Archicembalo in meantone 24: Renaissance fifthtone music (1 of 4)

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An archicembalo in 24-note meantone:
Renaissance fifthtone music on two 12-note keyboards
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(Part I of IV)

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.

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1. Parallel universes: The instrument and its music
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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]

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2. Historical comparisons: Vicentino and Colonna
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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

Most respectfully,

Margo Schulter
mschulter@value.net