Archicembalo in meantone 24: Renaissance fifthtone music (2 of 4)

--------------------------------------------------------
An archicembalo in 24-note meantone:
Renaissance fifthtone music on two 12-note keyboards
--------------------------------------------------------

(Part II of IV)

-------------------------------------------
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 raiesd 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.

Most respectfully,

Margo Schulter
mschulter@value.net