Re: Just Intonation definition (summing up)

Hello, there, and I would like, however distractedly, to comment on
the summing up of the just intonation (JI)/rational intonation (RI)
threads we have been experiencing.

For me, the debate is in some ways daunting, because I had supposed
that Gothic or neo-Gothic pieces with stable sonorities tuned at 2:3:4
and all intervals derived from integer ratios were manifestations of
"JI." At the same time, however, I was aware (although maybe I hadn't
considered the full implications) that JI itself seems to be a
post-medieval construct of European theory.

This is not a purely formal or theoretical debate for me, because over
the last few days I've been enmeshed in the process of writing a post
to describe a new (at least to me) tuning system which I was going to
describe as a kind of "JI," but I'm now calling "RI," although if
someone like Bill Alves described it indeed as JI, I would be
delighted.

Before getting to what I see and hear and feel as the musical heart of
the matter, at least from one point of view, I might comment that some
this debate seems to concern what I would call the "quantum" level of
JI.

For example, at some level, any synthesizer or the like which uses
integer ratios at some digital level might be described as JI -- and
any table or other solid object might be described as a shifting cloud
of elementary particles.

However, I'd prefer to focus on one issue which touches me in a
special way: the issue of what it means for an interval or sonority to
be "in tune."

From at least one JI perspective, JI implies a kind of "in-tuneness"
characterized by a locking in of partials, with the possible
implication that other kinds of intervals in such usual timbres or
settings are "not-thus-in-tune." Maybe we could call this an
"aesthetic of beatlessness."

Here I note Dave Keenan's important qualifications that in some
timbres numerically complex ratios may be "in tune" under this kind of
definition, and also that in some compositions of LaMonte Young, for
example, intervals with these ratios may occur in sonorities where
they are situationally "in tune," and thus "JI intervals."

Suppose we accept this definition of JI; the question now is, for
example, can integer-based Gothic/neo-Gothic music fit _comfortably_
within it. Or is there indeed something very precious that might be
compromised or lost by applying this kind of JI label?

What might be compromised is the democracy of intervals, the axiom
that each integer ratio and musical sonority counts.

Suppose, for example, we say that a traditional medieval Pythagorean
tuning is "JI" under a "harmonicity-based" definition because the
stable concords are indeed pure: 2:1, 3:2, 4:3 -- thus 2:3:4 or
sometimes 3:4:6. If we want to get "avant garde" and treat 1:3:9 or
1:6:9 as stable -- a possibility suggested by Jacobus of Liege around
1325 when he includes 9:1 as a "perfect concord" -- those sonorities
would be pure also.

However, what about such important sonorities as the 64:81:96 or
54:64:81 or 27:32:48 or 18:27:32 or 64:81:108? These unstable but
compatible sonorities are precious resources, coloristic and
cadential, and they have integer ratios fitting with the integral
structure of the tuning. To me, they are at once complex and "in
tune"; they are at their "correct" size -- although by no means the
only the pleasing or possible size.

Under a "harmonicity-based" JI definition, I might be in the position
of saying, "Yes, Pythagorean is a JI tuning because we can connect all
the notes with pure ratios of 3:2, even though most of the intervals
and sonorities are actually not in tune."

To me, this would be a kind of extraneous and invidious discrimination
against intervals which should enjoy equally "in tune" status, albeit
at different levels of complexity and stylistic "concord/discord."

Similarly, how about a tuning combining pure 3:2's and 4:3's and
2:3:4's with such "modern" integer-based sonorities as 28:33:42 or
14:17:21? To me, the numbers and sounds don't have to be bound by
tests of partials or harmonicity; they have their own beauty, and the
integer-based viewpoint is part of the total art, just as irrational
temperaments such as 29-tET have their own beauty and art.

One motivation for applying a label such as RI is to emphasize that
I regard simple and complex intervals alike as "in tune," all of them
in their own ways.

Please let me emphasize that this viewpoint is a musical as well as
mathematical one: intervals such as 32:27, 81:64, 27:16, and 16:9 are
a constant reality in the sound and texture of my music, and I want to
stay with definitions of what I am doing that affirm their dignity and
"in-tuneness." Curiously, Jacobus of Liege calls them compatible and
"consonant" (_consona_) to various degrees of "medial" or "imperfect"
concord, and I likewise call them "in tune," although at the same time
active and unstable (a dynamic virtue, not a vice or flaw); but some
more "modern" views focus on their "out-of-tuneness" or
"inharmonicity," or "nonjustness."

Such musicians as Prosdocimus say that a rightly intoned 81:64
is "fully perfected," striving to expand to a fifth, and this part of
my everyday musical experience; this means to me that the interval is
in tune, although not based on harmonics or locking-in partials. It
also has a conceptual beauty as a member of the community of intervals
arising from the integral powers of 3:2, and its musical and
mathematical logic reinforce each other, engaging at once the ear and
the intellect.

The debate may come down in large part to this: some of us see a
continuum of intervals which may be approached using either
integer-based or irrational ratios, typically producing somewhat
different formal and musical results, but with either approach
encompassing a spectrum of simple and complex intervals alike. The
integer-based side of this approach is the RI kind of viewpoint -- or,
even more expressively, Rational Continuum Intonation (RCI).

In contrast, others of us apparently seek to draw a contrast between
"JI" based on primarily on intervals with pure _harmonicity_, in
contrast to "nonjust" or "tempered" music with other types of
intervals, whether integer-based or otherwise.

For an RI enthusiast, the JI dichotomy of just/nonjust seems an
invidious distinction, a discriminatory measure imposed by some
foreign power on an equal polity of simple and complex ratios and
intervals -- although the "JI-as-harmonicity" exponent regards it
simply as a recognition of a paramount sensory and musical reality.

For a JI-as-harmonicity enthusiast, the RI concept seems to blur the
basic distinction between a just and a nonjust interval, contradicting
musical experience -- although the RI exponent routinely experiences
such intervals as 27:16 as musically natural, normal, and beautiful.

One important conclusion: the difference in viewpoints here may
reflect very audible differences in musical styles as well as fine
philosophical distinctions or mathematical niceties.

Either an integer-based Pythagorean tuning or an irrational
temperament like 29-tET fits neo-Gothic music, because both approaches
provide an attractive and stylistically right mixture of simple and
complex intervals (the simple intervals like 3:2 and 4:3 being pure in
Pythgorean and slightly impure in 29-tET, of course).

The main point may be the musical context: we have few stable and many
unstable intervals, and complexity is in style and flower for the
latter, whether based on integer or irrational mathematics.

In other styles, such as 16th-century European music, where more
intervals are stable and stable concords play a more pervasive role in
the texture, the "harmonicity-JI" paradigm may fit very nicely. The
difference between these viewpoints -- medieval vs. Renaissance, or RI
vs. JI today on this List -- is one of musical taste and context as
well as mathematical philosophy.

In modern physics, novel terms are sometimes chosen precisely to avoid
possible implications of more familiar terms that might be out of
place.

Following this precedent, I am leaning toward the RI category as a
description of integer-based music based on the "in-tuneness" of
simple and complex ratios alike, and potentially including any integer
ratio on the rational number line, often with a special delight in
the lore of large ratios.

If one follows the principle of the philosopher Grice that one should
strive to communicate as meaningfully and economically as possible,
then RI might be the most communicative term for such music. This is
not to repudiate a JI characterization, only to choose the term which
may maximize light and minimize possibly unnecessary definitional
friction or tension.

There is so much more to say, so many valuable contributions to
respond to, including the engaging scales of Jacky Ligon, which he has
described as RI.

If this thread marks the recognition of a new category of tuning, as
well as chronicling a diversity of views on JI as an established
category, then it may have served a creative purpose indeed.

Most respectfully,

Margo Schulter
mschulter@value.net

Margo wrote,

>To me, this would be a kind of extraneous and invidious discrimination
>against intervals which should enjoy equally "in tune" status, albeit
>at different levels of complexity and stylistic "concord/discord."

This discrimintation is no different that that offered by the outspoken
advocates of JI (no matter what definition they operate under), nor that of
medieval theorists like Jacobus with terms like "perfect" or "stable".

>Similarly, how about a tuning combining pure 3:2's and 4:3's and
>2:3:4's with such "modern" integer-based sonorities as 28:33:42 or
>14:17:21? To me, the numbers and sounds don't have to be bound by
>tests of partials or harmonicity; they have their own beauty, and the
>integer-based viewpoint is part of the total art, just as irrational
>temperaments such as 29-tET have their own beauty and art.

Well, that is the point where Dave Keenan's definition becomes a bit
problematic. For example, if you have a piece of music that uses the chords
above, and the notes involved cannot be fully connected with a chain of
consonant sonorities; but the piece has a provisional last movement where a
few additional pitches do allow this connection to happen -- would you then
say the piece is JI if the last movement is performed, but not JI if the
last movement is left out?

>Please let me emphasize that this viewpoint is a musical as well as
>mathematical one: intervals such as 32:27, 81:64, 27:16, and 16:9 are
>a constant reality in the sound and texture of my music, and I want to
>stay with definitions of what I am doing that affirm their dignity and
>"in-tuneness."

One objection that I like to bring up is, a lot of people hate numbers, so
why bring up numbers that are too high to be perceptually relevant? Clearly,
four 3:2 fifths linked together yield an 81:64 major third, but why actually
do the math? Isn't it better to just call it a Pythagorean major third, with
the explanation that Pythagorean tuning has pure 3:2 perfect fifths and the
chain of fifths is notated in the usual way? Or if you wish to be
mathematical, writing 3^4 or 3^4/2^6 seems much more appropriate in this
case -- who can hear the relationship between the 81st and 64th harmonics?

>Curiously, Jacobus of Liege calls them compatible and
>"consonant" (_consona_) to various degrees of "medial" or "imperfect"
>concord, and I likewise call them "in tune," although at the same time
>active and unstable (a dynamic virtue, not a vice or flaw); but some
>more "modern" views focus on their "out-of-tuneness" or
>"inharmonicity," or "nonjustness."

Well, the relevant point is that they cannot be precisely tuned by ear
without setting up much of the full chain of fifths leading from one note to
the other -- in sharp constrast to the normally considered "JI" intervals
like 5:4.

>Such musicians as Prosdocimus say that a rightly intoned 81:64
>is "fully perfected," striving to expand to a fifth, and this part of
>my everyday musical experience; this means to me that the interval is
>in tune, although not based on harmonics or locking-in partials.

As you know, anyone trying to equate "JI" with "in-tune-ness" gets promptly
shot down by me :), because melodic and motivic considerations often make JI
sound "out-of-tune". In some contexts, a meantone or some other non-JI
tuning might be the ideal of overall musical "in-tune-ness". Thus this
notion of "in-tune-ness" that you bring up doesn't really bear on the
definition of JI, it seems.

>It
>also has a conceptual beauty as a member of the community of intervals
>arising from the integral powers of 3:2, and its musical and
>mathematical logic reinforce each other, engaging at once the ear and
>the intellect.

Well, the intellect of very few medieval music-listers would be engaged by
these numbers, as they would have been totally unaware of them.

>One important conclusion: the difference in viewpoints here may
>reflect very audible differences in musical styles as well as fine
>philosophical distinctions or mathematical niceties.

May I suggest, Margo, that one reason you may see the argument in the way
you do may be that you've concerned yourself with tuning systems that all
begin from a Gothic model, in which there is really (in an octave-invariant
sense) only one type of stable consonant interval, thus the
JI-as-harmonicity exists in one dimension only -- and you haven't concerned
yourself as much with the case where there are two or more independent,
incommensurable stable consonant intervals . . . and thus there are a host
of issues relevant to this debate that you haven't spent a lot of time
thinking about?

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

> http://www.egroups.com/message/tuning/16353
>
> Margo wrote,
>
> > Please let me emphasize that this viewpoint is a musical
> > as well as mathematical one: intervals such as 32:27,
> > 81:64, 27:16, and 16:9 are a constant reality in the sound
> > and texture of my music, and I want to stay with definitions
> > of what I am doing that affirm their dignity and "in-tuneness."
>
> One objection that I like to bring up is, a lot of people
> hate numbers, so why bring up numbers that are too high to
> be perceptually relevant? Clearly, four 3:2 fifths linked
> together yield an 81:64 major third, but why actually do
> the math? Isn't it better to just call it a Pythagorean major
> third, with the explanation that Pythagorean tuning has pure
> 3:2 perfect fifths and the chain of fifths is notated in the
> usual way? Or if you wish to be mathematical, writing 3^4 or
> 3^4/2^6 seems much more appropriate in this case -- who can
> hear the relationship between the 81st and 64th harmonics?

Thanks, Paul! This kind of reasoning is a big part of why
I like to use prime-factor notation. I realized long ago
that 3^4 explained to me a lot more about the audible nature
of the 'Pythagorean major 3rd' than 81:64 did.

> > One important conclusion: the difference in viewpoints here
> > may reflect very audible differences in musical styles as
> > well as fine philosophical distinctions or mathematical
> > niceties.
>
> May I suggest, Margo, that one reason you may see the argument
> in the way you do may be that you've concerned yourself with
> tuning systems that all begin from a Gothic model, in which
> there is really (in an octave-invariant sense) only one type
> of stable consonant interval, thus the JI-as-harmonicity exists
> in one dimension only -- and you haven't concerned yourself as
> much with the case where there are two or more independent,
> incommensurable stable consonant intervals . . . and thus there
> are a host of issues relevant to this debate that you haven't
> spent a lot of time thinking about?

With all due respect, Margo, while I am in general agreement
with your conclusion, at the same time I think that Paul is
taking account of an important consideration here. The key
word is 'incommensurable': no 'in-tune' ratios with different
prime-factors will ever be an exact match mathematically.

Those other primes add multiple dimensions to the sonic
universe, and thinking in terms of these higher dimensions
is difficult at best.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

Margo and all,

I want to say that the real beauty of this RI definition is that it
is all encompassing of simple beatless JI intervals (at its core),
and allows a spectrum of complexity, which might not readily match
the classic JI label. This is such an appropriate term for a system
which considers and makes use of both of the philosophical poles of
tuning by ratios. I find an unbelievable expanse of common ground
within this post from Margo.

Thanks for this,

Jacky Ligon

--- In tuning@egroups.com, "M. Schulter" <MSCHULTER@V...> wrote:
> One motivation for applying a label such as RI is to emphasize that
> I regard simple and complex intervals alike as "in tune," all of
them
> in their own ways.
>
> The debate may come down in large part to this: some of us see a
> continuum of intervals which may be approached using either
> integer-based or irrational ratios, typically producing somewhat
> different formal and musical results, but with either approach
> encompassing a spectrum of simple and complex intervals alike. The
> integer-based side of this approach is the RI kind of viewpoint --
or,
> even more expressively, Rational Continuum Intonation (RCI).
>
> For a JI-as-harmonicity enthusiast, the RI concept seems to blur the
> basic distinction between a just and a nonjust interval,
contradicting
> musical experience -- although the RI exponent routinely experiences
> such intervals as 27:16 as musically natural, normal, and beautiful.
>
> In other styles, such as 16th-century European music, where more
> intervals are stable and stable concords play a more pervasive role
in
> the texture, the "harmonicity-JI" paradigm may fit very nicely. The
> difference between these viewpoints -- medieval vs. Renaissance, or
RI
> vs. JI today on this List -- is one of musical taste and context as
> well as mathematical philosophy.
>
> Following this precedent, I am leaning toward the RI category as a
> description of integer-based music based on the "in-tuneness" of
> simple and complex ratios alike, and potentially including any
integer
> ratio on the rational number line, often with a special delight in
> the lore of large ratios.
>
> There is so much more to say, so many valuable contributions to
> respond to, including the engaging scales of Jacky Ligon, which he
has
> described as RI.
>
> If this thread marks the recognition of a new category of tuning, as
> well as chronicling a diversity of views on JI as an established
> category, then it may have served a creative purpose indeed.
>
> Most respectfully,
>
> Margo Schulter
> mschulter@v...