Marchettus, the cadential diesis, and neo-Gothic tunings

Music or musicology, like politics, is often the art of redefining the possible. At the opening of a new millennium, this axiom may have a special relevance to questions of intonation, where the "hopelessly out of tune" or "impossible" intervals of one era or style may become the commonplaces of another.

A very notable example is Nicola Vicentino (1511-1576), who in
his treatise of 1555 entitled *Ancient Music Adapted to Modern
Practice* demonstrated how the chromatic and enharmonic genera
of Greek theory could be applied to the complex polyphony of
16th-century Europe.[1] His solution features a division of the
octave into 31 essentially equal parts, each defined as a fifth of
a whole-tone.

As a composer and theorist, Vicentino shows his genius in his grasp of common and uncommon practice alike, neatly adapting and extending a usual meantone temperament of the time to construct the framework for an extraordinary musical universe still largely awaiting exploration.

Evidently starting with a standard 12-note tuning in 1/4-comma
meantone, with pure 5:4 major thirds and slightly "blunted"
fifths, Vicentino extended this temperament to a complete cycle of
31 notes per octave, implemented on his *archicembalo* and
*arciorgano*, or superharpsichord and superorgan. Each
instrument ideally consisted of two manuals each with 19 notes per
octave, permitting the full 31-note cycle plus seven extra notes
on the upper manual tuned in pure fifths with the usual diatonic
notes on the lower manual to facilitate the just intonation of
certain common sonorities.

Masterfully combining the new and the familiar, Vicentino developed an "enharmonic" music featuring steps of 1/5-tone, comparable in size to the dieses of the Greek enharmonic genus. He mapped a novel world of intervals including a "proximate minor third" a fifthtone larger than minor or smaller than major, which he found rather consonant and described as having a ratio of approximately 5-1/2:4-1/2, or 11:9.[2]

In addition to using such radically new intervals, Vicentino explored modified versions of standard 16th-century progressions. In such variations the accustomed sizes of vertical concords often remain unaltered, but compact melodic semitones of approximately 2/5-tone (about 76.05 cents in 1/4-comma meantone) are substituted for the usual large diatonic semitone of about 3/5-tone (around 117.11 cents in 1/4-comma). In this manner, he fulfilled in compositional practice the teaching some 420 years later of another composer and theorist, Ivor Darreg, that meantone can be adorned by narrow cadential semitones a fifthtone smaller than the regular diatonic ones.[3]

Not so surprisingly, Vicentino's high art has evoked considerable controversy over the centuries. For an historian such as Charles Burney (1789), who judged 18th-century major/minor tonality as the consummation of earlier eras and styles, Vicentino was a competent musician who unfortunately foundered on "enharmonic rocks, and chromatic quicksands."[4] Certain 20th-century historians viewing the consummation of European intonational history as "equal temperament," meaning more specifically "12-tone equal temperament" (12-tET), may lean toward similar conclusions.

In the 16th century itself, Vicentino's enharmonic division of the whole-tone into five parts drew mixed reviews. One unfavorable critic was Vincenzo Galilei, a gifted lutenist and radical music theorist quite ready to challenge the conventional techniques and rules of counterpoint as expounded by his former teacher Gioseffo Zarlino. Both his independence of mind and his penchant for scathing polemics would be shared by his son the astronomer Galileo.

When it came to Vicentino's experimental music, however, Galilei as paraphrased by Karol Berger "expresses his disbelief in the supposed excellence of the enharmonic and argues that it cannot satisfy the ear because the number of intervals which are satisfying to it is limited and because the enharmonic diesis is contrary to the nature of singing and disproportionate with our sense of hearing."[5]

From a 21st-century perspective, we may evaluate Vicentino's music and theory very differently as one germinal source of a tradition encompassing the enharmonic meantone keyboards and music of Scipione Stella and Fabio Colonna in the early 17th century, the development of a mathematically refined concept of 31-tone equal temperament (31-tET) later in that century by Lemme Rossi and Christian Huygens, and the modern Netherlands school of 31-tone composition initiated by the physicist Adriaan Fokker and continued today by musicians and scholars such as Manuel op de Coul[6].

While Vicentino's fivefold division of the whole-tone provides
us with one famous example of redefining the possible in music, I
would like to focus here on another such fivefold division taking
us back some two centuries further in time to Padua in 1318, the
world of the innovative composer and theorist Marchettus, who in
that year completed his treatise entitled the *Lucidarium*.

Marchettus of Padua, unlike Vicentino, focuses in his
*Lucidarium* on the intonation of singers rather than the
tuning of fixed-pitch instruments; he especially advocates the use
of narrow cadential semitones or dieses, as he calls them, smaller
than the diatonic semitones of the usual Pythagorean
tuning of the Gothic era.

Like Vicentino, however - and perhaps even more so - Marchettus challenges the assumed limits of the intonationally "possible" not only in 1318, 1581, or 1789, but even in 1981 when his modern interpreter Jan W. Herlinger addressed the apparent consequences of his fivefold division of the tone.

In describing the extra-wide major sixth which he favors in cadences expanding to the octave by stepwise contrary motion, Marchettus notes that this interval should be midway in size between the fifth and the octave.

Before considering this and related teachings of Marchettus in more detail, let us observe the reaction of Herlinger, obviously an admirer of this composer and theorist: "The major sixth he places before an octave ..., at 955 cents ... is impossibly large - closer to the modern minor seventh than to the major sixth."[7]

Certainly a cadential interval of around 950 cents expanding to an octave, about halfway between a usual major sixth and minor seventh by medieval Pythagorean or most later Western European standards, is different - but does this make it impossible? With Marchettus, as with Vicentino, some actual musical experience can expand the realm of possibilities, whether in regard to 14th-century performance practices or to 21st-century tunings and music.

While Marchettus is describing the flexible intonation of singers, I will explore alternative 24-note tunings for keyboard approximately modelling two of the many possible interpretations of his system[8], taking note of some other modern tuning systems with kindred qualities.

Placing Marchettus in context, we find that his intonational theory is in keeping with the spirit of an age known for its innovation and questioning. His contemporary Dante Alighieri correctly explained the galaxy, or Milky Way, as a collection of stars too small to be seen individually by the naked eye, and noted the tendency of every culture and nation to consider its own language as the best or most natural - an observation not irrelevant to California in the 21st century.

Around 1316, just two years before the treatise of Marchettus,
the *Roman de Fauvel* used a mixture of poetic text and
musical pieces in various traditional and modern styles to tell the
story of Fauvel, a donkey symbolizing folly and corruption in Church
and State.

Even as Inquisitors such as Bernard de Gui plied their oppressive
and often lethal trade, an author named Marsilius of Padua in 1324
wrote a treatise entitled *Defensor Pacis*, or *Defender
of Peace*, arguing for the separation of Church and State, and
holding that a government might well choose to let peaceful Jews
or heretics live unmolested within its borders.

In the realm of mathematics and the natural sciences, 14th-century English scholars were formulating a new approach to motion, impetus, and acceleration. In the latter part of the century, Nicolas Oresme would explore the concept of rational exponents, a development more fully realized in such musical applications as the definition of equal temperaments with the introduction of logarithms around 1600.

Set against this cultural and intellectual background, Marchettus takes an approach to music and intonation mixing elements both traditional and new. His system is based in part on the established ratios of Pythagorean just intonation: the octave at a pure 2:1, the fifth at 3:2, the fourth at 4:3, and the major second or whole-tone at 9:8 (the difference between the pure consonances of the fifth and fourth).

In dividing the whole-tone into semitones, however, Marchettus
takes an approach differing from the traditional one defining a
usual diatonic semitone or *limma* at 256:243, or about 90.22
cents (e.g. E-F or B-C), and a chromatic semitone or *apotome*
at 2187:2048 or about 113.69 cents (e.g. F-F# or Bb-B).

Rather, he proposes the division of the whole-tone into *five*
parts, each called a diesis. In medieval theory, the term diesis
is sometimes used as a synonym for the usual diatonic semitone or
limma; in ancient Greek theory, it often refers to the enharmonic
diesis, an interval equal to half this size, or in Pythagorean
tuning about 2/9-tone (around 45.11 cents).

One enigma often debated by modern scholars is whether these five parts or dieses are equal, as some 15th-century interpreters assumed, or possibly unequal - whether derived from some division on a monochord, for example, or simply regarded as notional "parts" of a tone to guide singers rather than specific mathematical measurements.

However this may be, Marchettus explains that the usual limma or mi-fa semitone represents "two parts" of a tone, and the usual apotome "three parts." In his system, as in the conventional Pythagorean one, these two unequal semitones together form a 9:8 whole-tone.

In directed resolutions such as major third expanding to fifth or major sixth expanding to octave, Marchettus prescribes the use of a different division of the tone into a narrow cadential diesis equal to only "one part," and a wide chromatic semitone equal to "four parts."

The purpose of these special intervals, as Marchettus explains, is to let certain cadential major thirds or sixths approach "as closely as possible" to the stable intervals of the fifth and octave which they respectively seek by stepwise expansion.[9]

From this perspective, his intonational system may be seen as amplifying or accentuating certain features of standard Pythagorean tuning in the Gothic era, as also articulated by more conventional theorists including his outspoken critic Prosdocimus de Beldemandis about a century later.

In 13th-14th century practice and theory, stable octaves, fifths, and fourths contrast with unstable intervals representing various degrees of concord or discord. In 14th-century theory, thirds and sixths are typically described as "imperfect concords," unstable and at the same time somewhat blending; major seconds or ninths and minor sevenths are also sometimes regarded by Gothic composers or theorists as having a degree of "compatibility."

Directed Gothic cadences typically involve progressions from unstable to stable intervals by stepwise contrary motion; as Marchettus explains, both voices share in the tension of the unstable interval, and both participate by their motion in its resolution.[10]

In the 14th century, musicians especially favor progressions of this kind to the "nearest consonance," where one voice moves by a whole-tone and the other by a semitone. Thus a minor third typically contracts to a unison, while a major third expands to a fifth and a major sixth to an octave (m3-1, M3-5, M6-8). In some styles, the expansion of major second to fourth or the contraction of minor seventh to fifth (M2-4, m7-5) can also play a significant role.

Conventional Pythagorean tuning nicely fits the aesthetics of such progressions by providing a contrast between pure fifths and fourths (3:2, 4:3) and rather complex ratios for unstable thirds and sixths. Major thirds (81:64, ~407.82 cents) and sixths (27:16, ~905.87 cents) are wide, their complexity lending them a dynamic quality seeking efficient expansion respectively to the fifth and octave; minor thirds (32:27, ~294.13 cents) likewise can efficiently contract to the unison.

Melodically, these resolutions feature motions of a generous 9:8 whole-tone (~203.91 cents) in one voice and of a compact 90-cent diatonic semitone in the other. In Gothic music, as Mark Lindley has observed, a vertical taste for active thirds and sixths and a melodic taste for narrow semitones can happily harmonize[11]; the two dimensions are in a kind of equilibrium.

Marchettus is important both for giving us an early formulation of these 14th-century aesthetics of "closest approach," and for apparently carrying them a step further, seeking cadential major thirds and sixths yet wider than Pythagorean, and cadential semitones or dieses yet narrower.

Jan Herlinger, modern editor of the *Lucidarium*, seems
to share this general outlook on Marchettus even while arguing that
a cadential semitone or diesis of around 1/5-tone would lead to
"impossible" musical consequences.[12]

Here I would like to present two keyboard models which *might*
approximate the kind of early 14th-century vocal intonation which
Marchettus describes and recommends, the first a just intonation
system and the second a regular temperament which singers might
finely adjust in practice to obtain the pure fifths, fourths, and
9:8 whole-tones which he takes as axiomatic along with more
conventional theorists.

At the turn of the 21st century, this exploration may at once place Marchettus in a new light and suggest some possibilities of a trend in xenharmonic music which I term "neo-Gothic": the use of such systems as extended Pythagorean tunings and regular temperaments with fifths wider than pure to develop creative offshoots of 13th-14th century Gothic styles in the setting of an open intonational continuum.

Although Marchettus does not give specific ratios for his wide cadential major thirds and sixths, one intriguing possibility is to tune them at or near the pure and simple ratios of 9:7 and 12:7, or about 435.08 cents and 933.13 cents. These intervals are a septimal comma wider than their usual Pythagorean counterparts, a difference of 64:63 or about 27.26 cents.

We can neatly model this possible interpretation, and at the same time arrive at a very attractive 21st-century just intonation system, by placing two 12-note keyboards in identical Pythagorean tunings a septimal comma apart. Let us suppose we tune the first or lower keyboard with a chain of pure fifths from Eb to G#, an ideal choice for much 13th-14th century music. Then we tune the second or upper manual with the same chain of fifths, but a septimal comma higher.[13]

One system of notation reflecting the layout of the keyboards is to use a carat sign (^) to show a note on the upper manual, raised by a septimal comma from its position on the lower manual. Thus the major third E-G#^, with the E played on the lower manual but the G# on the upper manual, is a septimal comma wider than the usual Pythagorean major third at 81:64, forming a pure ratio of 9:7 or about 435.08 cents.

Similarly a major sixth E-C#^ is a septimal comma wider than the usual Pythagorean ratio of 27:16, forming a pure ratio of 12:7 at about 933.13 cents.

Conversely, if we were to play the lower note of a minor third on the upper manual and the upper note on the lower manual, e.g. E^-G, this interval would be a septimal comma narrower than the usual Pythagorean 32:27, forming a ratio of 7:6 at about 266.87 cents; and the minor seventh E^-D would likewise be a septimal comma narrower than the Pythagorean 16:9, or at a 7-based ratio of 7:4, about 968.83 cents.

Since Marchettus discusses the use of wide cadential major thirds and sixths before fifths and octaves, let us focus our discussion on these intervals and progressions.

If we resolve our 9:7 major thirds and 12:7 major sixths so as to use narrow semitones, as Marchettus urges, we arrive at progressions like the following; here I use a MIDI-like notation with C4 as middle C, and higher note numbers showing higher octaves:

G#^3 A3 C#^4 D4 E3 D3 E3 D3 M3 - 5 M6 - 8

Here the lower voice moves in each progression by a usual 9:8 whole-tone, with resolutions to a pure 3:2 fifth or 2:1 octave, thus fulfilling the feature of his system adopting these traditional Pythagorean ratios. The cadential semitone G#^3-A3 or C#^4-D4, however, is a septimal comma narrower than the familiar Pythagorean limma 256:243: it has a ratio of 28:27, or about 62.96 cents.

If we superimpose these two resolutions, we arrive at the favorite final cadence of the 14th century for three voices: an unstable sonority with major third and sixth above the lowest part, and a fourth between the upper voices, expanding to a complete stable sonority with an outer octave, lower fifth, and upper fourth.

Here I show this progression both in its conventional Pythagoeran tuning and in our possible "Marchettan" realization using 7-based ratios for the unstable thirds and sixths:

C#4 D4 C#^4 D4 G#3 A3 G#^3 A3 E3 D3 E3 D3 Conventional Marchettan(?)

Following the theorist Johannes de Grocheio around 1300, who
refers to a stable sonority such as D3-A3-D4 in these examples as
manifesting the *trina harmoniae perfectio* or "threefold
perfection of harmony" uniting octave, fifth, and fourth, we
can refer to this complete medieval harmonic unit in English as a
"trine." As another theorist of the same epoch explains,
this sonority follows the natural series of the concords as expressed
by the numbers 2-3-4, placing a 2:3 fifth below a 3:4 fourth. In
modern terms, the complete Gothic trine has a frequency ratio of
2:3:4.[14]

Both the standard Pythagorean version of this cadence and our version with 7-based thirds and sixths tune the trine at a pure 2:3:4. In the usual Pythagorean intonation, however, the unstable sonority E3-G#3-C#4 has a rather complex ratio of 64:81:108 (at a rounded 0-408-906 cents). In our possible "Marchettan" rendering, it has a simpler ratio of 7:9:12 (0-435-933 cents).

Either tuning has its own attractions. The Pythagorean contrast between complex and "beatful" thirds and sixths, and pure fifths and fourths, accentuates the musical tension of cadences; the smoother, more "streamlined" effect of an unstable sonority such as 7:9:12 might have a striking effect for either 14th-century or 21st-century ears.

Our 24-note keyboard model offers a choice of both types of cadences: each manual has the usual Pythagorean intervals, while 7-based intervals are available in sonorities mixing notes from the two manuals. Using regular Pythagorean sonorities at most points, but the wider 7-based major thirds and sixths for significant cadential progressions, might roughly approximate the kind of system which Marchettus recommends.

Is there any basis for a hypothesis that Marchettus and the performers whose practices he describes may have favored 7-based ratios? For JI theorists inclined to the view that singers generally lean toward simple ratios for important vertical intervals, the tuning of cadential major thirds and sixths at 9:7 and 12:7 might have the attraction of at once achieving simplicity and facilitating "superefficient" melodic semitonal motions of around 63 cents.

Marchettus himself states, as already noted, that the cadential
major sixth differs equally from a 3:2 fifth or a 2:1 octave. One
obvious reading, which will will shortly consider, is that this
interval defines a *geometric* mean between fifth and octave,
which we can calculate using modern methods by taking the average
of these two intervals in cents, getting a size of around 951 cents.
As we shall see, this interpretation nicely fits the model of an
equal fivefold division of the whole-tone.

A creative medievalist, however, might propose a different
interpretation: could the cadential major sixth represent an
*arithmetic* division on the monochord of the difference
between the string lengths for the fifth and octave of the note
produced by the full length?

Let us suppose that our monochord has a string length of 72, for example. Then 2/3 of this total length, 48, will produce a 3:2 fifth; 1/2 of this total length, 36, will produce a 2:1 octave. If we take the average of 48 and 36, we get a length of 42, and a ratio of 72:42, yielding a major sixth at a pure 12:7.

8 M6 5 |-----------------------------------------------------------| 0 36 42 48 72 2:1 12:7 3:2

How well does this interpretation fit the idea of a fivefold
division of the tone? If we accept for the moment Herlinger's
suggestion that this division be taken as an impressionistic guide
for performers rather than a mathematical formulation calling for
five *equal* parts, then our 7-based model does produce an
interesting kind of unequal fivefold division.

According to Marchettus, the narrow cadential diesis is equal to only "one part" of a tone - here defined at a 28:27 semitone of about 63 cents.

The regular limma or mi-fa step is equal to "two parts" - here the same 28:27 semitone plus a septimal comma at 64:63 or about 27 cents, yielding the standard Pythagorean 90-cent diatonic semitone at 256:243.

The regular apotome, e.g. Bb-B, is equal to "three parts" - here the preceding two parts plus a Pythagorean comma at 531441:524288 or about 23.46 cents, yielding the standard Pythagorean ratio of 2187:2048, about 113.69 cents.

The extra-large "chromatic semitone" of Marchettus is equal to "four parts," the preceding three parts plus another 64:63 septimal comma, producing an interval of 243:224, about 140.95 cents. This interval, plus the cadential diesis or semitone at 28:27, adds up to a 9:8 whole-tone.

203.91 9:8 |---------------------------------------------------------------| 62.96 27.26 23.46 27.26 62.96 |--------------------|---------|------|--------|----------------| 28:27 64:63 531441: 64:63 28:27 524288 |----------------------------------------------| "chromatic semitone" 243:224 (140.95) |-------------------------------------| usual apotome 2187:2048 (113.69) |------------------------------| usual limma 256:254 (90.22) |--------------------| cadential diesis

From a personal perspective, I find 14th-century cadences realized
with 7:9:12 sonorities often very attractive, reminding me of a
kind of intonation I seem to recall from at least one recorded
performance of Guillaume de Machaut's *Mass* some 30 years
ago.

Although no regular 12-note tuning can provide both pure 2:3:4 trines and pure 7-based ratios for unstable intervals such as thirds and sixths, one such tuning does in some ways approximate our 24-note JI model.

If we divide the octave into 22 equal parts, taking one step as a diatonic semitone at around 54.55 cents, and four steps as a whole-tone at around 218.18 cents, then we get regular major thirds and sixths at 8/22 octave and 17/22 octave respectively, or about 436.36 and 927.27 cents, quite close to 9:7 and 12:7.

Our usual 55-cent semitones - literally "diatonic quartertones," since four are equal to a whole-tone - serve both as regular steps and as compact cadential semitones in Marchettan progressions expanding from major third to fifth and from major sixth to octave.

As is often true in music, a tempered solution involves some very substantial compromises. First, fifths and fourths are impure by almost 7.14 cents respectively in the wide and narrow directions; also, this tuning does not provide or closely approximate the usual Pythagorean intervals which Marchettus adopts as a part of his system. Still, it has the charm of simplicity as well as a musical character intriguing in its own right.[15]

Suppose, however, that we prefer the obvious reading of an equal fivefold division, with slight adjustments by singers or players of flexible pitch instruments to maintain pure fifths, fourths, and melodic 9:8 whole-tones. As Herlinger notes[16], this division might suggest the modern model of 29-tET.

In this system, nicely fitting the scheme of Marchettus, the usual limma or mi-fa semitone is equal to 2/5-tone, about 82.76 cents; and the usual apotome to 3/5-tone, about 124.14 cents. The supercompact cadential diesis is equal literally to 1/5-tone, about 41.38 cents; and the extra-large "chromatic semitone" to 4/5-tone, about 165.52 cents (as it happens, very close to 11:10, as some JI enthusiasts might note).

On a fixed-pitch instrument, as Herlinger and others have noted, this equal division results in a slight tempering of the classic Pythagorean ratios: fifths and fourths are respectively wide and narrow by about 1.49 cents, and whole-tones larger than 9:8 by twice this amount, about 2.98 cents. Performers in flexible-pitch ensembles, however, would need to make only minute adjustments in order to achieve just ratios for these intervals.

We approximate this system on keyboard by placing two 12-note manuals in 29-tET (Eb-G# range) with a distance of a diesis or 1/29 octave between the two manuals. Each manual has regular 29-tET intervals, including the near-pure fifths and fourths, the semitones of 2/5-tone and 3/5-tone, and thirds and sixths with a moderately accentuated Pythagorean quality - major thirds and sixths a bit wider, minor thirds and sixths a bit narrower.

As with our 7-based JI system, mixing notes from the two manuals brings into play a new set of intervals and sonorities. Here I will use an asterisk (*) to show a note found on the upper keyboard, or in other words raised by a 41-cent diesis in relation to its counterpart on the first or lower manual. Let us consider a 29-tET version of the favorite 14th-century cadence with major third and sixth expanding to fifth and octave, here with a Marchettan melodic diesis literally equal to 1/5-tone:

C#*4 D4 G#*3 A3 E3 D3

Here we find that the cadential major third E3-G#*3 has a size of 11/29 octave or about 455.17 cents, a diesis larger than the regular interval of E3-G#3 at 10/29 octave or about 413.79 cents. In 29-tET, it is precisely midway between regular major third and fourth.

The cadential major sixth E3-C#*4, at 23/29 octave or about 951.72 cents, is likewise midway between a regular major sixth (22/29 octave, about 910.34 cents) and a minor seventh (24/29 octave, about 993.10 cents). While this interval is indeed strikingly distinct from usual medieval Pythagorean categories - or for that matter tha conventional categories of Renaissance-Romantic composition - that does not make it "impossible," only different.

Moreover, this 29-tET model fulfills an obvious literal reading of the statement of Marchettus that this major sixth has a size midway between that of the fifth and the octave, respectively 17/29 octave (about 703.45 cents, slightly larger than pure) and 29/29 octave; at 23/29 octave, it is the precise mean of these intervals. The very small adjustments which flexible pitch ensembles might make to achieve pure fifths, fourths, and melodic whole-tones at 9:8 would not substantially alter these mathematics.

One key passage in Marchettus seems ideally to fit a 29-tET kind of model, suggesting that the five dieses making up a tone are indeed of equal size and can be used as units in measuring and comparing melodic and vertical intervals. At the same time, this passage incorporates two musical themes which give Marchettus a special place in music history quite apart from his intonational system: direct chromaticism, and its use in deceptive cadential progressions.

Having explained that the division of the tone into the diesis of one part and the "chromatic semitone" of four parts is used in order to help an unstable interval approach more closely its stable goal - to widen major thirds before fifths or major sixths before octaves - he raises an interesting question concerning this "closest approach" principle.

Why is it, he asks, that the cadential major sixth more aptly seeks the octave rather than the fifth by stepwise contrary motion, when it is at an equal distance from either of these consonances?[17] He gives examples of both two-voice resolutions, comparing the number of dieses in each melodic motion of the voices. Here, following his exposition, I use numbers with plus and minus signs to show ascending or descending motions in each voice[18]:

(a) Regular M6-8 resolution (b) Special M6-5 resolution D4 -- -1 -- C#4 -- +1 -- D4 D4 -- -1 -- C#4 -- -4 -- C4 D3 -- +5 -- E3 -- -5 -- D3 D3 -- +5 -- E3 -- +2 -- F4 8 M6 8 8 M6 5

In the first example, as Marchettus notes, "the upper melody begins a diapason [octave] from the tenor and descends one diesis while the tenor descends a whole tone, which contains five dieses." This is a total contraction of six dieses, placing us at a cadential major sixth, which resolves by expansion back to the octave, "toward which the melodies are clearly tending."[19]

In the second example, as Marchettus remarks, the opening motion from octave to cadential major sixth is the same, and this interval places us six dieses from the fifth. "After this the upper melody descends a chromatic semitone, which contains four dieses," while the tenor ascends by a usual mi-fa semitone or limma, here E-F, which Marchettus interestingly terms an "enharmonic semitone, which has two dieses. This makes six," and thus we arrive this time at the fifth.[20]

Thus the cadential major sixth can either expand a total of six dieses to arrive at the octave, or contract the same distance of six dieses to arrive at the fifth. In our 29-tET keyboard model, the total expansion or contraction is in each case 6/29 octave, or about 248.27 cents.

Addressing the question of why the resolution to the octave is more pleasing, Marchettus observes in part that in this progression, one of the voices moves by the smallest possible melodic interval of a single diesis, namely the upper voice with its ascending diesis C#-D.

In the resolution by contraction to the fifth, however, neither voice moves by this smallest interval: the tenor or lower voice ascends by a usual semitone step of two dieses, E-F, while the upper voice descends by a chromatic semitone of four dieses, C#-C - the modern "#" sign here representing an accidental sign of Marchettus specifically showing a note raised by this large chromatic semitone, in contrast to the usual apotome of three dieses, of which more below.

He concludes that in order fully to realize the principle of closest approach, or of the "smallest distance" between an unstable interval and its stable goal, "the smallest part of the whole tone, which is the diesis, must be sung in one of the two melodies, above or below."[21]

Although he finds the special resolution of the cadential major
sixth to the fifth, with its direct chromaticism, less ideal,
Marchettus recognizes and approves the use of this kind of figure
in what he calls a *color fictitius* or "feigned
color," which we might freely translate as a deceptive inflection
or cadence.

Elsewhere, giving an example identical to that above, he advises whoever wishes to use a melodic figure such as D-C#-C to "feign in the first descending interval, which is a diesis," as if one "wished to return upward after this descent," but then to "descend a chromatic semitone."[22]

Here I should note, as Herlinger observes, that chromatic progressions of this kind in fact occur in certain early 14th-century Italian compositions. While the feigned color or deceptive cadence of Marchettus, as in the above example with the major sixth resolving to a fifth in the place of the expected octave, takes on a yet more dramatic quality in his intonational system, it can be very effective in the usual Pythagorean tuning also.

We have seen that 29-tET provides an ideal fixed-pitch approximation for the flexible system of Marchettus in its interpretation of an equal fivefold division of the tone. Other tunings, however, can also approximate some of its features, for example the popular 20th-century scheme of 24-tET.

In 24-tET, a single step of 1/24 octave or 50 cents can represent the cadential diesis of Marchettus, while vertical intervals of 9/24 octave or 450 cents and 19/24 octave or 950 cents can approximate his extra-wide cadential major thirds and sixths. As in 29-tET, the cadential major sixth is at precisely equal distances from the fifth and octave - here 14/24 octave or 700 cents and 24/24 octave or 1200 cents respectively.

Whether modelled in the ideal 29-tET, 24-tET, or some other fixed-pitch system, this interpretation of Marchettus indeed has radical consequences: the generation of new interval categories, also a feature of the different fivefold division of Nicola Vicentino some two centuries later. To what degree "microintervals" may have been a part of 14th-century Italian performance practice remains an open question, but early music ensembles are venturing to explore some possibilities.[23]

Like Nicola Vicentino in 1555, Marchettus of Padua in 1318 presented a system of notation reflecting his fivefold division of the tone. While Vicentino introduced a new accidental sign, a dot placed above a note to raise it by a diesis of 1/5-tone[24], Marchettus resourcefully proposed a distinction between two typically equivalent signs already in use, corresponding to our modern natural and sharp symbols.

In later medieval practice, either a "square-B" sign like our natural or a sign resembling more or less our sharp might be used to convey the same meaning: that the note so inflected should be sung "mi," the step below a semitone, often followed by a cadential or other semitonal ascent "mi-fa" (e.g. B-mi to C, i.e. B-C; G-mi to A, i.e. G#-A).

Both signs ultimately derived from the distinction between "square-B" or B-mi, the semitone below C in the medieval hexachord system, and "round-B" or B-fa, the semitone above A; the symbol for the latter form of the step B/Bb gives rise to the flat sign. Following scholars such as Peter Urquhart, we might refer to either of the first two signs as a "mi-sign," and to the round-B sign as a "fa-sign."

The two varieties of mi-signs were synonymous and largely interchangeable in much late medieval and Renaissance usage; but Marchettus cleverly introduced a distinction between the two signs in order to notate his different sizes of semitones and dieses.

For the division of the whole-tone into the familiar limma and apotome of "two parts" and "three parts" respectively, he uses the square-B sign. In our first or just intonation model of his system based on pure ratios of 3 and 7, these intervals are identical to the Pythagorean ones, roughly 4/9-tone and 5/9-tone; in our second model based on an equal fivefold division, they are equal to 2/5-tone and 3/5-tone. Thus the steps Bb-B-C involve an apotome Bb-B, followed by a limma B-C.

To show his division for the directed resolution of unstable intervals, where the tone is divided into a chromatic semitone of "four parts" and a diesis of "one part," he reserves the other version of the mi-sign for this specific purpose. Thus C-C#-D involves the large chromatic semitone C-C# - four of the five parts of the tone - followed by the diesis C#-D, the remaining single part of the tone.

This scheme with its two distinct mi-signs might be considered one of the earliest systems of microtonal notation for polyphonic music in Western Europe - in fact, the earliest of which I am presently aware. One feature of this system is that, to borrow a current term, it is "backward-compatible"; performers following conventional Pythagorean intonation can simply treat the two mi-signs as equivalent, alike dividing a tone into the usual limma and apotome.

For example, a chromatic progression such as that shown in the
following example of Marchettus, with an identical progression cited
by Herlinger from the composition *Sedendo all'ombra* by
Giovanni da Cascia[25]:

C4 C#4 D4 F3 E3 D3 5 M6 8

In a conventional Pythagorean reading, the step C4-C#4 in the upper voice would be a usual apotome, and C#4-D4 a usual limma. In a Marchettan reading, C4-C#4 would be the chromatic semitone of "four parts," and C#4-D4 the limma of "one part." Under the first of our two intepretations, these steps would be respectively a septimal comma larger and smaller than Pythagorean; under the second, they would be equal to 4/5-tone and 1/5-tone, much as in 29-tET; other readings would yield other sizes.

To sum up, Marchettus captures the excitement of early 14th-century
Italian music both in his intonational and notational refinements,
and in his chronicling of contemporary chromaticism. His
*Lucidarium* at once sheds light on xenharmonics in the
Gothic era and invites new experiments with 21st-century neo-Gothic
tuning systems and musics.

Nicola Vicentino,

*Ancient Music Adapted to Modern Practice*, tr. Maria Rika Maniates, ed. Claude V. Palisca (New Haven: Yale University Press, 1996), ISBN 0-300-06601-5. A fascimile of the 1555 edition is available:*L'antica musica ridotta alla modern prattica*, ed. Edward L. Lowinsky, Association Internale des Biblioteques Musicales, Documenta Musicologica, Erste Reihe: Druckschriften- Faksimiles 17 (Basel and New York: Barenreiter Kassel, 1959). On Vicentino's instrument and its tuning, see also Bill Alves, originally appearing in*1/1: Journal of the Just Intonation Network 5*(No.2):8-13 (Spring 1989), available at http://www2.hmc.edu/~alves/vicentino.html, with a helpful bibliography.Vicentino,

*Ancient Music Adapted to Modern Practice*, pp. 336-337, 436-437.For Vicentino's examples of four-voice enharmonic cadences, see ibid., pp. 207-209; for enharmonic musical settings and excerpts, pp. 209-221. Ivor Darreg, in "The Calmer Mood: 31 Tones/Octave,"

*Xenharmonic Bulletin*9 (January 1978), urges performers in 31-note equal temperament (31-tET) to overcome "the flatness of the leading-tone" by playing such notes "1/31 of an octave higher"; issue available at http://www.ixpres.com/interval/darreg/xhb9.htm. While Darreg urges such melodic adjustments regardless of vertical clashes with other voices - a liberty possibly analogous to that of augmented or diminished octaves and the like in 16th-century practice - Vicentino typically uses fifthtone adjustments to obtain narrow cadential semitones while maintaining usual vertical concords.Charles Burney,

*A General History of Music from the Earliest Ages to the Present Period*(Printed for the author, 1776-1789), vol. III, p. 162; quoted in Henry Kaufmann,*The Life and Works of Nicola Vicentino*(American Institute of Musicology, 1966), p. 174 and n. 199.Karol Berger,

*Theories of Chromatic and Enharmonic Music in Late Sixteenth Century Italy*(Ann Arbor, MI: UMI Research Press, 1980), ISBN 0835710653, p. 73.For a wealth of information on 31-tone music, see the resources of the Huygens-Fokker Foundation, http://www.xs4all.nl/~huygensf/; of special interest for English language materials are the Web pages http://www.xs4all.nl/~huygensf/english/theory.html and http://www.xs4all.nl/~huygensf/english/literature.html.

Jan W. Herlinger, "Marchetto's Division of the Whole Tone,"

*Journal of the American Musical Society*34(2):193-216 (1981), p.209.For a range of interpretations, see Herlinger, ibid.; David Lenson,

*Nonspecific Accidentals: A study in medieval temperament based on notation*(MA thesis, University of Western Ontario, 1987), Chapter 2; Joseph L. Monzo,*Speculations on Marchetto of Padua's "Fifth-Tones"*(1998), http://www.ixpres.com/interval/monzo/marchet.htm; and Jay Rahn, "Practical Aspects of Marchetto's Tuning,"*Music Theory Online*4.6 (1998), http://mto.societymusictheory.org/issues/mto.98.4.6/mto.98.4.6.rahn.html.See Marchettus of Padua,

*The Lucidarium of Marchetto of Padua*, Jan Herlinger, ed., Chicago: University of Chicago Press, 1985, pp. 148-153, 206-213; and Herlinger, n. 1 above, pp. 206-207 and n. 30. On the "closest approach" principle in standard Pythagorean tuning, see also Margo Schulter, "Ugolino's `Intelligent Organist' and the Seventeen-Note Octave, Part I: The Medieval Background,"*1/1*10, No. 3 (Fall 2000), pp. 1-15, 24, at 11-13.Marchettus of Padua, ibid., pp. 212-213.

Lindley, Mark, "Pythagorean Intonation and the Rise of the Triad,"

*Royal Musical Association Research Chronicle*16:4-61 (1980), ISSN 0080-4460, at p. 6, and "Pythagorean Intonation,"*New Grove Dictionary of Music and Musicians*15:485-487, ed. Stanley Sadie. Washington, DC: Grove's Dictionaries of Music. ISBN 0333231112.Herlinger, n. 1 above.

This scheme is very similar to a 24-note Pythagorean tuning with a chain of 23 pure fifths, except that the distance between the two 12-note manuals is slightly increased from a Pythagorean comma (531441:524288, ~23.46 cents) to a septimal comma (64:63, ~27.26 cents). If we use a carat sign (^) to show the notes on the upper manual raised by a septimal comma, then the fifth G#-Eb^ linking the two manuals is stretched by the difference between these two commas, 33554432:33480783 (about 3.80 cents), a ratio sometimes known as a septimal schisma. This wide fifth at about 705.76 cents has almost the same size as a fifth in 17-tET (about 705.88 cents or 3.93 cents wide). All other fifths in the chain remain pure.

Johannes de Grocheio's (or Grocheo's) treatise has variously been known as

*Theoria*,*De musica*, or*Ars musicae*. For Latin text, see E. Rohloff,*Der Musiktraktat des Johannes de Grocheo*(Leipzig 1943), with passage on*trina harmoniae perfectio*at p. 44; for English translation, see Albert Seay,*Johannes de Grocheo Concerning Music*(Colorado College Music Press Texts/Translations 1), Colorado Springs: Colorado College Music Press, 1967, at p. 6. For the passage deriving this sonority from the series of ratios 2-3-4, see*Jacobi Leodiensis Tractatus de consonantiis musicalibus, Tractatus de intonatione tonorum, Compendium de musica*, ed. Joseph Smits van Waesberghe, Eddie Vetter, and Erik Visser (Divitiae musicae artis, A/IXa), Buren: Knuf, 1988, 88-122 at 122; the Latin text is available on the World Wide Web, Thesaurus Musicarum Latinarum, Indiana University, http://www.chmtl.indiana.edu/tml/14th/JACCDM_TEXT.html.Of course, a neo-Gothic view of 22-tET gives only one perspective on this versatile tuning; for another outlook based on a very creative extension of 18th-century tonality to build a "decatonic" scale based on stable tetrads (4:5:6:7), see Paul Erlich, "Tuning, Tonality, and Twenty-Two-Tone Temperament,"

*Xenharmonikon*17:12-40 (Spring 1998). For the groundbreaking study and advocacy of this tuning by the Russian theorist Alexei Ogolevets, see Brian McLaren's paper presented as part of the proceedings of this Conference, "Alexei Ogolevets, Master Russian Microtonalist" (2001), and McLaren's "A Brief History of Microtonality in the Twentieth Century,"*Xenharmonikon*17:57-110 (Spring 1998), at pp. 74-75.Herlinger, n. 1 above, p. 215 n. 49.

Marchettus of Padua,

*Lucidarium*, n. 8 above, pp. 214-221.For my example (a), 8-M6-8, see ibid., pp. 214-215, ex. 9; for my example (b), 8-M6-5, see ibid, ex. 10.

Ibid. pp. 216-217.

Ibid. pp. 216-219.

Ibid. pp. 218-219.

Ibid. pp. 154-155, and Herlinger, n. 1 above, pp. 212-213, who takes this passage as an example of how Marchettus "often seems to be speaking directly to performers."

The ensemble Mala Punica has been noted for its use of "microintervals" in the performance of 14th-century Italian music.

Vicentino uses this diesis sign in connection with conventional 16th-century accidentals (Gb-B#) to specify the notes of his 31-tone meantone cycle. Using an asterisk (*) to represent his dot, we would thus specify the note a diesis above C as C*, and the note a diesis higher than Gb (or lower than A) as Gb*.

Herlinger, n. 1 above, p. 196, ex. 4(a).

Back to experimental tuning discussions index.

To some preliminary remarks on medieval chord structure

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Margo Schulter2 March 2001