Billbrpt write:
<<What gets me is that no matter how many notes these people can imagine
being
in a scale, it still can only be ET. Their whole world would explode if
anyone dared to think of something like a 31 Meantone.<<
There is no need to suffer the meantone restrictions in a scale with 31
notes to the octave. Harmonically, it lets us have it both ways, it is just
physically difficult to manipulate an instrument with that many keys. In
fact, the meantone tunings are simply subsets of the higher ET's. i.e. a 54
TET has all the notes found in 12 TET as well as those we form by moving from
ET to well-temperaments.
>> The greater
than 12 tone scale was being discussed and dismissed as foolish intellectual
pondering in the early 19th Century too.<<
I would like to post below a recent posting from Margo Schulter that
address the question in more detail. I don't think it was considered
foolish, and it certainly predates the early 19th century.
--------------------------------------------------
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
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