Yahoo Tuning Groups Ultimate Backup tuning Re: Renaissance meantone: "linear and selective JI"?

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Renaissance meantone: Linear and selective JI?
Reflections of a curious keyboardist
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Hello, there, and this article grows out of some recent reflections on
why as an enthusiast for medieval and Renaissance music I tend to
regard a 15-note Pythagorean or meantone tuning as "natural," but a
15-note tuning in 5-limit just intonation (JI) as "of mostly
theoretical interest." My warm thanks to Dave Hill, meantone and JI
pianist and advocate, for a correspondence which has made me reflect
on this question.

Here my purpose is _not_ to make any judgment about the merits of
playing Renaissance music on keyboards tuned in 5-limit JI[1], but
only to explain some of my own predilections and biases which may
incline me toward meantone. At the same time, I would like to suggest
a category which may describe both Renaissance meantone (specifically
the 1/4-comma variety) and my other favorite tuning, medieval
Pythagorean: linear or selective JI.

Please let me dedicate this article especially to Bill Alves, who has
inspired me both by his fine paper on Nicola Vicentino (with its
discussion of pragmatism in tuning), and by his many contributions to
this list. Of course, the responsibility for the predictable quirks
and possible errors here remain solely mine.

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1. Setting the context: live keyboard playing
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In defining the question of Renaissance meantone or 5-limit JI under
the rather narrow circumstances under which I am currently addressing
it, I would like not to limit others to the same perspective, but
indeed to invite a discussion of factors (involving both the nature of
the performance and technology, and the orientation of the performer)
which might lead to radically different conclusions.

The rather specific circumstances which I address are playing music in
16th-century styles (from the routine to the most "experimental") on a
TX-802 synthesizer supporting two conventional 12-note-per-octave MIDI
keyboards (technically speaking, "MIDI controller keyboards") with 49
keys (four conventional octaves) each.

Commonly I use the two keyboards as contrasting manuals, maybe playing
two voices of a four-voice texture on each manual. For example, in an
"organ" style, one manual might have a conventional "organ-like"
sound, and the other a "crumhorn-like" or "regal-like" sound --
crumhorns and regal (reed) organs having a distinctive "buzzy"
sound. In a "harpsichord" style, one manual might have a fuller
harpsichord-like sound, the other an effect more like a lute stop.

Using the "part-tuning" feature, it is easy to tune each manual (or,
more precisely, the "instrument(s)" mapped to it) to a different
arrangement of the same tuning: for example, 1/4-comma meantone in
Eb-G# for the lower manual, and F-A# for the upper manual. This
example happens to be the actual 14-note tuning I use in practicing
the Prologue of Orlando di Lasso's _Prophetiae Sibyllarum_.

Thus options range from a standard 12-note tuning with all notes on
the two manuals in unison, to a 24-note tuning where none of the notes
are in unison -- assuming that each keyboard has repeating 12-note
octaves. The TX-802 synthesizer itself does _not_ impose such an
assumption, which may result more from my "conventionality," or
limited keyboard skills in coping with even a standard layout <grin>.

Since my main practical and theoretical orientation is to medieval and
Renaissance music, notes outside of the typical Eb-G# range seem
"unusual" to me -- so a bit of extra effort to use these "special"
notes on a keyboard is for me rather expected. In contrast, I regard
the notes of the standard medieval/Renaissance gamut, the seven
diatonic notes plus Bb (also an integral part of this gamut, or
_musica recta_), as "home." The other "routine" accidentals -- Eb, F#,
C#, G# -- are for me somewhere in between.

As stated at the beginning of this section, my purpose is to suggest
some of the circumstances in either the technological setting of the
performance or the orientational "set" of the performer which might
lead to a different viewpoint on the "Renaissance meantone or JI"
question than that I am about to express.

For example, how about a performer using software to obtain a just
tuning "on the fly"? How about a performer who routinely plays pieces
using all kinds of signatures with multiple sharps and flats, or who
uses some method other than live performance to realize Renaissance
music in an historical tuning (be it meantone or 5-limit JI)?

Also, of course, how about a performer using non-standard keyboards
designed to support more than 12 notes per octave, and maybe even with
a layout especially modelled for JI?

Anyway, having announced some of my circumstances and predilections, I
would like to consider first two factors which might make me lean
toward meantone, and then a curious kinship between 1/4-comma meantone
and Pythagorean tuning in relation to the "JI" phenomenon.

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2. Meantone attractions: WYSIWYG and safety "at home"
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For an advocate of meantone in a Renaissance context, one line of
argument might be to invoke a principle associated with the UNIX
operating system: "When you get 90% of the benefit for 10% of the
effort, a compromise may be a good thing."

However, meantone as well as 5-limit JI can involve nontrivial
complications for Renaissance music, specifically experimental music
such as the extended chromaticism of Lasso or even the enharmonicism
of Vicentino.[2] In either tuning system, for such music we must in one
way or another step beyond the usual 12 notes per octave.

To explain why the complications of a meantone tuning beyond 12 notes
seem "natural" to me, while 5-limit JI seems "gratuitously difficult,"
I might focus on two personal preferences, the first of which I
humorously call WYSIWYG ("What You Spell Is What You Get"), and the
second, "safety at home among the _musica recta_ notes."

By WYSIWYG, in a "meantone vs. 5-limit JI" context, I mean the
principle that intervals normally spelled as concords should actually
sound as concords on the keyboard. That is, for example, D-A should be
a concordant fifth -- but G#-Eb or C#-F, a diminished sixth and
diminished fourth respectively, need not provide an acceptable fifth
or major third respectively, a result hardly advertised by their
spellings.[3]

Another side of this preference is the principle of "safety at home"
-- that is, among the _musica recta_ notes (diatonic notes plus Bb)
there should not be unpleasant "surprises" such as a D-A Wolf fifth.
Such exciting complications are to be expected as one adventures
beyond Eb or G# -- but hardly when a playing an interval which
continually occurs in almost any mode.[4]

From a practical view, of course, I might observe that D-A comes up in
Renaissance music far more frequently than an Ab or D#, let alone one
of Vicentino's intriguing enharmonic steps beyond the first 19 notes
of 1/4-comma meantone or 31-tet.

The "manual choreography" of playing a 14-note meantone scheme on two
12-note keyboards requires some planning, but the complications
reinforce the sense of playing an audacious piece, sharing in a way in
the composer's adventure. Negotiating an experimental piece by Lasso,
like climbing a mountain or journeying into space, may require some
"special gear."

In contrast, a Wolf fifth at D-A feels like a booby trap in my own
livingroom. Each time the interval comes up -- as the final of Dorian,
and in a cadential role at final or internal cadences in other modes
-- I must be sure in 5-limit JI to play all D's on the right keyboard.

One might say that with meantone, the tuning tends for me to reinforce
a certain worldview of musical "distance" and "adventure": _musica
recta_ is safe home territory, a secure base from which to venture out
into chromatic or even enharmonic spaces. Further, the keyboard
reflects the note spellings on the page: G# and Ab are two notes (as
advertised), while D is a single note which can be trusted to form a
concordant fifth D-A.

This raises a question, of course, of how musicians from different
backgrounds and tuning perspectives might define an "adventurous"
interval or composition. For someone coming from a background of
playing 18th-19th century music in all 12 keys -- as opposed to
Renaissance music with its 12 modes -- the idea of _musica recta_ and
_musica ficta_ notes might be more of a historical curiosity than an
everyday worldview. A Wolf at G#-Eb might seem just as much an
"unwelcome" complication as one at D-A.

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3. Pythagorean/meantone affinities: linear or selective JI?
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Interestingly, Renaissance meantone in its 1/4-comma version (with
pure major thirds at 5:4) shares the attractions (for me) of WYSIWYG
and safety at home with my other favorite tuning system: medieval
Pythagorean, or 3-limit JI.

In Pythagorean, as in meantone, What You Spell Is What You Get: D-A is
a concordant (and indeed pure) fifth, while G#-Eb is, as advertised, a
diminished sixth. The _musica recta_ notes are safe, with "adventure"
agreeably reserved for the territory beyond the usual Eb-G# range
which some experimental pieces do explore.

From a structural or analytical viewpoint, these tunings share a style
which I might call "linear JI": the choicest concord of the system is
made pure and beatless (a 3:2 fifth in Pythagorean, a 5:4 major third
in 1/4-comma meantone), but other intervals (apart from the 2:1
octave) are tuned "up the line," as it were, with all regular fifths
having the same size.

In Pythagorean, of course, these fifths are themselves the prime
concords to be made pure; in 1/4-comma meantone, each fifth is
tempered or narrowed 1/4 syntonic comma so as to achieve pure major
thirds. In each case, the other, "non-choice" intervals fall into
place.

To describe this approach of tuning "up the line" based on the ratio
of a single pure concord (plus the octave, of course), I suggest the
term "linear JI." Here I am indebted to Paul Erlich -- if I have the
attribution correct -- who replied to my question on this Tuning
List about a "JI lattice" for Pythagorean by saying that it would be
simply a line.

In the case of Pythagorean, we have a tuning which is not only
"linear" but also "3-limit JI" in a traditional sense: not only the
choicest concords of medieval polyphony (fifths at 3:2), but all
_simpler_ concords (octaves at 2:1, and unisons at 1:1), are just.
Further, of course, all stable and unstable intervals in Pythagorean
are derived as integer ratios, another hallmark of traditional JI.

With 1/4-comma meantone, however, we do _not_ have "JI" in this sense,
because while the choice major thirds at 5:4 are pure, the simplier
fifths are not (being 1/4 comma or ~5.38 cents narrower than 3:2) --
and apart from the major thirds and octaves, all intervals are
tempered, having irrational ratios.

However, the term "selective JI" might communicate this kind of
arrangement, where the _choicest_ concord is indeed pure, but other
concords participating in stable sonorities (here the minor third, as
well as the fifth and fourth) are impure or tempered.

Following this proposed typology, Pythagorean or 3-limit is the one
traditional JI system which is also linear, being based, aside from
octaves, on a single line of pure 3:2 fifths.

In contrast, 1/4-comma meantone is not only linear (deriving from a
line of 5:4 major thirds) but "selective," because the simpler stable
concord of the fifth as well as the complementary concord of the minor
third are neither pure nor based on integer ratios.

While applying the term "linear and selective JI" to a tuning such as
1/4-comma meantone with mostly tempered intervals may seem a rather
free poetic license, I would urge that this term might be more
descriptive than "non-just, non-equal, tuning." Above all, the fact
that the beloved 5:4 major thirds _are_ pure seems to me worthy of
notice, as well as the way that this "natural" landmark on the
meantone spectrum serves as an attractant for the tuning.

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5. Lines and lattices
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Traditionally, "JI" has tended to mean a system where all stable
concords are pure or beatless, and where all intervals are derived
from integer ratios.

Here I would like to propose as one variant a looser interpretation
holding simply: "The choicest concord in a given system of polyphony
should be pure." Both Pythagorean and 1/4-comma meantone are tuning
systems based on the "loose JI principle" in this sense, with
Pythagorean (3-limit) additionally meeting the classical test.

From a theoretical and practical viewpoint, the "linear" quality of
both Pythagorean and 1/4-comma meantone stands in contrast to the
intricate lattices of 5-limit, 7-limit, and higher JI systems. One
might ask what musical or cultural ramifications -- tuning theory
being also a kind of culture -- stem from this choice of line or
lattice as the realization of a quest for pure concords.

Also, might the recent theoretical viewpoint describing Pythagorean
as "zero-comma meantone" be something more than an ingenious
contradiction of historical common sense, which defines meantone
precisely as the compromising of fifths for the sake of more blending
thirds? Might this terminology point to the simplicity of a tuning "up
the line," as contrasted to the intricacies and delicacies of a
5-limit or higher lattice for classic JI?

From this point of view, both 0-comma meantone (classic 3-limit, pure
fifths) and 1/4-comma (selective 5-limit, pure major thirds) mark
points where the meantone spectrum intersects with the world of "JI."

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6. Linear JI: beatlessness and beatfulness
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From a musical point of view, linear JI systems (classic 3-limit, and
1/4-comma meantone or selective 5-limit) involve a mixture of
beatlessness and beatfulness.

In Pythagorean, the stable medieval concords and the complete
three-voice harmony of the trine (outer octave, lower fifth, upper
fourth, e.g. D3-A3-D4) are of course beatless, while unstable
sonorities involving intervals such as 81:64 major thirds or 32:27
minor thirds are "beatful." Here I use a MIDI-style notation where
middle C is C4, and higher numbers represent higher octaves.

In 1/4-comma meantone, stable Renaissance combinations dividing a
fifth into two adjacent thirds (e.g. F3-A3-C4) mix a beatless major
third with a mildly beatful minor third and a somewhat more
prominently beatful fifth -- the latter two intervals both tempered by
~5.38 cents, but the fifth generally considered to be more sensitive
to such a narrowing.

A number of 16th-century keyboard pieces end on a sonority with the
major third but without the fifth, e.g. D2-D3-F#3-D4. Might the
omission of the fifth (which Vicentino and Zarlino teach is required
along with the third for complete or perfect harmony) represent a
choice to conclude with a beatless sonority (at least in a 1/4-comma
tuning)?

In other settings, curiously, I somehow find that meantone fifths seem
to add to the flavor of much Renaissance music, such as Spanish pieces
around 1500 (e.g. Juan del Encinas) where it is still common to begin
and end on a sonority of octave, fifth, and upper fourth -- thirds
being still in the process of getting established as concluding
intervals. For buoyant, dancelike, pieces, the beating of those fifths
seems to add to liveliness of the music, and to introduce a kind of
acoustical paradox: the more perfectly concordant fifths are less pure
than the more imperfect (and now pervasive) major thirds.[5]

Maybe this kind of contrast around 1500, in the early meantone period,
is a kind of balance or obverse face to medieval Pythagorean tuning.

In conclusion, I wonder if "partly beatless and partly beatful"
tunings, including Renaissance 1/4-comma meantone and medieval
Pythagorean or 3-limit JI, might be a useful category.

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Notes
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1. This question has been debated since Renaissance times, with
Vicentino and Zarlino leaning toward meantone temperament (although
the latter also describes a 5-limit JI keyboard), and Salinas
reportedly favoring a 24-note JI system. Today this debate, as
directed to the playing of Renaissance music on keyboards, might
include the issue of "authenticity": does the evident prevalence of
meantone in the 16th century make it more "period-appropriate"? Or
might one argue that, at least when realizing vocal music of the
period, keyboards should approach the Renaissance vocal ideal of
5-limit JI as closely as technology permits, an argument advanced by
the presence (however uncommon) of JI keyboards in the 16th century
itself? Are the alternating 9:8 and 10:9 whole-tones of 5-limit JI to
be considered a melodic refinement, or a compromise of what some
current theorists call an ideal of "melodic consonance" in which
similar intervals of a scale should preferably have the same tuning?
At least one Renaissance theorist, Giovanni Battista Benedetti, also
raises the question of "pitch drift" because of cumulative comma
adjustments, a topic addressed by Paul Erlich's "adaptive JI." A
discussion might touch on these and other issues.

2. Here it may be noted that much audacious chromaticism is quite
possible within a standard 12-note meantone range such as Eb-G#, as
Easley Blackwood points out.

3. An advocate of 5-limit JI might very fairly observe, of course,
that just as "form follows function," so should musical spellings. If
one wishes to achieve pure fifths as well as pure major thirds in
Renaissance music, then the distinction between "D0" and "D-1" (using
whatever numbering system is desired to show syntonic commas) might be
considered no more "artificial" than that between Eb and D#.

4. With either Pythagorean or meantone, I tend to regard the step from
16 notes per octave to 17 as a significant leap, because adding A#
means that one of the keyboards is without a Bb, a _musica recta_
note. Psychologically, however, this complication is compensated for
by a sense of "high adventure" -- after all, A# is an outer limit of
the 17-note Pythagorean schemes proposed in the early 15th century.
As mentioned above, A# can also occur in some Renaissance pieces using
fewer than 17 notes, e.g. Lasso's Prologue to the _Prophetiae
Sibyllarum_ which uses the standard Eb-G# plus D# and A#.

5. However, a 5-limit JI advocate might point out that such
four-voice ensemble pieces were likely performed mostly by singers and
players of non-fixed-pitched instruments, where the technical factors
prompting a meantone solution for keyboards would not apply.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Renaissance meantone: "linear and selective JI"?

🔗M. Schulter <mschulter@xxxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>

🔗Joe Monzo <monz@xxxx.xxxx>

🔗Paul H. Erlich <PErlich@xxxxxxxxxxxxx.xxxx>