Yahoo Tuning Groups Ultimate Backup tuning Re: JI as tuning by multitudes (integer ratios) -- Hammond too

[Please note that this is a response to posts from last week, and that
more recent remarks by David Finnamore and others raise all kinds of
nuances and viewpoints. What follows may indeed by a "Platonic" (or
Boethian?) perspective, and yes, at least one person on this List is quite
ready to say that the Hammond organ is an ingenious example of a JI tuning
emulating a tempered system. Likewise, Monz's 64:75:96 is to me JI simply
because it involves all integer ratios; one of my early reactions was that
64:75 is rather close to the regular 22-tET minor third of 5/22 octave, or
vice versa. There are many more new points, and what follows is merely a
response to one stage of the dialogue which I hope may still be relevant.]

Hello, there, everyone, and I would like to respond to four
contributors who have posted recently on the topic of just intonation
(JI), thanking them all very warmly both for their invaluable insights
and for their stimulating provocations to further discussion.

--------------------------------------------------------------
1. To Dave Keenan: JI as tuning by multitudes (integer ratios)
--------------------------------------------------------------

Please let me begin, Dave, by expressing my admiration for your
eloquent statement of a vital viewpoint on the nature of just
intonation and of music in general with a very rich tradition behind
it.

Here, as is so often the case, our differences may reflect our
divergent philosophical approaches and first axioms. We are both
strongly oriented to historical traditions and concepts, but select
different aspects of history on which to focus. In explaining my own
viewpoint, I hope that I may help to bring these differences into
better perspective while promoting mutual understanding.

From my largely medievalist perspective, I might define JI in the
broadest sense as "tuning by multitudes," that is, integer ratios, in
contrast to "tuning by magnitudes" including intervals with irrational
ratios. Thus my definition _is_ a mathematical one, in line with the
medieval European quadrivium (arithmetic, geometry, astronomy,
music). We might say that JI involves arithmetic ratios, while
tempered systems involve geometric ones.

However, especially when placed in historical perspective, this
formally mathematical definition of JI normally has some concrete
musical implications, because typical JI systems feature some _small_
integer ratios as well as other more complex ones.

In Pythagorean tuning, for example, both medieval and modern theorists
recognize such simple ratios as 2:1, 3:2, 4:3, and 9:8, as well as
other notably complex ones. We have valleys (2:3:4, 3:4:6, 6:8:9,
etc.) as well as plateaux (e.g. 54:64:81) and indeed summits of
complexity (e.g. 512:729:972, Perotin's g-c#'-f#' or G3-C#4-F#4).

Similarly, LaMonte Young's intricate JI systems feature valley ratios
such as 3:2 or 7:6, as well as other very complex ones. We might say
that JI systems characteristically, although not universally, feature
a contrast between simple and complex ratios, both of which are part
of a given intonational whole.

Possibly many of our differences stem from our each subscribing to a
time-honored concept of JI:

(1) Just integer ratios as opposed to irrational tempered ratios;
(2) Just or "pure" intervals as opposed to "impure" or complex ones.

Please let me say that I consider either definition a valid viewpoint,
and that your able advocacy in favor of the second definition reminded
me of Pietro Aaron's description of meantone temperament in 1523.

Addressing the musical beginner setting out, possibly for the first
time, to tune a harpsichord or the like, he directs the reader to make
the major third C-E "as just and sonorous as possible," evidently
describing a pure 5:4 ratio in audible rather than mathematical terms.

From your perspective, I would gather, this kind of "justness" is not
only a characteristic of JI but its very essence.

From my perspective, JI embraces integer ratios of all kinds, whether
valley, plateau, or summit: 3:2, 32:27, and 243:128 are all part of
the Pythagorean "family," and partake of its "justness" in the generic
sense. At the same time, among these ratios, 3:2 additionally is a
Just or pure fifth, a valley interval making Pythagorean tuning not
only formally but _characteristically_ a "JI system."

This brings us to the issue of what I would describe as less
characteristic or more marginal JI systems such as that of the Hammond
organ which you discuss, where we have _only_ complex integer ratios,
as opposed to simple ones (3:2, 7:4, 11:8) or "intermediate" ones
(e.g. 13:11, 17:14, 19:16).

From my own viewpoint, I would say that the Hammond organ illustrates
the use of a formal JI system to emulate a system of tempered tuning,
here 12-tone equal temperament (12-tET). Similarly, Kirnberger's
"schisma fourth" at 10935:8192 (or a "schisma fifth" at 16384:10935)
represents what Owen Jorgensen has described as tuning 12-tET (or
actually an extremely close approximation) by "just intonation
techniques."

Given that we can approximate an irrational ratio such as 700 cents as
closely as desired using just or integer ratios (the "multitudes" of
medieval European theory), and likewise can approximate an integer
ratio as closely as desired using irrational or tempered ratios (as we
do when stating approximate values for just intervals in cents), such
marginal cases seem to me natural as well as inevitable.

Recognizing marginal cases of JI such as the Hammond organ -- and
these cases have a mathematical and musical beauty of their own --
need not prevent those of us who take a "JI as tuning by ratios"
perspective from acknowledging that valley ratios are characteristic
of most JI systems, and likewise the co-occurrence of valley,
plateaux, and summit ratios.

As mentioned in a previous article, I might suggest a distinction
between the uppercase Just ("pure") and lowercase just ("having an
integer ratio"). Alternatively, we might refer to the latter and
broader category of intervals simply as "JI intervals and ratios,"
reserving "just" for valley ratios or possibly also intermediate
ratios. Thus 81:64 is "a JI ratio," but not a "Just" or "pure" one.

While my definition of JI is formally mathematical, this approach is
not without its aesthetic motivations. Specifically, I find that
complex JI intervals and sonorities may not too infrequently actually
seem more "concordant" or "harmonious" to my ears than intervals or
sonorities featuring only valley ratios a:b where a*b <= 105, Paul
Erlich's proposed limit for a "simple" ratio.

For example, in some timbres, I find the complex JI sonority 64:81:96
in Pythagorean tuning, e.g. f-a-c' or F3-A3-C4, to seem more
"blending" or "concordant" than the simpler and purely tuned 14:18:21,
the latter seeming more like a "special effects" sonority, and a very
striking one. Musically, the first sonority might seem more apt as a
_relatively_ concordant point of pause, the latter as a most memorable
point of cadential tension. To me, both are aspects of "JI," and a JI
system might well embrace and treasure both sonorities for their
unique qualities.

With regard to LaMonte Young, you raise an interesting point: certain
complex ratios, in certain special settings, may take on an aural
"primacy" or "quasi-simplicity" normally associated with "valley"
ratios only. Thus under your approach to JI, these complex ratios
might have a "JI" quality if and only if these unusual circumstances
are satisfied (e.g. extremely prolonged durations, special timbres),
while under mine, something like 32:27 is "a JI interval" simply by
virtue of being an integer ratio.

Whatever our differences stemming from the historical traditions of
"just vs. irrational" and "just vs. impure or complex," please let me
warmly concur with your distinction between pure and near-pure or
"quasi-pure" intervals.

For example, we agree that while a Pythagorean interval of 16 fifths
up minus nine octaves rather closely approximates a pure 9:7 (being
~3.80 cents narrow), this interval is "near-pure" or "quasi-pure"
rather than pure. Similarly, I would say that 10/31 octave defines a
"near-pure" 5:4, and 9/31 octave a "near-pure" 11:9, etc.

From my perspective, 16 fifths up minus nine octaves, or eight 9:8
whole-tones minus an octave (43046721:33554432), is additionally a "JI
interval" in its own right, since it is an integer ratio, but one
distinct from the pure (or Just) 9:7.

Above all, I see no reason why the term "just" cannot mean either
"integer-based" or "pure," depending on the viewpoint of the speaker
or the context.

Either viewpoint has its own historical and intuitive appeal, and also
its complications. If we take an "integer-based" approach, we must
acknowledge JI systems audibly indistinguishable from tempered
tunings, e.g. Kirnberger's realization of 12-tET or the Hammond
organ. If we take a "purely and perfectly in tune" approach, we must
recognize "JI systems including non-just intervals," and also "just
tunings" which may sound less concordant than "non-just tunings" of
the same musical categories of intervals (e.g. 14:18:21 and 64:81:96).

Might I hazard a guess that the "tuning by integers" approach might
reflect a medieval view, and the "purely in tune" approach might
reflect, for example, a Renaissance perspective when 5-limit concords
are pervasive and more complex sonorities restricted to a somewhat
"incidental" or at least cautious treatment (the suspension dissonance
having a particular expressiveness and power when set against this
general norm of homogenous concord)?

While favoring the "tuning by integer ratios" view, I would emphasize
that your view may reflect an equally important side of music and
history, and that such an exchange of perspectives adds much to the
common dialogue of this forum.

---------------------------------------------------
To Monz: Music, intellect, and audible distinctions
---------------------------------------------------

Your recent statement that tuning can involve more than music prompts
me to a somewhat different statement but one possibly kindred in its
intent: that in tuning and in music more generally, not every concept
and detail need involve audible distinctions.

It is a proper role of music (including its intonational aspects) to
engage the intellect and imagination as well as the sense of hearing.
Sometimes the reach of the intellect may exceed the firm grasp of the
senses, and this stretching or striving may itself be a high form of
musical art and science.

For example, while the intellect can readily grasp a ratio such as
14:11, some skilled experimenters have concluded that the ear cannot
distinguish this interval as a simple or primary ratio. To use this JI
interval, at once pure to the intellect but somewhat complex to the
ear, may thus be motivated at once by its tangible musical beauty and
its allegorical meaning.

Also, in a JI system combining pure 3:2 fifths and 14:11 major thirds,
a wonderfully complex ratio of 12544:9801 arises differing from the
ratio of 32:25 by a kalisma of 9801:9800, a proportion to delight the
intellect although too small to be perceived by the ear. The very
imperceptibility of this distinction makes it all the more rare and
engaging, a cherished gem in the array of the sonorous numbers.

In other areas of music, also, we have these artful subtleties
delighting the speculative or practical musician although without any
obvious perceptible consequences. Consider, for example, the canons or
ingenious and often elusive rules for deriving one part from another,
as used by Josquin and other composers: the audible result is no
different than if all parts were written out plainly, but the puzzle
is part of the total art.

Here I would add that the intellect should indeed be informed by the
senses, although not circumscribed by their limits. Further, as you
have often most eloquently emphasized, we should beware of taking the
restrictions of a given musical style or tuning system as defining
some universal standard.

-----------------------------------------------------
3. To Dan Stearns, on intonational poetry and passion
-----------------------------------------------------

As what I might term a _musico-mathematical poet_ of high repute, you
provide much inspiration as well as concrete substance for lovers of
JI and tempered systems alike. Thanks to you I am now familiar with
the Stern-Brocot Tree, even if my concept of a "periodicity" built
from such a basis is yet uncertain.

Some of your comments move me especially to affirm the _poetry_ of JI,
the beauty of the complex as well as the simple.

A standard physics text has a saying which nicely fits my JI
philosophy: "Do not disdain what the equations cast in your path." One
of the special pleasures of (re-)inventing a JI system is to discover
some new (at least to me) and complex ratio arising quite
serendipitously from simpler ones.

You and Jacky Ligon have pointed to "two T's" intimately intertwined
with just tunings: timbre and tempo. While Bill Sethares (and earlier
Ivor Darreg) has brought timbre to the fore, your remarks about tempo
or duration are also very important.

In fast tempos, I might agree if I read your comments correctly, we
tend to perceive intervals more "categorically" (e.g. "some kind of
major third or minor seventh") than specifically ("this sounds like an
81:64, this more of a 9:7 or 13:10").

Of course, there is also the vital variable of timbre: I am fascinated
by how a pure 12:14:18:21 (e.g. e-g-b-d' or E3-G3-B3-D4) can sound so
beguilingly suave and smooth in one timbre, and yet quite strident in
another.

Also, within a single JI system as well as between systems, there is
the factor of "contextual normality," rather analogous to the
principle of Newtonian physics that we perceive acceleration rather
than uniform velocity. Heard as a usual tuning of the minor third, a
32:27 has its own perfection and "naturalness"; when compared to a
pure 7:6, I find that this same interval has a certain "complexity" or
"cloudiness." Possibly the contrast is a bit like travelling outside
the atmosphere of a familiar planet, and realizing how much
atmospheric haze is a part of the surface environment often taken for
granted, indeed, a shaping feature of the ecosystem.

Similarly I find that either a more complex 27:32:48 or a simpler
12:14:21 (e.g. d-f-c' or D3-F3-C4) can be a "sweet" and pleasing
tuning of this unstable but compatible sonority. Heard in itself,
either version makes an apt intonational standard; having both
versions available enriches the expressiveness of a JI system.

What your tuning systems, Phi-based and otherwise, communnicate to me
especially is a love of diversity and its passionate cultivation.
Thank you for your mathematical expertise and musical poetry alike.

----------------------------------------
To Robert Walker, on JI and transitivity
----------------------------------------

If I read your remarks on "transitivity" and "intransitivity"
correctly, you have very aptly expressed one of my primary motivations
for defining "just" or "JI" intervals generally as those having
integer ratios, but "Just" intervals as those having _simple_ integer
ratios directly tuneable by ear or at least arguably distinguishable
as "primary acoustical colors," so to speak.

By "transitivity," I take you to mean that a JI interval plus a JI
interval will always produce another JI interval: e.g. 9:8 plus 3:2
gives us 27:16, another integer ratio, and therefore a "just" or JI
interval also.

By "intransitivity," I take you to be observing, however, that while
3:2 or 9:8 may be described as a "pure" or "Just" or "valley" interval
falling within the Erlich limit of a*b <= 105, here 3*2=6, 9*8=72,
they combine to generate 27:16, where a*b = 432, going outside this
Erlich limit of simplicity or "Justness."

Here it seems musically as well as mathematically elegant to me to say
that 3:2, 9:8, and 27:16 are all "JI intervals," and more specifically
part of the JI set derived from powers of 3:2, while recognizing the
different degrees of complexity.

Incidentally, as I remarked earlier in this article, since the
distinction in capitalization between "just" and "Just" may be an
overly fine one, maybe the adjective "JI" would be preferable for the
first or general concept: "27:16 is a JI interval, but not a `just'
one in the sense of simplicity or direct tuneability by ear."

As you nicely sum up, this gives JI intervals, like integer fractions,
a property of closure for operations like addition (multiplication of
ratios) or subtraction (division of ratios).

Most appreciatively,

Margo Schulter
mschulter@value.net

--- In tuning@egroups.com, "M. Schulter" <MSCHULTER@V...> wrote:

> http://www.egroups.com/message/tuning/16007
>
> ...
>
> From my largely medievalist perspective, I might
> define JI in the broadest sense as "tuning by
> multitudes," that is, integer ratios, in contrast
> to "tuning by magnitudes" including intervals with
> irrational ratios. Thus my definition _is_ a
> mathematical one, in line with the medieval European
> quadrivium (arithmetic, geometry, astronomy, music).
> We might say that JI involves arithmetic ratios,
> while tempered systems involve geometric ones.
>

Thanks very much for that reference, Margo. As a
theorist who is *very* concerned with historical issues,
let it go on record that this respect for the _quadrivium_
informs a lot of *my* opinions on exactly what concerns
tuning (in general) and this List (in particular); hence
my firm belief that tuning involves many other disciplines
besides only music.

>
> Possibly many of our differences [Margo's and Dave
> Keenan's] stem from our each subscribing to a
> time-honored concept of JI:
>
> (1) Just integer ratios as opposed to irrational
> tempered ratios;
>
> (2) Just or "pure" intervals as opposed to "impure"
> or complex ones.
>
> Please let me say that I consider either definition
> a valid viewpoint, and that your able advocacy in
> favor of the second definition reminded me of
> Pietro Aaron's description of meantone temperament
> in 1523.
>
> Addressing the musical beginner setting out, possibly
> for the first time, to tune a harpsichord or the like,
> he directs the reader to make the major third C-E "as
> just and sonorous as possible," evidently describing
> a pure 5:4 ratio in audible rather than mathematical
> terms.
>

I wanted to point out that while the actual term
'just intonation' is never used in the work of Aristoxenus,
I would say that these comments describe quite accurately
his conception of tuning: totally ignoring rational
calculations while basing his methods strictly on *audible*
perceptions.

> Above all, I see no reason why the term "just" cannot
> mean either "integer-based" or "pure," depending on
> the viewpoint of the speaker or the context.
> ...
> While favoring the "tuning by integer ratios" view,
> I would emphasize that your [Dave Keenan's] view may
> reflect an equally important side of music and history,
> and that such an exchange of perspectives adds much
> to the common dialogue of this forum.
>

And here I give you a resounding 'amen-a', Margo.

Given the controversy and, even more so, the *emotionality*
surrounding the arguments in this forum about the
definition of JI, I think this is an eminently sensible
solution to the problem; define your terms accurately
*and comprehensively*, and indicate which of any possible
multiple definitions is in effect under given circumstances.

> ... in tuning and in music more generally, not
> every concept and detail need involve audible
> distinctions.
>
> It is a proper role of music (including its
> intonational aspects) to engage the intellect and
> imagination as well as the sense of hearing.
> Sometimes the reach of the intellect may exceed
> the firm grasp of the senses, and this stretching
> or striving may itself be a high form of musical
> art and science.
>

Another 'amen-a'!

I simply must agree with you here, Margo. In addition
to the medieval-centric examples You give, I'd like
to point out that there are hundreds (thousands?) of
12-tET serial composers (especially concerning the
body of work created between c. 1925 and 1980, and on
second thought, not necessarily restricted to serialists
using 12-tET), who would subscribe wholeheartedly to
this statement.

Many present-day tuning theorists (Brian McLaren comes
immediately to mind) are fond of pointing out that
so much of the compositional construction in these serial
pieces cannot be audibly perceived, but IMO, that
doesn't necessarily negate the significance of those
techniques. I firmly believe that there are many
as-yet unrevealed means of perceiving data that *do*
have an effect on the reception of any given experience,
particularly in some kind of wholistic sense. In other
words, just because a listener can't immediately *hear*
that a 12-tone row is being used in inversion or
retrograde doesn't mean categorically that s/he is
not picking up that information in some other way.

Again, as I just posted the other day, a listener's
familiarity with the score or with pre-compositional
procedures can have a big effect on his perception
of the musical experience, even affecting his *belief*
of what he is hearing.

I might also point out that Schoenberg - the acknowledged
originator of serial technique - believed strongly in
'mysterious' aspects of musical masterpieces which defy
rigorous analysis. (See several of his essays in _Style
and Idea_.)

> Further, as you [me, monz] have often most eloquently
> emphasized, we should beware of taking the restrictions
> of a given musical style or tuning system as defining
> some universal standard.

Absolutely! [pun intended] Yet another 'amen-a'.

If there's anything I've learned from years of
historical tuning research, it's that uncovering
'universal standards' in tuning is at best difficult,
and may in fact be impossible. Again, as I've done
many times here before, I heartily recommend Richard
Norton's _Tonality In Western Culture_ as an eye-opener
to the importance of recognizing the role of
subjectivity in determining standards of musical
aesthetic values.

(And thanks for the nice compliment.)

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

Re: JI as tuning by multitudes (integer ratios) -- Hammond too

🔗M. Schulter <MSCHULTER@VALUE.NET>

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Monz <MONZ@JUNO.COM>

🔗David Finnamore <daeron@bellsouth.net>

🔗D.Stearns <STEARNS@CAPECOD.NET>