source file: m1580.txt Date: Thu, 12 Nov 1998 20:21:28 -0800 (PST) Subject: Re: Marchettus of Padua and "7-limit" intervals From: "M. Schulter" --------------------------------- Marchettus of Padua's tuning and Gothic cadential aesthetics --------------------------------- 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. -------------------------------------- 1. Marchettus, semitones, and cadences -------------------------------------- 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) -------------------------------------- 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). -------------------------------------- 3. Marchettus as a Gothic Aristoxenian -------------------------------------- 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