Tuning Theory and False Reification

The following essay will be published in 1/1 11:1 (currently at the printer). Since it was inspired (if that be the right word) mainly by postings on this list, I am posting it here for the benefit of list members who do not subscribe to 1/1.

--DBD

On Tuning Theory and False Reification

The tendency has always been strong to believe that whatever received a name must be an entity or being, having an independent existence of its own. And if no real entity answering to the name could be found, men did not for that reason suppose that none existed, but imagined that it was something peculiarly abstruse and mysterious.--John Stuart Mill[1]

To reify (from Latin res: a thing) is to treat an idea or abstraction as if it had concrete or material existence. False reification, then, is the error of assuming that a pattern of data represents a "thing" in the real world, when there is no evidence for the existence of the thing other than the pattern. One of the oldest and best know examples is the Zodiac. In ancient times, people in various cultures identified patterns of stars along the ecliptic with figures of animals, mythical beings, and the like (with different cultures identifying different patterns). Then, for reasons beyond the scope of this essay, people developed the belief, still held by many today, that the positions of the sun, moon, and planets relative to these figures at the time of a person's birth determine his or her character, personality, and fate, in a way that reflects the attributes of the animals or mythical beings identified with the star patterns and the planets.[2] Yet the patterns themselves exist only relative to the position of an observer on earth-some stars in a constellation may be a few dozen light years away, others millions-an observer elsewhere in the universe would not see the same patterns. And the identities of the constellations are clearly creations of the human imagination, and some rather vivid imaginations at that.

I first encountered the concept of false reification in Stephen Jay Gould's excellent book, _The Mismeasure of Man_.[3,4] The reification error that most concerns Gould is that scores from IQ tests represent a real-world phenomenon called "general intelligence" (and, further, that this commodity is heritable). However, I find the concept equally applicable to tuning theory and, by extension, to music theory in general. And, although no misconception in music theory is likely to result in such serious social consequences as those produced by the misapplication of intelligence testing, it can still lead to the expenditure of much time and energy on the formulation, presentation, and debating of theories that ultimately have little bearing on how any actual music might be composed or heard. Typically, these theories are based on mathematical models or constructive methods abstracted from scales or other musical structures.

Of course, everyone who has worked with Just Intonation is aware that numbers can represent real, significant musical phenomena. The very basis of Just Intonation is that musical intervals that can be represented by small integer frequency ratios are a distinct class, readily distinguished from (and, arguably, more beautiful than) those represented by more complex ratios. Clearly, this distinction extends to chords (triads, tetrads, pentads, and beyond) that can be represented as low-numbered subsets of a harmonic series. Given these undoubted facts, it would seem reasonable to suppose that the numerical properties of larger musical structures-scales, gamuts, melodies, harmonic progressions, and so on-would be similarly descriptive or predictive of their perceivable musical properties. To a certain extent, this is undoubtedly true. A scale or gamut comprising a network of closely connected consonant intervals based on low prime-limit ratios will certainly sound different and have different musical properties than one comprising large number and/or high prime-limit ratios (or irrational intervals); a scale with equal tetrachords will have different melodic properties than one with unequal tetrachords, and so on. These properties are obvious to the ear and do not require sophisticated mathematical analysis to discover.

Given that some characteristics of a scale, tuning system, or composition can be represented mathematically, it might seem a reasonable conclusion, or at least a reasonable speculation, that all significant features can be so represented. Perhaps so, although the truth of this statement is far from self-evident. From this arises the corollary that all features of a tuning system (or of a musical composition) that can be represented mathematically are musically significant and that there is a rather precise correspondence between mathematical and musical properties. Continuing down this path, we arrive at the idea that structures that are mathematically interesting, elegant, beautiful, or sophisticated necessarily have similar musical properties, or that the more sophisticated the mathematics required to describe or generate a tuning, the better will be the resulting music. Here we are deep in the territory of false reification. The computation, manipulation, and examination of the mathematical representations of tunings and other musical structures easily becomes an end in itself, far removed from the creation of actual music; in other words, it becomes a kind of recreational mathematics.

Part of the problem is the inherent difficulty of proving the truth of any proposition of music theory (or, by extension, of aesthetics) whatever. Because they are working with tables of numbers, geometric lattices, equations, and the like, tuning theorists may conclude that their theories can be proven on paper (or on a computer screen) like any other mathematical theorem. But to the extent that the numbers are meant to represent musical (that is, aesthetic), as opposed to merely acoustic phenomena, this cannot be so, at least not with our current level of understanding of human perception and emotional response. The proof of music must be in the hearing. Thus, to test any prescriptive theory, actual music based on the theory must be composed and heard. But this results in further difficulties. Different composers, given the same material, will produce vastly different results; indeed the same composer, working with the same material at different stages in his or her career (or even on successive days), may produce vastly different results. And, of course, different listeners may have very different reactions to the same composition and performance. How, then, can we determine to what extent the aesthetic impression created by a musical composition results from the theoretical properties of the raw material, the musicality, creativity, or stylistic preferences of the composer, or the musical tastes of the listeners?

Supposing that much of what passes for tuning theory is, in fact, recreational mathematics, is there any harm in this? Recreational mathematics is, no doubt, an enjoyable, blameless hobby that can occasionally result in mathematical discoveries of real value. And it is certainly possible that composers, in experimenting with the structures devised by mathematicians, may discover new musical possibilities that might otherwise go unnoticed. The harm, if any, is of two sorts. The first is that talented composers (or potential composers) may be seduced by the beauty of the numbers and spend too much time in mathematical exercises, to the neglect of composing, or be led to the false conclusion that music can be understood from the manipulation of numbers, without the intervention of the ears. A second and worse harm would result if it were believed that a high degree of mathematical sophistication was a prerequisite for composing in Just Intonation, where in fact no more than an understanding of elementary arithmetic is required.

Should the mathematical study of tuning be abandoned, then? I do not propose this, nor do I think that such a proposal would be taken very seriously by those engaged in this activity. What I do suggest, however, its that those who are interested in composing music resist the temptation to be seduced into recreational mathematics and instead devote more time to the aural study of tunings. Spend more time with your instruments and voice and less with computers and calculators, and better music will almost certainly be the result. The ear is and must ever be the best and final judge of music. A modest amount of math is necessary for the understanding of Just Intonation, but more is not necessarily better.

Notes:

1. _A System of Logic, Ratiocinative and Inductive, Being a Connected View of the Principles of Evidence and the Methods of Scientific Investigation_, 2 volumes. London, Parker: 1843.

2. In an odd twist, most Western astrologers are not even concerned with the real positions of the constellations as they appear in the sky today, but with fixed "signs" based on their positions in the second century C.E.., established by our old friend Claudius Ptolemy.

3. New York and London, W.W. Norton & Company: 1981; revised and expanded, 1996.

4. Gould died on May 20, 2002, as I was finishing this essay. As a paleontologist, he made important contributions to the theory of evolution. He was a skilled and prolific writer, known to many for his essays in the magazine _Natural History_, and for books such as _Ontogeny and Phylogeny_ (1977), _Wonderful Life_ (1989), and _Full House_ (1996), in addition to _The Mismeasure of Man_, cited above. I will miss him.

--
==========================================================================

David B. Doty jinetwk@dnai.com

Just Intonation Network http://www.dnai.com/~jinetwk

Editorial:(650) 694-4727 535 Stevenson Street

JI Store: (415) 864-8123 San Francisco, CA 94103

FAX: (415) 864-8726

--- In tuning@y..., Just Intonation Network <JINETWK@D...> wrote:

/tuning/topicId_38399.html#38399

> The following essay will be published in 1/1 11:1 (currently at the
> printer). Since it was inspired (if that be the right word) mainly
> by postings on this list, I am posting it here for the benefit of
> list members who do not subscribe to 1/1.
>
> --DBD
>

***Quite frankly, I see this as a rather self-evident and silly
essay, not up to the general level of articles I have seen in 1/1
over the years...

Joseph Pehrson

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_unknown.html#38407

> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > but it looks to me like a lot of people shriveled in fear after
> > the "professor"'s big rampage last year, and are looking
to "point
> > fingers" or "offer advice" so as to appear "above" his
criticisms.
>
> hmm . . . maybe this was his intent . . . breed suspicion and
> misunderstanding through lies, lies, innuendo, and more lies,
> create "division in the ranks", and watch amusedly while the whole
> community becomes a "rankled" mess. perhaps he gets a feeling
> of "coming out on top" as a result. i don't know, it's all pretty
> sick, but there's little disagreement that this is a rather sick
> individual . . . i just hope we can all rise above
> the "pettiness" . . . ok, enough, i guess any more should go on
> metatuning . . .

***Yes, regrettably, I think it has been more *his* problem than
anything. More on MetaTuna..

JP

Paul,

I first started a real long reply to this, and then I thought "oh, hell, there's enough noise as it is". DBD has just put out one more opinion, and if the mathematical work he seems to be talking 'against' has any validity at all, you and everyone else have nothing to worry about. There are plenty of voices on all sides of the issue, and no side is losing.

And why do grown-up people keep saying "nutty professor"?

Regards,
Jon

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:

/tuning/topicId_38399.html#38410

> Paul,
>
> I first started a real long reply to this, and then I thought "oh,
hell, there's enough noise as it is". DBD has just put out one more
opinion, and if the mathematical work he seems to be
talking 'against' has any validity at all, you and everyone else have
nothing to worry about. There are plenty of voices on all sides of
the issue, and no side is losing.
>
> And why do grown-up people keep saying "nutty professor"?
>
> Regards,
> Jon

***Jon, do you really want Brian McLaren, Brian McLaren, BRIAN
MCLAREN, bRiAn mClArEn, brian MCLAREN, ..... .... etc., etc.

Joe

Joe,

--- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> ***Jon, do you really want ...

What I want is for people to act like adults.

Jon

--- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:
/tuning/topicId_38399.html#38412

> Joe,
>
> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > ***Jon, do you really want ...
>
> What I want is for people to act like adults.
>
> Jon

***The reason, Jon, that at least *I* use that little "shorthand" is
due to the fact that, personally, I belirve that more than half of
what the gentleman in question does is to get attention. So calling
out his name all over the place is *exactly* serving his purposes, if
you know what I mean...

Joe

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_unknown.html#38414

> --- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:
> > Paul,
> >
> > I first started a real long reply to this, and then I
thought "oh,
> >hell, there's enough noise as it is". DBD has just put out one
more
> >opinion, and if the mathematical work he seems to be
> >talking 'against' has any validity at all, you and everyone else
> >have nothing to worry about.
>
> i don't think it even exists! well, maybe a few of marion's posts a
> couple of years ago bear some resemblence to what he's portraying,
> but that's about it . . . at least when you take into account the
> kind of material that *has* been published in 1/1, and assume that
> he's not referring to anything like that . . .
>

***My presumption is that for Doty, any involved mathematics that
pertains to JUST INTONATION is JUST fine, but if somehow it involves
equal temperaments or "micro temperaments" than it's unnecessary,
sophistic and a waste of time!

Here we have Julia Werntz on one side, Doty on the other side...
whoaboy!

Joseph

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

/tuning/topicId_unknown.html#38420

> --- In tuning@y..., "jpehrson2" <jpehrson@r...> wrote:
> > --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> >
> > /tuning/topicId_unknown.html#38414
> >
> > > --- In tuning@y..., "jonszanto" <JSZANTO@A...> wrote:
> > > > Paul,
> > > >
> > > > I first started a real long reply to this, and then I
> > thought "oh,
> > > >hell, there's enough noise as it is". DBD has just put out one
> > more
> > > >opinion, and if the mathematical work he seems to be
> > > >talking 'against' has any validity at all, you and everyone
else
> > > >have nothing to worry about.
> > >
> > > i don't think it even exists! well, maybe a few of marion's
posts
> a
> > > couple of years ago bear some resemblence to what he's
> portraying,
> > > but that's about it . . . at least when you take into account
the
> > > kind of material that *has* been published in 1/1, and assume
> that
> > > he's not referring to anything like that . . .
> > >
> >
> > ***My presumption is that for Doty, any involved mathematics that
> > pertains to JUST INTONATION is JUST fine, but if somehow it
> involves
> > equal temperaments or "micro temperaments" than it's unnecessary,
> > sophistic and a waste of time!
>
> but some (albeit comparatively primitive) material on these latter
> subjects *has* been published in 1/1 . . . are you familiar with
> harald waage's articles, for instance?
>
> > Here we have Julia Werntz on one side, Doty on the other side...
> > whoaboy!
>
> you're sure feeling feisty these days, joseph? is it the heat?
> perhaps some of us would rather be playing football but are using
the
> tuning list as our arena?

***I find Julia Werntz articles and attitudes absolutely annoying.
So this is the last that I am going to say at all on this subject.

Joseph

Hi there,

JUst a few thoughts on this subject.

Mathematicians of course have long been used to studying pure
structures, such as geometrical ones, and ones that are even more
abstract than those. They feel no need at all to tie them down
to physical things, but study them for the pure beauty of them.
This then is a form of recreation in a way, as it has no practical
aspect of things driving it - the pure study of maths anyway as
carried out by a mathematician who does it for a love of the subject
rather than in order to get papers published.

Also there's another connection there as well. Recreational
maths is a pursuit of amateur and non mathematicians, but
also a pursuit of professional mathematicians as recreation
from their main field of study. A very famous example is
Penrose's non periodic set of tiles. He came up with these
in his recreation time between his actual work which is the
study of gravitation theory, on the face of it not much
related to tilings.

These tilings fascinated a whole generation of geometers, and
also actually turned out to be closely related to the
structure of actual crystals made in laboratories, and
now if you look on the shelves of a research science
library you'll find many books and journal articles
on the structure of these Penrose tiling related crystals.

So you never know do you.

Of course it has been a long running debate in maths, about
whether mathematical structures are invented or discovered.
If invented then false re-ification might be an appropriate
word for the way they appear to have an independent existence
when you study them. But it could be that they actually do
in some sense have a life of their own.

I was brought up myself as a mathematician and so as a
platonist in this respect (as most are). Then became
seriously involved in philosophy and veered towards
constructivism, as for some reason many philosophers
have strong tendencies towards - as your essay
rather reflects. Now I'm perhaps between the two
and see them as different aspects. My logical work
was something of a fusion betweenthe two approaches
as it had elelments of both classical and constructivist
logic.

Perhaps one of the strongest platonist philosophers of
maths in recent times was Godel - he of the incompleteness
theorem. He held very platonist views.

Turning to music, I see some of this mathematical beauty
of form for its own sake in some of the work we come across here,
but with an interesting fusion between maths and music which
one could perhaps call musical geometry. I'm thinking
particularly here of the CPS sets.

At the moment perhaps we await new discoveries or techniques
to really see the music and the geometrical aspects of things
come together, but I think somehow that time may come.
Maybe we will even live to see it.

Imagine a geometrical structure that you can explore both
mathematically and musically at the same time...

Perhaps as you touch surfaces of the structure, music
springs to life related to the geometry of the structure.
Not just chords, but melodies and counterpoint too.

Maybe as you wonder through the structure you can sing
in response to it, and your songs become part of the
structure, echoing away into the distance and
getting taken up into the counterpoint...

Robert

Hello Paul!
You have in the past and continue to refer to Erv Wilson as a "was" when in fact he is quite active.
Much of his material has not even been discussed on these list being outside the roads travelled here.
it appears the list has beewn able to assimulate about the first three months of material he gives his students.
Even with the CPS we see a few hexanies but only one new Eikosany. This is just the ground floor.
there is alot of material outside this that is there for the picking. Possibly if you lived here
and saw the amount of work this man produces on a weekly basis you would have a better idea.
Since you don't, i won't hold it against you, but will try to impress that alot is still going on there.
His work has little to due with Partch and much of it predates his meeting.
--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:

> it's really too bad this has happened, it's put somewhat of a damper
> on the development of ideas through cooperation that characterized
> the best days in the field (think of when erv wilson was hanging out
> with harry partch).

--- In tuning@y..., "banaphshu" <kraiggrady@a...> wrote:

> You have in the past and continue to refer to Erv Wilson as a "was" when in fact he is quite active.

It would be nice if he were to show up here.

--- In tuning@y..., "banaphshu" <kraiggrady@a...> wrote:

> Hello Paul!
> You have in the past and continue to refer to Erv Wilson as
>a "was" when in fact he is quite active.

all i said was that he "was hanging out with partch". he's clearly
not hanging out with partch anymore. so what's wrong with that?

> Much of his material has not even been discussed on these list
being outside the roads travelled here.
> it appears the list has beewn able to assimulate about the first
three months of material he gives his students.
> Even with the CPS we see a few hexanies but only one new Eikosany.

one new eikosany?

> This is just the ground floor.
> there is alot of material outside this that is there for the >
picking.

i'd bet!

> Possibly if you lived here
> and saw the amount of work this man produces on a weekly basis you
would have a better idea.
> Since you don't, i won't hold it against you, but will try to
impress that alot is still going on there.
> His work has little to due with Partch and much of it predates his
meeting.

right -- that's why i said "when erv wilson was hanging out with
partch" rather than "when partch was hanging out with erv wilson".

> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> > it's really too bad this has happened, it's put somewhat of a
damper
> > on the development of ideas through cooperation that
characterized
> > the best days in the field (think of when erv wilson was hanging
out
> > with harry partch).

Hello Robert!
Erv Wilson has shown one way these tilings can be used musically.
On a personal level I designed my own variation on such tiles for my yard for stepping stones. I have to admit i enjoy these little fragments out of the whole tiling more than using solid areas. After all its to go between the plants. Other places i have taken sections out of larger blocks.
Your point is well taken in that one cannot know where good musical material might come from.
Being not of the mathematical inclination of many of you , i still wish those that are might try looking at something besides ETs, and i don't mean JI. I believe there are more than two ways to feed a cat

>
> From: "Robert Walker" <robertwalker@ntlworld.com>
> Subject: Tuning Theory and False Reification
>
>
> These tilings fascinated a whole generation of geometers, and
> also actually turned out to be closely related to the
> structure of actual crystals made in laboratories, and
> now if you look on the shelves of a research science
> library you'll find many books and journal articles
> on the structure of these Penrose tiling related crystals.
>
> So you never know do you.
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

>

Hello Gene!
He doesn't even have a computer. Calculators yes. He also does quite a bit of work with plants, these days he
back with corn, i believe after the disastrous genetic spill in Mexico.

>
> From: "genewardsmith" <genewardsmith@juno.com>
>
>
> > You have in the past and continue to refer to Erv Wilson as a "was" when in fact he is quite active.
>
> It would be nice if he were to show up here.
>
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

> Being not of the mathematical inclination of many of you , i
>still wish those that are might try looking at something besides
>ETs, and i don't mean JI. I believe there are more than two ways to
>feed a cat

sounds like we're in perfect agreement then!

search for this phrase: "the middle path"

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:

> Being not of the mathematical inclination of many of you , i
>still wish those that are might try looking at something besides
>ETs, and i don't mean JI. I believe there are more than two ways to
>feed a cat

kraig, i wonder: who were you thinking of when saying "you" in this
message? is it any member of the tuning-math list? if so, i wish
you'd deign to even take a look over at that list -- pick a random
post from its archives. the vast majority of the discussion about
specific tunings is about non-et, non-ji tunings.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

/tuning/topicId_38399.html#38435

> --- In tuning@y..., "banaphshu" <kraiggrady@a...> wrote:
>
>
> > You have in the past and continue to refer to Erv Wilson as
a "was" when in fact he is quite active.
>
> It would be nice if he were to show up here.

***For all his math prowess, I don't believe he's much into "computer
stuff..."

Does he even *own* a computer..??

Joseph

banaphshu wrote:

> Hello Paul!
> You have in the past and continue to refer to Erv Wilson as a "was" when in fact he is quite active.
> Much of his material has not even been discussed on these list being outside the roads travelled here.
> it appears the list has beewn able to assimulate about the first three months of material he gives his students.
> Even with the CPS we see a few hexanies but only one new Eikosany. This is just the ground floor.
> there is alot of material outside this that is there for the picking. Possibly if you lived here
> and saw the amount of work this man produces on a weekly basis you would have a better idea.
> Since you don't, i won't hold it against you, but will try to impress that alot is still going on there.
> His work has little to due with Partch and much of it predates his meeting.

Any chance of enrolling as a distance learning student? Seriously - I'm fascinated by his work and would like to
learn more.

Kind Regards