Yahoo Tuning Groups Ultimate Backup mills-tuning-list Marchettus of Padua and "7-limit" intervals

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Marchettus of Padua's tuning
and Gothic cadential aesthetics
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Recently Joe Monzo raised the issue of "7-limit" intervals as
described or implied in the theory of Marchettus of Padua (1318), as
well as the possibility of "5-limit" intervals as implied in this
often original and unconventional theorist's _Lucidarium_.

Here I would like to suggest that Marchettus's approach to the
division of the whole-tone and the tuning of unstable intervals at
cadences indeed points to the use of "super-Pythagorean" major thirds
and sixths possibly quite close to ratios of 9:7 and 12:7. In a
14th-century setting, such intervals can be seen as an accentuated
expression of the usual cadential aesthetics as discussed by more
conventional theorists such as Prosdocimus (1413), and realized in
actual compositions of the period.

For these exponents of standard Pythagorean tuning, as for Marchettus,
thirds and sixths are "tolerable" but active intervals which "strive"
for resolution to stable 3-limit intervals by stepwise contrary motion
including efficient semitonal progressions (e.g. m3-1, M3-5, M6-8).
While a tuning based on the usual Pythagorean mathematics very nicely
serves this Gothic aesthetic, the "superwide" major thirds and sixths
of Marchettus and the "supernarrow" leading-tones suggested by his
division of the tone can be seen as a more dramatic manifestation of
the same musical feeling.

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1. Marchettus, semitones, and cadences
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Like conventional Gothic theorists, Marchettus expounds a
sophisticated technique of polyphony in which the octave (2:1), fifth
(3:2), and fourth (4:3) are stable concords, while the whole-tone
(9:8) is a basic element of melody. Thirds and sixths -- which many
theorists of the era describe as "imperfect concords" or "intermediate
concords," and Marchettus calls "tolerable discords" -- strive for
resolution by directed contrary motion to stable concords.

Both for conventional theorists of the epoch and for Marchettus, an
unstable third or sixth should approach as closely as possible the
stable concord toward which it tends. Traditional Pythagorean tuning
nicely serves this artistic ideal: it provides active thirds and
sixths, and narrow leading-tones for cadential resolutions such as
m3-1, M3-5, and M6-8.

In this system, the diatonic semitone or _limma_ (e.g. e-f or b-c') is
a compact 256:243 or ~90.22 cents, while the larger chromatic semitone
or _apotome_ is 2187:2048 or ~113.69 cents (e.g. c-c#, eb-e). Such a
tuning makes for efficient and expressive cadences. In the following
diagrams, numbers in parentheses show vertical intervals in cents, and
signed numbers show ascending or descending melodic motions:

e -- -204 -- d g# -- +90 -- a f#' -- +90 -- g'
(294) (0) (408) (702) (906) (1200)
c# -- +90 -- d e -- -204 -- d a -- -204 -- g

m3 - 1 M3 - 5 M6 - 8

In these progressions, the unstable m3 (32:27, ~294.13 cents), M3
(81:64, ~407.82 cents), or M6 (27:16, ~905.87 cents) is relatively
concordant, but has a definite "edge" adding impetus to the
resolution. One voice moves by what Mark Lindley[1] has aptly called
an "incisive" limma of 90 cents, and the other by a generous 204-cent
whole-tone (9:8).

Marchettus may have been motivated to devise a new division of the
whole-tone both in order to make these cadences even more dramatic and
efficient, and to make the intricate Pythagorean mathematics simpler.

He proposes a division of the 9:8 tone into "five parts," a proposal
which seems to permit of two interpretations. The first reading,
favored by Jan Herlinger, divides the tone into five _equal_ portions
of about 41 cents each.[2]

The second reading is suggested by a passage where Marchettus suggests
that the tone with its 9:8 ratio might be divided into "five parts" as
follows: 1, 3, 5, 7, 9. This could be taken as a division actually
based on _1/9-tones_, very close as it happens to the ~22.64-cent
steps of 53-tone equal temperament (53-tet):

First reading (5 equal parts): 1/5 1/5 1/5 1/5 1/5
Second reading (5 unequal parts): 1/9 2/9 2/9 2/9 2/9

David Lenson, in a study of 14th-century manuscript accidentals and
intonations available on the World Wide Web[3], favors this
interpretation.

As the more mathematically orthodox Prosdocimus will emphatically
assert a century later (1413, 1425), such even divisions of a ratio
such as 9:8 are _in no way_ possible using the integer mathematics of
Pythagorean just intonation -- or other integer ratio systems, one
might add. Rather, such divisions seem akin to the tradition of
Aristoxenos, who pragmatically recognized intervals such as
1/6-octaves although they cannot be expressed as integer ratios..

As flexible in his terminology as in his arithmetic, Marchettus
borrowed the names of the three Greek genera to describe three kinds
of semitones resulting from his division of the tone.

An _enharmonic_ semitone, corresponding with the usual Pythagorean
diatonic semitone found at mi-fa (e.g. e-f), has "two parts" out of
the five. In the 1/5-tone reading, this normal semitone would be equal
to 2/5-tone or ~81.56 cents; in the 1/9-tone reading, it might be
equal to two "parts" of 2/9-tone each, thus yielding a 4/9-tone
interval (~90.63 cents) virtually identical to the usual Pythagorean
limma of ~90.22 cents).

A _diatonic_ semitone, corresponding to the chromatic Pythagorean
semitone, has "three parts." If these parts are equal 1/5-tones, then
this semitone would be 3/5-tone or ~122.35 cents; in the 1/9-tone
reading, if we take these three parts as 1/9-tone, 2/9-tone, 2/9-tone,
then a 5/9-tone interval of ~113.28 cents, almost identical to the
Pythagorean apotome, results.

A _chromatic_ semitone, by which Marchettus prescribes a note should
be inflected before a cadence (e.g. amount by which cadential g# is
raised from g), has "four parts" out of five. In the 1/5-tone reading,
this means an interval of 4/5-tone or ~163.13 cents, leaving a
cadential leading-tone (e.g. g#-a) of only 1/5-tone or ~40.78 cents.
In the 1/9-tone reading, if we take one of the "four parts" as the
1/9-tone portion (2/9 + 2/9 + 2/9 + 1/9), we get an interval of
7/9-tone or ~158.60 cents, leaving a leading-tone of 2/9-tone or
~45.31 cents.

Even in the latter reading, this scheme if taken literally results in
cadential semitones of only about 45 cents, and cadential major thirds
and sixths of about 453 cents and 951 cents respectively -- about
midway between a Pythagorean M3 and fourth, or M6 and m7!

While we have no way of being sure that such an intonation was not at
times used, one tempting interpretation is that Marchettus was
providing an easy-to-remember scheme for singers rather than an exact
mathematical formula. If we take his "enharmonic" and "diatonic"
semitones as essentially equivalent to the Pythagorean limma and
apotome, then his narrow cadential leading-tone might be somewhat
narrower than the usual 90-cent limma but not quite so narrow as
1/5-tone or 2/9-tone.

----------------------------
2. A "quasi-7-limit" reading
----------------------------

By following the 1/9-tone reading of Marchettus's "five part" division
of the tone -- but defining the parts as 3/9, 1/9, 1/9, 1/9, 3/9 -- we
can arrive at an interesting result where cadential semitones are
equal to about 1/3-tone, and major thirds and sixth to ratios not far
from 9:7 and 12:7.

Thus our new "fivefold division" of the tone would be

"Parts": 1 2 3 4 5
3/9 | 1/9 | 1/9 | 1/9 | 3/9 |
----------------------------------------------------------------
cents: (Pyth) 67 90 114 137 204
(53-tet) 68 91 113 136 204

----------------------
3/9 = cadential (limma - comma)

------------------------------

Topic No. 11
4/9 = limma (mi-fa)

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5/9 = apotome (e.g. c-c#)

-----------------------------------------------
6/9 = superapotome (e.g. c-c# in c-c#-d)

In this scheme, we get regular diatonic and chromatic semitones of
4/9-tone and 5/9-tone, almost identical either to the usual
Pythagorean limma and apotome, or to 53-tet semitones of 4/53 octave
and 5/53 octave. Additionally, we get a "supercompact" cadential
semitone of 3/9-tone, equivalent either to a limma minus a Pythagorean
comma or to 3/53 octave.

Let us see how the standard two-voice progressions discussed by
Marchettus and illustrated in his musical examples look when we apply
our leading-tone of 3/9-tone:

e -- -204 -- d g# -- +67 -- a f#' -- +67 -- g'
(271) (0) (431) (702) (929) (1200)
c# -- +67 -- d e -- -204 -- d a -- -204 -- g

m3 - 1 M3 - 5 M6 - 8

Here we arrive at a minor third of ~271 cents, close to 7:6; a major
third of ~431 cents, close to 9:7; and a major sixth of ~929 cents,
close to 12:7. These intervals, about a Pythagorean comma or 1/9-tone
closer to their stable goals than even the usual Pythagorean
intervals, seem to fit Marchettus's accentuated version of usual
14th-century cadential aesthetics.

In practice, of course, singers and players of non-fixed-pitch
instruments would doubtless vary in their cadential intonations. From
a xenharmonic perspective, here are some tunings suggesting the range
of possibilities:

cadential m2 m3 (to 1) M3 (to 5) M6 (to 8)
-------------------------------------------------------------------
Pyth +/- comma 66.76 270.67 431.28 929.33
limma-comma 32:27-comma 81:64+comma 27:16+comma
-------------------------------------------------------------------
7-limit 62.96 266.87 435.08 933.13
28:27 7:6 9:7 12:7
-------------------------------------------------------------------
53-tet 67.92 271.70 430.19 928.30
3/53 oct 12/53 oct 19/53 oct 41/53 oct
-------------------------------------------------------------------
17-tet 70.59 282.35 423.53 917.64
1/17 oct 4/17 oct 6/17 oct 13/17 oct
-------------------------------------------------------------------

These are only some convenient landmarks on the xenharmonic continuum;
for example, a more subtle stretching of a Pythagorean major third
cadencing to a fifth might result in an interval mentioned by Charlie
Jordon in Tuning Digest 1577 -- 14:11 (~417.51 cents), about 10 cents
wider than a usual 81:64 (~407.82 cents).

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3. Marchettus as a Gothic Aristoxenian
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In building his system, Marchettus (like Vicentino 230 years later)
uses both traditional integer ratios and a pragmatic division of the
tone into a small number of equal parts, whether we read this system
to be based on 1/5-tone or 1/9-tone.

This system, interestingly, appears to fit and indeed to accentuate
the aesthetic qualities of Gothic intervals and cadential progressions
as presented in actual compositions of the time as well as in the
writings of more orthodox Pythagorean theorists. Active major thirds
and sixths become yet more active in cadences arranged with the
supernarrow Marchettan leading-tones, and the release of their tension
in expansion to the stable fifth and octave becomes yet more efficient
and dramatic.

Thus the musical feeling of 13th-14th century Continental European
polyphony, while nicely agreeing with the mathematics of the
Pythagorean system, may be expressed through other schemes including
53-tet and the system of Marchettus (just possibly close
equivalents).

As art changes, the same basic mathematical scheme may also be
modified to serve new aesthetic requirements. Thus the division of the
tone into portions of 4/9-tone and 5/9-tone is a quite practical model
for either Gothic or Renaissance polyphony -- but with the normal
diatonic semitone shifting from the smaller to the larger portion as
vertical thirds and sixths become smoother, and leading-tones
consequently become wider (at or near 16:15).

Whether or not the "superwide" cadential major thirds and sixths which
Marchettus seems to favor actually approximated "7-limit" values,
considering this possibility may direct our attention to some of the
qualities of the Gothic harmonic aesthetic also nicely (albeit less
drastically) expressed by the usual Pythagorean values. The
exploration of the artistic territory sketched out by Marchettus --
however uncertain the interpretation of his division of the tone -- is
one of the main themes of neo-Gothic and Xeno-Gothic music.

-----------------------------
Notes
-----------------------------

1. Lindley, Mark, "Pythagorean Intonation and the Rise of the Triad,"
_Royal Musical Association Research Chronicle_ 16:4-61 (1980), ISSN
0080-4460, pp. 6-7, 45; see also a convenient summary in Lindley,
Mark, "Pythagorean Intonation," _New Grove Dictionary of Music and
Musicians_ 15:485-487, ed. Stanley Sadie. Washington, DC: Grove's
Dictionaries of Music (1980), ISBN 0333231112.

2. Jan W. Herlinger, "Marchetto's Division of the Whole Tone," _Journal
of the American Musical Society_ 34(2):193-216 (1981).

3. David Lenson, _Nonspecific Accidentals: A study in medieval
temperament based on notation_ (MA thesis, University of Western
Ontario, 1987), http://home.on.rogers.wave.ca/dlenson/Thesis/
-- especially Chapter 2.

Most respectfully,

Margo Schulter
mschulter@value.net

Marchettus of Padua and "7-limit" intervals

🔗"M. Schulter" <mschulter@...>

🔗Paul Hahn <Paul-Hahn@...>