-------------------------------------------------------- An archicembalo in 24-note meantone: Renaissance fifthtone music on two 12-note keyboards -------------------------------------------------------- While many people have discussed the justly famed _archicembalo_ or "superharpsichord" of Nicola Vicentino (1511-1576) with five ranks of keys dividing the octave into 31 more or less equal intervals of approximately 1/5-tone, plus a sixth rank for the just intonation of certain fifths[1], actual 16th-century music using direct intervals of 1/5-tone may be less familiar. In this article, I would like to describe a "reduced" 24-note archicembalo which may fit the musical structure of known compositions by Vicentino and his younger contemporary Anthoine de Bertrand (c. 1530-1540 - c. 1580-1582) using these "fifthtone" intervals, as I shall call them in a friendly fashion.[2] In 16th-century terminology, such intervals and compositions are often known as "enharmonic," being seen as a revival of the ancient Greek enharmonic genus dividing a semitone into two dieses, and I use the term here in this sense. Our 24-note instrument consists of two 12-note keyboards each tuned in an identical 1/4-comma meantone temperament with pure major thirds -- but with the second manual a diesis or fifthtone of 128:125 (~41.06 cents) higher. This diesis is equal to the difference in 1/4-comma meantone between G# and Ab, for example, and also to the difference between three pure major thirds at 5:4 and a pure octave at 2:1.[3] Such an instrument can provide a gateway to the world of Renaissance fifthtone music, a style combining the beauty and fluidity of the 16th century with awesome "diesis jumps," "proximate minor thirds" near 11:9, and familiar progressions stunningly transformed by "enharmonic" inflections. Whether implemented as a two-manual harpsichord, for example, or as a digital synthesizer such as the Yahama TX-802 mapping independent "part-tunings" to two standard MIDI keyboards, our archicembalo can be used either for playing 16th-century fifthtone pieces, or for new compositions and improvisations in this kind of style. In what follows, I introduce the instrument and its music (Section 1), and compare it (Section 2) to the larger archicembalo of Vicentino and the _Sambuca Lincea_ of Fabio Colonna (1618). Then I consider the "fine structure" of a 24-note meantone tuning and its division of the tone, suggesting some possible notational conventions (Section 3); and briefly address issues of "historical tunability" using 16th-century techniques (Section 4). Last but not least (Section 5), I discuss a complication of making do with only 24 notes which will emerge early in this presentation: the issue of meantone ranges, retuning between pieces, and transposition, including an intriguing dilemma in choosing an ideal keyboard arrangement for pieces such as Vicentino's _Musica prisca caput_ which call for a range of Bb-D# plus its counterpart a fifthtone higher. --------------------------------------------------- 1. Parallel universes: The instrument and its music --------------------------------------------------- Taking the lower manual as the "standard" one, we may diagram the archicembalo as follows, with an asterisk (*) used as an ASCII equivalent of Vicentino's dot placed above a note to show its raising by a diesis or fifthtone. Here the basic meantone gamut is the most typical Eb-G#: (C#*) (F#*) (G#*) C* Db D* Eb* E* F* Gb Ab A* Bb* B* ---------------------------------------------------------------------- C# Eb F# G# Bb C D E F G A B C In this arrangement, the two manuals represent as it were "parallel universes," each a standard 12-note meantone gamut including all of the usual accidentals called for by the 16th-century system of modes.[4] Since the 128:125 diesis between the two keyboards is precisely equal to the difference between G#/Ab, etc., some notes of the upper manual invite two alternate spellings: either Db or C#*, Gb or F#*, Ab or G#*. Both conventions have 16th-century precedents, and we may choose either spelling as seems convenient. Consider, for example, the following examples using a MIDI-like notation in which C4 is middle C and higher numbers show higher octaves: Ab4 E*4 E*4 Eb4 B*3 B*3 C4 G#*3 Ab3 Ab3 E*3 or E*3 In the first example, the spelling "Ab3" fits the role of this note as a concordant fifth below Eb4; in the second example, the spelling "G#*3" might more clearly convey its role as a concordant major third above E*3, although using "Ab3" here also may have an advantage of familiarity and consistency. As it happens, Vicentino's conventions in this second case would call for Ab3 and Bertrand's for G#*3. Bertrand's spelling can be useful in conveying the idea of a sonority such as E*3-G#*3-B3*-E4* as a counterpart or "twin" to E3-G#3-B3-E4. Making a "quantum leap" from such one sonority to the other is one form of what I term a "diesis jump," as in Vicentino's complete Latin secular motet _Musica prisca caput_ from his treatise of 1555[5]: E4 E*4 G3 C*4 C4 G*3 C3 C*3 Aside from the voice-crossing, not in evidence if one plays this progression on a keyboard instrument, we have two versions of the same sonority a diesis or fifthtone apart. On our keyboard, we negotiate this "diesis shift" by jumping from the lower manual to the same notes on the upper manual. Such leaps or shifts also occur in the opposite direction from the "alternate" to the "normal" 12-note universe, as in Bertrand's single known "enharmonic" or fifthtone piece, _Ie suis tellement amoureux_ from his chanson collection of 1578[6]: F*4 Bb4 Bb*3 F4 D*4 D4 Bb*2 Bb2 Both examples feature parallel tenths by enharmonic inflections in two voices (the outer pair in the first example, here the lowest pair) -- as well as melodic motions by what Vicentino calls "proximate" fourths or fifths a diesis wider or narrower than the usual intervals. An example of a four-voice enharmonic cadence from Vicentino's treatise demonstrates another aspect of fifthtone music, the availability of new vertical intervals such as the "proximate minor third" which Vicentino describes as having a ratio of roughly 5-1/2:4-1/2 (i.e. 11:9). Here the symbol "r" shows a rest[7]: 1 & 2 | 1 2 | 1 C4 C[*]4 C4 Ab3 G*3 Ab3 Ab*3 A3 F3 C*3 F3 r F3 E*3 F3 In addition to the inflected melodic intervals, we have motion of the alto in the last part of the cadence from the minor third Ab3 above F3 through the "proximate minor third" Ab*3 (a diesis wider than a minor third) to the major third A3. Vicentino regards this middle interval as relatively concordant, having a quality somewhere between that of the minor and major third[8]. This example illustrates the problem of meantone ranges already mentioned: it calls for a tuning based on Ab-C#, with this 12-note range for the lower manual and Ab*-Db (=Ab*-C#*) for the upper manual, rather than our tuning of Eb-G# and Eb*-Ab (=Eb*-G#*) shown at the beginning of this section. To change tunings, we would need to retune G# on the lower keyboard to Ab, and Ab (=G#*) on the upper keyboard to Ab*. Alternatively, by transposing Vicentino's example up a fifth or down a fourth, we could map it to our original Eb-G#/Eb*-Ab range. Section 5 discusses such issues in more detail. A final example from Bertrand[9] may illustrate a familiar cadence subtly transformed by the use of altered melodic intervals: C5 D[*]5 G4 F#*3 G3 A*3 Eb3 D[*]3 The brackets show accidentals not in the original which Bertrand _may_ have intended in order to achieve normal vertical fifths and thirds. If so, the result is a typical Renaissance cadence with downward semitonal motion in a "Phrygian" fashion -- but with descending semitones a diesis narrower than usual, and ascending whole-tones a diesis wider. In place of the usual _major_ semitones Eb3-D3 and G4-F#4 (3/5-tone), we have the minor semitones Eb3-D*3 and G4-F#*4 (2/5-tone); and in place of the normal whole-tones G3-A3 and C4-D5 (5/5-tone), we have the large whole-tones G3-A*3 and C5-D*5 (6/5-tone). Under this reading, we have a cadence with the accustomed smooth concords of the Renaissance, but contrasting narrow semitones and extra-wide whole-tones so as to give the progression a new flavor. Such is the unique beauty of this music.[10] ------------------------------------------------ 2. Historical comparisons: Vicentino and Colonna ------------------------------------------------ From an historical perspective, our 24-note archicembalo represents a subset of the larger archicembalo of Nicola Vicentino (1555) or the _Sambuca Lincea_ (1618) of Fabio Colonna (c. 1567-1640), the latter design also being claimed by Scipione Stella (c. 1558 or 1559-1622) as borrowed from his own _pentorgano_[11]. These instruments implement a complete 31-note division of the octave, whether taken as the subtly unequal division of 1/4-comma meantone or the precise symmetry of 31-tone equal temperament (31-tet). Vicentino's archicembalo has two manuals with six ranks, the first five ranks dividing the octave into 31 dieses or fifthtones, and the sixth rank apparently providing pure rather than meantone fifths for a few frequently used sonorities. Here an asterisk (*) shows notes raised by a diesis, while an apostrophe (') shows notes raised by a "comma," a term which in Vicentino may mean the amount by which the fifth is tempered in meantone (~5.38 cents in a 1/4-comma tuning): Rank 6: D' E' G' A' B' Rank 5: Db* Eb* Gb* Ab* Bb* Rank 4: C* D* E* F* G* A* B* C* Manual 2 ---------------------------------------------------------------------- Rank 3: Db D# E# Gb Ab A# B# Manual 1 Rank 2: C# Eb F# G# Bb Rank 1: C D E F G A B C The first two ranks, as Vicentino notes, are tuned as on usual keyboard instruments, and provide a typical Eb-G# meantone range. The third rank supplies seven "unusual" accidentals, three flats and four sharps. Taken alone, the first manual could serve as a 19-note "chromatic harpsichord" of a kind described by 16th-century theorists and favored in some parts of Italy (e.g. Naples) around 1600.[12] The fourth rank is tuned like the first, but a diesis higher; the fifth rank supplies the remaining enharmonic accidentals needed to complete the 31-note division of the octave. The "comma keys" of the sixth rank, under the interpretation proposed above, provide certain sonorities with just 3:2 fifths and 6:5 minor thirds as well as 5:4 major thirds (e.g. C3-E3-G'3; G3-B3-D'4).[13] While Vicentino's keyboard provides a fully circulating temperament of 31 notes, the 16th-century fifthtone compositions of Vicentino and Bertrand use no more than 24 notes each, and 26 notes in all. Absent from these pieces are the three extreme sharps of Vicentino's third rank (A#, E#, B#), and two enharmonic accidentals of the fifth rank (Gb*, Db*). In 1618, Colonna uses a different design for his Sambuca Lincea, _sambuca_ being a kind of ancient Greek triangular harp which a harpsichord might resemble, and the adjective _Lincea_ or "Lynxian" likely referring to the Accademia dei Lincei of which he was a member (along with such colleagues as Galileo Galilei).[14] Further, he includes in his treatise an "example of circulation": a composition cadencing on all 31 steps of the instrument in a series of descending fifths or ascending fourths, and "returning on the 32rd repetition" to where it commenced, the first and final cadences setting a mode of C Ionian.[15] Seeking a systematic organization for his keyboard, Colonna has six ranks of keys, each tuned 1/5-tone higher than the last, with some of the 31 notes per octave duplicated to permit easier fingering. Here I use the letter "h" (the German notation for B-natural) to show his use of a natural sign for a note raised by 4/5-tone[16]: Rank 6: D E F# G A B C# D (+5/5) Rank 5: Ch Dh Eh Fh Gh Ah Bh Ch (+4/5) Rank 4: Db Eb F Gb Ab Bb C Db (+3/5) Rank 3: C# D# E# F# G# A# B# C# (+2/5) Rank 2: C* D* E* F* G* A* B* C* (+1/5) Rank 1: C D E F G A B C (0) Barbour[17] compares this radical scheme to the "generalized keyboards" of the 19th-century theorist Bosanquet and followers. In contrast, Vicentino's archicembalo is an outgrowth of the usual 12-note meantone keyboard (Eb-G#) making up its first two ranks; the third rank mixes "unusual" flats and sharps (e.g. Ab and A#) in a pattern similar to that found on more modest Renaissance keyboards with split accidentals (e.g. G#/Ab and Bb/A#, with the "unusual" note in back). The second or enharmonic manual is somewhat more systematic: the seven notes of the fourth rank are a diesis above those of the first, while the accidentals of the fifth rank can each be seen as a 4/5-tone division of one of the five whole-tones in a diatonic octave (C-Db*-D, D-Eb*-E, F-Gb*-G, G-Ab*-A, A-Bb*-B). Our 24-note instrument combines Vicentino's approach of making the first two ranks of his archicembalo identical to those of a standard 12-note meantone keyboard, and Colonna's approach (borrowed from Stella?) of having the ranks of his Sambuca Lincea systematically arranged in spacings of 1/5-tone. The following diagram may illustrate these points, while mapping the five accidentals of the second manual to their ranks on Vicentino's instrument: V.3 V.5 V.3 V.3 V.5 Rank 4: Db Eb* Gb Ab Bb* (Rank 2 +1/5) Rank 3: C* D* E* F* G* A* B* C* (Rank 1 +1/5) = V.4 ------------------------------------------------------------------------ Rank 2: C# Eb F# G# Bb = V.2 Rank 1: C D E F G A B C = V.1 ------------------------------------------- 3. The "fine structure" of 24-note meantone ------------------------------------------- An intriguing quality of 24-note meantone with pure major thirds is its subtly unequal division of the tone, featuring two slightly different sizes of fifthtones. This nuance raises some fine points of tuning and of note spelling, and makes available two slightly different versions of certain enharmonically inflected intervals such as Vicentino's proximate minor third near 11:9 (e.g. C-Eb*, D*-F#). As in a usual 12-note tuning, so in extended 24-note or 31-note tunings, the division of a pure major third at 5:4 (~386.31 cents) into a chain of four equal fifths of ~696.58 cents -- each tempered or made narrower than a pure 3:2 by 1/4 syntonic comma or ~5.38 cents -- determines the rest of the structure. Each regular whole-tone is a "meantone" equal to precisely half of a pure major third (~193.157 cents), or to the "mean" or average of the unequal whole-tones of 9:8 and 10:9 together forming a pure major third in 16th-century systems of just intonation. This tone is divided into a large diatonic semitone (e.g. E-F, C#-D, Bb-A) of ~117.11 cents and a small chromatic semitone (e.g. C-C#, Bb-B) of ~76.05 cents. This contrast between the two semitones lends a distinctive flavor to chromatic progressions on a 12-note or extended meantone keyboard. At a finer or enharmonic level of structure, the difference between these semitones defines the diesis of 128:125 or ~41.06 cents, the distance for example between G# and Ab, and also between any note on our lower manual and its counterpart on the upper manual (e.g. D-D*, Eb-Eb*), see Section 1, and the following keyboard diagram: 8:5 117.11 351.32 620.53 813.69 1047.90 Db/C#* Eb* Gb/F#* Ab/G#* Bb* ----------------------------------------------------------------------- C* D* E* F* G* A* B* C* 41.06 234.22 427.37 544.48 737.64 930.79 1123.95 1200.00 32:25 Manual 2 -------------------- Manual 1 25:16 76.05 310.26 579.47 772.63 1006.84 C# Eb F# G# Bb ---------------------------------------------------------------------- C D E F G A B C 0 193.16 386.31 503.42 696.58 889.74 1082.89 1200.00 5:4 Looking at the fine structure more closely, we find that in addition to this diesis or "fifthtone" there is a second flavor of fifthtone, a slightly smaller interval (e.g. C*-C#) equal to about 34.99 cents, or the difference between a chromatic semitone and our diesis. Our 24-note tuning divides each whole-tone into two "larger fifthtones" (LF), a "smaller fifthtone" (SF), and a chromatic semitone (CS): LF SF LF CS |------------|----------|------------|---------------------| C C* C# Db D (Db*) |------------|----------|------------|-----------|---------| 0 ~41.06 ~76.05 ~117.11 ~158.17 ~193.16 ~41.06 ~34.99 ~41.06 ~41.06 ~34.99 |------------|----------|------------|-----------|---------| (D#) D D* Eb Eb* E |------------|-----------------------|-----------|---------| LF CS LF SF This chart shows two types of patterns: one for diatonic whole-tones divided in a 12-note tuning of Eb-G# by a sharp (e.g. C-C#-D), the other for those divided by a flat (e.g. D-Eb-E). Our 24-tone tuning more generally splits _any_ regular whole-tone (e.g. D*-E*, Gb-Ab) into some pattern of the four adjacent intervals CS-LF-SF-LF, although the order may fit into any of four "whole-tone species."[18] A full 31-note tuning such as Vicentino's or Colonna's (Section 2), if realized in 1/4-comma meantone, would divide each tone into five adjacent intervals: some pattern of three large and two small fifthtones (e.g. LF-SF-LF-LF-SF). The central portion of the diagram shows this fivefold division, naming in parentheses the extra notes which would split an undivided chromatic semitone into two unequal fifthtones (LF + SF = CS). Our fourfold division of the tone happily supports some of the same striking effects as a full fivefold division. Consider Colonna's remarkable example of a cadence with "fourth and minor sixth" progressing by a _strisciata di voce_ or "sliding" in fifthtones to a major sixth an extra diesis wide before resolving to the octave[19]: (|) C5 C*5 | C#5 Db5 D5 G4 G4 G4 G4 A4 E4 E4 E4 E4 F#4 E3 E3 E3 E3 D3 Colonna's even minims or half-notes and his one barline "|" suggest a reading in a fluent 2/2; I have added a barline "(|)" before the prolonged final sonority, a breve. A three-voice version in which the right hand takes the highest part only, moving freely between manuals, may be much more practical on some keyboards (including mine). Intervals formed by the upper part with the other two are shown both as rounded cents and as approximate integer ratios, possibly of special interest since Colonna admits interval ratios such as 7:6, 11:8, and 17:12 on his monochord[20]: 814( 8:5) 855(~18:11) 890(~5:3) 931(~12:7) 1200(2:1) 503(~4:3) 544(~11:8) 579(~7:5) 620(~10:7) 814(8:5) +41 +35 +41 +76 C5 C*5 | C#5 Db5 (|) D5 G4 G4 G4 G4 F#4 E4 E4 E4 E4 D4 The signed numbers above the notes show the melodic steps of the upper voice as measured in our 24-note tuning. Starting at a usual minor sixth and fourth above the lower voices, it ascends 41 cents to form what Vicentino calls a "proximate minor sixth" (between minor and major) and a more-than-fourth (about 7 cents narrower than 11:8). The next 35-cent fifthtone brings us to a usual major sixth and augmented fourth, inviting a Renaissance-style resolution by contrary motion to octave and sixth respectively (e.g. E4-G4-C#5 to D4-F#4-D5). First, however, the major sixth is enlarged by another 41-cent fifthtone to a diminished seventh, Vicentino's "proximate major sixth," and the augmented fourth to a diminished fifth. Without affecting the routine progression of the lower voices, this increases the cadential tension and narrows the final resolving motion of the upper voice from a usual diatonic semitone (C#5-D5) to a 76-cent chromatic semitone (Db5-D5). This cadence, which may reflect the vertical liberties of around 1600 rather than the more measured style of Vicentino or Bertrand[21], is available in our standard 24-note tuning of Eb-G#/Eb*-Ab to five of the seven diatonic notes: D (as above), F, G, A, and C. These are the cadential centers where, in usual 16th-century practice, a major sixth expands to the octave with ascending semitonal motion. In each case, subtracting the bracketed notes of Colonna's formula, we are left with such a routine cadence, the outer voices expanding M6-8 while the upper voices resolve an augmented fourth to a sixth: LF SF LF CS LF SF LF CS LF SF LF CS [C4 C*4] C#4 [Db4] D4 [Eb4 Eb*4] E4 [E*4] F4 [F4 F*4] F#4 [Gb4] G4 G3 F#3 Bb3 A3 C4 B3 E3 D3 G3 F3 A3 G3 LF SF LF CS LF SF LF CS [G4 G*4] G#4 [Ab4] A4 [Bb4 Bb*4] B4 [B*4] C5 D4 C#4 F4 E4 B3 A3 D4 C4 Here, as in the usual 16th-century modal system, the E center is special, calling for the _descending_ semitonal approach (F-E) so characteristic of Phrygian. If we "adapt" Colonna's formula to this mode by placing a _downward_ melodic sequence of LF-SF-LF in the _lowest_ voice, we get a different set of vertical intervals and conclude with a normal 117-cent diatonic semitone (DS) rather than a chromatic one as above. This cadence might also be transposed to the less common cadential center of B, the middle voice ending on the fifth F# or the minor third D[22]: D5 E5 A4 B4 A4 G#4 E4 D4/F#4 [Gb4 F#4 F*4] F4 E4 [Db4 C#4 C*4] C4 B4 LF SF LF DS LF SF LF DS -41 -35 -41 -117 -41 -35 -41 -117 A subtler kind of asymmetry involves slight variations in the size of intervals such as Vicentino's "proximate minor thirds" because of the two sizes of fifthtones: |-------------------------- 351.32 -----------------------| 34.99 | Eb* C--------|------------------------------------------------|-------E 0 C* 386.31 | 41.06 |---------------------- 345.25 --------------------------| The interval C-Eb* is equal to the pure major third C-E minus the small fifthtone Eb*-E, a size of about (386.31 - 34.99) or 351.32 cents. With C*-E, however, we have C-E minus the large fifthtone C-C*, about (386.31 - 41.06) or 345.25 cents. Alternatively, we could show that C-Eb* is equal to the meantone minor third C-Eb (~310.26 cents) plus the large fifthtone Eb-Eb*, and C*-E to the minor third C#-E plus the small fifthtone C*-C#. These slightly unequal flavors of proximate minor thirds compare with Vicentino's integer approximation of 11:9 (~347.41 cents), and the single size of 9/31 octave (~348.39 cents) occurring in 31-tet.[23] While the size of our two flavors of fifthtones results from the general properties of 1/4-comma meantone, the specific placement of these large and small fifthtones is a matter of choice. This choice is determined by whether we tune an enharmonic note such as C* or Eb* on the flat or the sharp side of the chain of fifths. As it happens, our tuning treats all enharmonic notes as flats; but in Vicentino's (Section 2), following his suggested tuning order, almost all are treated as sharps. Thus if we realized his tuning as 1/4-comma meantone, C-C* would be a small fifthtone and Eb*-E a large fifthtone, the converse of what we have just seen above. A helpful rule is that each flat lowers a note by a chromatic semitone, about 76.05 cents in a 1/4-comma tuning, while each sharp raises it by the same amount. Thus double flats or sharps lower or raise a note by ~152.1 cents, and the triple flats or sharps found in some 31-note tunings (e.g. Vicentino, G*=F###) by ~228.3 cents. Thus C*, when tuned on the flat side of the chain of fifths as Dbb, is located at a distance of two chromatic semitones below D (Dbb-Db-D), or about 152.10 cents. In 1/4-comma meantone, a regular whole-tone such as C-D is about 193.16 cents, so the fifthtone C-C* (C-Dbb) is equal to roughly (193.16 - 152.10) or 41.06 cents, our larger fifthtone: C* G* D* A* E* B* Tuning chain: ...Dbb-Abb-Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F... |---41---|------------- 152 ------------| | |------ 76 -----|----- 76 -----| | |---------------|--------------| C Dbb=C* Db D |--------|---------------|--------------| 0 41 117 193 When tuned on the sharp side of the chain as B##, however, C* is located at a distance of two chromatic semitones above B (B-B#-B##), again ~152.10 cents. In 1/4-comma tuning, a diatonic semitone such as B-C is equal to about 117.11 cents, so C-C* is equal to roughly (152.10 - 117.11) or 34.99 cents, our smaller fifthtone. Gb* Db* Ab* Eb* Bb* F* C* Tuning chain: ...B-F#-C#-G#-D#-A#-F##-C##-G##-D##-A##-E##-B##... |----- 76 -----|----- 76 -----| |--------------|--------------| B B# C B##=C* |--------------|-------|------| 0 76 117 152 |--35--| For these asymmetries in the placement of large and small fifthtones depending on whether C* is tuned as Dbb or B##, for example, I am tempted to use the term _chirality_ or "handedness."[24] The metaphor of a "mirrorlike asymmetry" like that between a pair of hands or gloves may become clearer if we compare the fivefold division of a whole-tone such as C-D in full 31-note tunings of contrasting chirality. Note that the ascending sequence LF-SF-LF-LF-SF in the first tuning is identical to the _descending_ sequence in the second tuning: |---- 41 ----|--- 35 ---|---- 41 ----|---- 41 ----|--- 35 ---| C C*=Dbb C# Db Db*=Ebbb D |------------|----------|------------|------------|----------| LF SF LF LF SF |--- 35 ---|---- 41 ----|---- 41 ----|--- 35 ---|---- 41 ----| C C*=B## C# Db Db*=C## D |----------|------------|------------|----------|------------| SF LF LF SF LF The choice of chirality for an archicembalo tuning may depend in part on the size of the tuning (e.g. 24 or 31 notes), in part upon the kind of music most likely to be played, and in part on sheer convention. Vicentino, for example, tunes B* as Cb, but the other 11 notes of his fourth and fifth ranks (Section 2) from Gb* up to E* as F##-D###. For our usual 24-note tuning of Eb-G#/Eb*-Ab, tuning all 12 notes of the second manual as flats has the convenience of a single chain of fifths for the whole instrument: Eb* Bb* F* C* G* D* A* E* B* Fbb-Cbb-Gbb-Dbb-Abb-Ebb-Bbb-Fb-Cb-Gb-Db-Ab-Eb-Bb-F-C-G-D-A-E-B-F#-C#-G# Manual 2 | Manual 1 In this arrangement, the "*" sign always raises a note by the large fifthtone of about 41 cents, the 128:125 diesis (e.g. C-C*, Eb-Eb*). Apart from its conceptual simplicity, this convention may have a musical advantage for the fifthtone compositions of Vicentino and Bertrand. In such compositions, the three flats Gb, Db, and Ab often serve as major thirds in "alternate universe" sonorities with enharmonic notes such as E*3-Gb3-B*3, A*3-Db4-E*4, and D*4-Gb4-A*4, also spellable as E*3-G#*3-B*, A*3-C#*3-E*3, and D*4-F#*4-A*4 (see Section 1). In our tuning, a spelling of the second type such as E*3-G#*3-B*3 accurately suggests a sonority with vertical intervals identical to those of E3-G#3-B3, with all three notes raised by the same distance: the 128:125 diesis or large fifthtone found between G# and Ab(=G#*). |B3 |B*3 = B + 41.06 | 310.27 | 310.27 696.58 |G#3 696.58 |Ab = G# + 41.06 | 386.31 | 386.31 |E3 |E*3 = E + 41.06 In each case we have usual meantone concords, a pure 5:4 major third plus a fifth and minor third both 1/4-comma (~5.38 cents) narrower than 3:2 and 6:5 respectively. We could also verify that E*3-Ab-B*3 has the expected meantone intervals by confirming its regular spelling on our chain of fifths shown above: Fb-Ab-Cb. Suppose, however, that E-E* and B-B* were small fifthtones: |B3 |B*3 = B + 34.99 | 310.27 | 304.20 696.58 |G#3 696.58 |Ab = G# + 41.06 | 386.31 | 392.38 |E3 |E*3 = E + 34.99 Since Ab remains a large fifthtone above G#, the major third E*3-Ab becomes about (41.06 - 34.99) or 6.07 cents wider than pure, while the minor third Ab-B*3 is narrowed by this amount beyond its usual tempering, or about (5.38 + 6.07) or 11.45 cents narrower than pure. The spelling for such a sonority on the chain of fifths, D###-Ab-A###, might also suggest a regular fifth but somewhat altered thirds. In a full 31-tone tuning in 1/4-comma, some sonorities of this kind are inevitable, and they hardly seem to constitute musical calamities even by 16th-century standards relishing pure or near-pure thirds. With only 24 notes, however, tuning all enharmonic notes on the flat side provides pure major thirds and other regular meantone intervals in the expected places. ------------------------------------------------ 4. Historical issues: "tunability" and chirality ------------------------------------------------ Like the larger archicembalo of Vicentino and Sambuca Lincea of Colonna (Section 2), a 24-note archicembalo for fifthtone music raises interesting issues regarding an "historically appropriate" or "authentic" tuning.[25] We might divide the tuning question into two basic parts. First, should we use 1/4-comma meantone with its subtly unequal fifthtones, or 31-tet? Here the arguments may be mostly the same for a 24-note instrument as for the complete 31-note or larger keyboards of Vicentino and Colonna. Secondly, if we choose a 1/4-comma tuning, the question of chirality (Section 3) arises: which enharmonic notes should be tuned as flats or sharps? Here a 24-note tuning may invite a simpler answer than a 31-note tuning, where arbitrary convention, the nature of the music to be played, and intriguing questions of intonational aesthetics may all play a part.[26] As to the first or general question, the beauty of 1/4-comma tuning is that it happily approximates a division by equal fifthtones[27] while fitting the advice of both Vicentino and Colonna that the usual notes on their instruments are tuned "as on ordinary keyboards," serving as a foundation for the rest of the tuning. Vicentino tells us that the 12 notes of his first two ranks (see Section 2) are tuned as on usual instruments, with the fifths "somewhat blunted, as the good masters do it."[28] Colonna similarly states that his ranks of basic diatonic notes, sharps (+2/5-tone), and flats (+3/5-tone) are tuned as on "common chromatic harpsichords" (_Cembali Chromatici communi_) of 19 notes, the remaining enharmonic ranks then being accorded with these.[29] This latter remark may reflect the popularity of such 19-note keyboards among Neapolitan musicians around 1600. As a well-documented 16th-century temperament quite practical to tune by ear, 1/4-comma meantone with its pure major thirds would seem to fit these descriptions better than a precise 31-tet division. In 1523, Pietro Aaron directs that at least one major third (C-E) be tuned "sonorous and just, as blending as possible," while Zarlino (1571) and Salinas (1577) define the 1/4-comma tuning in systematic and mathematical terms. While Zarlino calls this "a new temperament," Salinas claims that he was regarded as its inventor during his youthful stay in Rome, where he arrived in 1538.[30] In practice, the 1/4-comma tuning approximates Vicentino's or Colonna's model of apparently equal fifthtones much as the Earth approximates, but does not precisely conform to, the shape of a sphere. When carried to 31 notes, the result is a circulating temperament of the kind described by these authors, albeit with subtle asymmetries -- quite possibly overshadowed by those resulting from the vagaries of the tuning process itself.[31] If we do choose a 1/4-comma meantone, then the second question of chirality arises. Here both the tuning process itself and the intended repertory may influence the decision. Curiously, the use of "authentic historical approaches" might lead to a different tuning of certain notes on a 24-note instrument than on Vicentino's, for example. With our 24-note instrument intended mainly for the Renaissance fifthtone compositions of Vicentino and Bernard, or for new music in similar styles, tuning all enharmonic notes at the flat end of the chain is an obvious choice. From an "historical tunability" perspective, this approach seems the natural one because it tunes all 24 notes as a single chain of fifths (Eb*-G#). One might say: "Simply place the first manual in a usual Eb-G# tuning with pure major thirds, and then, starting at Eb, carry the tuning an additional 12 fifths down on the second manual." One could carry out the tuning of the second manual as a series of pure major thirds: first Ab-C, Db-F, and Gb-C; then the remaining enharmonic notes, B*(Cb)-Eb, E*(Fb)-Ab, A*(Bbb)-Db, etc. Musically, this tuning provides usual meantone major and minor thirds for all regularly spelled sonorities within its range; and tuning all enharmonic notes as flats also has this advantage in the Ab-C#/Ab*-Db or Bb-D#/Bb*-Eb ranges called for by Vicentino's fifthtone pieces. While this flat-end tuning seems the natural one for our 24-note archicembalo, it yields some intervals differing from those in a 1/4-comma realization of Vicentino's scheme, which generally places enharmonic notes on the sharp end of the chain. Specifically, proximate minor thirds will vary, as in this variety of cadential flourish discussed in Section 1 where the outer voices stand at the fifth while the middle voice ascends from minor to major third: Vicentino: Eb* = D## -------------------- G3 (386.31) (351.32) (310.26) Eb3 -- +34.99 -- Eb*3=D## -- +41.06 -- E3 (310.26) SF (345.25) LF (386.31) C3 24-note: Eb* = Fbb ------------------ G3 (386.31) (345.25) (310.26) Eb3 -- +41.06 -- Eb*3=Fbb -- +34.99 -- E3 (310.26) LF (351.32) SF (386.31) C3 In Vicentino's sharp-end tuning, the middle voice moves by a small fifthtone followed by a large fifthtone (SF-LF), while our flat-end tuning has the converse sequence -- and also a converse arrangement of the slightly unequal proximate minor thirds in the second sonority. Note that the smaller of these thirds, ~345.25 cents, is closer to Vicentino's approximate ratio of 11:9 (~347.41 cents); the larger, ~351.32 cents, is almost identical to a pure 49:40 (~351.34 cents). While matching Vicentino's chirality might seem ideally "authentic," on a 24-note instrument it would involve the unlikely procedure of two disconnected chains of fifths: Cb(B*)-G# and D##(Eb*)-D###(E*). Tuning along a single chain of fifths, Eb*-G#, seems far more in keeping with 16th-century procedures such as Vicentino's: natural variations may be more authentic than absolute uniformity. ----------------------------------------------- 5. Meantone ranges, retuning, and transposition ----------------------------------------------- As discussed in Section 1, our 24-note archicembalo fits the musical requirements of 16th-century fifthtone compositions by Vicentino and Bertrand fitting into a "dual meantone" gamut such as Eb-G#/Eb*-Ab. Fortunately, each of Vicentino's and Bertrand's pieces fit into such a 24-note range. Possibly not quite so fortunately, different pieces use different 24-note ranges: Eb-G#/Eb*-Ab, Ab-C#/Ab*-Db, or Bb-D#/Bb*-Eb. This situation is closely analogous to the more usual predicament of using a 12-note meantone keyboard to play Renaissance pieces having different 12-note ranges (Eb-G#, Ab-C#, Bb-D#). The obvious solution is to retune between pieces, choosing the versions of G#/Ab and Eb/D# needed for a specific piece. This same solution is the obvious one for our 24-note archicembalo, with a few complications added by the factor of _two_ keyboards. Linking our three 24-note ranges to specific pieces of Vicentino and Bertrand may make issues more concrete. Since Eb-G# enjoys a status as the usual 12-note range in Renaissance theory and practice, it seems natural to regard an Eb-G#/Eb*-Ab tuning as "standard." This tuning fits Bertrand's single known enharmonic chanson _Ie suis tellement amoureux_ (1578), and also Vicentino's example in his treatise of 1555 giving the opening of his madrigal _Soav'e dolc'ardore_[32]: (C#*) (F#*) (G#*) C* Db D* Eb* E* F* Gb Ab A* Bb* B* ---------------------------------------------------------------------- C# Eb F# G# Bb C D E F G A B C Vicentino's first part of his madrigal _Dolce mio ben_ from the same treatise, and also some of his four-voice examples of enharmonic cadences, may be accommodated with an Ab-C#/Ab*-Db tuning[33]: (C#*) (F#*) C* Db D* Eb* E* F* Gb Ab* A* Bb* B* ---------------------------------------------------------------------- C# Eb F# Ab Bb C D E F G A B C This tuning, like the standard one, invites a perfectly symmetrical arrangement of the two keyboards. In this respect, our third tuning presents a bit of a dilemma. Vicentino's complete Latin secular motet _Musica prisca caput_, and also his first part of the madrigal _Madonna, il poco dolce_, call for a Bb-D#/Ab*-Db tuning[34]. Mapping this to another purely symmetrical arrangement, we would have: (C#*) (D#*) (F#*) C* Db D* Eb E* F* Gb Ab* A* Bb* B* ---------------------------------------------------------------------- C# D# F# Ab Bb C D E F G A B C While this mapping is quite practice, it presents the curious situation where D# appears as a "usual" note on the lower keyboard, and Eb as an "unusual" note on the upper keyboard. If we wish to follow the convention of treating Eb as usual and D# as unusual, as on Vicentino's keyboard, then a slight break in symmetry is required: (C#*) (F#*) C* Db D* D# E* F* Gb Ab* A* Bb* B* ---------------------------------------------------------------------- C# Eb F# Ab Bb C D E F G A B C For a harpsichord, one advantage of this arrangement is that it makes it possible to move from Eb-G#/Eb*-Ab to Bb-D#/Bb*-Eb by retuning only a single note per octave: changing Eb* on the upper manual to D# at the opposite extreme of the 24-note chain of fifths. On a digital instrument where the retuning is accomplished by tables, the choice may be more of a matter of pure taste. If retuning between pieces is inconvenient or impractical, then an alternative would be to transpose where necessary into our standard range of Eb-G#/Eb*-Ab. Pieces calling for a range of Ab-C#/Ab*-Db could be transposed up a fifth or down a fourth, and pieces calling for Bb-D#/Bb*-Eb likewise transposed up a fourth or down a fifth. Most respectfully, Margo Schulter [56]mschulter@... ----- Notes ----- 1. For a complete translation of Vicentino's treatise, see Nicola Vicentino, _Ancient Music Adapted to Modern Practice_, tr. Maria Rika Maniates, ed. Claude V. Palisca (New Haven: Yale University Press, 1996), ISBN 0-300-06601-5. A facsimile of the 1555 edition is available: _L'antica musica ridotta alla modern prattica_, ed. Edward L. Lowinsky, Association Internale des Biblioteques Musicales, Documenta Musicologica, Erste Reihe: Druckschriften-Faksimiles 17 (Basel and New York: Barenreiter Kassel, 1959). On Vicentino's instrument and its tuning, see also Bill Alves, originally appearing in _1/1: Journal of the Just Intonation Network 5(No.2):8-13 (Spring 1989), available at [56]http://www2.hmc.edu/~alves/vicentino.html, with a helpful biography; Henry W. Kaufmann, "More on the Tuning of the _Archicembalo_," _Journal of the American Musicological Society_ 23:84-94 (1970); and Marco Tiella, _L'Archicembalo_, available at [57]http://www.infosys.it/pamparato/ima/ma/ma81/Tiella.html. 2. The term "fifthtone" may be more specific than "diesis," which can refer among other things to a Pythagorean diatonic semitone at 256:243 (the usual limma). At the same time, it is more accurately descriptive than "quartertone" for Renaissance intervals equal to about 1/5-tone. 3. This 128:125 interval is sometimes known more specifically as the "lesser diesis," the "greater diesis" being the difference between four pure 6:5 minor thirds and a 2:1 octave, 648:625 or ~62.57 cents. In the Greek enharmonic genus, a semitone is divided into two dieses, ~45.11 cents each if we take the semitone as a Pythagorean limma of 256:243 or ~90.22 cents. 4. See, for example, Mark Lindley, "Temperaments," _New Grove Dictionary of Music and Musicians_ 18:660-674, ed. Stanley Sadie, (Washington, DC: Grove's Dictionaries of Music, 1980), ISBN 0333231112, at 666. In the standard 16th-century modal system, G# serves for example as a major third above the final of E Phrygian, while Eb permits a fluid degree E/Eb in G Dorian with Bb signature, a transposed equivalent of B/Bb in D Dorian. 5. Transcribed in Maniates and Palisca, n. 1 above, at pp. 218-222, mm. 30-31; and facsimile, Lowinsky, n. 1 above, at 69v-70v. 6. Transcribed in Henry Expert, ed., _Anthoine de Bertrand, Second livre des Amours de Pierre de Ronsard_, Monuments de la Musique Francaise au temps de la Renaissance_ 6 (New York, Broude Brothers, n.d.), 27-30, at p. 30, 3rd system, m. 4, at text _Plus il est contraint_. 7. Transcribed in Maniates and Palisca, n. 1 above, at p. 208. For Vicentino's approximate ratio of 11:9, see ibid. p. 437; for facsimile of musical examples, see Lowinsky, n. 1 above, at 66v. 8. Maniates and Palisca, ibid., pp. 337, 436-437. 9. Expert, n. 6 above, at p. 30, second system, mm. 2-3, at text _navre a` tort_. 10. From a modern perspective oriented to medieval as well as Renaissance music, this use of a small cadential semitone might suggest the Pythagorean tuning of the 13th-14th centuries, where relatively concordant but active and unstable thirds and sixths "strive" toward stable intervals such as fifths and octaves. Combining such compact semitones with pure or near-pure thirds, however, requires "unusual" intervals such as the 6/5-tone steps in this example. 11. Fabio Colonna, _La Sambuca Lincea, overo Dell'Istromento Musico Perfetto, con annotazioni critiche manoscritte di Scipione Stella (1618-1622)_, ed. Patrizio Barbieri, Musurgiana 24 (Lucca: Libreria Musicale Italiana, 1991), ISSN 1121-0508, ISBN 88-7096-026-9. This facsimile edition includes a very helpful commentary in Italian and English. 12. On such 19-note instruments and their music around 1600, see Barbieri's commentary, ibid. pp. XLIX-L. In this music based on a range of Gb-B#, an occasional enharmonic note may arise at either end of the chain. Thus Gesualdo, in one of his vocal madrigals, uses a Cb (Vicentino's B*), while a keyboard composition by Giovanni Maria Trabaci from the same era has an optional F## as a major third to D#, with the performer invited to substitute a minor third F# on a usual "chromatic" instrument. On Gesualdo's accidentals, see Glenn Watkins, _Gesualdo: The Man and His Music_ (Chapel Hill: University of North Carolina Press, 1973), pp. 194-196. Stella entered Gesualdo's service in 1594, see Barbieri, ibid. p. XXXIV. 13. If this is the correct interpretation, then Vicentino's use of these keys, or of all 17 keys of the second manual in an alternate tuning based on "just fifths" with the basic 19 notes of the first manual (e.g. Maniates and Palisca, n. 1 above, 333-334 and commentary at xlix-l, and Tiella, n. 1 above, Fig. 5), might fit the concept of "adaptive just intonation" discussed by modern theorists such as Paul Erlich. In such a system, the melodic steps of the gamut are tuned in a scheme such as meantone, but concords with pure ratios are placed above these steps. 14. Colonna, n. 12 above, Barbieri's commentary at XXXV. 15. Colonna, ibid., at 103-110 (110 numbered as 100), and transcription at LVII-LXII. 16. Ibid. at 72, and Barbieri's interpretation at XLVI-XLVII and Fig. 2; my diagram is based on ranks consistently 1/5-tone apart, with Eh (E+4/5) equivalent to Vicentino's F*, and Bh to C*. If Colonna's Cb and Fb on Rank 4 are emended to C and F, as Barbieri and I have done, this keeps the symmetry of the ranks. 17. J. Murray Barbour, _Tuning and Temperament: A Historical Survey_ (East Lansing: Michigan State College Press, 1953), p. 119. 18. E.g. C-C*-C#-Db-D (LF-SF-LF-CS); G*-G#-Ab-A-A* (SF-LF-CS-LF); E-E*-F-F*-F# (LF-CS-LF-SF); and Gb-G-G*-G#-Ab (CS-LF-SF-LF). 19. Colonna, n. 12 above, at 101-102, and Barbieri's commentary at LI and transcription at LXVI-LXVII. 20. Ibid., XL-XLIII, where Barbieri explains Colonna's principle that a ratio is admissible if its smaller term forms an acceptable concord with the difference of the two terms, e.g. 17:12, 12:(17-12) = 12:5 (a just minor tenth); prime factors include "3, 5, 7, 11, 13 and 17." 21. See Maniates and Palisca, n. 1 above, pp. 336-340, where Vicentino expresses the viewpoint that while the proximate minor thirds and sixths (here E4-C*5) are acceptable concords, proximate major thirds and sixths (here E4-Db5) are "less tolerable" or even "harsh," although he does not categorically exclude them. In his partial madrigal _Madonna, il poco dolce_, he has a passing sonority of minim or half-note duration B3-F#3-B4-Eb4, ibid. at 214-217 at 216 m. 31, discussed at lv; facsimile in Lowinsky, n. 1 above, 68v-69r. This sonority includes both the proximate major third or tenth B3-Eb4 and the proximate major sixth F#3-Eb4, and occurs at the word _pianger_, "to weep." Colonna's G-C*, a diesis wider than a usual fourth, does seem to me a vertical interval uncharacteristic of Vicentino. 22. While I find this example pleasant, and milder than Colonna's (the fourths between the upper voices remaining unaltered), Vicentino says that the "minimal third" a diesis smaller than minor (here Gb-A, first sonority) is generally "discordant" and should be "set aside" like the proximate major sixth, pp. 339-340. To my ears, this interval (~269.21 cents) near 7:6 (~266.87 cents) is rather concordant; here it is combined with the augmented fifth Gb-D, a not too unusual interval in 14th-century music. Since the LF-SF-LF motion in the lowest part (Gb-F#-F*-F) leads to the diatonic semitone F-E, the total descent Gb-E is a curious 6/5-tone, used as a direct melodic interval in Bertrand's cadence cited in Section 1 and n. 9. 23. See nn. 7-8. 24. This term is borrowed from physics and chemistry. As used here, "chirality" in 1/4-comma meantone involves a distinction equal to that between 31 tempered fifths and a pure 2:1 octave, or between the large and small fifthtones, ~6.07 cents. The "handedness" metaphor may suggest the relationship between pairs of intervals such as the large fifthtone (12 fifths down) and the small fifthtone (19 fifths up); or the larger proximate minor third (15 fifths down) and the smaller one (16 fifths up). 25. My special thanks to Ed Foote, musician, advocate, and tuner extraordinaire, for inspiring this section. 26. On Vicentino's archicembalo tuning, views have varied. Barbour, n. 17 above, pp. 117-119, describes this tuning as "a clever method of extending the usual meantone temperament of 1/4 comma," and asserts that this temperament, the "common practice" of the time, "is undoubtedly what Vicentino used." Tiella also takes this view, and gives values in cents for the notes of the instrument based on this tuning. However, Maniates and Palisca, n. 1 above, Appendix VII, pp. 452-453, give rounded values in cents for 31-tet, and Kaufmann, n. 1 above, pp. 87-88, 93-94, and Tables I and II, adopts the view of Lemme Rossi, _Sistema musico_ (1666), that Vicentino's tuning is 31-tet rather than the very closely related 1/4-comma. The scale file vicentino1.scl from scales.zip, an archive of scale data for Manuel op de Coul's Scala computer program, likewise gives the tuning as 31-tet. Among those favoring 1/4-comma, views of chirality vary, although Vicentino's own tuning procedure, Maniates and Palisca, pp. 332-333, specifies Cb-D###, with B* tuned a fifth down from Gb, but all other notes of the enharmonic fifth and fourth ranks up from B# (Gb*-E*), the fifth rank (Gb*-Bb*=F##-A##) being tuned _before_ the remaining notes of the fourth (F*-E*=E##-D###). Barbour, p. 118, says that the fourth rank adds flats (i.e. B*-F*=Cb-Gbb), while the fifth rank adds sharps (i.e. Gb*-Bb*=F##-A##), for a tuning of Gbb-A## -- following Vicentino's scheme for the fifth rank and for B*=Cb, but not for the other notes of the fourth rank. Tiella, in his Fig. 4, shows values in rounded cents which would define a 1/4-comma tuning with all 12 notes of the fourth and fifth ranks tuned as flats (Gb*-B*=Abbb-Cb), or Abbb-B#. This scheme would be identical to the 24-note tuning Eb*-G# described here plus Gb*-Ab*(Abbb-Bbbb) and D#-B# to complete a 31-note division; corresponding notes on the two manuals, as Tiella's diagram shows, are consistently a 128:125 diesis or large fifthtone apart (a rounded 41 cents). While it is gratifying to find an earlier precedent for my 24-note scheme with its flat chirality, Vicentino's scheme is different. 27. Fifths are tempered by ~5.38 cents in 1/4-comma, and ~5.18 cents in 31-tet. This excellent approximation is noted, for example, by Barbour, n. 17 above, pp. 117-119, citing also the calculations of the 17th-century Netherlands theorist Christian Huyghens that the two tunings differ by no more than "1/110 comma." Kaufmann, n. 1 above, pp. 87-88 and 93-94, cites Lemme Rossi on the similarities, and noting "how closely the two tunings are related," states at p. 87: "The 1/4 comma temperament will thus be used for the tuning of the fifths in the remaining orders of the _archicembalo_" (although values in cents are given for 31-tet). Barbieri's commentary in Colonna, n. 12 above, at XLVII and XLIX, refers to 1/4-comma as "the most common tuning in the early seventeenth century" and notes that 31-tet is "excellently approximated" by this tuning. 28. Maniates and Palisca, n. 1 above, pp. 331-332. 29. Colonna, n. 12 above, e.g. at 69, and Barbieri's commentary at XLIX. 30. Lindley, n. 4 above, at 662, and his "Early Sixteenth-century Keyboard Temperaments, _Musica Disciplina_ 28:129-151 (1974), with pp. 139-144 on Aaron, and a statement at p. 150 and n. 23 that "one should not assume _a priori_ that it was 1/4-comma meantone in particular that Nicola Vicentino necessarily had in mind" for the standard tuning of the first two ranks of his instrument. In my view, one can propose 1/4-comma as an historically likely realization of Vicentino's 31-note division while joining Lindley in recognizing the variety of Renaissance meantone tunings. Incidentally, at 150 n. 23, Lindley cites Barber's puzzlement with part of Vicentino's scheme; Barber's perplexity may have resulted from a confusion in his sources of Vicentino's tuning based on a 31-note division of the octave and his alternative tuning with the notes of the second manual providing just fifths to those of the first (see n. 13 above). 31. Vicentino describes intervals of a "minor diesis" (1/5-tone), "major diesis" (2/5-tone, a chromatic semitone), and "major semitone" (3/5-tone, a diatonic semitone); and Colonna, n. 12 above, at 103, prefaces the music for his "example of circulation" with a statement that this composition demonstrates that "our division of the tone into five intervals" is true. In fact, Vicentino's model can be as useful in extended 1/4-comma as in 31-tet for many purposes (e.g. counting a "proximate minor third" such as C-Eb or C*-E as 9/5 of a tone); and either tuning permits Colonna's full circulation. Playing Colonna's piece on his instrument or Vicentino's in 1/4-comma, we would encounter a few sonorities of the kind described near the end of Section 3, with thirds of "opposite chirality" (major thirds 27 fifths down instead of four fifths up, ~392.38 cents; minor thirds 28 fifths up instead of three fifths down, ~304.20 cents). Intriguingly, we would also encounter one fifth of opposite chirality (30 fifths down rather than one fifth up), an almost pure interval of ~702.65 cents, wider than a just 3:2 (~701.955 cents) by the difference between the almost perfectly balanced factors of chirality and usual tempering, (~6.07 - ~5.38) or ~.69 cents. With Vicentino's tuning scheme of Cb-D###, for example, we would encounter this near-pure fifth D###3-Cb4, and likewise the near-pure fourth Cb4-D###4, at the sonority E*3-B*3-E*4-Ab4 (Colonna, ibid. at 105, m.2 of this page), realized as D###3-Cb4-D###4-Ab4, with the major tenth and third D###3-Ab4 and D###4-Ab4 also of opposite chirality, but the major sixth Cb4-Ab4 a usual meantone interval (as the spelling suggests). How noticeable these variations of chirality would be among those of the tuning process over 31 notes remains an interesting question, but in any case they seem no barrier to Colonna's full "circulation." 32. For Bernard's piece, see n. 6 above; since this piece does not use G#, one could also play it in Ab-C#/Ab*-Db. For Vicentino's example, see Maniates and Palisca, n. 1 above, pp. 209-210, with commentary at lii-liii, and facsimile, Lowinsky, n. 1 above, 67r.; this excerpt likewise does not use G#. 33. Maniates and Palisca, ibid., pp. 211-213, with commentary at liii-lv, and facsimile, Lowinsky, ibid., 67v-68r. In a pinch, one might attempt this piece in Eb-G#/Eb*-Ab by disregarding the enharmonic inflections in all parts at m. 14 of the transcription, changing Ab*2-Ab*3-Eb*4-C*4 to Ab2-Ab3-Eb4-C4 (upper voices crossed). Vicentino's own remarks about optionally performing in "mixed" genera would seem to permit such an expedient, and here I see no obvious vertical or melodic contraindications. 34. For _Madonna, il poco dolce_, see Maniates and Palisca, ibid., pp. 214-217, and facsimile, Lowinsky, n. 1 above, 68v-69r; for _Musica prisca caput_, see n. 5 above. Commentary on both pieces appears in Maniates and Palisca, lv-lviii. Most respectfully, Margo Schulter [58]mschulter@... Originally posted on Yahoogroups Tuning list 19-23 October 1999, in four parts Messages #5609, #5621, #5646, #5670 This text version 4 July 2014 mschulter at calweb dot com