledge the presence of the harmonic overtones. Descartes wrote his only work on music, Compendium musicae, at the age of 22, shortly after graduating from law school and joining the French army so he could travel and think. He stated that “we never hear any sound without its upper octave.” The work was only circulated in manuscript during Descartes’ life, and was published in 1650, fire year he died. In his Abrégé de musique, Descartes observes “of the two terms required to form a consonance, the lower ... in some way includes the other. This is manifest on the strings of the lute. When one of these is plucked, those an octave or a fifth higher vibrate and sound by themselves.”