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'''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.

Aside from the diesis, there are three types of intervals smaller than the whole tone, found respectively at two, three, and four times the size of the diesis.  The two-diesis semitone is known as a limma, and is the small step of the diatonic scale, which Marchettus notes is used in plainchant.  The three-diesis semitone is the apotome, the counterpart to the limma, and the difference between a large and small step in the diatonic scale; Marchettus notes that its larger size can create discordance when employed as a step, and hence it is not often used in plainchant.  The final semitone is the counterpart of the diesis: Marchettus simply calls this interval a ''chroma'', from the Greek word for ''color''.  This interval is used, as he claims, to "color" consonances for specific melodic effects.

Marchettus claims that the chroma is used in ascending melodies when two semitones occur in a row; for instance, a melodic pattern that travels F - F♯ - G will be sung as though F♯ were a chroma higher than F, and G a diesis higher than F♯.  This claim may be representative of the tendency of vocal music to squish leading tones, especially at the end of a phrase.

== Implications for modern tuning theory ==
The ideas provided by Marchettus in the ''Lucidarium'' collection of treatises have many applications in modern tuning theory and xenharmonic composition, including construction of and composition in equal temperaments and regular temperaments of multiple kinds.

The only precise ratios outside the 3-limit cited in ''Lucidarium II'' are 18/17 for the limma and 17/16 for the apotome, which represent an otonal division of 9/8 as the trichord 16:17:18.  If we assume that these semitones are tuned justly, then the diesis does not indicate an exact ratio, but rather a size range which is used to estimate the sizes of each part.

However, Marchettus does discuss the diesis as a consistent interval from which other intervals are formed, which would make this analysis quite strange.  A likely alternative possibility is that Marchettus borrowed the ratios 18/17 and 17/16 from the works of Boethius and of his successors, without regard for the exact sizes of those intervals, and thus the diesis does indeed represent a consistent step size of which five produce a justly-tuned 9/8.  This analysis would measure the size of the diesis as approximately 41 cents.

=== Equal temperaments ===
Because all intervals in conventional theory of the time are described by a formula of the whole tone and the limma, and because both such intervals are divisible by a diesis, one can determine the exact size of an interval measured in dieses based on these values.  For instance, the perfect fourth is made up on twelve dieses, the perfect fifth is seventeen, and the octave is twenty-nine.  Under this analysis, Marchettus' paradigm can be thought of in modern terms as [[29edo]] with a compressed octave to ensure that 9/8 is tuned exactly just.

Indeed, the diatonic scale in 29edo is made out of whole tones of 5\29, and semitones of 2\29, precisely consistent with the numbers provided by Marchettus.  Knowing this, Marchettus' description of how to use other types of semitones to "color" the consonances may be applied to composition in 29edo.

=== Regular temperaments ===
One may construct a [[regular temperament]] based on Marchettus' description, with the diesis representing the difference between 18/17 and 17/16, and five of those dieses representing 9/8.  The easiest way to do this is by equating a stack of three 18/17s with a stack of two 17/16s, as each of them create an interval of six dieses; the ratio for this comma is 1492992/1419857.

In the minimal [[subgroup]] of 2.3.17, this structure is supported by edos 29, 31, 33, and 35; if we restrict this to 2.9.17, then edos 30 and 32 additionally support it.

=== The diesis in other structures ===
Additionally, the fivefold division of the whole tone can be applied to meantone tunings as well, if we take the limma to be three dieses and the apotome to be two (the opposite of how Marchettus defined them).  This is of note because quarter-comma meantone is practically indistinguishable from [[31edo]], and thus the diesis structure can be used as a means of understanding that system.  Vicentino's [[L'Antica Musica|L'antica Musica]] explores a system not dissimilar from this one, even adopting the word diesis to describe his small step, though he does not define intervals by a number of dieses.