In describing the musical practice of their time, theorists such as Johannes de Garlandia (c. 1240?), Franco of Cologne (c. 1260?), and Jacobus of Liege developed sophisticated scales of vertical tension. (For a fuller account, see also Thirteenth-century polyphony.)

Fifths and fourths are the most complex stable intervals; major and minor thirds are relatively blending but unstable; major seconds and minor sevenths, along with major sixths, are rather more tense but somewhat compatible; and minor seconds, major sevenths, and tritones, often along with minor sixths, are regarded as strong discords. Here's a table showing the Pythagorean ratios of intervals on this spectrum:

---------------------------------------------------------------------- Stability Category Intervals ---------------------------------------------------------------------- Purely blending 1 (1:1), 8 (2:1) Stable ----------------------------------------------------- Optimally blending 5 (3:2), 4 (4:3) ______________________________________________________________________ ---------------------------------------------------------------------- Relatively blending M3 (81:64), m3 (32:27) ----------------------------------------------------- Unstable Relatively tense M2 (9:8), m7 (16:9), M6 (27:16) ----------------------------------------------------- Most tense m2 (256:243), M7 (243:128), A4 (729:512), d5 (1024:729), m6 (128:81) ______________________________________________________________________ ----------------------------------------------------------------------

It bears emphasis that this theoretical scheme is based not only on mathematical logic but on pragmatic musical experience. Thus while the stable concords have the simplest ratios, it should be noted that relatively blending major and minor thirds (81:64, 32:27) actually have more complex ratios than the rather more tense major second and minor seventh (9:8, 16:9).

In exploring how Pythagorean tuning interacts with and enhances harmonic style in this period, we may consider first the simple intervals, then "coloristic" harmony involving mildly unstable combinations, and finally the melodic and vertical aspects of cadential action.

Pythagorean tuning involves three main nuances nicely tying in with the 13th-century spectrum of intervallic tensions: it makes fifths and fourths ideally euphonious, unstable but relatively blending major and minor thirds somewhat more tense, and relatively tense major seconds and minor sevenths somewhat more blending.

Being based on just or pure fifths, Pythagorean tuning optimizes fifths (3:2) and fourths (4:3), making it possible to obtain these intervals in their most harmonious ratios. Given that these intervals, with their opulence and clarity, define the stable basis of Gothic polyphony and lend their prevalent color to the texture, Pythagorean intonation well concords (pun intended) with this artistic style.

In addition to presenting fifths and fourths in their ideal just ratios, Pythagorean tuning makes mildly unstable major thirds (81:64) and minor thirds (32:27) somewhat more active or tense. In a style where these intervals represents points of instability and motion standing in contrast to stable fifths and fourths, this extra bit of tension may be seen not only as a tolerable compromise, but indeed as an expressive nuance.

The major sixth (27:16) and minor sixth (128:81) also take on a bit of an extra "edge," adding emphasis to cadential resolutions involving these intervals.

Pythagorean tuning makes the relatively tense major seconds (9:8) and minor sevenths (16:9) as blending as possible by presenting them in their ideal just ratios. Thus a major second is equal to precisely two pure (3:2) fifths less an octave, while a minor seventh is equal to precisely two pure (4:3) fourths.

As early as the 11th century, Guido d'Arezzo recognizes M2 as a useful interval for polyphony, and both M2 and m7 play a prominent role in 13th-century practice. While theorists of the period typically describe M2 and m7 as "imperfect discords" having some degree of "compatibility," Jacobus of Liege actually classifies them as "imperfect concords." By presenting these intervals in an ideal just ratio, Pythagorean tuning tends to bring out their more "compatible" or "concordant" side.

By making major and minor thirds a bit more tense, and major seconds and minor sevenths a bit more blending, Pythagorean tuning also affects the quality of some typical 13th-century multi-voice sonorities built by combining these intervals. The following points might be read in connection with a more general discussion of these combinations.

Please note that symbols such as "8/5" or "5/4" are used here in their continuo meaning to identify a sonority as a set of intervals in relation to the lowest part, rather than to specify tuning ratios (here written as 8:5, 5:4, etc.).

In 13th-century polyphony, the unit of complete three-voice harmony is the 8/5 trine consisting of an outer octave, lower fifth, and upper fourth. In the variant 8/4 form, the two adjacent intervals are arranged "conversely," with the fourth below and the fifth above.

Pythagorean tuning yields ideal ratios for all intervals in these stable combinations. Using the medieval approach of string ratios, we have 6:4:3 for 8/5 and 4:3:2 for 8/4; using modern frequency ratios, we have 2:3:4 and 3:4:6 respectively.

Thus Pythagorean tuning in a 13th-century context, like the rather different systems of just intonation favored in the Renaissance, presents the stable harmonies of the period in their most pure and restful aspect.

By adding a bit of extra tension to simple thirds
(Section 3.1.2), Pythagorean intonation also
influences the color of one of the most popular
unstable combinations, the *quinta fissa* or "split
fifth" of Jacobus with its outer fifth "split" by
a third voice into a major third below and minor third above, or
vice versa (5/M3 or 5/m3).

While the outer fifths of these relatively blending combinations will be a just 3:2, the Pythagorean thirds somewhat accentuate the sense of instability: we get string ratios of 81:64:54 (5/M3) and 96:81:64 (5/M3), or frequency ratios of 64:81:96 and 54:64:81 respectively.

Jacobus finds these combinations pleasing when aptly used, and
Pythagorean tuning tends to bring out their *relatively*
concordant but active nature.

While making the "split fifth" more tense, Pythagorean intonation makes another group of mildly unstable sonorities somewhat more blending: quintal/quartal sonorities combining fifths and/or fourths with a relatively tense M2, m7, or M9.

All intervals in these relatively blending combinations will be presented in their just ratios: not only the ideally euphonious fifths (3:2) and fourths (4:3), but the unstable M2 (9:8), m7 (16:9), and M9 (9:4).

Thus for 9/5 and 7/4, we get respective ratios of 9:6:4 and 16:12:9 (here the string and frequency ratios are the same); for 5/4, string and frequency ratios of 12:9:8 and 6:8:9; and for 5/2, ratios of 9:8:6 and 8:9:12.

These sonorities, common in practice and endorsed by Jacobus in theory (both he and Anonymous I, possibly Jacobus at an earlier age, much recommend 9/5), present their most concordant face under a Pythagorean system of intonation.

As in many periods and styles of European music, cadences in Gothic polyphony have both a melodic and a vertical dimension. Especially in progressions by contrary motion such as m3-1, M3-5, and M6-8, Pythagorean tuning affects both dimensions to make the resolution more dramatic and "incisive," as Mark Lindley has pointed out.

For example, let us consider a very popular 13th-century cadence which becomes the most common close of the 14th century:

e'-f' b -c' g -f M6-8 M3-5 (M6-8 + M3-5)

From a melodic point of view, Pythagorean tuning makes the descending major second (9:8) in the lowest voice handsomely wide, while also providing decisively narrow semitones (256:243) in both upper voices.

At the same time, in the vertical dimension, the wide unstable intervals of M3 (81:64) and M6 (27:16) not only gain a bit of extra tension but "stretch" more closely toward the fifth and octave respectively. Both the heightened sense of instability and the incisive resolution to a complete trine contribute to the total effect.

While in this case the unstable intervals expand, Pythagorean tuning also enhances cadences where such intervals contract, as in the following resolution of a seventh combination common in the 13th century and also used by Machaut in the following century:

d'-c' b -c' e -f m7-5 5 (m7-5 + m3-1)

In this case the lower two voices both ascend by decisive Pythagorean semitones, while the upper voice descends by a generous whole tone. The outer minor seventh (16:9) efficiently contracts to a stable fifth, while the upper minor third (32:27) likewise contracts to a unison. As it happens, the tuning characteristically makes the minor third more tense and the minor seventh more mild, while letting both intervals resolve incisively to stability.

Thus while Pythagorean tuning in the West far predates the advent of the sophisticated multi-voice cadential formulae of the 13th and 14th centuries, it admirably fits the expressive nature of these progressions.

To Section 4 - Pythagorean tuning in more detail.

Margo Schulter