_O Europae_ Concluding Portion (mm. 33-41) in JI Including a fluid JI version of the entire piece With much catalytic encouragement and inspiration from Mykhaylo Khramov and his MIDI tuning in JI for my composition _O Europae_, originally composed in May 2004 in the Peppermint temperament, I have in previous articles presented a JI tuning first for the middle portion of the piece (mm. 22-32) with its highly chromatic style, and then for the opening (mm. 1-21) mostly in an everyday _musica recta_ style. In this paper I will first present a JI realization for the concluding portion of the piece (mm. 33-41), and then combine this solution with the realizations in the previous articles so as to give a JI version of the entire piece. Between the first paper on the chromatic middle portion of the piece, and the second on the opening mostly _musica recta_ portion, I slightly shifted my methodology. In the first paper, where the just ratios intended for sonorities were generally quite clear to me, I measured the sizes of melodic steps and vertical intervals, and the positioning of notes, not only in terms of integer ratios and cents, but with reference to their positions in the near-just Peppermint system. In the second paper, however, while continuing to describe note locations in terms of ratios and cents, I shifted my focus for an additional measurement from comparisons with Peppermint to a specification of note frequencies based on Mykhaylo's standard for his JI MIDI realization of F3=176 Hz. My purpose is not to propose this or any other standard of absolute pitch as best, since my medieval orientation leads me to consider various local standards or customs as equally valid, but simply to illustrate how Mykhaylo's F3=176 is often very convenient for the kind of fluid JI here addressed, where many common note positions conveniently have frequencies in Hertz expressed as integers. Also, while deviations from Peppermint in a fluid JI setting of _O Europae_ may say some very interesting things about Peppermint and its compromises -- compromises which, as Zarlino (1558) notes in a very different musical context, may be highly practical for fixed-pitch instruments -- I have quickly confirmed the obvious observation of Mykhaylo and others than it is deviations from Pythagorean intonation that may be most significant. Let us now proceed to a fluid JI realization of the conclusion of _O Europae_, and then a complete JI realization of the entire piece. ------------------------------------------------ 1. Realizing the conclusion of _O Europae_ in JI ------------------------------------------------ Like the opening portion of _O Europae_, the conclusion is mostly in a usual _musica recta_ style based on the diatonic notes and Bb, with the step B/Bb fluid and both forms ("hard B" or B-natural and "soft B" or Bb) often appearing in a Lydian modality, as in the F Lydian setting of this piece. As it happens, in the brief conclusion, only B-natural is used. This may be true, in part, because Bb often occurs as a step in the lowest voice descending to A, for example in the remissive cadence Bb3-D4-G4 to A3-E4-A4 (e.g. mm. 5-6, 14-15), or in the descending pentachord C4-Bb3-A3-G3-F3 (lowest voice, mm. 7-9, 16-21). Other typical applications are to obtain a direct leap of a perfect fourth, F3-Bb3 (lowest voice, m. 3); or a frequent although not invariable preference for Bb as the highest note of a phrase approached and left by step (e.g. A4-Bb4-A4 in the highest voice, mm. 4-5). In the concluding portion of this piece, however, the lowest voice descends over the limited range of A3-G3-F3. Note that the immediately proceeding middle chromatic section of _O Europae_ has ended by returning to a usual _musica recta_ context with the remissive cadence on A, Bb4-D4-G4 to A3-E4-A4, which leads to A3-E4-A4 repeated as the first sonority of the final section. Thus just as Bb is often associated with a remissive focus on A, in this section B-natural is associated with a focus on the cadential major third G3-B3 and its ultimate expansion to the stable fifth on the final of F3-C4, as one aspect of the intensive final cadence G3-B4-E4 to F3-C4-F4. Intonationally, a complicating factor, as at the end of the opening section (mm. 16-21), is an intervening intensive cadence on G, which receives a Marchetan nuance. Here, it is conveniently possible to present this cadence in its optimal form and then gently shift pitch levels down to the desired locations for the final cadence. 33 34 35 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 8:9 27:28 7:8 8:7 28:27 3267;2912 -203.910 -62.961 -231.174 231.174 62.961 199.147 A4 G4 Gv4 E4 Gv4 G4 28/11 224/99 24/11 21/11 24/11 224/99 1617.508 1413.598 1350.637 1119.463 1350.637 1413.598 448 Hz 398.222 Hz 384 Hz 336 Hz 384 Hz 398.222 Hz 8:9 27:28 7:8 8:7 28:27 363:364 -203.910 -203.910 -231.174 231.174 62.961 -4.763 E4 D4 Dv4 B3 Dv4 D4 21/11 56/33 18/11 63/44 18/11 56/33 1119.463 915.553 852.592 621.418 852.592 915.553 336 Hz 298.667 Hz 288 Hz 252 Hz 288 Hz 298.667 Hz 8:9 363:364 -203.910 -4.763 A3 A3 A3 A3 G3 14/11 14/11 14/11 14/11 112/99 417.508 417.508 417.508 417.508 213.598 224 Hz 224 Hz 224 Hz 224 Hz 199.111 Hz Ut prae- va- le- at 4 16 12 12 12 4 3 12 9 9 9 3 2 9 7 8 7 2 36 37 38 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 8:9 351:352 -203.910 -4.925 A4 G3 G3 33/13 88/39 9/4 1612.745 1408.835 1403.910 397.128 Hz 397.128 Hz 396 Hz 273:242 208.673 D4 D4 E4 22/13 22/13 21/11 910.790 910.790 1119.463 297.846 Hz 297.846 Hz 336 Hz 351:352 -4.925 G3 G3 G3 44/39 44/39 9/8 208.835 208.835 203.910 198.564 Hz 198.564 Hz 198 Hz con - cor - 9 4 66 6 3 56 4 2 33 39 40 41 | 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 99:112 8:9 21:22 22:21 -213.598 -203.910 -80.537 80.537 E4 D4 C4 B3 C4 21/11 27/16 3/2 63/44 3/2 1119.463 905.865 701.955 621.418 701.955 336 Hz 297 Hz 264 Hz 252 Hz 264 Hz 8:9 -203.910 G3 G3 G3 F3 9/8 9/8 9/8 1/1 203.910 203.910 203.910 0 198 Hz 198 Hz 198 Hz 176 Hz - - di- a. 66 32 112 9 56 4 56 27 99 8 42 3 33 18 66 6 33 2 At measures 33-35, the stable trine A3-E4-A4 leads to an intensive cadence on G with the Marchetan nuance. The upper voices move in parallel fourths through the sonority A3-D4-G4 or 14/11-56/33-21/11 (9:12:16) with a 16:9 minor seventh divided into two concordant 4:3 fourths, to the accentuated major sixth sonority A3-Dv4-Gv4 or 14/11-18/11-24/11 at 7:9:12, with a pure 9:7 major third and 12:7 major sixth (0-435.084-933.129 cents). This momentarily decorated by descents in the upper voices by large 8:7 tones (231.174 cents) to another mildly unstable sonority of A3-B3-E4 or 14/11-63/44-21/11 at 8:9:12, with a pure fifth divided into a just lower 9:8 tone and upper 4:3 fourth, then returning to the cadential major sixth sonority A3-Dv4-Gv4 at 7:9:12 In a classic style of resolution, the lowest voice descends by a 9:8 tone (A3-G3, 14/11-112/99), while each upper voice ascends by the 28:27 thirdtone of Archytas (Dv4-D4, 18/11-56/33; Gv4-G4, 24/11-112/99), thus arriving at the 2:3:4 trine G3-D4-G4. Having brought about this cadence with pure vertical sonorities and the chosen ideal here for melodic steps in realizing the Marchetan nuance (9:8, 28:27), we now face a small complication. In completing this cadence (measures 34-35), the lowest has moved down by precisely a 9:8 tone from A3 at 14/11 (417.508 cents) to G3 at 112/99 (213.598 cents). For the final intensive cadence to our modal final or 1/1 at F3, however, we would like it very slightly lower, at G3 tuned as 9/8 (203.910 cents), so it can descend by another cadential 9:8 step as the upper voices ascend by steps of 22:21, thus G3-B3-E4 (9/8-63/44-21/11) to F3-C4-F4 (1/1-3/2-2/1). This means somehow shifting the pitch of the lowest voice, which remains on G3, in effect as a kind of pedal point or drone, until the final resolution to F3 (measures 40-41), gently downward from 112/99 to 9/8, a difference of 896:891 or 9.688 cents. The situation at measures 35-38 permits the accomplishment of this small "midcourse correction" unobtrusively, with two vertical events providing occasions for downward shifts in pitch of not quite 5 cents each. The first small shift occurs at measures 35-36, as the resolving trine G3-D4-G4 has the highest voice move to form a mildly unstable and relatively blending sonority much recommended by Jacobus in the _Speculum musicae_ and related writings: G3-D4-A4 at 4:6:9, an outer major ninth at 9:4 (1403.910 cents) divided into two pure 3:2 fifths G3-D4 and D4-A4. As Jacobus notes, and some modern musicians agree, the 9:4 major ninth is actually a bit milder than 9:8, and the rather simple ratio of 4:6:9 for the three-voice sonority as a whole suggests its considerable degree of blend or concord. Although Jacobus finds the 9:4 ninth definitely unstable, he adds that its outer voices "seems to concord better" when combined with the richly euphonious fifths in this sonority. We to take this opportunity to shift subtly from G3-D4-G4 at 112/99-56/33-224/99 (112/99-56/33-224/99, 213.593-915.553-1413.593 cents) to the mildly unstable G3-D4-A4 at 44/39-22/13-33/13 (208.835-910.790-1612.745 cents), a shift downward in each of the lowest two voices of 364:363 (4.763 cents), not far from the Just Noticeable Difference (JND) of around 3 cents for melodic distinctions. At the same time, the upper voice must move from G4 at 224/99 (213.593 cents) as the 2:1 octave of G3-D4-G4 (measure 35) to A4 at 33/13 (412.745 cents) as the 9:4 major ninth of G3-D4-A4 with the minute shift down of the two lower voices (measure 36). This altered melodic step of G4-A4, or 224/99-33/13, is smaller than a usual 9:8 tone by the amount that the lower voices shift down, or 364:363, and thus 3267:2912 or 199.147 cents. In the variety of neomedieval JI we are considering, this is a special situation where tones slightly smaller than the usual 9:8 may occur, in contrast to typical use of tones at 9:8 and often larger (e.g. 273:242 at 208.673 cents; 44:39 at 208.835 cents; 112:99 at 213.598 cents; and 8:7 at 231.174 cents). The relatively blending G3-D4-A4 sonority at 4:6:9 has a common oblique resolution of the highest voice to the octave of a stable 2:3:4 trine, G3-D4-G4. Then motion of the middle voice to another unstable stable, G3-E3-G4 (measure 38), with a lower major sixth and upper minor third, provides an opportunity for another subtle downward shift of the outer voices from 44/39 and 88/39 to our desired 9/8 and 9/4 (203.910-1403.910 cents), an adjustment of 352:351 or 4.925 cents. The middle voice here moves D4-E4 at 22/13-21/11, a tone at 273:242 (208.673 cents). Vertically, we have a colorful major sixth sonority at 1:1-56:33-2:1 (0-915.553-1200 cents), which establishes our desired pitch level for the approach to the final cadence, and also in a sense anticipates this cadence, which will have outer voices at this major sixth G3-E3 (9/8-21/11) resolving to the final and its octave, F3-F4 (1/1-2/1). At measure 39, this sonority is a starting point that leads to brief ornamental sonorities of G3-D4-F4 (9/8-27/16-2/1), an outer 16:9 minor seventh divided into lower 3:2 fifth and upper 32:27 minor third, or 18:27:32; and G3-D4-E4 (9/8-27/16-21/11), wither outer 56:33 major sixth, lower 3:2 fifth, and upper 112:99 major second (66:99:112). As noted in the discussion of the opening section of this piece, G3-D4-E4 lends itself to a direct resolution to F3-C4-F4 with the outer major sixth expanding to an octave, and the upper major second to a fourth; but here it a transient event, albeit one nicely pointing toward the resolution soon to occur of G3-B3-E4 to F3-C4-F4. At measures 40-41, we have the same 13th-14th century formula ending the opening section. A mildly unstable G3-C4-D4 sonority at 9/8-3/2-27/16, with outer 3:2 fifth, lower 4:3 fourth, and upper 9:8 major second or tone (6:8:9) leads to the penultimate major sixth sonority of G3-B3-E4 at 9/8-63/44-21/11, with its 14:11 major third and 56:33 major sixth (0-417.508-915.553 cents). The lowest voice then descends by a 9:8 tone (G3-F3, 9/8-1/1) as the upper voices ascend by 22:21 semitones (B3-C4, 63/44-3/2; E3-F4, 21/11-2/1), arriving at the closing trine on the final, F3-C4-F4 (1/1-3/2-2/1). When I composed _O Europae_ and its approach to the final cadence, I may have been influenced by the text _ut praevaleat concordia_, "so that concord may prevail," in this approach with a repeated G3 in the lowest voice and a variety of relatively concordant or "compatible" sonorities above it (e.g. 4:6:9, 6:8:9) leading up to the concluding directed resolution of G3-B3-E4 to F3-C4-F4. In translating musically from Peppermint to fluid JI, these sonorities also provide an opportunity to negotiate two subtle downward shifts in pitch level, with the lowest voice at G3 moving from 224/99 to 44/39 (by 364:363 or 4.763 cents), and then from 44/39 to 9/8 (by 352:351 or 4.925 cents). These shifts, small in themselves, are made yet more discreet by changes of vertical color to divert the ear's attention. --------------------------------------------------------------------- 2. A quick note on a fixed JI tuning for _musica recta_ in this style --------------------------------------------------------------------- While a fluid or flexible JI style of intonation is required to obtain the pure vertical concords (especially 4:3 fourths and 3:2 fifths) and superparticular melodic steps often featured in this version of _O Europae_, it is interesting to ask how a tuning of the usual _musica recta_ notes in this style might be arranged on a keyboard instrument, for example. Here we are interested in the eight-note gamut of Bb-B. Based on some commonly occurring locations in the setting above, the solution that follows might be a good starting point. Note that, curiously, if we take Mykhaylo's suggested absolute pitch of F3=176 Hz, then the frequencies in Hertz of all notes except for Bb as a 4/3 fourth come out neatly as integers. note ratio cents frequency ---------------------------------- F4 2/1 1200.000 352.000 Hz E4 21/11 1119.463 336.000 Hz D4 27/16 905.865 297.000 Hz C4 3/2 701.955 264.000 Hz B3 63/44 621.418 252.000 Hz Bb3 4/3 498.045 234.667 Hz A3 14/11 417.508 224.000 Hz G3 9/8 203.910 198.000 Hz F3 1/1 0 176.000 Hz ---------------------------------- This tuning has tetrachords with a 4:3 fourth divided into steps of 9:8 (usual tone, 203.910 cents); 112:99 (larger tone, 213.598 cents); and 22:21 (diatonic semitone or limma, 80.537 cents). Thus we have, for example, using the symbols Tr for the regular 9:8 tone, Tl for the larger 112:99 tone, and S for the 22:21 semitone: Tr Tl S Tl S Tr S Tr Tl C4 D4 E4 F4 || G3 A3 Bb3 C4 || B3 C4 D4 E4 3/2 27/16 21/11 2/1 9/8 14/11 4/3 3/2 63/44 3/2 27/16 21/11 1:1 9:8 14:11 4:3 1:1 112:99 32:27 4:3 1:1 22:21 33:28 4:3 0 203.9 417.5 498.0 0 213.6 294.1 498.0 0 80.5 284.5 490.0 9:8 112:99 22:21 112:99 22:21 9:8 22:21 9:8 112:99 203.9 213.6 80.5 213.6 80.5 203.9 80.5 203.9 213.6 This tuning also ideally realizes the final cadence in _O Europae_, the intensive cadence on the final F, with an unstable major sixth sonority G3-B4-E4 with a 14:11 third and 56:33 sixth (33:42:56), and resolving steps of 9:8 and 22:21. E4 --- 22:21 -- F4 21/11 80.5 2/1 1119.5 1200 B3 --- 22:21 -- C4 63/11 80.5 3/2 621.4 702.0 G3 --- 8:9 --- F4 9/8 -203.9 1/1 203.9 0 The remissive cadence on A is quite serviceable although not quite so ideal: G4 -- 112:99 -- A4 9/4 213.6 28/11 1403.9 1617.5 D4 -- 112:99 -- E4 27/16 213.6 21/11 905.9 1119.5 Bb3 -- 21:22 -- A3 4/3 80.5 14/11 498.0 417.508 Here Bb3-D4-G4 forms a major sixth sonority with a Pythagorean third at 81:64 and sixth at 27:16 (64:81:108) rather than the somewhat larger 14:11 and 56:33; the semitone step, now descending (Bb3-A) remains at 22:21, while the upper voices ascend by larger tones at 112:99. These usual medieval thirds are quite apt, but make for a resolution not quite as efficient as with the 33:42:56 sonority in the previous example. While such nuances of intonation or touches of "modal color" might add variety, the obvious flaw in this scheme as a simple fixed-pitch tuning is the impure fourth at A3-D4 or 14/11-27/16, at 297:224 or 488.357 cents, narrow of 4:3 by 896:891 or 9.688 cents. For many purposes, this fourth and the vitally important fifth D-A here equally impure at 448:297 or 711.643 cents could be considered "semi-wolf" intervals, unlikely to be heard in harmonic timbres as musically interchangeable with a stable 4:3 fourth or 3:2 fifth. Since the fifth D-A is so important, for example in the trine D3-A3-D4 on the final of D Dorian, some solution is needed. Here the disparity between the pure 3:2 or 4:3 and the wide fifth D3-A4 or narrow fourth A3-D4 is 896:891, also the difference between the two sizes of tones at 9:8 and the larger 112:99. This is analogous to the problem that occurs in Zarlino's simple fixed-pitch version of Ptolemy's Syntonic Diatonic, with regular tones at 9:8, smaller tones at 10:9 (182.404 cents), and diatonic semitones at 16:15 (111.731 cents). note ratio cents ---------------------- C4 2/1 1200.000 B3 15/8 1088.269 Bb3 16/9 996.010 A3 5/3 884.359 G3 3/2 701.955 F3 4/3 498.045 E3 5/4 386.314 D3 9/8 203.910 C3 1/1 0 ---------------------- Here an analogous anomaly likewise occurs at D3-A3 tuned as 9/8-5/3, with a narrow fifth at 40:27 (680.449 cents), and wide fourth A3-D4 at 27:20 (519.551 cents), impure by the syntonic or Didymic comma of 81:80 (21.506 cents) also defining the difference between the regular 9:8 and smaller 10:9 tones. For fixed-pitch JI keyboards, the solution is to have at least two versions of certain notes at an 81:80 comma apart: e.g. separate keys for D3 at 9/8 (a pure 4:3 fourth below G3 at 3/2) or 10/9 (a pure 3:2 fifth below A3 at 5/3). While it is possible to use such a keyboard, and Zarlino describes an arrangement with 16 notes per octave, he adds that generally temperament seems a far more practical solution. With our neomedieval JI system, a tenable although not perfect solution is available using only one note on the keyboard for each _musica recta_ step: note ratio cents frequency ---------------------------------- F4 2/1 1200.000 352.000 Hz E4 21/11 1119.463 336.000 Hz D4 22/13 910.790 297.846 Hz C4 3/2 701.955 264.000 Hz B3 63/44 621.418 252.000 Hz Bb3 4/3 498.045 234.667 Hz A3 14/11 417.508 224.000 Hz G3 9/8 203.910 198.000 Hz F3 1/1 0 176.000 Hz ---------------------------------- We simply move the note D4 from 27/16 to 22/13 (910.790 cents), a position higher by 352:351 (4.925 cents). Rather than a single impure fifth D3-A3 at 896:891 or 9.688 cents wide, we now have two "virtually tempered" fifths: G3-D4 at 9/8-22/13 with a size of 176:117 or 706.880 cents, wide by 352:351; and D4-A4 at 22/13-28/11 with a size of 182:121 or 706.718 cents), wide by the almost identical amount of 364:363 or 4.763 cents. These fifths G-D and A-E are comparable in impurity at not quite 5 cents wide to fifths tempered at 1/4 syntonic comma (5.377 cents) narrow, either in a regular 1/4-comma meantone with pure 5:4 major thirds, or in various irregular temperaments of the late 17th and 18th centuries in Europe. It must be said that this degree of impurity may be more notable where fifths and fourths are the primary stable concords, unlike a meantone style where stable thirds and sixths may in part distract from and effectively mitigate the tempering of fifths and fourths. However, this virtually equal _participatio_ or sharing of the 896:891 comma between two fifths tempered at 352:351 and 364:363 does permit the use of a fixed JI tuning with all _musica recta_ fifths acceptable at least for more relaxed neomedieval styles, where the medieval Pythagorean ideal of pure fifths and fourths is open to some substantial compromise. Sharing the comma between two fifths in this way also sets the minor third A3-C4 at 22/13-2/1 to a just 13:11 (289.210 cents), introducing another relatively simple ratio into the system. In Pythagorean tuning, or a regular temperament, we would expect the _musica recta_ system Bb-B to present itself as a chain of seven identically tuned fifths, either at a pure 3:2 in Pythagorean, or at some smaller or larger size in a regular temperament (e.g. 704.096 cents in Peppermint, 2.141 cents wide). Here, we have a chain including five pure 3:2 fifths, plus two impure ones "tempered by ratio" at almost identical sizes of 176:117 and the infinitesimally smaller 182:121: 3:2 3:2 3:2 176:117 182:121 3:2 3:2 701.955 701,955 701.955 706.880 706.718 701.955 701.955 pure pure pure +4.925 +4.763 pure pure Bb ------ F ------ C ------ G ------ D ------ A ------ E ------ B 4/3 1/1 3/2 9/8 22/13 14/11 21/11 63/44 498.045 0 701.955 203.910 910.790 417.508 1119.463 621.418 We may quickly note that in a regular temperament such as Peppermint, this scheme is simplified by distributing the 896:891 comma and its near-equal parts of 352:351 and 364:363 among all the fifths or fourths, so that each is quite subtly impure in the wide direction, here by 2.141 cents: 704.096 704.096 704.096 704.096 704.096 704.096 704.096 +2.141 +2.141 +2.141 +2.141 +2.141 +2.141 +2.141 Bb ------ F ------ C ------ G ------ D ------ A ------ E ------ B 4/3 1/1 3/2 9/8 22/13 14/11 21/11 63/44 495.904 0 704.096 208.191 912.287 416.382 1120.478 624.574 498.045 0 701.955 203.910 910.790 417.508 1119.463 621.418 -2.141 0 +2.141 +4.281 +1.497 -1.126 +1.015 +3.156 In a regular temperament like this we have a simplification and standardization of diatonic step sizes: all _musica recta_ tones are at 208.191 cents (4.281 cents wide of 9:8), and all diatonic semitones at 79.522 cents, just over a cent narrow of 22:21 (80.537 cents). As a modern fixed-pitch tuning, this kind of temperament may in a sense be less far from an ideal medieval Pythagorean ethos than the "virtually tempered" JI system above, since the compromise in the fifths is subtle and uniform, and may call less attention to itself than a contrast between pure 3:2 fifths and other usual fifths at not quite 5 cents wide. However, it is a _musica recta_ system without any pure fifths or fourths, which is itself a significant compromise. Fluid JI, where it is practical, offers the pure vertical fifths and fourths of Pythagorean intonation as well as a variety of just steps and intervals. -------------------------------- 3. A setting of the entire piece -------------------------------- We can now consolidate the portions of _O Europae_ presented in each of these three papers in order to provide a complete version of the piece in fluid JI. a musical "translation" of the Peppermint version into just steps and intervals, both simple and complex. Here the chromatic middle portion (measures 22-31) is given, like the opening and conclusion, with frequencies for note locations based on F3=176 Hz, as suggested by Mykhaylo Khramov's MIDI version in JI. Again, this is not a prescription on how performers should use their discretion to set absolute pitch, only an interesting possibillity. The option of F3=176 Hz places A3, in its often occurring location at 14/11 above F3, at 224 Hz, thus implying A4=448 -- or for its alternative position of 33/13 (lower by 364:363), A4=446.769 cents -- which might be compared with a measured average pitch for some 16th-century European wind instruments at A=466 or so, but with wide variations in both directions. 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 22 23 24 25 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 27:28 28:27 22:21 21:22 -62.961 62.961 80.537 -00.537 E4 E4 Ev4 E4 F4 F4 F4 21/11 21/11 81/44 21/11 2/1 2/1 2/1 1119.463 1119.463 1056.502 1119.463 1200 same same 336 Hz 336 Hz 324 Hz 336 Hz 352 Hz same same 9:8 8:9 9:7 203.910 -203.910 435.084 A3 A3 B3 A3 Dv4 Dv4 Dv4 14/11 14/11 63/44 14/11 18/11 18/11 18/11 417.508 417.500 621.418 417.508 852.592 same same 224 Hz 224 Hz 252 Hz 224 Hz 288 Hz same same 27:28 28:27 -62.961 62.961 A3 A3 Av3 A3 A3 A3 A3 14/11 14/11 27/22 14/11 14/11 14/11 14/11 417.508 417.500 354.547 417.508 417.508 same same 224 Hz 224 Hz 216 Hz 224 Hz 224 Hz same same Ex du- me- to Il- la- que- 9 11 11 11 3 3 7 3 9 9 9 2 2 6 2 7 7 7 26 27 28 29 30 | 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 2 3 4 5 6| 22:21 27:28 28:27 80.537 -62.961 62.961 E4 F4 Fv4 Fv4 Fv4 F4 21/11 2/1 27/14 27/14 27/14 2/1 1119.463 1200 1137.039 1137.039 1137.039 1200 336 Hz 0 339.429 Hz 339.429 Hz 339.429 Hz 352 Hz 33:32 8:9 27:28 28:27 8:9 14:11 53.273 -203.910 -62.961 62.961 -203.910 417.508 Dv4 Dv4 D4 C4 Cv4 C4 Bb3 18/11 18/11 27/16 3/2 81/56 3/2 4/3 852.592 852.59 905.865 701.955 638.994 701.955 498.045 288 Hz same 297 Hz 264 Hz 254.571 Hz 264 Hz 234.667 Hz 99:98 28:27 17.576 62.961 A3 A3 Bbv3 Bbv3 Bbv3 Bb3 14/11 14/11 9/7 9/7 9/7 4/3 417.508 417.508 435.084 435.084 435.084 498.045 288 Hz same 226.286 Hz 226.286 Hz 226.286 Hz 234.667 Hz an- te bel - - - - li 21 11 24 9 12 9 18 9 21 7 9 7 3 14 7 16 6 8 6 2 31 32 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 112:99 9:8 213.598 203.910 F4 F4 G4 A4 2/1 2/1 224/99 28/11 1200 1200 1413.598 1617.508 352 Hz 352 Hz 398.222 Hz 448 Hz 9:8 203.910 D4 D4 D4 E4 56/33 56/33 56/33 21/11 915.553 same same 1119.463 298.667 Hz same same 336 Hz 21:22 80.537 Bb3 Bb3 Bb3 A3 4/3 4/3 4/3 14/11 498.045 same same 417.508 234.667 Hz same same 224 Hz du- ci- te nos 33 33 56 4 28 28 42 3 22 22 33 2 33 34 35 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 8:9 27:28 7:8 8:7 28:27 3267;2912 -203.910 -62.961 -231.174 231.174 62.961 199.147 A4 G4 Gv4 E4 Gv4 G4 28/11 224/99 24/11 21/11 24/11 224/99 1617.508 1413.598 1350.637 1119.463 1350.637 1413.598 448 Hz 398.222 Hz 384 Hz 336 Hz 384 Hz 398.222 Hz 8:9 27:28 7:8 8:7 28:27 363:364 -203.910 -203.910 -231.174 231.174 62.961 -4.763 E4 D4 Dv4 B3 Dv4 D4 21/11 56/33 18/11 63/44 18/11 56/33 1119.463 915.553 852.592 621.418 852.592 915.553 336 Hz 298.667 Hz 288 Hz 252 Hz 288 Hz 298.667 Hz 8:9 363:364 -203.910 -4.763 A3 A3 A3 A3 G3 14/11 14/11 14/11 14/11 112/99 417.508 417.508 417.508 417.508 213.598 224 Hz 224 Hz 224 Hz 224 Hz 199.111 Hz Ut prae- va- le- at 4 16 12 12 12 4 3 12 9 9 9 3 2 9 7 8 7 2 36 37 38 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 1 2 3 4 5 6 | 8:9 351:352 -203.910 -4.925 A4 G3 G3 33/13 88/39 9/4 1612.745 1408.835 1403.910 397.128 Hz 397.128 Hz 396 Hz 273:242 208.673 D4 D4 E4 22/13 22/13 21/11 910.790 910.790 1119.463 297.846 Hz 297.846 Hz 336 Hz 351:352 -4.925 G3 G3 G3 44/39 44/39 9/8 208.835 208.835 203.910 198.564 Hz 198.564 Hz 198 Hz con - cor - 9 4 66 6 3 56 4 2 33 39 40 41 | 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 99:112 8:9 21:22 22:21 -213.598 -203.910 -80.537 80.537 E4 D4 C4 B3 C4 21/11 27/16 3/2 63/44 3/2 1119.463 905.865 701.955 621.418 701.955 336 Hz 297 Hz 264 Hz 252 Hz 264 Hz 8:9 -203.910 G3 G3 G3 F3 9/8 9/8 9/8 1/1 203.910 203.910 203.910 0 198 Hz 198 Hz 198 Hz 176 Hz - - di- a. 66 32 112 9 56 4 56 27 99 8 42 3 33 18 66 6 33 2 Margo Schulter 14 June 2017