Yahoo Tuning Groups Ultimate Backup tuning Re: Gothic/Neo-Gothic "JI" -- or "RI"? (reply to Dave Keenan)

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Just Intonation (JI) and Rational Intonation (RI)
Reply to David Keenan on Gothic/neo-Gothic RI
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Hello, everyone, and I would like to respond to a recent article by
Dave Keenan prompted by some offlist correspondence. In reply to his
article, I would like to propose a new category of "rational
intonation" (RI) for Gothic/neo-Gothic tuning systems based
exclusively on integer ratios, but not on the kind of "harmonically
based" intonation which Dave ably argues is a basic feature of "just
intonation" (JI).

In the first section of this response, I consider the interesting
conclusion that "JI" may be an anachronistic term when used to
describe Gothic or neo-Gothic music in rational tunings indeed based
on integer ratios both simple and complex, but not based on a
systematic use of or preference for the harmonic or subharmonic
series.

If great theorists such as Jacobus of Liege (c. 1325) or Prosdocimus
(1413) could get along very nicely without, to my knowledge, invoking
a term such as "just intonation," maybe I would do best to follow
their example and avoid a Renaissance and later term which may invite
spurious associations. To distinguish Gothic and neo-Gothic tunings
based exclusively on integer ratios from neo-Gothic temperaments, the
term RI serves nicely while avoiding such associations.

In the second section, I briefly consider an interesting possible
ramification of Dave's approach: that some musics may be _both_ just
and tempered (e.g. a composition in a temperament with relevant
intervals within 0.5 cents of pure); and others may be _neither_ just
nor tempered (e.g. Gothic and neo-Gothic pieces in rational intonation
systems not "harmonically based").

There I propose the term "nuancing" to describe the neo-Gothic concept
of a continuum of ratios, RI or irrational, as opposed to either the
harmonic-series-based outlook of JI or the calculated compromising of
pure intervals involved in temperament.

Much of what follows might best be read in connection with my series
of articles offering "A Gentle Introduction to neo-Gothic
Progressions," where such relevant concepts as sonorities, cadences,
and intonational flavors are presented at length.

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)

Please let me add that this process of dialogue has persuaded me to
describe a new (to me) integer-based tuning system I shall soon be
posting about as RI rather than "JI."

It's very exciting to be involved with such tunings and discovering
(or rediscovering) all kinds of delightful surprises, and the idea of
a new (or at least less familiar) category of rational tunings adds to
the excitement of this process.

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1. Gothic/neo-Gothic "JI": An infelicitous anachronism?
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Dave, the 1811 definition of just intonaton you have quoted both
publicly and in our e-mail messages seems an ideal expression of your
sense of "JI" as a system based on "harmonic tuning."

From my point of view, definitions dating to 1211 or 1411 might be more
relevant to my own musical approach, and this brings us to an interesting
and possibly humorous quirk of this whole dialogue which maybe leads to a
curious conclusion.

The term "JI" might arguably be more of an anachronism than a historically
appropriate term when applied either to medieval European music or to my
neo-Gothic endeavors involving systems using integer ratios (or the
closest approximations I can get in practice).

The problem is that JI may be too _modern_ a concept to apply very well to
music not based systematically on the harmonic series. Both 13th-14th
century theorists and I agree that 32:27 is a regular and relatively
concordant (although unstable) interval, actually somewhat more concordant
as a bare interval than the simpler 9:8 -- without any consideration of
such things as the 27th and 32nd harmonics, which I would consider quite
irrelevant in this connection.

The intonational world of 1811 -- or 1529 or 1558, for that matter (the
5-limit systems of Fogliano and Zarlino, which we would agree _are_ JI) --
has very different musical assumptions than the world of 1211 or 1311 or
1411 which is a starting point for my own Gothic/neo-Gothic integer-based
endeavors (and endeavors involving temperaments also).

This raises an interesting question: does the term "just intonation" get
used by medieval theorists at all? If not, might not I do better just
using integer ratios and leaving "JI" to people involved with styles where
"lower integer ratios" and "harmonically based music" are fitting
concepts?

Please let me add that I do consider barbershop singing as JI, because it
is based on pure integer ratios or very close approximations -- low
integer ratios and harmonically locking intervals are synonymous for the
applicable timbres. Just as it is impossible to draw a line with zero
thickness, so realized JI will involve some variance from precise integer
ratios -- but the idea of integer ratios doesn't seem inconsistent to me
with such necessary variances, nor to require that singers tuning pure
intervals know the mathematical ratios involved.

However, I might add, your example of barbershop singing is more than
an ingenious (and eloquently argued) debating point. The issue is not
merely whether and how we should address numbers and sounds, but
exactly what those numbers stand for. To you, 4:7 is evidently not so
much an integer ratio as a beatless locking of partials 4 and 7. To
me, 7:4 is an integer ratio like 16:9 representing one possible nuance
upon the general theme of a minor seventh inviting contraction to a
stable fifth by stepwise contrary motion.

As the numbers get higher, the difference between our conceptions (if
I interpret yours correctly) becomes more and more obvious. To me,
32:27 or 13:11 or 27:23 is a possible nuance for a minor third which
might cadentially contract to a unison. To you, I suspect, describing
these ratios as "JI" would imply relations between _locking-in
partials_ of 27:32, 11:13, or 23:27 -- something I do not mean to
imply, and evidently foreign to medieval European theory also.

Please let me also add that I regard the Hammond organ as a rational
intonation system in a purely formal sense, not a characteristic or
exemplary one. Really, my main point is just that I'm inclined to be
inclusive, but to recognize that some cases are quite formal or
marginal, rather than to take the risk of excluding something that
maybe I'd want to include.

From your point of view, however, I suspect that the absurdity of such
an instrument as "JI" goes beyond the use of rational ratios to
emulate tempered ones (or sometimes possibly vice-versa). For example,
in a neo-Gothic style, a pure (i.e. exact) 14:11 is almost identical
to 16/46 octave, and likewise a 127:100 to 10/29 octave. In either RI
or tempered (i.e. irrational) form, these ratios and intervals are
cherished for their intriguing complexity, not as incidental offshoots
of simpler ratios but as regular and basic forms in their own right.

The point is that the Hammond has no pure or "harmonically based"
intervals other than the octave. From an RI perspective with its
nuanced gradations on a dense rational number line, this is merely a
curious marginal case; from your JI perspective, it is additionally a
grating musical anomaly.

Anyway, to state my own musical perspective positively, I tend to view
RI as following and further developing an older approach to integer
ratios not based on the harmonic series as a basic criterion, an
approach embracing the continuum of intervals seen as a rational
rather than real number line. Actually I suspect that the RI and
tempered neo-Gothic systems I most commonly use in practice are quite
similar -- more similar to each other than either is to "JI."
Specifically, I find that I lean very strongly in practice toward
regular or virtually regular tunings, either RI (i.e. Pythagorean) or
tempered (e.g. 29-tET, Noble Fifth, e-based, etc.).

Why do I approach things in this way? Because it seems to fit the music I
am drawn to in theory and practice. Were I drawn to 16th-19th century JI
as a main priority, your definition might very nicely fit my focus.

Anyway, to answer your question as to motivation, I follow the "tuning
by integer ratios" concept because it seems to fit my own medieval
musical roots and tradition, and also the music I'm actually
producing. From your viewpoint, of course, much of what I do might be
described simply as using rational ratios to come up with various
kinds of tempered intervals -- or better, "non-harmonically-based"
or "nuanced" ones.

Maybe something like "rational continuum intonation" (RCI) might
describe my own viewpoint. One sign of this might be that when Joe
Monzo was discussing 75:64, my immediate reaction was nothing to do
with 5-limit ratios -- it was, "this is an interesting place on the
continuum between 32:27 and 7:6, sort of moving toward the outskirts
of the 7:6 valley, and 75:64 is very close to 5/22 octave, the regular
minor third of 22-tET."

Up to now, my implicit reasoning has been rather like the following:

1. Gothic/neo-Gothic integer-based tunings are indeed based on integers.
2. JI tunings are based on integer ratios (or especially "valley" ones).
3. Therefore Gothic/neo-Gothic integer-based tunings are a form of "JI."

However, if the "harmonically-based" concept of JI is the correct or
generally accepted one, then I might wisely abandon the use of the
term and invent a new one -- as I have done with "Rational Intonation"
or RI.

To me, either 9:8 or 32:27 is a primary rational ratio; the dichotomy
harmonically-based/not-harmonically-based doesn't, for me, fit with a
Pythagorean tuning system, because both of these intervals are somewhat
compatible but neither is stable, and in fact I agree with medieval
theorists (and composers in practice) that the bare 32:27 is somewhat
_more_ concordant than the bare 9:8.

Note that 32:27 is some distance from any "valley," if I'm correct -- an
81:80 from 6:5, and a 64:63 from 7:6. Yet I regard it as basic and
relatively concordant, although unstable -- the idea of a distinction
between "harmonically-based" and "other" intervals is something that seems
foreign to me in medieval or neo-medieval theory. If this is what JI
implies, then the term might well be avoided for this music.

We would agree that most traditional "JI" (and RI) systems include
_some_ valley ratios: I would be unlikely to describe any Gothic or
neo-Gothic RI system that didn't include a pure 3:2 and 4:3. However,
a pure 2:3:4 or 6:8:9, etc., doesn't imply a systematic "harmonically
based" outlook; and I would say that 54:64:81 or 14:17:21 expresses my
outlook on RI just as much as such simple ratios. Indeed the contrast
between simplicity and complexity is a large part of the point, as we
discussed in our joint paper on Noble Mediants of complexity.

In short, I am open to an argument which might be consistent with both of
our musical approaches: that simply because I am using integer ratios,
this does not necessarily mean that I am following the rather _modern_
concept of JI.

Trying to outline a possible case for this viewpoint, I might argue that
indeed "the Just scale" by default refers to the 5-limit scale of Fogliano
or Zarlino, and marks a point where (as you pointed out on the List) the
_senario_ (1,2,3,4,5,6) becomes a basis for vertical textures generally,
even if the harmonic series itself is still to be discovered in the early
17th century.

A concept where 3:2, 9:8, and 32:27 or 27:16 all represent a spectrum of
intervals not divided into "harmonically-based/not-harmonically-based" --
although they certainly have differing degrees of concord -- is maybe best
not confused with JI in this historical sense, especially since I am
unaware of any medieval theorist calling it "JI."

In other words, such an argument might conclude that my concepts follow my
historical roots in practice and theory, but that the label of "JI" with
its harmonic-based implications is an anachronism which may mainly
confuse, rather like trying to use Rameau's terminology to describe
14th-century vertical technique. Carl Dahlhaus wrote a book about the
misconceptions which can result from such dubious use of language, and
as an advocate of Gothic and neo-Gothic tunings and music, I have
every reason to favor definitional clarity rather than confusion.

We all have our own theoretical approaches, and maybe RI is as good a
term as any for mine. A very enjoyable offshoot of the 75:64 discussion
for me was determining that this ratio represents a weighted intermediate
between 6:5 and 7:6 where y/x=9/2:

(7*9 + 6*2) (63 + 12) 75
----------- = --------- = --
(6*9 + 5*2) (54 + 10) 64

Maybe a lot of my approach might be called not so much tuning by
integer ratios as _nuancing_ by integer ratios -- and we certainly
agree that either the rational or real number line has the property of
density.[1]

Given the nature of my approach, maybe I should recuse myself from
further attempts to define JI, focusing on defining RI as a distinct
approach differing from either JI or temperaments involving irrational
numbers, although in some ways resembling either (pure valley ratios
plus lots of complex and finely nuanced ones).

Anyway, that 1811 quote really made a point for me: if this definition
doesn't seem to fit my concept which I've been calling "JI," then maybe I
should find a quote from 1211 or 1311 or 1411 which might -- but can I
find _any_ definition of "just intonation" from this era? If not, maybe
it's an extraneous concept which I shouldn't borrow just for the sake of
familiarity or popularity at the expense of clear communication.

A special advantage of avoiding the whole "JI" area of terminology is
minimizing the risk that someone might confuse neo-Gothic RI with JI
aesthetics which _are_ harmonically based. In other words, I might do
well to avoid approaching a position where I appear to be proposing
definitions based on 13th-14th century or derivative music for
application also to "JI" music in 19th-century style, say -- as
unfortunate as the opposite situation, with which I am all too
familiar.

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2. Just and/or tempered -- or both, or neither?
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From your JI perspective based on a locking-in of partials, you raise
the interesting possibility that a tempered tuning may nevertheless
achieve this "locking-in" for the relevant set of just intervals in a
given context, and therefore be both "just" and "tempered."

For example, if 1:2 octaves and 2:3 fifths were the only relevant pure
ratios (note that I follow your ascending order of terms, to suggest
relations between partials), and we adopt your proposal of an
acceptable variance of 0.5 cents for a "just" interval, then 41-tET
(fifth 24/41 octave, ~702.44 cents, ~0.48 cents wide) would be a just
_and_ tempered tuning, or indeed a "just temperament." As you noted,
53-tET would be a yet more accurate example (fifth 31/53 octave,
~701.89 cents, ~0.07 cents narrow).

If we are considering 5-limit JI, then as you pointed out, 118-tET
(fifth 69/118 octave, ~701.69 cents, ~0.26 cents narrow; major third
38/118 octave, ~386.44 cents, ~0.13 cents wide) would be a just
temperament.

In contrast, I might argue that with a Pythagorean tuning used in a
Gothic or neo-Gothic context where something like 64:81:96, for
example, is considered as relatively blending or "concordant" although
unstable, we have neither a "just" (harmonically based) nor a
"tempered" (irrational or deliberately compromised) tuning, but a
_nuanced_ rational intonation (RI).

The same conclusion may apply to neo-Gothic RI systems featuring pure
3:2 fifths and 14:11 major thirds and 33:28 minor thirds -- these
thirds are neither harmonically-based nor tempered, but "nuanced"
using integer ratios. A neo-Gothic 28:33:42 is a point on a continuum
of flavors which might be seen as variations on the traditional
Pythagorean 54:64:81 -- the basic theme is an unstable sonority where
the lower minor third invites contraction to a unison, for example,
while the upper major third invites expansion to a fifth.

Similarly, either 54:64:81:96 or 12:14:18:21 is a possible nuance or
intonational variation on the basic theme of a sonority, unstable but
"compatible," with an outer minor seventh inviting contraction to a
stable fifth. The former is the "classic Gothic" flavor of Pythagorean
tuning, while the latter is a more "modern" or "streamlined" variant;
but they are considered equally "in tune," representing fine gradients
sharing a similar position on the continuum of concord/discord. To
borrow a term from Jacobus of Liege which he uses to describe the
16:9, either form of minor seventh sonority might be called an
"imperfect concord," appreciably tense but with some degree of blend
also.

To conclude, I might remark that I have adopted the term "JI" for my
integer-based Gothic and neo-Gothic efforts largely by inertia: "Since
this tuning doesn't have any irrational ratios, and has pure trines at
2:3:4, it seems to fit in the 'just' category."

However, maybe the categories "just" and "tempered" weren't designed
mainly with Gothic/neo-Gothic music in mind. Why not a new concept of
"nuanced intervals," rational (RI) or irrational (neo-Gothic
temperaments), which better fits the continuum of intonational flavors
I've attempted to describe in my "Gentle Introduction?"

Thanks to you, Dave, and the many others who have contributed to this
dialogue leading me to consider proposing a new category of tunings.

----
Note
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1. My longer term "rational continuum intonation" (RCI) emphasizes
that not only is the system based exclusively on integer ratios (as
are many "harmonic-tuning-oriented" JI systems), but that it favors a
continuum of intonational nuances and flavors as defined by such
ratios of any desired complexity on a dense number line, rather than a
tuning set derived mainly from harmonic/subharmonic relations.

Most respectfully,

Margo Schulter
mschulter@value.net

Re: Gothic/Neo-Gothic "JI" -- or "RI"? (reply to Dave Keenan)

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Monz <MONZ@JUNO.COM>

🔗ligonj@northstate.net