Re: Adaptive JI/RI tuning and neo-Gothic valleys (Part 2)

-------------------------------------------------------
Adaptive rational intonation and neo-Gothic valleys
A 24-note scheme a la Vicentino
(Essay in honor of John deLaubenfels)
Part II: Cadences and comparative adaptive structure
-------------------------------------------------------

---------------------------------------------------------
2. Streamlined cadential valleys, Archytas, and al-Farabi
---------------------------------------------------------

Having taken an overview of our 24-note adaptive tuning and some basic
patterns of 3-flavor and 7-flavor neo-Gothic cadences supported by
this tuning (Part I, http://www.egroups.com/message/tuning/16640), we
now look at the fine intonational structure of these cadences in more
detail.

For a fuller, step-by-step presentation on neo-Gothic cadences and
flavors, please see the series "A gentle introduction to neo-Gothic
progressions":

http://www.egroups.com/message/tuning/15038 (1/Pt 1)
http://www.egroups.com/message/tuning/15630 (1/Pt 2A)
http://www.egroups.com/message/tuning/15685 (1/Pt 2B)
http://www.egroups.com/message/tuning/16134 (1/Pt 2C)

Here we focus especially on the "streamlined" quality of pure 7-flavor
resolutions, optimizing "sensory consonance" and cadential efficiency,
and the affinity of these progressions to the 7-based tetrachords and
scales of the Greek theorist Archytas and the Arabic theorist
al-Farabi with their melodic steps of 9:8, 8:7, and 28:27.

----------------------------------------------
2.1. Cadential streamlining and fine structure
----------------------------------------------

In traditional Pythagorean or 3-flavor cadences of our tuning,
unstable intervals and sonorities with rather complex ratios resolve
to stability by efficient progressions featuring compact diatonic
semitones of 256:243 (~90.22 cents).

Illustrating both points, the following diagrams show standard
3-flavor resolutions with vertical intervals between a given voice and
each higher voice shown in parentheses as rounded values in cents, and
melodic intervals shown with signed numbers indicating ascending
(positive) or descending (negative) motion.

3-flavor resolutions

Expansive/Intensive Contractive/Intensive

E4 ----- +90 ----- F4 D4 ----- -204 ----- C4
(204) (498) (294) (0)
D4 ----- -204 ----- C4 B3 ----- +90 ----- C4
(498,294) (498,0) (702,408) (702,702)
B3 ----- +90 ----- C4 G3 ----- -204 ----- F3
(906,702,408) (1200,702,702) (996,702,294) (702,702,0)
G3 ----- -204 ----- F3 E3 ----- +90 ----- F3

Expansive/Remissive Contractive/Remissive

E4 ----- +204 ----- F#4 D4 ----- -90 ----- C#4
(204) (498) (294) (0)
D4 ----- -90 ----- C#4 B3 ----- +204 ----- C#4
(498,294) (498,0) (702,408) (702,702)
B3 ----- +204 ----- C#4 G3 ----- -90 ----- F#3
(906,702,408) (1200,702,702) (996,702,294) (702,702,0)
G3 ----- -90 ----- F#3 E3 ----- +204 ----- F#3

Unstable major thirds (81:64, ~407.82 cents), minor thirds (32:27,
~294.13 cents), and major sixths (27:16, ~905.87 cents) have an
acoustically rather complex and tense quality, adding to the dynamic
quality of the cadence.

In terms of "harmonic entropy" theory, these intervals are located
along "plateaux" at some distance from forms with simple "valley"
ratios (M3 at 5:4 or 9:7; m3 at 6:5 or 7:6; M6 at 5:3 or 12:7). Such
complex ratios for regular thirds and sixths are characteristic both
of standard medieval Pythagorean intonation and of various neo-Gothic
temperaments.[1]

Releasing their tension in directed cadential action, these complex
intervals and sonorities (expansive quad 64:81:96:108; contractive
quad 54:64:81:96) resolve quite efficiently by way of two-voice
progressions involving motion by a usual 9:8 whole-tone in one voice
and an incisive 256:243 semitone in the other (m3-1, M3-5, M6-8, M2-4,
m7-5). These resolutions bring us to the pure "valley" sonority of a
complete 2:3:4 trine or its prime interval of the 3:2 fifth.

A convenient measure of "cadential efficiency" is the total distance
an unstable interval must expand or contract by stepwise contrary
motion in order to reach stability. Here that distance is equal to the
usual Pythagorean minor third of 32:27 or a rounded 294 cents -- one
voice moving by a 204-cent whole-tone, the other by a 90-cent diatonic
semitone.

In 7-flavor resolutions, we find that unstable sonorities as well as
stable trines take on pure "valley" ratios, while at the same time
resolutions become yet more efficient. Here the carat sign (^), as in
Part I, shows a note raised by a septimal comma from its usual
Pythagorean position:

7-flavor resolutions

Expansive/Intensive Contractive/Intensive

E^4 ----- +63 ----- F4 D4 ----- -204 ----- C4
(231) (498) (267) (0)
D4 ----- -204 ----- C4 B^3 ----- +63 ----- C4
(498,267) (498,0) (702,435) (702,702)
B^3 ----- +63 ----- C4 G3 ----- -204 ----- F3
(933,702,435) (1200,702,702) (969,702,267) (702,702,0)
G3 ----- -204 ----- F3 E^3 ----- +63 ----- F3

Expansive/Remissive Contractive/Remissive

E^4 ----- +204 ----- F#^4 D4 ----- -63 ----- C#^4
(231) (498) (267) (0)
D4 ----- -63 ----- C#^4 B^3 ----- +204 ----- C#^4
(498,267) (498,0) (702,435) (702,702)
B^3 ----- +204 ----- C#^4 G3 ----- -63 ----- F#^3
(933,702,435) (1200,702,702) (969,702,267) (702,702,0)
G3 ----- -63 ----- F#^3 E^3 ----- +204 ----- F#^3

Here all unstable intervals have simple valley ratios, forming
expansive quads of 14:18:21:24 and contractive quads of 12:14:18:21
(m3 at 7:6; M3 at 9:7; M6 at 12:7; vertical M2 at 8:7; m7 at 7:4).
These "streamlined" sonorities, while representing tunings of maximal
consonance or minimal harmonic entropy, still remain complex enough to
present a satisfying contrast to stable 2:3:4 trines or 3:2 fifths.

Along with this optimized consonance, we find superefficient
resolutions of our unstable "valley" intervals involving a 9:8
whole-tone motion in one voice and a superincisive 28:27 semitone in
the other. These resolutions (again m3-1, M3-5, M6-8, M2-4, m7-5)
require a total expansion or contraction of only 267 cents, the size
of our pure 7-flavor minor third at 7:6 (a 204-cent whole-tone plus a
63-cent semitone) -- or, more precisely, ~266.87 cents.[2]

To explain this increase in both sensory consonance and cadential
efficiency, we might note that 7-based intervals expanding to
stability (M2, M3, M6) are a septimal comma (64:63, ~27.26 cents)
_wider_ than their usual Pythagorean counterparts, while intervals
contracting to stability (m3, m7) are a septimal comma _narrower_.
Thus they can expand (M2-4, M3-5, M6-8) or contract (m3-1, m7-5) by a
septimal comma less in order to attain their stable goal -- a total
distance of 267 cents (7:6) rather than 294 cents (32:27).[3]

Each flavor has its own musical charm: the classic Pythagorean with
its complex plateau intervals, and the streamlined 7-flavor with its
pervasive valley intervals. Certain hybrid progressions may mix these
flavors, often by way of striking septimal comma shifts (Section 4).

-----------------------------------------------------------
2.2. Cadences and scales: the Archytas/al-Farabi connection
-----------------------------------------------------------

Our pure 7-flavor cadences may be derived from tetrachords and octave
scales of a kind described in Greek and Arabic theory, notably by
Archytas of Tarentum (c. 428-350 BC) and Abu al-Nasr al-Farabi
(870-950). These tetrachords and scales feature whole-tones
alternating between ratios of 9:8 (~203.91 cents) and 8:7 (~231.17
cents), and semitones of 28:27 (~62.96 cents).

The basic septimal tetrachord (i.e. four-note pattern) of Archytas
divides the interval of a 4:3 fourth into these three steps, with some
possible arrangements of whole-tones (T) and semitones (S) like these:

S T T T T S T S T
E F G A C D E F D E F G
28:27 9:8 8:7 9:8 8:7 28:27 8:7 28:27 9:8

These tetrachords have an appealing melodic and mathematical logic,
with all steps derived from superparticular ratios or epimores fitting
the pattern (n+1:n) -- 9:8, 8:7, 28:27.

As documented in the scales archives of Manuel op de Coul's Scala
program, al-Farabi described an octave scale (identified in the file
al-farabi_g3.scl as the "Greek genus primum conjunctum") featuring
these steps and tetrachords, which I here notate as an octave F-F:

1:1 9:8 9:7 81:56 3:2 12:7 27:14 2:1
F3 G3 A3 B3 C4 D4 E4 F4
0 204 435 639 702 933 1137 1200
9:8 8:7 9:8 28:27 8:7 9:8 28:27
204 231 204 63 231 204 63

Using our 24-note keyboard notation, we can find this scale at the
notes F3-G3-A^3-B^3-C4-D^4-E^4-F4, the carat (^) indicating notes on
the upper manual a septimal comma above those of the lower manual.

In a neo-Gothic setting, we can generate a scale almost identical to
al-Farabi's by beginning with two sonorities together forming a
standard cadence: on our 24-note keyboard, the stable trine F3-C4-F4
and the unstable 7-flavor expansive quad G3-B^3-D4-E^4. Adding the
step A^3 to fit the pattern of adjacent whole-tones with alternating
ratios of 9:8 and 8:7, we arrive at this scale in the octave F-F,
which might be described in medieval European terms as a version of
the Lydian mode or Mode V:

1:1 9:8 9:7 81:56 3:2 27:16 27:14 2:1
F3 G3 A^3 B^3 C4 D4 E^4 F4
0 204 435 639 702 906 1137 1200
9:8 8:7 9:8 28:27 9:8 8:7 28:27
204 231 204 63 204 231 63

Here the only difference from al-Farabi's scale is the placement of D4
at a 27:16 rather than a 12:7 above F3, with the upper tetrachord
C4-D4-E^4-F4 arranged 9:8-8:7-28:27 rather than 8:7-9:8-28:27. This
arrangement permits a pure fifth G3-D4 in the quad G3-B^3-D4-E^4,
rather than G3-D^4, a septimal comma wide (32:21, ~729.22 cents).

More generally, taking the lowest note of any 2:3:4 trine as a local
reference note "1/1," we can define all steps but one of this variant
on al-Farabi's scale from our trine (1/1-3/2-2/1) plus its 7-flavor
expansive/intensive quad at 14:18:21:24 (9/8-81/56-27/16-27/14).

To demonstrate the musical use of the remaining third degree "9/7," we
consider cadences like the following involving an M2-5 resolution by
near-conjunct contrary motion (stepwise motion in one voice, thirdwise
motion in the other):

E^3 F4 E^4 F4
B^3 C4 A^3 C4
A^3 F3 G3 F3

(M2-5) (M6-8 + M2-5)

The first progression features the resolution of a mildly unstable
"quintal/quartal" sonority (8:9:12), while the second involves a
7-flavor form of expansive sixth sonority (7:8:12). All vertical
fifths and fourths of these sonorities are pure, and all melodic
semitones are a supercompact 28:27.

We can derive a similar octave scale from any 2:3:4 trine plus its
7-flavor expansive/remissive quad, for example E^3-B^3-E^4 and
F3-A^3-C4-D^4:

1:1 28:27 7:6 4:3 3:2 14:9 16:9 2:1
E^3 F3 G3 A^3 B^3 C4 D^4 E^4
0 63 267 498 702 765 996 1200
28:27 9:8 8:7 9:8 28:27 8:7 9:8
63 204 231 204 63 231 204

This scale might be described in medieval European terms as a 7-flavor
version of the Phyrgian mode or Mode III (E-E). Again taking the
lowest note of the stable trine as a local "1/1," we now have a trine
of (1/1-3/2-2/1) plus an expansive quad of (28/27-4/3-14/9-16/9) --
sonorities with the desired ratios of 2:3:4 and 14:18:21:24.

Interestingly, if we add a third degree G3 or "7/6" so as to maintain
the pattern of alternating 9:8 and 8:7 whole-tones, we arrive at a set
of pitch classes identical to those of al-Farabi's scale -- enumerated
in a range of F3-F4, F3-G3-A^3-B^3-C4-D^4-E^4-F4.

Placing the third degree at G3 facilitates a beautiful resolution of
the mildly unstable F3-G3-C4 (8:9:12) combining melodic motions of a
descending 28:27 semitone in the two outer voices (F3-E^3, C4-B^3)
with an ascending 9:7 major third (G3-B^3) in the middle voice[4]:

C4 B^3
G3 B^3
F3 E^3

(M2-5)

Some other types of cadences, however, raise the complication of the
impure fifths or fourths in these scales: A^3-D4 in our F-F scale, and
G3-D^4 in our E^-E^ scale with pitch classes identical to al-Farabi's.
A ready solution in our 24-note tuning is to vary the form of either
note of such an interval (e.g. D4 or D^4, G3 or G^3) so as obtain pure
vertical fifths and fourths[5]:

E^4 F4 D^4 E^4
D^4 C4 G^3 B^3
A^3 F3 F3 E^3

(M2-4) (M6-8 + M2-5)

In the first cadence on F, a mildly unstable quintal/quartal sonority
A^3-D^4-E^4 (6:8:9) resolves to the trine F3-C4-F4 with the unstable
upper major second expanding to a fourth (M2-4); and the highest voice
progresses by a 28:27 semitone (E^4-F4). Here D^4 is required in the
middle voice for the pure fourth A^3-D^4 of the first sonority.

In the second cadence on E^, the sixth sonority F3-G^3-D^4 (7:8:12)
resolves to E^3-B^3-E^4, the outer 12:7 major sixth F3-D^4 expanding
to the octave E^3-E^4 with a descending 28:27 semitone F3-E^3 in the
lowest voice. Here G^3 is needed in the middle voice for the pure
fifth G^3-D^4 of the first sonority.

The pure 7-flavor progressions of neo-Gothic music thus have a certain
affinity with these septimal tetrachords and scales of Archytas and
al-Farabi, just as the 5-limit just intonation systems of the
Renaissance have an affinity with the syntonic diatonic of Ptolemy.
Common features include superparticular ratios for melodic steps,
unequal whole-tones, and commatic distinctions between multiple
versions of the same note (e.g. D4 and D^4 in the above examples).

Such a union of pure ratios and melodic patterns in complex polyphony
is an art subtle and yet not impossible, at once intricate and
beautiful.

--------------------------------------------------
3. Adaptive structure: a comparison with Vicentino
--------------------------------------------------

While realizing tetrachords and scales like those of Archytas and
al-Farabi in a "classic JI" manner, our 24-note adaptive scheme also
somewhat resembles the 38-note system of Vicentino for obtaining
vertical concords with pure ratios. Here we focus on some similarities
and differences between these two adaptive systems, from the viewpoint
of the keyboard performer as well as the tuning theorist.

Vicentino's system for obtaining both "perfect fifths" and "perfect
thirds," as mentioned at the opening of Part I, involves two 19-note
manuals evidently each tuned in 1/4-comma meantone (Gb-B#) with pure
major thirds at a distance of 1/4 syntonic comma (~5.38 cents) apart.

A likely arrangement for the keyboards of Vicentino's _archicembalo_
or _arciorgano_ is the following, with the usual five accidental keys
of each manual split front-to-back (e.g. G#/Ab, Eb/D#), and extra
small keys for E# and B#. Here Vicentino's comma sign (') -- for
Vicentino, the term "comma" may refer to various small intervals --
serves to show the raising of the notes on the second manual by 1/4
syntonic comma:

Db' D#' Gb' Ab' A#'
C#' Eb' E#' F#' G#' Bb' B#'
C' D' E' F' G' A' B' C'
-----------------------------------------------
Db D# Gb Ab A#
C# Eb E# F# G# Bb B#
C D E F G A B C

Using this instrument, the player can attain pure ratios for complete
5-limit sonorities of the 16th century extolled both by Vicentino and
Gioseffo Zarlino (1558) as manifesting "perfect" harmony (Zarlino's
_harmonia perfetta_).

Here are just realizations on Vicentino's keyboard for four forms of
these complete sonorities regarded as concordant in practice and
theory, and also two forms discussed by Zarlino[6] combining a sixth
with a fourth above the lowest part, commonly subject to many of the
same restrictions as more unequivocal discords. Interval arrangements
for each sonority are shown by the notation (outer|lower + upper), and
the just tuning is shown using Zarlino's string ratios, more modern
frequency ratios, and rounded cents with respect to the lowest voice:

-----------------------------------------------------------------------
Example Intervals String ratio Freq ratio Cents
-----------------------------------------------------------------------
C3-E3-G3' (5|M3 + m3) 15:12:10 4:5:6 0-386-702
D3-F3'-A3' (5|m3 + M3) 6:5:4 10:12:15 0-316-702
E3-G3'-C4 (m6|m3 + 4) 24:20:15 5:6:8 0-316-814
F3'-A3'-D4 (M6|M3 + 4) 5:4:3 12:15:20 0-386-884
.......................................................................
G3'-C4-E4 (M6|4 + M3) 20:15:12 3:4:5 0-498-884
A3'-D4-F4' (m6|4 + m3) 8:6:5 15:20:24 0-498-814
-----------------------------------------------------------------------

Pure major thirds and minor sixths are played with both notes on the
same keyboard. Fifths and minor thirds, tempered narrow by Vicentino's
"comma" of 5.38 cents on either keyboard, are restored to their just
proportion by playing the lower note on the lower manual, and the
upper note on the upper manual (e.g. D3-F3'-A3'). Fourths and major
sixths, tempered wide by this same amount, are played in their just
forms with the lower note on the upper keyboard and the upper note on
the lower keyboard (e.g. F3'-A3'-D4).

As Vicentino's notation shows, narrow intervals are thus enlarged by
this "comma," and wide intervals reduced by it, achieving pure ratios.
In his circular of 1561 advertising the arciorgano, this "perfect
diatonic music" is described along with its marvellous effects[7]:

First there are obtained perfect fifths above the
white keys of the common organ, which make a
wonderful sound; then two kinds of thirds, one
major, one minor, and similarly, two kinds of
sixths, in which case it happens that whenever
perfect fifths are struck together with perfect
thirds, they fill the ears with such harmony that
no better can be heard on earth.

Both Vicentino's adaptive tuning and the adaptive Xeno-Gothic scheme
appear to have the following properties in common:

(1) Both systems make it possible to realize a complete set of
sonorities featuring pure ratios: Vicentino's stable sonorities based
on 5-limit ratios of 2-3-5, or stable and unstable Gothic/neo-Gothic
sonorities based on ratios of 2-3-7.

(2) Each of the two manuals in itself represents a standard regular
tuning: Vicentino's 19-note meantone (Gb-B#), or a Gothic/neo-Gothic
12-note Pythagorean (Eb-G#).

(3) The notes of the upper keyboard are shifted by a slight interval
to obtain pure ratios for vertical sonorities: by 1/4 syntonic comma
(~5.38 cents) in Vicentino's scheme, or a septimal schisma (~3.80
cents) in the neo-Gothic scheme.

While both schemes yield pure vertical sonorities combining multiple
prime factors (2-3-5 or 2-3-7), Vicentino's is based on a regular
temperament, evidently 1/4-comma meantone: melodic intervals, and also
vertical intervals in unstable sonorities, typically have irrational
meantone ratios. Indeed the beauty and elegance of this system lies in
its synthesis between the flexibility and regularity of a Renaissance
temperament and the purity of just vertical concords.

In contrast, the adaptive neo-Gothic scheme is entirely based on
integer ratios, a much easier solution in a medieval or neo-medieval
setting than in a 5-limit Renaissance setting.[8] Such an "adaptive
rational" scheme satisfies two additional properties:

(1) The regular tuning on each keyboard is also a just tuning based on
integer ratios, namely Pythagorean intonation; and

(2) The small interval of adjustment between the two keyboards is
itself rational, here the septimal schisma of 33554432:33480783 (~3.80
cents), so that all ratios in the scheme remain integer-based.

In one obvious respect, Vicentino's system shares the melodic
smoothness and regularity of meantone while our neo-Gothic adaptive
system exemplifies the intricacy or unevenness of classic multi-prime
JI systems: the matter of commas and comma shifts.

-------------------------------------------
3.1. Commas and melodic evenness/unevenness
-------------------------------------------

Vicentino's system, like a usual meantone temperament, disperses the
syntonic or 3-5 comma (81:80, ~21.51 cents) by narrowing each regular
fifth by 1/4 of this amount, equal to the 5.38-cent adjustment between
the two manuals. In progressing from one pure vertical sonority to the
next, the usual meantone melodic intervals need vary by only this
small amount, rather than by a full syntonic comma as often happens in
classic Renaissance JI.

In contrast, the adaptive neo-Gothic system features the classic
septimal or 3-7 comma (64:63, ~27.26 cents); notably unequal
whole-steps of 9:8 and 8:7, and sometimes direct melodic progressions
by a septimal comma (Section 4), are characteristic, whether regarded
as blemishes or adornments.

As discussed in Part I, Section 1, this latter system has effectively
exchanged one comma for another: the Pythagorean or 2-3 comma which
would normally obtain between the manuals is enlarged by a septimal
schisma to a septimal comma, making pure 7-based sonorities possible.

Here we might add that Vicentino himself ardently advocates another
kind of striking melodic shift: the enharmonic shift of a meantone
diesis, equivalent to a 2-3 comma (in 1/4-comma tuning, typically
128:125 or ~41.06 cents). However, such shifts are an optional and
expressive effect providing a prime motivation for his other
archicembalo/arciorgano tuning dividing the octave into 31 essentially
equal steps of 1/5-tone, rather than an integral and necessary feature
of his adaptive JI system.[9]

The question of the commas invites a comparison of the practical
challenges and opportunities which keyboardists may encounter using
these two systems in the lively musical settings for which they are
designed.

-----------------------------------------------------
3.2. Commas and stable concords: A keyboardist's view
-----------------------------------------------------

While the presence of an unexpurgated septimal comma in the neo-Gothic
system makes it in some ways more intricate, Vicentino's system may
actually offer a more arduous task for the performer who wishes to
maintain pure concords wherever possible. This is true in part because
of the distinctions of vertical stability/instability involved in each
case.

In Gothic or neo-Gothic music, the complete stable sonority is the
three-voice trine (e.g. D3-A3-D4, 2:3:4). In our adaptive scheme, as
in a usual Pythagorean tuning, all regularly fifths and fourths on
each keyboard -- and therefore complete trines -- are pure. One need
only play them in the usual manner.[10]

In Vicentino's 16th-century system, however, playing any full 5-limit
concord (Zarlino's _harmonia perfetta_) requires mixing notes from the
two keyboards in order to obtain pure fifths (or fourths) and minor
thirds (or major sixths).

Of course, such mixing of notes from the two manuals is required in
the neo-Gothic scheme to obtain pure 7-flavor versions of unstable
sonorities. However, the typical vertical spacing of such medieval or
neo-medieval sonorities can simplify this task. In many three-voice or
four-voice progressions, the widest vertical interval negotiated by a
single hand is characteristically a fifth or fourth:

E^4 F4 F4 E^4
E^4 F4 D4 C4 F4 E^3 D^4 E^4 D4 E^4 F4
B^3 C4 B^3 C4 Bb3 A^3 Bb3 A^3 C4 B^3 C4
G3 F3 G3 F3 G^3 A^3 G^3 A^3 G3 F3

While keeping the right hand on the right manual at the right time
remains a challenge, the keyboardist is at least assured of a friendly
"landing" on a trine or fifth conveniently located on a single manual.

In contrast, let us see what is involved in some routine Renaissance
progressions at Vicentino's keyboard if we wish to maintain pure
concords for any length of time:

G4 F'4 B4 C'5 A4 B4 A'4 G#4 G4 F#4
D'4 D4 G4 A4 E'4 G4 D4 E4 D4' D4
B3 A'3 D'4 F4 C'4 D'4 A'3 B'3 Bb'3 A'3
G3 D3 G3 F3 A3 G3 F'3 E3 G2 D3

As these examples may illustrate, we can indeed progress between pure
vertical sonorities with smooth and virtually regular melodic steps,
but with a complication more generally noted for the archicembalo by
Ercole Bottrigari in 1594[11]:

"[T]he player must many times press and hold down
down with one hand some keys of both keyboards
at the same time, and occasionally do this with
both hands at once."

Possibly new and more ergonomic keyboard designs may alleviate the
mechanical difficulties of needing to mix notes from both keyboards,
frequently within a single hand, "whenever perfect fifths are struck
with perfect thirds," to borrow Vicentino's phrase.

Happily, both adaptive systems provide a "safety valve" for the
intrepid keyboardist, who is free to revert to a regular and
stylistically felicitous tuning system readily at hand as a subset of
the full scheme: 19-note Renaissance meantone for the player of
Vicentino's instrument, or 12-note medieval Pythagorean for the player
of the neo-Gothic instrument.

This flexibility might serve not only as a concession to the
beleaguered performer, but as a musical virtue: the opportunity to mix
and contrast pure sonorities with more complex ones. In a neo-Gothic
setting, complex Pythagorean versions of unstable sonorities are
valued and cultivated in their own right alongside pure 7-flavor
versions. In Vicentino's system, for example, the playing of certain
passages with a subtle contrast between pure major thirds and the
tempered minor thirds in a usual meantone fashion could add a welcome
note of musical variety[12], alongside other passages featuring pure
fifths and thirds.

In short, both adaptive systems may have the advantage of combining
the musically routine with the extraordinary, giving the performer the
creative scope for choice in seeking out effects both old and new.

-----
Notes
-----

1. See, e.g., Margo Schulter and David Keenan, "The Golden Mediant:
Complex ratios and metastable musical intervals," Tuning Digest 810:3,
18 September 2000; http://www.egroups.com/message/tuning/12915. Note
that the term "Noble Mediant" might now be preferred in the title, as
suggested by Dave Keenan.

2. For these standard forms of 7-flavor cadences with melodic steps of
9:8 and 28:27, the general rule is that intensive progressions (with
ascending semitones) resolve to a trine or fifth on the lower manual,
while remissive progressions (with descending semitones) resolve to
the upper manual. Another way of stating this pattern: melodic
whole-tones are played on the same keyboard (e.g. G3-F3, E^4-F#^4);
ascending semitones move from the upper to the lower keyboard
(e.g. B^3-C4, E^4-F4); and descending semitones from the lower to the
upper keyboard (e.g. F3-E^3, C4-B^3).

3. In contrast, the 15th-century shift in Western Europe from active
Pythagorean thirds and sixths to 5-limit "valley" tunings resulted in
larger cadential semitones and less efficient resolutions. Major
thirds at 5:4 (~386.31 cents) and major sixths at 5:3 (~884.36 cents)
are a syntonic comma (81:80, ~21.51 cents) _narrower_ than their
Pythagorean counterparts, while minor thirds at 6:5 (~315.64 cents)
are _wider_ by this same amount. The m3-1, M3-5, and M6-8 resolutions
thus require this extra amount of expansion or contraction, a total
distance of a rounded 316 cents (the size of the 6:5 minor third).
Cadential semitones in a pure 5-limit tuning are typically 16:15
(~111.73 cents), again a syntonic comma larger than Pythagorean. On
early Renaissance tension between the musical values of smoother
thirds and incisive cadential semitones, see also Mark Lindley,
"Pythagorean Intonation and the Rise of the Triad," _Royal Musical
Association Research Chronicle_ 16:4-61 at 45 (1980), ISSN 0080-4460;
and "Pythagorean Intonation," _New Grove Dictionary of Music and
Musicians_ 15:485-487, ed. Stanley Sadie, Washington, DC: Grove's
Dictionaries of Music (1980), ISBN 0333231112.

4. While the musical style is derived from that of Gothic Europe, the
mixture of melodic semitones and major thirds might somehow evoke for
me also the Balinese or Javanese pelog and Japanese pentatonic scales
featuring these intervals.

5. In the right setting, these narrow fourths (21:16, ~470.78 cents) and
wide fifths (32:21, ~529.22 cents) might present an opportunity for
special effects rather than a problem. Keenan Pepper, for example, has
extolled the "crunchiness" of 21:16, and neo-Gothic styles favor many
varieties of altered or "less usual" intervals.

6. Gioseffo Zarlino, _The Art of Counterpoint: Part Three of Le
Istitutioni harmoniche, 1558_, trans. Guy A. Marco and Claude Palisca
(W. W. Norton, 1976), ISBN 0-393-00833-9, Chapter 60, pp. 190-193. On
Zarlino's approach to multivoice sonorities, see also my article,
http://www.egroups.com/message/tuning/15397, and also emendations in
http://www.egroups.com/message/tuning/15434 in response to valuable
corrections by Paul Erlich also available as part of that thread, e.g.
http://www.egroups.com/message/tuning/15400.

7. Henry W. Kaufman, "Vicentino's Arciorgano: An Annotated
Translation," _Journal of Music Theory_ 5:32-53 (1961) at 34-35.

8. In a neo-Gothic JI system (factors of 2-3-7) based entirely on
integer ratios, stable trines (2:3:4 or 3:4:6) are free from the
complications of the septimal or 3-7 comma at 64:63. In contrast, in a
Renaissance JI system (factors of 2-3-5), stable sonorities combining
Vicentino's "perfect fifths and perfect thirds" (4:5:6 or 10:12:15)
squarely confront the problem of the syntonic or 3-5 comma at 81:80.
As we shall see in Section 3.2, even Vicentino's system, while it
succeeds in dispersing the syntonic comma, bears some sign of this
greater complication by requiring the performer to mix notes from both
manuals whenever such a "perfect" Renaissance sonority is desired.

9. Vicentino's adaptive system offering sonorities where all fifths
and thirds are "perfect," as well as his first or circulating 31-note
scheme for his archicembalo and arciorgano, would nicely fit a base
tuning of 1/4-comma meantone for the first 12 notes of the lower
manual, which Vicentino simply directs should be tuned as on usual
keyboard instruments, with the fifths slightly narrowed or "blunted."

10. In an "avant-garde" neo-Gothic style which acted on the view of
Jacobus of Liege that 9:1 (a major 23rd, e.g. D3-E6) is a "perfect
concord" by treating sonorities such as D3-A5-E6 (1:6:9) as stable,
these 3-prime-limit sonorities would also be available in their usual
locations on either keyboard.

11. Ercole Bottrigari, _Il Desiderio, 1594_, trans. Carol MacClintock,
Musical Studies and Documents 9 (American Institute of Musicology,
1963), p. 51, where it is noted that tuners and organists may be
rather intimidated by "the keys separated, as I have said, into two
keyboards with the usual black semitones divided in two and others
added."

12. On such a contrast in 1/4-comma meantone "between pure major 3rds
and tempered minor 3rds," see Mark Lindley, "Temperaments," _New Grove
Dictionary of Music and Musicians_ (n. 3 above), 18:660-674 at 663.

Most respectfully,

Margo Schulter
mschulter@value.net