Lucidarium II
Lucidarium II was the second musical theory treatise in the Lucidarium, published in the late 1310s by choral singer Marchettus of Padova. While Padova was not a composer by trade, his experience singing seemed to beget a great interest in understanding why music is what it is; as such, the Lucidarium treatises represent a very unique yet well-learnèd perspective on music theory of the Renaissance era.
Whereas the first book of the Lucidarium discussed the meaning of music and hypotheses on its origins, the second book begins to discuss the practice of how to make music, particularly as regards tuning systems.
On the Whole Tone
The first several chapters cover the concept of the whole tone; according to Marchettus, this interval is the primary basis of musical tuning because it is the main type of step used in singing. He states that the 9/8 ratio for the whole tone is taught as common knowledge, but questions the origin of this knowledge, and thus sets out to determine why this ratio is used and not any other.
Marchetto begins with arithmetic on splitting a continuum, such as a string, into parts: 3, he declares, is the first nontrivial number of divisions, as 1 represents not splitting, and 2 represents a point which cannot be further divided, whereas a set of 3 divisions can be arranged as 2/3 + 1/3. Marchettus extends this by dividing the threefold division itself by three, thus producing nine arithmetic divisions of the continuum, and describes that this pattern can be continued indefinitely.
Marchettus then notes that by dividing divisions, one can make the trivial divisions become nontrivial; if the half is itself split into halves, then the full continuum is split into fourths, which can be arranged in patterns such as 1/4 + 3/4 or 2/4 + 2/4. Because relating two powers from the same series will always produce elements from that series (that is, 3^a : 3^b will reduce to 3^c for all a and b such that c = a - b), these two series must be related to one another to form nontrivial ratios.
What Marchettus describes here can be summarized in modern tuning theory term as the 3-limit: each ratio is some power of 3 compared to some power of 2. Because the 3-limit is here taken as the basis for tuning, the primary interval of music must be a 3-limit ratio, and 9/8 is the first such nontrivial ratio (3/2 is apparently trivial to consider in this case because it is made up solely of the original primes 3 and 2, rather than their powers).
While questionable, this long-winded explanation does follow a consistent and intriguing logic. The 9/8 basis is used throughout the rest of the treatise as the basis for all other notes, intervals, and scale forms.
On the Semitones
While the whole tone is the main type of step used in singing, and the basis of musical tuning according to Marchettus, it is not indivisible. In practice, the whole tone can be divided into several types of smaller parts known as semitones.
Marchettus notes that the 9th harmonic, being an odd number, cannot be split into two equal parts, and thus the reduced 9/8 ratio cannot be divided into equal semitones. However, 9 can be split up into five equidistant parts by way of the odd series: 1, 3, 5, 7, and 9 being its constituents. Thus, the whole tone, being the reduced ninth harmonic, can be divided into five equal parts, each of which Marchettus calls a diesis. Any number of these dieses fewer than five will constitute an interval smaller than a whole tone, and thus a semitone.
This explanation is noted by many later authors as rather questionable. It is unclear how the odd series as divisions of the number nine translate to actual division of a continuum, and it is further unclear how dividing the ninth harmonic into constituents will divide its octave-reduced counterpart into the same number.
