_O Europae_ Opening Portion (mm. 1-21) in JI By generously focusing his attention on my composition _O Europae_, Mykhaylo Khramov has raised some basic questions about one "translates" a piece of music from a tempered system to JI. Here I would like to suggest some general principles and patterns, more specifically in this article for mostly routine diatonic passages. While I am accustomed to thinking in terms of integer ratios, cents, savarts (which are used in some of the literature on Persian or more generally Iranian music), and sometimes 1024-ed2 as a synthesizer tuning, here I will follow Mykhaylo's approach in also thinking in terms of a possible absolute frequency standard. Following his suggestion, I will take the modal final of the piece in F Lydian, the note F3, as having a frequency of 176 Hz. As it happens, this number is very convenient for many of my purposes; thus I will show notes both as ratios with reference to F3 as the 1/1, and as frequencies based on F3=176. Interestingly, in the context of this piece and the kind of JI realization that the Peppermint temperament in which it was composed may imply, this suggests A4=448 as a convenient but not rigid or invariable standard for this piece, based on a usual major third F-A at a just 14/11 (417.508 cents). Another likely tuning of a usual major third, for example, would be at 33/26 (412.745 cents), which occurs in Mykhaylo's realization; if used at F-A, this would imply, for the moment, A4=446.769. Either tuning in a usual diatonic context for this piece is just about ideal, with small variations one of the attractions of fluid JI. In fact, the realization I am about to present of the first 21 measures of _O Europae_ uses major thirds at both 14/11 and the slightly smaller 33/26 -- here the difference is quite small, being equal to a comma of 364:363 (4.763 cents). Generally I would say that such subtle distinctions of shading are fully acceptable if not desirable, while one should reserve larger comma shifts for some expressive purpose, especially in a usual context where they are not required by the diatonic JI system or pattern one is using. ------------------------ 1. Some basic principles ------------------------ To begin, _O Europae_ is a piece in the Lydian mode on F, written without any accidental signature. However, I might have followed a custom common in 13th-14th century European practice: using a Bb signature for the lowest voice, which in the mainly diatonic sections of the piece (mm. 1-21, 33-41) consistently uses the Bb; and no signature in the middle voice, which consistently uses B-natural (apart from one cadence to Bb in the middle chromatic section, at measure 30). The highest voice has one occurrence of Bb (measure 4), and none of B-natural, so that one might also use a Bb signature for this part. However, if I follow the modern convention of using the same signature for all parts, then for this variety of F Lydian I much prefer using no signature, since it signals that B/Bb is indeed a fluid degree. In 13th-century pieces where Bb is more or less consistently preferred in all voices, a Bb signature in all parts occurs in the medieval notation, if I am correct, and is helpful in modern scores. From a medieval perspective, chants or polyphonic pieces in F with consistent use of Bb represent a variety of Lydian or tritus; from the 16th-century perspective of Glareanus and his followers, of course, they represent the distinct Ionian mode. Here B-natural is important, and indeed essential to final cadences on F, so I have preferred not to use a signature, and to mark Bb wherever it occurs. The next consideration on which I would focus, quite important from a JI perspective, is the scale of concord/discord in use for a given style. This can make a difference in terms of melodic as well as vertical structure, for example in shaping the sizes of tones and semitones. Clearly the style here is later medieval European, for the most part, with 14th-early 15th century idioms combined with certain 13th-century touches. This means that the most complex stable sonority is 2:3:4 or 1:1-3:2-2:1, which in JI has a just 2:1 octave, 3:2 fifth, and upper 4:3 fourth. Thirds and sixths, by comparison, are in this kind of style unstable and rather complex, both musically and intonationally. Historically, medieval Pythagorean intonation nicely realizes this kind of style, and indeed a straightforward setting in Pythagorean JI would be quite fine for the opening and closing sections of _O Europae_ -- apart from nuances for a couple of inflected cadences on G which I will address later in this discussion. Pythagorean intonation for these portions mainly using _musica recta_ -- that is, staying mostly within the regular gamut built by a chain of fifths Bb-F-C-G-D-A-E-B, with Bb and B-natural as fluid versions of the same step, and each playing a very important role in this piece -- would have the advantage of simplicity and what I will call congruity. Thus each melodic step of a tone would be tuned at 9:8 (203.910 cents), and each diatonic semitone (e.g. E-F, B-C, A-Bb) at 256:243 (90.225 cents). Congruity is my name for a concept I borrow from Zarlino: it means that the same intervals used as vertical intervals are also used more or less consistently as melodic steps. That is, if 33:26 is favored as a major third, then we might expect that a melodic third such as F-A or Bb-D should also have a size at or near 33:26 at 412.745 cents, a bit larger than the classic Pythagorean major third or ditone equal to precisely twice a 9:8 tone, or 81:64 (407.820 cents). Here 33:26 is larger than the Pythagorean 81:64 by 352:351 (4.925 cents), but, as mentioned above, smaller than 14:11 by 364:363. Small shifts between such subtly different ratios may sometimes be helpful, for example in maintaining pure vertical fifths and fourths at 3:2 and 4:3. Thus if, above F, the position of A sometimes varies between 33/26 and 14/11, in a _musica recta_ style where these two sizes of major thirds are typical, we would still have a considerable degree of congruity. I say "in a _musica recta_ style," because in chromatic or enharmonic contexts it is common to have comma or diesis shifts, unexpected dissonances, and similar effects -- either to express the words of a text, as musicians such as Vicentino and the Monteverdi brothers have noted, or at the discretion of the composer of a piece for instruments without a text. In usual _musica recta_ contexts favoring mostly diatonic steps, however, congruity is highly desirable. In translating music for a temperament such as Peppermint into JI, fortunately, we have the advantage that such gentle temperaments, as they are sometimes styled, do not for most of their usual diatonic intervals deviate too far from Pythagorean with its ideal congruity. Thus the compromises involved in gaining pure vertical intervals (especially fourths and fifths), while optimizing certain melodic progressions, will generally allow a rather high degree of congruity, with variations typically on the order of only a few cents. Another general principle is that often we should be willing to accept small variations in the placement of notes, or sizes of melodic intervals, in order to achieve pure vertical concords -- especially stable fifths and fourths, and also certain mildly unstable sonorities such as 4:6:9, 9:12:16, 6:8:9, 8:9:12, that frequently occur in a _musica recta_ context in a style of neomedieval JI based on a temperament such as Peppermint. This observation as to the desirability of just vertical sonorities applies to some other especially euphonious sonorities which may occur in a more chromatic or enharmonic style, as well as in contexts based on Greek or Near Eastern modes: e.g. 6:7:9, 4:6:7, 12:14:18:21, 7:9:12, 7:9:11, 8:11:13, 7:9:11:13, etc. Indeed the concept of congruity may apply, for example, to an Arab Rast or Persian Shur used as the basis for polyphony (not itself an element of maqam or dastgah music with its purely melodic focus): that is, we should strive to choose stable and unstable vertical intervals that fit the melodic structure of the given mode. In such a "fusion" style, for example, we might expect to find a variety of smaller or larger middle or Zalzalian thirds (often called neutral thirds) expanding to fifths or contracting to unisons, with these two-voice resolutions often involved the step of a tone in one voice, and of some kind of middle or Zalzalian second in the other voice. In a highly chromatic or enharmonic style, congruity may not be so relevant, since variability and unpredictability are often virtues in this kind of setting, although one may strive for a certain musical logic and economy in achieving one's expressive goal. With typical _musica recta_ passages, however, congruity is an important element of art. There are situations where considerable compromises here may be unavoidable, as with classic 5-limit JI, where obtaining 5:4 major thirds from unequal steps of 9:8 (203.910 cents) and 10:9 (182.404 cents) requires that these tones differ by the syntonic or Didymic comma of 81:80 (21.506 cents). And the absolutely exquisite Diatonic of Archytas, with its 7:6 minor thirds, requires yet more disparate steps at 9:8 and 8:7 (231.174 cents), a difference of a septimal or Archytan comma at 64:63 (27.264 cents). However, for the _musica recta_ portions of a piece like _O Europae_, the likely variations or disparities are much smaller, involving differences like 352:351 (4.925 cents), e.g. 81:64 versus 33:26; or 364:363 (4.763 cents), e.g. 33:26 versus 14:11. Thus the ideal of JI congruity can apply with only minor compromises. While pursuing in this mostly diatonic context both pure vertical concords and melodic congruity, we may note that small variations tend to be more noticeable in vertical intervals than in melodic steps or precise locations of notes. Consider, for example, the vertical concord of a fifth tuned at a pure 3:2 (701.955 cents), as compared to one tuned at 352:351 or not quite 5 cents wider, at 176:117 (706.880 cents). The first fifth will sound ideally smooth and just, while the second will have in many timbres quite noticeable beating, This might not be so noticeable in a sonority including other more tense intervals; but generally, especially in a neomedieval style where fifths and fourths are the most rich stable concords, one strives in JI to keep these vertical intervals pure. In contrast, suppose that above F as the 1/1, the position of C changes slightly from 3/2 to 176/117, again a difference of not quite 5 cents. Here, the difference may be almost imperceptible, since around 3 cents is often regarded as the Just Noticeable Difference for melodic steps. Especially if this slight melodic variability facilitates the fine-tuning of pure vertical concords, it may be well worthwhile without substantially compromising congruity. ------------------------------------------------------ 2. Some guidelines for congruity: _Musica recta_ notes ------------------------------------------------------ In a regular temperament like Peppermint, the size of diatonic tones and semitones from usual chains of fifth is generally fixed. Thus in Peppermint, each regular tone is 208.191 cents, larger than 9:8 by 4.281 cents, and almost as large as 44:39 (288.835 cents), a ratio exceeding 9:8 by a comma of 352:351 (4.925 cents). Each regular diatonic semitone or limma is 79.522 cents, close to 22:21 (80.537 cents), a ratio narrower than the Pythagorean limma at 256:243 (90.225 cents) by 896:891 (9.688 cents). In getting a sense of what a JI realization of _musica recta_ based on Peppermint might look like, we can also helpfully consider the sizes of regular major and minor thirds. Here, as in regular temperaments generally, a regular major third is equal to two regular tones (e.g. C-D-E), or 416.382 cents. We find that this is a bit larger than 33:26 (412.745 cents), and very close to 14:11 (417.508 cents). This tempered third might represent either 33:26 or 14:11, and in JI we are free to use either ratio as convenient. A regular minor third, equal to regular tone plus a limma, is in Peppermint 287.713 cents. This is a bit larger than 33:28 (284.447 cents), which together with a 14:11 major third forms a 3:2 fifth; and very close to 13:11 (289.210 cents), which together with 33:26 likewise forms a 3:2 fifth. While the precise sizes of tempered intervals are compromises, and indeed a main reason for translating a piece from a tempered system to JI is to obtain pure or just sizes where possible for both vertical concords and melodic steps, these tempered sizes can also serve as a guide to the just ratios we may be seeking for vertical and melodic intervals alike. Here the tempered diatonic semitone at 79.52 cents suggests a just ratio of 22:21, or more generally a semitone step that will be somewhat smaller than Pythagorean. Another way we can get a sense of likely diatonic semitone sizes is to take a just 4:3 fourth and subtract the size of a major third. Here 4:3 less a 14:11 third leaves a semitone of 22:21, used by Ptolemy in his Intense Chromatic. If we take 4:3 less a 33:26 major third, this leaves a slightly larger semitone at 104:99 (85.300 cents) -- smaller than Pythagorean by a 352:351, and larger than 22:21 by a 364:363. Thus in routine _musica recta_ music we may start by expecting compact semitones a bit smaller than Pythagorean, with 22:21 and 104:99 as likely sizes; tones at 9:8, or possibly a bit larger in certain situations (e.g. 44:39); major thirds around 33:26 or 14:11, a bit larger than the Pythagorean 81:64; and minor thirds around 33:28 or 14:11, slightly smaller than a Pythagorean 32:27 (294.135 cents). Within this general framework, choices of the precise JI ratio may vary based on context. For example, suppose we are dividing a perfect fifth into a sonority with a major and a minor third, the mildly unstable _quinta fissa_ or "split fifth" sonority given this name by Jacobus, the author of the _Speculum musicae_ (c. 1330?). Since we seek where possible to use pure fifths and fourths, this means that if the lower interval is a minor third at 13:11, for example, then the upper interval will be a major third at 33:26, which together form a 3:2 fifth. And if the lower interval is a 14:11 major third, the upper interval will be a minor third at 33:28. Indeed our general framework of ratios and frequencies may take note of such choices by sometimes suggesting more than one position for a given _musica recta_ note. We need not attempt, in advance, to make an exhaustive list of all such possibilities, which are best discovered by going through a piece and addressing each situation on its own merits. However, having a general framework does provide a sense of usual sizes for steps and intervals, and some of the small adjustments which may routinely occur. Starting with Mykhaylo's suggested frequency of F3=176 Hz, we might propose a tentative _musica recta_ framework like the following: ====================================== F4 2/1 1200.000 cents 352.000 Hz -------------------------------------- E4 21/11 1119.463 cents 336.000 Hz 99/52 1114.700 cents 335.077 Hz -------------------------------------- 56/33 915.553 cents 298.667 Hz 22/13 910.790 cents 297.846 Hz D4 27/16 905.865 cents 297.000 Hz -------------------------------------- 176/117 706.880 cents 264.752 Hz C4 3/2 701.955 cents 264.000 Hz -------------------------------------- 56/39 626.343 cents 252.718 Hz B3 63/44 621.418 cents 252.000 Hz -------------------------------------- Bb3 4/3 498.045 cents 234.667 Hz 117/88 493.120 cents 234.000 Hz -------------------------------------- A3 14/11 417.508 cents 224.000 Hz 33/26 412.745 cents 223.385 Hz -------------------------------------- 44/39 208.835 cents 198.564 Hz G3 9/8 203.910 cents 198.000 Hz -------------------------------------- F3 1/1 0.000 cents 176.000 Hz ====================================== It will be seen that a number of these likely ratios have even frequency values: thus 9/8 (198 Hz); 14/11 (224 Hz); 63/44 (252 Hz); 3/2 (264 Hz); 27/16 (297 Hz); 21/11 (336 Hz); and 2/1 (352 Hz). Additionally, a slightly narrow fourth at 117/88, or 352:351 lower than 4/3, also has the even frequency value of 234 Hz. While we would prefer not to use the ratio of 117:88 for a vertical fourth (which would have the effect of tempering 4:3 narrow by not quite 5 cents), we might well shift the location of Bb3 to 117/88 in certain situations, e.g. for a descending semitone of Bb3-A3 placed at 117/88-33/26 where this might be a musically convenient choice. Although not mean to be exhaustive, this framework does provide a general guide for fine-tuning the _musica recta_ sections of a piece such as _O Europae_. Let us now see how this framework may apply in practice. -------------------------------------------------- 3. The framework in practice: The opening measures -------------------------------------------------- The piece _O Europae_ begins with a "departure and return" structure: sounding the stable trine or complete 2:3:4 sonority on the final of F Lydian, F3-C4-F4, with its outer octave, lower fifth, and upper fourth; then moving to an unstable sonority G3-B3-E4, followed by a directed resolution back to F3-C4-F4, affirming this trine or complete three-voice consonance as a modal center. 1 2 3 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3... 21:22 22:21 27:32 112:99 22:21 -80.537 80.537 294.135 213.598 80.537 F4 F4 E4 F4 D4 E4 F4 2/1 2/1 21/11 2/1 27/16 21/11 2/1 1200 1200 1119.463 1200 905.86 1119.463 1200 352 Hz 352 Hz 336 Hz 352 Hz 297 Hz 336 Hz 352 Hz 21:22 22:21 -80.537 80.537 C4 C4 B3 C4 3/2 3/2 63/44 3/2 701.955 701.955 621.418 701.955 264 Hz 264 Hz 252 Hz 264 Hz 9:8 8:9 203.910 203.910 F3 F3 G3 F3 1/1 1/1 9/8 1/1 0 0 203.910 0 176 Hz 176 Hz 198 Hz 176 Hz O Eu- ro- pae 4 4 56 176 33 56 4 3 3 42 126 28 42 3 2 2 33 99 22 33 2 Here the main contrast is between a stable trine F3-C4-F4 tuned at a pure 2:3:4 or 1/1-3/2-2/1, as we might expect in JI; and the unstable major sixth sonority G3-B3-E4 tuned at 1:1-14:11-56:33 (0-417.508-915.553 cents), with a 14:11 major third and 56:33 major sixth, located at 9/8-63/44-21/11. After a bit of ornamentation, this sonority resolves back to F3-C4-F4, with the lower 14:11 third expanding to 3:2, and the outer 56:33 expanding to 2:1. Each directed two-voice resolution involves the lowest voice descending by a 9:8 tone (G3-F3), while an upper voice ascends by a 22:21 semitone (B3-C4, 63/44-2/1; E4-F4, 21/11-2/1). All things being equal, my preference is often for directed resolutions involving superparticular melodic steps, here 9:8 and 22:21. This is a cadence in the intensive manner, with ascending whole-tone motion and descending semitonal motion. My leaning to superparticular cadential steps, and a desire to keep vertical perfect fifths and fourths pure, leads to this solution. Looking at the vertical ratios shown at the bottom of the musical notation may underscore the usual medieval European or neomedieval contrast between complete stable sonorities at 2:3:4 and mildly unstable ones involving usual thirds and sixths at considerably more complex ratios such as 33:42:56 (1:1-14:11-56:33). In directed progressions, this tension is typically released by resolutions of unstable intervals (here Maj6-8, Maj3-5) involving stepwise contrary motion. While this statement is true for the entire 13th-14th century era, by around 1300 there is a special predilection for cadences of this type involving compact melodic semitonal motion, either ascending (intensive manner), or descending (remissive manner). The Pythagorean semitone at 256:243 (90.225 cents) is already quite incisive, with modern ratios such as 22:21 as a mild accentuation of this trait. While the resolution from 9/8-63/44-21/11 to 1/1-3/2-2/1 sets the structure for this passage, the ornamentation at measure 2 in the highest voice illustrates some fine point of intonation. Here the melodic figure is E4-F4-D4-E4, leading to the resolution on F4. We start with E4 at 21/11, a 56:33 major sixth above the lowest voice at 9/8, and a 4:3 fourth above the middle voice at 63/44 (a 14:11 major third above the lowest voice). The highest voice next moves to F4 at 2/1, momentarily forming a sonority G3-B3-F4 with the relatively tense but somewhat "compatible" interval of a 16:9 minor seventh G3-F4 (996.090 cents) with the outer voice; and an acutely tense diminished fifth B3-F4 at 88:63 (578.582 cents) with the middle voice. The quite complex ratio for this transient sonority is 99:126:176 (0-417.508-996.090 cents). It then moves down by a third to D4, placed at 27/16, a pure 3:2 above G3 at 9/8; this step of F4-D4 is tuned at 32:27, a regular Pythagorean minor third (294.135 cents) which can happily coexist with the somewhat smaller minor thirds at 33:28 and 13:11 typical in this style. This produces for the moment a mildly unstable split fifth sonority of G3-B3-D4 at 1:1-14:11-3:2 (9/8-63/44-27/16), with lower 14:11 and upper 33:28 thirds, or 22:28:33 (0-417.508-701.955 cents). From D4 the highest voice now returns to E4, or 21/11, ascending by a rather large tone of 112:99 or 213.598 cents, D4-E4, setting the stage for the directed resolution of E4-F4 by a 22:21 step at the beginning of measure 3, 21/11-2/1. In order to maintain pure vertical fourths and fifths at 4:3 and 3:2, and also superparticular cadential steps at 9:8 and 22:21, one main compromise is involved from a certain statement: the use of melodic whole-tone steps at somewhat different sizes: thus 1/1-9/8-1/1 (F3-G3-F3) in the lowest voice (as in Pythagorean intonation); but at one point 27/16-21/11 (D4-E4) in the highest voice, a tone at 112:99, larger than 9:8 by 896:891 (9.688 cents). Steps of this size are routine in some styles of Byzantine and Persian music, and represent the intonational diversity and intricacy that fluid JI can introduce. A curious quirk is that in this brief example, all notes have locations with frequencies expressed, with reference to an F3=176 Hz pitch standard, as even numbers of Hz. This quick exercise provides a kind guide for _musica recta_ styles of this kind. We expect tones to be at 9:8, or sometimes a bit larger; and semitones to be a bit narrower than the Pythagorean 256:243. Likewise, major thirds and sixths are typically a bit larger than the Pythagorean 81:64 (407.820 cents), with 81:64 itself not out of place if musical exigencies favor it; and minor thirds typically a bit smaller than 32:27, which is itself available as a melodic or vertical interval if the situation calls for it. Thus, for example, if in this kind of _musica recta_ style and setting we arrive at tones smaller than 9:8, or diatonic semitones larger than 256:243, or major thirds smaller than 81:64, or minor thirds larger than 32:27 (either vertically or melodically), then we should seriously question what is happening. Such things might, for example, occur if a composer learning this style is accustomed to writing in 5-limit JI, where larger semitones (e.g. 16:15, 111.731 cents) and small tones (e.g. 10:9, 182.404 cents) are standard, together with the smaller 5:4 major third (386.314 cents) and larger 6:5 minor third (315.641 cents). In line with the concept of "congruity," the vertical and melodic elements of intonation in each style may agree with its ethos. Thus in a setting of 16th-century polyphony, where 5-limit JI or meantone are characteristic, the larger semitones and smooth regular thirds fit well with an ethos of carefully controlled shades of tension, with the rather subtle idiom of the suspension dissonance as a favorite treatment of instability, which punctuates a texture generally based on a smooth flowing between 4:5:6 or 10:12:15 sonorities. In a medieval or neomedieval style, in contrast, favoring Pythagorean JI or its accentuated variants including temperaments with wide fifths such as Peppermint and related systems of JI (fixed or, as here, fluid), narrow semitones and rather complex and active regular thirds agree with an ethos of often bold and dramatic contrasts between stable 2:3:4 sonorities and a great variety of unstable intervals and sonorities ranging from relatively concordant to acutely tense and discordant. Zarlino, in 1558 and later, of course masterfully codifies and explains the 16th-century approach, including its intonational ramifications (in 5-limit JI or the temperaments now generally known as meantone). Beyond this, however, he expresses a larger guideline of congruity, or the use where practical of the same interval sizes for vertical and melodic purposes, which also is a desideratum for the _musica recta_ aspects of a neomedieval style. --------------------------------------------------------- 4. Realizing the full first section in JI (measures 1-21) --------------------------------------------------------- Beginning with the opening we have already translated to JI, we shall now add the rest of the first section of _O Europae_. This is only one possible solution, but I hope it will be a useful and illustrative one. 1 2 3 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 21:22 22:21 27:32 112:99 22:21 44:39 272:243 -80.537 80.537 -294.135 213.598 80.537 208.835 208.673 F4 F4 E4 F4 D4 E4 F4 G4 2/1 2/1 21/11 2/1 27/16 21/11 2/1 88/39 1200 1200 1119.463 1200 905.86 1119.463 1200 1408.835 352 Hz 352 Hz 336 Hz 352 Hz 297 Hz 336 Hz 352 Hz 397.128 Hz 21:22 22:21 44:39 272:243 -80.537 80.537 208.835 208.673 C4 C4 B3 C4 D4 3/2 3/2 63/44 3/2 22/13 701.955 701.955 621.418 701.955 910.790 264 Hz 264 Hz 252 Hz 264 Hz 297.845 Hz 9:8 8:9 4:3 9:8 203.910 203.910 498.045 203.910 F3 F3 G3 F3 Bb4 1/1 1/1 9/8 1/1 4/3 0 0 203.910 0 498.045 176 Hz 176 Hz 198 Hz 176 Hz 234.667 Hz O Eu- ro- pae do- 4 4 56 176 33 56 4 44 3 3 42 126 28 42 3 33 2 2 33 99 22 33 2 26 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 22:21 21:22 99:112 8:9 14:11 80.537 -80.537 -213.598 -203.910 417.508 A4 Bb4 A4 G4 G4 F4 A4 28/11 8/3 28/11 9/4 9/4 2/1 28/11 1617.508 1698.045 1617.508 1403.910 same 1200 1617.508 448 Hz 469.333 Hz 448 Hz 396 Hz same 352 Hz 448 Hz 22:21 21:22 99:112 112:99 80.537 -80.537 -213.598 213.598 E4 F4 E4 D4 E4 21/11 2/1 21/11 27/16 21/11 1119.463 1200 1119.463 905.865 1119.463 336 Hz 352 Hz 336 Hz 297 Hz 336 Hz 112:99 99:112 39:44 21:22 213.598 -213.598 -208.673 -00.537 C4 D4 C4 Bb3 A3 3/2 56/33 3/2 4/3 14/11 0 915.553 701.955 493.282 417.508 264 Hz 298.667 Hz 264 Hz 234 Hz 224 Hz mus com- mu- - nis 56 44 56 39 108 96 4 42 33 42 33 81 81 3 33 28 33 26 64 64 2 7 8 9 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 8:9 99:112 21:22 99:112 112:99 22:21 -203.910 -213.598 -80.537 -213.598 213.598 00.537 A4 G4 F4 E4 D4 E4 F4 28/11 224/99 2/1 21/11 27/16 21/11 2/1 1617.508 1413.598 1200 1119.463 905.865 1119.463 1200 448 Hz 398.222 Hz 352 Hz 336 Hz 297 Hz 336 Hz 352 Hz 8:9 99:112 21:22 22:21 -203.910 -213.598 -80.537 80.537 E4 D4 C4 B3 C4 21/11 56/33 3/2 63/44 3/2 1119.463 915.553 701.955 621.418 701.955 336 Hz 298.667 Hz 264 Hz 252 Hz 264 Hz 8:9 21:22 99:112 8:9 -203.910 -80.537 -213.598 203.910 C4 Bb3 A3 G3 F3 3/2 4/3 14/11 9/8 1/1 701.955 498.045 417.508 203.910 0 264 Hz 234.667 Hz 224 Hz 198 Hz 176 Hz am- pli- fi- ca- ta, 56 56 44 56 33 56 4 42 42 33 42 28 42 3 33 33 28 33 22 33 2 10 11 12 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 9:8 8:9 21:22 99:112 32:27 203.910 -203.910 -80.537 -213.598 294.135 F4 G4 F4 E4 D4 F4 F4 2/1 9/4 2/1 21/11 27/16 2/1 2/1 1200 1403.910 1200 1119.463 910.620 1200 1200 352 Hz 396 Hz 352 Hz 336 Hz 297 Hz 352 Hz 352 Hz 9:8 8:9 21:22 22:21 203.910 -203.910 -80.537 80.537 C4 D4 C4 B3 C4 C4 3/2 27/16 3/2 63/44 3/2 3/2 701.955 905.685 701.955 621.418 701.955 701.955 264 Hz 297 Hz 264 Hz 252 Hz 264 Hz 264 Hz 33:26 44:39 9:8 14:11 412.745 208.835 203.910 417.508 F3 A3 G3 G3 F3 A3 1/1 33/26 9/8 9/8 1/1 14/11 0 412.745 203.910 203.910 0 417.508 176 Hz 223.385 Hz 198 Hz 198 Hz 176 Hz 224 Hz O Gal- li- a vox 4 52 4 16 56 33 4 44 3 39 3 12 42 28 3 33 2 33 2 9 33 22 2 28 13 14 15 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 9:8 8:9 39:44 33:26 203.910 -203.910 -208.835 412.745 G4 A4 A4 G4 G4 F4 A4 88/39 33/13 33/13 88/39 88/39 2/1 33/13 1408.835 1612.745 1612.74 1408.835 same 1200 1612.745 397.128 Hz 446.769 Hz 446.769 Hz 397.128 Hz same 352 Hz 446.769 Hz 9:8 8:9 9:8 203.910 -203.910 203.910 D4 E4 D4 D4 E4 22/13 99/52 22/13 22/13 99/52 910.790 1114.700 910.790 910.790 1114.700 297.846 Hz 335.077 Hz 297.846 Hz 297.846 Hz 335.077 Hz 9:8 8:9 99:104 13:11 203.910 -203.910 -85.300 289.210 Bb3 C4 Bb3 Bb3 A3 4/3 3/2 4/3 4/3 33/26 498.045 701.955 498.045 498.045 412.745 234.667 Hz 264 Hz 234.667 Hz 234.667 Hz 223.385 Hz no- bi- lis- si- ma 44 44 99 44 44 39 4 33 33 66 33 33 33 3 26 26 52 26 26 26 2 16 17 18 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 8:9 117:121 121:117 351:352 -203.910 -58.198 58.198 -4.925 A4 G4 Gv4 G4 33/13 88/39 24/11 88/39 1612.745 1408.835 1350.637 1408.835 446.745 Hz 397.128 Hz 384 Hz 397.128 Hz 8:9 117:121 121:117 351:351 -203.910 -58.198 58.198 -4.925 E4 D4 Dv4 D4 99/52 22/13 18/11 22/13 1114.700 910.790 852.592 910.790 335.077 Hz 297.846 Hz 288 Hz 297.846 Hz 8:9 21:22 242:273 351:352 -203.910 -80.537 -208.673 -4.925 C4 Bb3 A3 G3 3/2 4/3 14/11 44/39 701.955 498.045 417.508 208.673 264 Hz 234.667 Hz 224 Hz 198.564 Hz que- - ru- lae 44 44 12 4 33 33 9 3 26 26 7 2 19 20 21 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 || 8:9 21:22 99:112 112:99 22:21 -203.910 -80.537 -213.598 213.598 80.537 G4 F4 E4 D4 E4 F4 9/4 2/1 21/11 27/16 21/11 2/1 1403.910 1200 1119.463 905.865 1119.463 1200 396 Hz 352 Hz 336 Hz 297 Hz 336 Hz 352 Hz 112:99 99:112 8:9 21:22 22:21 213.598 -213.598 -203.910 -80.537 80.537 D4 E4 D4 C4 B4 C4 27/16 21/11 27/16 3/2 63/44 3/2 905.865 1119.463 905.865 701.955 621.418 701.955 297 Hz 336 Hz 297 Hz 264 Hz 252 Hz 264 Hz -203.910 8:9 G3 G3 F4 9/8 9/8 1/1 203.910 203.910 0 198 Hz 198 Hz 176 Hz pa- cis: 4 66 32 112 9 56 4 3 56 27 99 8 42 3 2 33 18 66 6 33 2 A few considerations shaping the intonational choices are pure vertical concords, especially 4:3 fourths and 3:2 fifths; pitch stability of the modal final F3 (1/1), and also the fifth C4 (3/2), with some other steps more flexible; and a general preference at cadences for classic resolutions where voices move either by 9:8 tones or by preferred semitone steps (e.g. 22:21, 104:99; or for the Marchetan nuance we shall address at measures 17-18, ideally 28:27, although here with a subtle compromise). After the opening departure and return progression of F3-C4-F4 to G3-B3-E4 to F3-C4-F4 described in Section 3, we move to a passage with a series of parallel mildly unstable sonorities with an outer sixth, lower third, and upper fourth, the kind of texture called fauxbourdon that occurs in some 14th-century Continental European pieces, but is more common in England during that era, and also on the Continent in the earlier 15th century. Maintaining pure vertical fourths, and ratios for major and minor thirds such as 14:11 or 33:26, and 13:11 or 33:28, raises some choices in terms of precise melodic steps. 3 4 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | F4 G4 A4 Bb4 2/1 88/39 28/11 8/3 C4 D4 E4 F4 3/2 22/13 21/11 2/1 F3 Bb4 C4 D4 1/1 4/3 3/2 56/33 pae do- mus com- Here, for example, after the stable F3-C4-F4 sonority (1/1-3/2-2/1) at the beginning of measure 3, the middle voice progresses C4-D4-E4 at 3/2-22/13-21/11, with melodic steps of 3/2-22/13 or 44:39 (208.835 cents, wider than 9:8 by 352:351 or 4.925 cents); and 22/13-21/11 or the minutely smaller 273:242 (208.673 cents, wider than 9:8 by 364:363 or 4.763 cents). Vertically, we move from a 3:2 fifth at F3-C4 to a 33:26 major third Bb4-D4 at 412.745 cents (4/3-22/13) to a 14:11 major third C4-E4 at 417.508 cents (3/2-21/11). The highest voice, moving with the middle voice in parallel fourths, follows the same melodic steps to move from the 2:1 octave F3-F4 to the 22:13 major sixth Bb3-G4 (910.790 cents) to the 56:33 major sixth C3-A4 at 56:33 (915.553 cents). Taking all three voices into account, we thus move from F3-C4-F4 at a pure 2:3:4 to Bb3-D3-G4 at 26:33:44 to C4-E4-A4 at 33:46:56. There is thus a contrast between the richly stable 2:3:4 trine on F, and mildly unstable sonorities with outer sixths which are at once sweet and yet rather active and complex. Melodically, the lowest voice has to this point maintained regular Pythagorean steps (thus F3-Bb4-C4 as 1/1-4/3-3/2), while each upper voice has ascended by two almost identical steps of a tone (44:39, 273:242) not quite 5 cents larger than 9:8, which together equal a 14:11 major third (3/2-21/11 or 2/1-28/11). The next sonority in measure 4 is D4-F4-Bb4 at 56/33-2/1-8/3, with a lower 33:28 minor third (284.447 cents) and an outer minor sixth at the relatively simple ratio of 11:7 (782.492 cents), or 28:33:44. For each upper voice, moving to this sonority completes a melodic tetrachord or division of a 4:3 fourth: thus in the middle voice, 3/2-22/13-21/11-2/1 or 44:39-273:242-22:21 (208.835-208.673-80.537 cents); and in the upper voice, 2/1-88/39-14/11-4/3, with identical steps. The melodic division of a 14:11 third into two virtually equal tones is much like what happens in a regular temperament like Peppermint, where two identical tones at 208.191 cents produce a near-14:11 third at 416.391 cents, very slightly narrow of a just 14:11. In moving from C4-E4-A4 to D4-F4-Bb4, the lowest voice deviates appreciably from Pythagorean locations and step sizes, using a larger tone C3-D4 (3/2-56/33) at 112:99 or 213.598 cents, larger than 9:8 by 896:891 or 9.688 cents. Thus in this voice, the 14:11 major third Bb3-D4 (4/3-56/33) is divided into a usual 9:8 step at Bb3-C4 followed by a 112:99 at C4-D4, a less even than the 44:39-273:242 division of this third found in each upper voice. This implies a tetrachord of 9:8-112:99-22:21 (203.910-213.598-80.537 cents), which seems to parallel Greek and Near Eastern patterns with a 9:8 tone, a larger (as here) or smaller tone, with some superparticular semitone completing the 4:3 fourth (e.g. the 9:8-8:7-28:27 steps of Archytas at 203.910-231.174-62.961 cents). At measure 5 we return to C4-E4-A4 at 3/2-21/11-28/11 (1:1-14:11-56:33) or 33:44:56, and move into a cadential approach where constraints of maintaining pure vertical fifths and fourths offer the convenient solution of using some Pythagorean steps and vertical thirds and sixths: 5 6 | 1 2 3 4 5 6 | 1... A4 G4 G4 F4 A4 28/11 9/4 9/4 2/1 28/11 E4 D4 E4 21/11 27/16 21/11 C4 Bb3 A3 3/2 4/3 14/11 mu- - nis The progression here is to a remissive cadence on A3, with the lowest voice descending by a semitone (Bb3-A3) and the middle voice ascending by a tone (D4-E4), with the similar ascent in the upper voice (G4-A4) ornamented in characteristic late medieval fashion by a momentary descent from the major sixth to the fifth followed by the upward leap of a third to the octave of the resolving trine A3-E4-A4 (G4-F4-A4). Here, while C4-E4-A4 is tuned at a typical neomedieval 33:44:56, the ornamental notes in the upper voice of G4 at beat 3 and F4 at beat 6 raise questions of vertical concord. At beat 3, we have C4-E4-G4, with C4 at a usual 3/2, so that G4 must be placed at 9/4 (the usual Pythagorean step) in order to attain a pure vertical fifth C4-G4 at 3:2. Unless we indulge in some small melodic shift, the repeated G4 at beat 4 must therefore also be at 9/4, which requires that D4 in the middle voice be placed at a Pythagorean 27/16 in order to observe a pure 4:3 vertical fourth. The resulting cadential sixth sonority Bb3-D4-G4, or 4/3-27/16-9/4, has a Pythagorean tuning of 1:1-81:64-27:16 or 64:81:108 (0-407.820-905.865 cents), with a curious quirk in its resolution. Here it seemed convenient to conclude the remissive cadence on A at the steps 14/11-21/11-28/11, which tie nicely into the next phrase. Thus the lowest voice descends 4/3-14/11, a compact semitone at 22:21 (80.537 cents). However, the lower two voices at 4/3-27/16 form a usual Pythagorean major at 81:64 (407.820 cents) rather than the wider 14:11 we might expect (417.508 cents) to resolve, in a remissive cadence, by a descending 22:21 step and an ascending 9:8 step (e.g. 4/3-56/33 to 14/11-21/11). Thus the middle voice, moving 27/16-21/11, must ascend by a wider tone of 112:99 or 213.598 cents. This motion of 112:99, plus 22:21 in the lowest voice, together represent an expansion of 32:27 (294.135 cents) from the Pythagorean major third Bb3-D4 at 81:64 to a pure fifth at A3-E4. It may be worthwhile quickly to show two alternative approaches to this cadence, one of them compromising the principle of pure vertical fourths and fifths, and the other involving a small pitch shift in a repeated note: 5 6 | 1 2 3 4 5 6 | 1... * A4 G4 G4 F4 A4 28/11 88/39 88/39 2/1 28/11 E4 D4 E4 21/11 22/13 21/11 C4 Bb3 A3 3/2 4/3 14/11 mu- - nis In this solution, the cadential major sixth sonority is 4/3-22/13-88/39 or 1:1-33:26-22:21 (0-412.745-910.790 cents), or 26:33:44. Here the complication is that the momentary fifth C4-G4 at beat 3, 3/2-88/39, is equal not to a pure 3:2 but to 176:117 (706.880 cents), wider than pure by 352:351 or 4.925 cents. On keyboards for this variety of JI, such "virtually tempered" fifths at ratios such as 176:117, not quite five cents wide, are common, with the result a tuning system having pure fifths in some locations and tolerably impure ones at others, as in some irregular 18th-century European temperaments, for example. With fluid JI, however, it is possible to maintain pure 3:2 fifths and 4:3 fourths as in Pythagorean intonation -- with some other compromises. For many purposes, an impurity of this kind in a momentary fifth should be quite acceptable, so that to choose one solution is not to exclude others. 5 6 | 1 2 3 4 5 6 | 1... A4 G4 G4 F4 A4 28/11 27/16 88/39 2/1 28/11 E4 D4 E4 21/11 22/13 21/11 C4 Bb3 A3 3/2 4/3 14/11 mu- - nis In this solution, pure fifths and fourths are maintained, while tuning the cadential major sixth sonority at 4/3-22/13-88/39 or 26:33:44, by having the highest voice at beat 3 tune G4 as 27/16 (a 3:2 above C4 at 3/2), but then at beat 4 shift the repeated G4 very slightly higher at 88/39, a shift upward of 352:351 or 4.925 cents, not much larger than the estimated Just Noticeable Difference (JND) on the order of 3 cents for melodic changes. In either this alternative solution or the last, the remissive cadence to A3 features a descending 22:21 semitone in the lowest voice, and an ascending tone in the middle voice at 22/13-21/11, or 273:242 (208.673 cents). My simpler Pythagorean solution was chosen in part to illustrate how usual Pythagorean thirds (81:64, 32:27) are quite acceptable in a neomedieval style, and can nicely coexist with their accentuated modern equivalents at 33:26 and 13:11, or 14:11 and 33:28. In a polyphonic Lydian modality of the 13th-14th centuries of the kind illustrated here, it is very characteristic to have intensive cadences on F and remissive cadences on A, the former more conclusive and the latter often used as internal cadences -- in a medieval terminology, respectively _clos_ and _ouvert_ or "closed" and "open" cadences (French); or _chiuso_ and _overto_ (Italian). This intensive/remissive contrast is as basic to later medieval European polyphony as the major/minor contrast is to Classic or Romantic European tonality. Comparing the two cadences we have encountered so far, intensive on F at measures 2-3, and remissive on A at measures 5-6, will reveal another important contrast of much Lydian modality: || E4 F4 G4 A4 B3 C4 D4 E4 G3 F3 Bb3 A3 Intensive cadences on F of this kind feature the major third G3-B4 expanding to the fifth F3-C4, with the ascending semitone step B4-C4 calling for B-natural. Remissive cadences on A, however, have the major third Bb3-D4 expanding to the fifth A3-E4, with the descending semitone step Bb3-A3 calling for Bb. This contrast between B-natural or "hard B" in an ascending semitone B-C and Bb or "soft B" in a descending semitone Bb-A also occurs in monophonic chants and secular songs, but in polyphony frequency underscores the contrast between intensive cadences on F and remissive cadences on A. It is noteworthy that these intensive and remissive cadences share alike the directed resolutions by stepwise contrary motion of major third to fifth (Maj3-5) and major sixth to octave (Maj6-8), but with a different quality of "manner" of cadencing based on the contrast between ascending and descending semitonal motion. The next passage, at measures 7-9, shows how the fluidity of the B/Bb step can reveal itself in other contexts also: 7 8 9 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | A4 G4 F4 E4 D4 E4 F4 28/11 224/99 2/1 21/11 27/16 21/11 2/1 E4 D4 C4 B3 C4 21/11 56/33 3/2 63/44 3/2 C4 Bb3 A3 G3 F3 3/2 4/3 14/11 9/8 1/1 am- pli- fi- ca- ta, While the lowest voice with its descend through the lower pentachord of Lydian, C4-Bb4-A3-G3-F3, prefers Bb3 (a perfect fourth above the final), the middle voice prefers the ascending semitone B3-C4, which often occurs in chant as well as polyphony in such figures as F3-A3-B3-C4. Here the contrast between Bb3 in the lower voice, and B3 in the middle voice, is a feature of the polyphonic texture. As tuned in this solution, the upper voices from the beginning of measure 7 to the beginning of measure 8 describe melodic descends through a 4:3 fourth, E4-D4-C4-B3 (21/11-56/33-3/2-63/44; 28/11-224/99-2/1-21/11) with steps of 9:8-112:99-22:21. To identify the direction of these melodic steps as descending, we might also write, as in the full notation showing these motions, 8:9-99:112-21:22. Theoretically, this style of melodic notation suggests that the numbers represent string lengths or wavelength, with a larger number showing a greater length and thus a note lower in frequency. The lowest voice at the same time descends through the tetrachord C4-Bb3-A3-G3 (3/2-4/3-14/11-9/8), or 8:9-22:21-99:112 -- the same steps as in the upper voices, but in a different ordering or permutation. Vertically, the three voices move smoothly through sonorities with 14:11 major thirds and 56:33 major sixths (33:42:56), or a 33:28 minor third and 11:7 minor seventh (A3-C4-F4, 28:33:44). Thus at the opening of measure 8, we reach the cadential major sixth sonority G3-B3-E4 at 9/8-63/44-21/11, again 33:42:56, which is ornamented by the figure E4-D4-E4 in the upper voice, and then resolves by stepwise motion in all voices to F3-C4-F4, with a descending 9:8 step (G3-F3) in the lowest voice and ascending superparticular semitone steps of 22:21 in the upper voices (63/44-3/2, 21/11-2/1). The main intonational adjustment here is that the momentary D4 in the highest voice must be placed at 27/16 in order to form a pure vertical fifth with the lowest voice, G3-D4 (9/8-27/16). From a melodic perspective, this stretches the descending fifth A4-D4 (28/11-27/16) and fourth G4-D4 (224/99-27/16) in the highest voice by 896:891 beyond the expected 3:2 and 4:3 respectively, producing sizes of 448:297 (711.643 cents) and 3584/2673 (507.733 cents). As vertical intervals, such fifths and fourths would be highly problematic in many timbres; but as melodic intervals they seem more acceptable. Apart from this D4, the melodic steps in each part suggest a classic type of tetrachord division with steps of 9:8-22:21-112:99. There are some notable "diagonal" contrasts in the positioning of the same note in different voices: thus, for example, G4 at 224/99 in the highest voice (measure 7, beat 4), and G3 at 9/8 in the lowest voice (measure 8, beat 1); or D4 at 56/33 in the middle voice (measure 7, beat 4), and in the upper voice at 27/16 (measure 8, beat 3). We have now finished the setting of the words _O Europae domus communis amplificata, "O common house of Europe expanded," with this text suggested by an expression of Mikhail Gorbachev often translated in English as "common house of Europe" or "common European home." From an intensive cadence on F, we moved to a remissive cadence on A, and then back to an intensive cadence on F. From measure 10 to the beginning of measure 12, we have the setting of the words _O Gallia_, "O France": 10 11 12 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 F4 G4 F4 E4 D4 F4 2/1 9/4 2/1 21/11 27/16 2/1 C4 D4 C4 B3 C4 3/2 27/16 3/2 63/44 3/2 F3 A3 G3 F3 1/1 33/26 9/8 1/1 O _ Gal- li- a We start at the trine F3-C4-F4 on the final of the mode, with the lowest voice then moving to A3, here placed at 33/26 (412.745 cents), for a minor sixth sonority of A3-C4-F4 at 1:1-13:11-52:33 (0-289.210-787.255 cents), or 33:39:52. In what might be considered a secondary directed progression, this minor sixth sonority resolves to the trine G3-D4-G4 at 2:3:4, with the minor third A3-C4 expanding to the fifth G3-D4, and the outer minor sixth A3-F4 expanding to the octave G3-G4. Each of these two-voice resolutions is in what I term the "omnitonal" manner, meaning simply that each voice moves by a tone. Here, the lowest voice descends 33/26-9/8, a step of 44:39 (208.835 cents), while each upper voice ascends by a step of 9:8 (203.910 cents). Together, these step sizes in contrary motion produce a total expansion of (44:39 x 9:8) or 33:26, which is the amount by which a 13:11 minor third must expand to reach a 3:2 fifth, or a 52:33 minor sixth to reach a 2:1 octave. In a 13th-century style, this progression frequently serves as a final or internal cadence, and it has a stately quality. The 14th-century guideline of "closest approach" prefers cadences with semitonal motion in one or more of the voices, either ascending (intensive manner) or descending (remissive manner). A question raised by some theoretical sources is whether this preference should apply to any potentially directed progression in a piece -- which might here call for some form of A3-C#4-F#4 to G3-D4-G4 (intensive), or possibly Ab3-C4-F4 to G3-D4-G4 (remissive) -- or may be treated as an optional matter, especially when the progression seems relatively incidental. Here I take the second and more flexible approach, finding that the omnitonal progression adds a pleasing element of diversity. The setting of A3 at 33/26 rather than the slightly higher 14/11 -- thus tuning A3-C4-F4 at 33:39:52 rather than 28:33:44 -- is also a decision I reached in good part for the sake of variety. The lowest voice descends from this note to G3 at 9/8, the superfinal or step above the final, preparing the way for an intensive cadence on F3. Above G3 (9/8), the upper voices begin on D4 (27/16) and G4 (9/4), forming a 2:3:4 trine, and then each descend a 9:8 tone to form the momentary sonority G3-C4-F4 at 9:12:16 (0-498.045-996.090 cents), a just Pythagorean minor seventh at 16:9 divided into two pure 4:3 fourths. This is a mildly stable and relatively concordant sonority favored, for example, by Perotin around 1200. Each descending by a 22:21 semitone, the upper voices then arrive at the directed cadential sonority of G3-B3-E4 at 1:1-14:11-56:33 or 33:42:59, each then ascending by a 22:21 semitone (B3-C4 or 63/44-3/2, E4-F4 or 21/11-2/1) as the lowest voice descends 9/8-1/1 to complete the cadence to F3-C4-F4. The highest voice has a brief ornament moving from the major sixth to the fifth before progressing to the octave of the resolving trine (E4-D4-F4), a figure often associated with Francesco Landini (c. 1325-1397) and termed a "Landini cadence." It appears in the 13th century, and is in widespread use during the 14th and 15th centuries, and in some 16th-century compositions also. As in the first portion of the opening (measures 1-3, 3-6), an initial intensive cadence on F3 leads next to a contrasting and less conclusive remissive cadence on A3: 12 13 14 15 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 F4 G4 A4 A4 G4 G4 F4 A4 2/1 88/39 33/13 33/13 88/39 88/39 2/1 33/13 C4 D4 E4 D4 D4 E4 3/2 22/13 99/52 22/13 22/13 99/52 F3 A3 Bb3 C4 Bb3 Bb3 A3 1/1 14/11 4/3 3/2 4/3 4/3 33/26 a vox no- bi- lis- si- ma Here the cadence to F3-C4-F4 is followed by motion of the lowest part of A3 at 14/11, producing a minor sixth sonority of A3-C4-F4 this time at 1:1-33:28-11:7 or 28:33:44, a small variation on the same basic sonority at measure 10 with A3 at 33/26, and thus 33:39:52. We then have parallel major sixth sonorities with outer sixth, lower third, and upper perfect fourth, Bb3-D4-G4, C4-E4-A4, and again Bb3-D4-G4, leading to a remissive cadence on A3-E4-A4. All these major sixth sonorities are tuned 1:1-33:26-22:13 or 26:33:44 (0-412.745-910.790 cents), with a remissive cadence Bb3-D4-G4 to A3-E4-A4 fitting a classic model with the upper voices ascending by 9:8 tones, and the lowest descending by a semitone of 104:99 (85.300 cents). While this is not a superparticular ratio like the slightly smaller 22:21 (80.537 cents), it seems a very attractive one, a bit narrower and more incisive than the already quite compact Pythagorean limma or regular diatonic semitone at 256:243 (90.225 cents). I would like very warmly to thank Mykhaylo for using this precise cadence, with its steps of 9:8 and 104:99, at another place in his MIDI setting of my piece, and thus inspiring me to use it somewhere in my own setting. Given the most noble nature of this progression, I find that the words _vox nobilissima_, "most noble voice," provide such a happy occasion. This passage is is varied by a couple of ornamental touches. At beat 1 of measure 14, we have an idiom related to the Renaissance suspension where A4 is repeated from the previous C4-E4-A4 while the lower voices move, producing a momentary Bb3-D4-A4 at 4/3-22/13-33/13 or 1:1-33:26-99:52 (0-412.745-1114.700 cents), a rather tense major seventh sonority with lower major third at 33:26 and an upper 3:2 fifth. The major seventh at 99:52, like the Pythagorean 243:128 (1109.775 cents) and indeed slightly wider (by 352:351 or 4.925 cents), is acutely tense; major seventh sonorities are used quite boldly in the 13th century, and also here and there in the 14th century. Here the effect is more transient, with the major seventh A4 moving to the usual major sixth of Bb3-D4-G4, with the remissive cadence to A3-E4-A4 following. This cadence is again ornamented by the Landini idiom in the highest voice, G4-F4-A4. In this passage, the upper voices consistently adhere to the steps C4-D4-E4 at 3/2-22/13-99/52, or F4-G4-A4 at 2/1-88/39-33/13, a pattern with melodic steps of 44:39-9:8 (208.835-203.910 cents) together forming a 33:26 major third. The lowest voice has a subtle distinction between ascending A3-Bb3 at measures 12-13 (14/11-4/3), a step of 22:21, and descending Bb3-A3 at measures 14-15 (4/3-33/26). a slightly larger step of 104:99. It would, of course, be easy to place the first A3 at 33/26 also, but the small difference may add interest. Here the descending figure C4-Bb3-A3 (3/2-4/3-33/26) in the lowest voice divides the 13:11 minor third C4-A3 into steps of 9:8 and 104:99; if we continued the descent through to G3 at 9/8, we would have a classic variety of tetrachord with a 4:3 fourth (G3-A3-Bb3-C4) at 9/8-33/26-4/3-3/2 divided into three steps of 44:39-104:99-9:8 (208.835-85.300-203.910 cents). This tetrachord has two sizes of minor thirds: a Pythagorean 32:27 (294.135 cents) at G3-Bb3 or 9/8-4/3; and a 13:11 at A3-C4 or 33/26-3/2 (289.210 cents). To this point in the piece, we have stayed within the _musica recta_ gamut consisting of the diatonic notes plus Bb, with the fluidity of B/Bb an integral part of the system. The next phrase, however, represents a brief departure from this order, optionally marked, as here, by what I term a Marchetan nuance in intonation: 15 16 17 18 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | A4 A4 G4 Gv4 G4 33/13 33/13 88/39 24/11 88/39 E4 E4 D4 Dv4 D4 99/52 99/52 22/13 18/11 22/13 A3 C4 Bb3 A3 G3 33/26 3/2 4/3 14/11 44/39 ma que - ru- lae The Latin _querela_ can refer either to a complaint or grievance, or to the cooing of a dove, as in the famous treatise of Erasmus, _Querela pacis_ or "The Complaint of Peace," whose title is here quoted -- or almost, since _querulae pacis_ uses the related adjective _querula_, "complaining, lamenting." This was evidently a small mishap in which _querulae_ should be corrected to _querelae_. This word evokes an expressive musical setting. After the trine A3-E4-A4 from the previous remissive cadence, the lowest voice ascends to C4 at 3/2, forming a major sixth sonority C4-E4-A4 at 3/2-99/52-33/13 or 1:1-33:26-22:13 (26:33:44), with the voices all descending by 9:8 tones to another identical sixth sonority Bb3-D4-G4 at 4/3-22/13-88/39. Now we encounter the Marchetan nuance, where conventional notation would show the inflected sonority A3-C#4-F#4 resolving intensively to the trine G3-D4-G4. This nuance, described and advocated by Marcheto of Padua in his _Lucidarium_ (1317-1318), calls for directed progressions involving ascending semitones with inflections which in modern terms would involve sharps to have these semitones considerably narrower than usual diatonic ones (e.g. C#3-D4 or F#4-G4 smaller than a usual B3-C4 or E4-F4 or Bb3-A3, etc.). In his approach, a usual semitone or limma might have the size of around a Pythagorean 256:243 (90.225 cents), or possibly a bit narrower, with our 104:99 (85.300 cents) or 22:21 (80.537 cents) as JI realizations of this "a bit smaller than Pythagorean" region. Some of his language has suggested to Jay Rahn, for example, that the smaller cadential steps involving sharps might be around 37:36 (47.434 cents), just a bit more than half of a usual Pythagorean semitone (which would be 45.112 cents, known to Boethius, citing a Greek theorist named Philolaus, as a diaschisma, and now one of the meanings of that term). Here, adopting a rather more moderate solution, I take this Marchetan diesis as ideally realized by the just thirdtone of Archytas, 28:27 (62.961 cents). Using a notation based on Peppermint and similar tempered systems, i use the spellings in the upper voices of D4-Dv4-D4 and G4-Gv4-G4 to show motion down and then up by a thirdtone step, here actually somewhat narrower than 28:27 because of other exigencies involving events that will follow this cadence. In part for the sake of simplicity, from Bb3-D4-G4 I have the lowest voice descend Bb3-A3 at 4/3-14/11, with the other voices thus forming an accentuated or Marchetan cadential sonority at A3-Dv4-Gv4 or 14/11-18/11-24/11, a septimal major sixth sonority based on simple ratios of primes 2-3-7, 1:1-9:7-12:7 or 7:9:12 (0-435.084-933.129 cents). This 7:9:12 sonority is at once simpler or less complex than typical diatonic major sixth sonorities (e.g. 26:33:44 or 33:42:56 or the customary medieval Pythagorean 64:81:108, as at measure 5), and at the same time yet more dynamic and "streamlined" in its forward-going quality because of the very wide major third and sixth at 9:7 and 12:7. While the vertical 7:9:12 sonority is pure, its resolution involves a melodic compromise of sorts because of issues of pitch stability. From A3-Dv4-Gv4 at 14/11-18/11-24/11 we wish to make an intensive cadence on G3, ideally placed at 9/8, and leading to the conclusion of this section with the expected intensive cadence on F3-C4-F4 (1/1-3/2-2/1). If we were only considering the Marchetan cadence on G3 by itself, then this would be the classic and obvious solution. Gv4 G4 24/11 224/99 Dv4 D4 18/11 56/33 A3 G3 14/11 112/99 The just 7:9:12 sonority resolves to 2:3:4 with the lowest voice descending by a 9:8 tone, and each upper voice ascending by a 28:27 thirdtone or small semitone. The problem with this solution, in our present context, is that it would leave the voice at G3 tuned as 112/99 (213.598 cents), a full 896:891 or 9.688 cents above its desired location at 9/8 (203.910 cents) for the immediately following approach to the final cadence. A reasonable compromise is to place G3 at 44/39 or 208.835 cents, leaving it at a smaller 352:351 or 4.925 cents above its desired location for the next measures at 9/8. This descending step of 14/11-44/39 has a size of 273:242 (208.673 cents), larger than 9:8 by 364:363 (4.763 cents). As a result, to move from a just 7:9:12 sixth sonority to a just 2:3:4 trine, each upper voice must ascend by a step yet a bit smaller than 28:27 by this same difference of 364:363, or 121:117 (58.198 cents). Note the melodic steps of 9:8 and 28:27 in the classic version, or 44:39 and 121:117 in our compromise version, add up to a minor third at 7:6 (266.871 cents), the total amount by which a 9:7 major third must expand to arrive at a 3:2 fifth, or a 12:7 major sixth to arrive at a 2:1 octave. Gv4 G4 24/11 88/39 Dv4 D4 18/11 22/13 A3 G3 14/11 44/39 Curiously, this compromise closely parallels the melodic steps involved for this progression in the Peppermint temperament, where A3-G3 is a regular tone at 208.191 cents, and Dv4-D4 or Gv4-G4 a spacing interval of 58.680 cents. The main distinction, of course, is that the vertical sonorities of 7:9:12 and 2:3:4 are just in this JI version, but slightly tempered in Peppermint apart from the pure 12:7 major sixth. Thus A3-Dv4-Gv4 at (0-437.225-933.129 cents) has 9:7 at 2.141 cents wide, and the fourth between the upper voices (495.904 cents) narrow by this same amount; in G3-D4-G4, the fourth is again 2.141 cents narrow, and the fifth (704.096 cents) wide by this same amount. With this compromise, a very small shift in the pitches of all three voices, down by 352:351 to a repeated G3-D4-G4 at 9/8-27/16-9/4 at measure 19, situate us for the approach to the final cadence. This example, starting with the Marchetan nuance at measures 16-17, may provide perspective: 17 18 19 20 21 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 Gv4 G4 G4 F4 E4 D4 E4 F4 24/11 88/39 9/4 2/1 21/11 27/16 21/11 2/1 Dv4 D4 D4 E4 D4 C4 B4 C4 18/11 22/13 27/16 21/11 27/16 3/2 63/44 3/2 A3 G3 G3 G3 F3 14/11 44/39 9/8 9/8 1/1 ru- lae pa- cis The Marchetan intensive cadence on G3-D4-G4 brings us to 44/39-22/13-88/39, followed by a downward 352:351 shift in each voice bringing us to a repeated G3-D4-G4 at 9/8-27/16-9/4. In measure 19, ornaments bring about momentary sonorities of G3-E4-G4 (9/8-21/11-9/4) at beat 3, a lower 56:33 major sixth, upper 33:28 minor third, and outer octave; G3-D4-F4 at beat 4 at 9/8-27/16-2/1, a typical medieval European minor seventh sonority with an outer 16:9 minor seventh, lower 3:2 fifth, and upper 32:27 minor third (18:27:32), common in Perotin and Machaut, for example; and a major sixth sonority G3-D4-E4 at 9/8-27/16-21/11 with an outer 56:33 major sixth, a lower 3:2 fifth, and an upper 112:99 tone (0-701.955-915.553 cents), or in this tuning 66:99:112. This major sixth sonority with lower fifth and upper major second, as here G3-D4-E4, often is used in the 13th century in directed progressions to F3-C4-F4, with the outer sixth expanding to an octave and the upper major second to a 4:3 fourth; the progression also occurs in Machaut and some other 14th-century composers. Here G3-D4-E4 is quite transient and decorative, but may help point toward the eventual resolution to F3-C4-F4. At the beginning of measure 20, we encounter a mildly unstable sonority of the 13th century, G3-C4-D4 (9/8-3/2-27/16) at 6:8:9, with an outer 3:2 fifth, a lower 4:3 fourth, and an upper 9:8 tone. The vertical 9:8 tone adds tension and excitement, while the 3:2 fifth and 4:3 fourth are ideally and richly concordant, with the ratio of 6:8:9 reflecting the relatively simple and blending nature of this sonority. In this cadential idiom, the upper voices move from G3-C4-D4 to the directed major sixth sonority G3-B3-E4, here tuned 9/8-63/44-21/11 or 33:42:56 (1:1-14:11-56:33). We then have a final intensive cadence to F3-C4-F4, with the lowest voice descending by a 9:8 tone (9/8-1/1), and the upper voices ascending by 22:21 semitone steps (63/44-3/2, 21/11-2/1). Looking again at this last example, we can see a role that the step A3 plays in a polyphonic Lydian modality. In previous passages, it has served as the goal of the remissive cadence Bb3-D4-G4 to A3-E4-A4. Here, at measure 17, it has the different role of the step at a tone above G3, and thus the foundation for the unstable A3-Dv4-Gv4 (notation to signal a Marchetan realization of A3-C#4-F#4) resolving in an intensive cadence to G3-D4-G4. Then G3, as the step above the final F3, or superfinal, in turn provides the foundation for an intensive cadence to the final, G3-B3-E4 to F3-C4-F4. In this closing chain of cadences, one might say that A3 acts first as the usual goal of a remissive cadence Bb3-D4-G4 to A3-E4-A4, with the lowest voice descending by a semitone Bb3-A3 (measures 14-15); and then as "the superfinal of the superfinal" in the intensive cadence, here with a Marchetan nuance, of A3-Dv4-Gv4 to G3-D4-G4, with the lowest voice descending a tone to the resolution on the superfinal (A3-G3), followed by the expected intensive cadence G3-B3-E4 to F3-C4-F4, with the lowest voice descending again by a tone, now from superfinal to final (G3-F3). --------------------------------------------------------------- 5. Alternative possibilities: A caution on the Marchetan nuance --------------------------------------------------------------- While the intensive cadence on G fits the text _querula_ (which should be corrected to _querela_), an alternative version where this inflected cadence is replaced with a simple omnitonal cadence is in many contexts quite pleasant, and could be tuned as follows: 17 18 19 20 21 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 F4 G4 G4 F4 E4 D4 E4 F4 2/1 9/4 9/4 2/1 21/11 27/16 21/11 2/1 C4 D4 D4 E4 D4 C4 B4 C4 3/2 27/16 27/16 21/11 27/16 3/2 63/44 3/2 A3 G3 G3 G3 F3 14/11 9/8 9/8 9/8 1/1 With version, we have a minor sixth sonority A3-C4-F4 at 14/11-3/2-2/1, 1:1-33:28-11:7 or 28:33:44 (0-284.447-782.498 cents) resolving in an omnitonal manner (a whole-tone step in each voice) to G3-D4-G4 at 9/8-27/16-9/4, with the lowest voice descending by a larger 112:99 tone (14/11-9/8), and each upper voice ascending by a usual 9:8 tone (3/2-27/16, 2/1-9/4). After this stately omnitonal approach to the superfinal G3, the usual intensive cadence to F3 follows. All voices move in this version of the passage by tones of 9:8 and 112:99, with semitone steps at 22:21. Note that in this version, there is no motivation to have the first G3, at measure 18, higher than its Pythagorean position at 9/8. In our inflected intensive cadence with the Marchetan nuance, the motivation for placing it a bit higher (with 44/39 suggested) was to minimize the compromise in what were deemed to be the ideal melodic steps in resolving A3-Dv4-Gv4 at 7:9:12 to G3-D4-G4, 9:8 and 28:27. If we had the lowest voice in that progression move 14/11-9/8, this would have been the result: 17 18 | 1 2 3 4 5 6 | 1 33:32 53.273 Gv4 G4 24/11 9/4 33:32 53.273 Dv4 D4 18/11 27/16 99:112 213.598 A3 G3 14/11 9/8 In fact, although this realization of the Marchetan nuance has melodic steps deviating by 896:891 (9.688 cents) from the 9:8 tone and 28:27 thirdtone, with the tone larger (112:99, 213.598 cents) and the thirdtone (or quartertone) narrower at 33:32 (53.273 cents), it is an open question whether the 33:32 step might be closer to the very small "diesis" step of which Marcheto speaks, with Jay Rahn, as noted above, suggesting a monochord ratio of 37:36 (47.434 cents). Also, while 7:9:12 may be a very attractive just tuning for a wide major third and sixth, where either Marcheto or 14th-century singers were attracted to this ratio is a very open question. If Rahn's suggestion of a cadential diesis at 37:36 is correct, and we assume 9:8 tones and regular Pythagorean steps for the sake of simplicity, this suggests a yet more "radical" realization like the following, outside the scope of a temperament like Peppermint but certainly possible in JI or a variety of neomedieval temperament supporting this variety of intonation: 17 18 | 1 2 3 4 5 6 | 1 37:36 47.434 Gv4 G4 81/37 9/4 37:36 47.434 Dv4 D4 243/148 27/16 8:9 203.910 A3 G3 81/64 9/8 Here the accentuated major sixth sonority A3-Dv4-Gv4 is tuned at 81/64-243/148-81/37 or 1:1-48:37-64:37 (37:48:64) at 0-450.611-948.656 cents, with resolving motions of a descending 9:8 tone (81/64-9/8) and ascending dieses at 37:36 (243/148-27/16, 81/37-9/4). How Marcheto or other 14th-century flexible-pitch musicians (whose practice Marcheto at once describes and endorses) actually tuned the Marchetan nuance is thus a question where there are different possibilities but no clear answer. Christopher Page suggests that singers will ideally tune each such progression as seems right at the time, preferring not to be constrained by fixed-pitch instruments. What I would very much like to avoid is any implication that because I often prefer an approach approximating whole-tone motions at around 9:8 and narrow semitones or thirdtones at around 28:27 (a taste George Secor shares with me), that this is necessarily a privileged solution, much less a definitive solution supported by any medieval documentation from Marcheto or any writer on European polyphony. ------------- 6. Conclusion ------------- The above version of _O Europae_ in JI, like the version in the near-just Peppermint temperament from 2004, is merely one neomedieval realization, and one illustrating a set of melodic and vertical preferences. Our main focus has been on usual diatonic, or better _musica recta_, steps and progressions, with the term _musica recta_ nicely emphasizing the fluidity of the step B/Bb and the inclusion of both B-natural and Bb in the regular medieval gamut. The Marchetan nuance adds some drama and color, but mostly we have been dealing with melodic motions, vertical sonorities, and directed progressions involving the usual modal steps (including fluid B/Bb). Melodically, the Lydian mode may be built from a lower pentachord F-C (often F-G-A-B-C ascending and C-Bb-A-G-F descending) and an upper tetrachord C-F (C-D-E-F). While melody often follows these genera, more or less, directed vertical progressions tend to involve stepwise motion in all voices. In a 14th-century idiom, intensive resolutions by stepwise contrary motion involving ascending semitonal and descending whole-tone motion tend to be the most compelling and conclusive (e.g. G3-B3-E4 to F3-C4-F4), with remissive cadences featuring descending semitonal and ascending whole-tone motions (e.g. Bb3-D4-G4 to A3-E4-A4) typically less conclusive, and often favored as internal cadences. Omnitonal cadences with all voices moving by whole-tone steps (e.g. A3-C4-F4 to G3-D4-G4) are popular in the 13th century, and have for me a stately quality. Various 14th-century sources express a preference for "closest approach" progressions, either intensive or remissive, involving semitonal motion in one voice and whole-tone motion in another (e.g. min3-1, Maj3-5, Maj6-8; and also Maj2-4 and min7-5). However, despite some categorical statements that closest approach resolutions are to be preferred, omnitonal resolutions were likely now and then used in practice, with a piece by the great French composer known as Solage concluding with the cadence D3-F3-Bb3 to C3-G3-C4, with min3-5 and min6-8 (a common 13th-century formula). While these progressions nicely fit a medieval Pythagorean intonation, what prevails in a tempered system like Peppermint, or the JI realization given here, is a moderately "accentuated" variation on Pythagorean tuning, with major intervals tending to be a bit wider, and minor intervals a bit smaller. Thus the standard Pythagorean 9:8 tone or major second sometimes expands slightly to 44:39 or 112:99, while the compact 256:243 diatonic semitone or limma generally becomes a bit narrower and yet more incisive at 104:99 or 22:21. These "accentuated" tunings notably apply to imperfect concords: thus the Pythagorean ditone or major third at 81:64 usually expands a bit to 33:26 or 14:11, while the minor third at 32:27 contracts to 13:11 or 33:28. Major sixths likewise expand from 27:16 to 22:13 or 56:33, while minor sixths contract from 128:81 to 52:33 or 11:7. Applying a 14th-century interval ethos of the kind expressed by Marcheto, tuning a major third or sixth a bit wider allows it "more closely to approach" in size the stable fifth or octave, respectively, to which it "strives" to expand by directed motions of a tone in one voice and an incisive semitone in another. Likewise making a minor third a bit narrower brings about its "closer approach" to the unison to which it seeks to contract by stepwise contrary motion. - There are additional motivation for step and interval sizes such as usual thirds at 13:11 and 14:11, and semitone steps such as 22:21, which apply especially where Near Eastern styles are in use: thus the intent of a temperament like Peppermint is what has been termed a "bimusical" system which can support either medieval European or Near Eastern music, or some fusion between elements of both. Thus a 13:11 minor third may be divided into middle second steps of 13:12:11 (1/1-13/12-13/11, 0-138.573-289.210 cents) or 11:12:13 (1/1-12/11-13/11, 0-150.637-289.210 cents). A system of this kind can also nicely realize Qutb al-Din al-Shirazi's tuning of a Hijaz tetrachord around 1300 at 1/1-12/11-14/11-4/3 (0-150.673-417.508-498.045 cents), a permutation of Ptolemy's Intense Chromatic with the steps here ordered 12:11-7:6-22:21 (150.637-266.871-80.537 cents). However, _O Europae_ focuses on a medieval European context, and in the mostly diatonic opening and closing sections mostly on fairly routine idioms, apart from the optional use of the Marchetan nuance for the intensive cadence on G at measures 17-18, a nuance which alters the usual intonational fabric in a way which further underscores the "closest approach" ethos (with yet wider major thirds and sixths at 9:7 and 12:7, and narrower semitones or thirdtones at 28:27 or even a bit narrower). Here, in one common pattern for the Lydian mode as expressed in 13th-14th century polyphony, we have intensive final cadences like G3-B4-E4 to F3-C4-F4, and internal remissive cadences like Bb3-D4-G4 to A3-E4-A4. This is only one pattern, but a popular one, which can be woven and adorned in many ways. As we approach the end of the first section, the Latin word _querela_ (written in the text of the PDF version as _querula_) prompts an inflected cadence in the intensive manner to G3, realized with a Marchetan nuance as A3-Dv4-Gv4 to G3-D4-G4. Then follows the final cadence G3-B3-E4 to F3-C4-F4, another intensive progression with the same basic two-voice resolutions (Maj6-8, Maj3-5), but a usual diatonic resolution. This chain of two intensive resolutions, as the lowest voice moves A3-G3 and then G3-F3, creates an interesting role for the step A3 that might be called "superfinal of the superfinal." While such a terminology of roles for modal steps in medieval or neomedieval polyphony might or might not be the most useful approach -- often I might simply speak of "a chain or vertical sequence of two intensive cadences at A3-G3 and G3-F3") -- what I wish to convey is that medieval progressions, and schemes of polyphony organization, have their own logic which is coherent and compelling. In realizing the kind of "moderately accentuated" variations on Pythagorean intonation followed in the _musica recta_ portions of this piece, I find that either a temperament like Peppermint or fluid JI provides an attractive solution. The process of achieving pure vertical 3:2 fifths and 4:3 fourths (the medieval Pythagorean standard), while maintaining pitch stability and including some subtly varying shadings of intervals such as thirds and sixths, is an intriguing discipline and art of which I have attempted to explore a few essentials in this article. Margo Schulter 11 June 2017