Set theory

On 8 Apr 2001 tuning@yahoogroups.com wrote:

(Paul Erlich wrote:)
>
> Please be aware, you'll find that there are
> extremely conflicting views on how to use
> math in microtonality, beyond the most
> pedestrian calculations. For example, some
> (like Pierre Lamothe) feel that pitch heights
> should correspond to simple ratios; others
> (like myself) subscribe instead to the view
> that intervals should correspond as closely as
> possible to simple ratios; others (like Stephen
> Soderberg) don't care about ratios at all and
> are interested instead in the "set theory" math
> of pitch classes (this is not related to the "set
> theory" you'd find offered in a mathematics
> department; it relates instead to the numerics
> of musicians like Forte and Babbitt). We'll
> probably debate these things endlessly (if
> given the opportunity) -- you should come to
> your own conclusions based on some serious
> and honest listening on your own (and of
> course a means of producing microtonal
> musical examples).

Thanks for the mention, Paul. Though maybe a little more explanation is
needed here, lest others leave with the impression that I could care less
about how things are tuned. My recollection is that, yes, I have said (on
more than one occasion) that "how you tune it" is "irrelevant." Paul took
strong exception to that stance when I said it in passing on the tuning
list, and I should have made more effort at that time to explain the
context. So let me try to make up for that now -- Paul and I are not as
far apart here as he might believe (the confusion is my fault).

First, I have to admit to going for a little shock value in order to get
others to think *beyond* tunings into the region of what compositional
strategies might be implied (or maybe even compelled) by a tuning choice.
To get to that point, one should first realize that, *however* you define
a scale (whether a specific scale you want to use as a compositional basis
or the general notion of "scale") it's first and foremost a collection of
pitches (a set) *together with* an "index set" (some way of putting the
collection in "order").

If you have some 7-note scale C-D-E-F-G-A-B you can tune these 7 "objects"
with respect to one another in a variety of ways (acoustically relating n
objects -- the broadest definition of tuning (and composition) I can come
up with -- is a lot of what the list is about); and ratios are almost
always involved. But without indexing those 7 notes -- (pre)assigning
order, importance, etc. -- you have no way of finding your way around or
locating significant structures once you're into the music (again, a
random walk isn't necessarily bad, just limiting). IMO, composition has
as much or more to do with playing with the index set as it does with the
tuning of the underlying objects (tones, notes, pitches)... but with some
very important qualifications.

Let's say I want to write a piece based on "quartertones." (There are
good logical and aesthetic reasons to pick qts beyond the fact that
they've been somewhat maligned around here in the past.) So I start out
thinking in 24tet. Because of the fact that 12tet is imbedded in 24tet,
and because I see some very interesting sonorities to be had by tuning
pairs of instruments a qt apart (to say nothing of making it easier on the
traditional classical performer), the "look" of the score has patches of
12tet juxtaposing and interleaving with patches of 12tet a qt higher or
lower. Now, using my computer's sound card to get a VERY rough idea of
what all this sounds like, I soon realize that 24 EQUAL is producing a lot
of "off" intervals to my ears. So I start to play with the MIDI's equal
setting by shading the quartertones down more and more, i.e., each time
the interface reads, say, "play this B a qt flat," the qt gets flatter.
What I eventually settle on is not 24 EQUAL at all, but two 12tet scales,
S1 and S2, tuned somewhere between a 1/5-tone and a 1/6-tone apart.

So now, via my notation software, I have two complete compositions, C1 and
C2, side by side. In C1, S1 and S2 are as close to an "equal" qt apart as
possible, whereas in C2, S1 and S2 are, say 1/5.29 of a "whole
tone" apart.

The question then arises, are C1 and C2 the SAME composition or are they
different compositions? I would answer (my point to all this) BOTH. In
abstract set-theoretic terms, where it doesn't matter what the elements of
a set actually are, both C1 and C2 "hang together" compositionally the
exact same way. Is there a climax at a formal "golden mean"? - if so it's
still there. Are some verticles connected by common tones? - if so, they
still are. And the rhythmic elements, voicing, and tessituras are
identical between C1 and C2; and so on. [All of these things can
profitably be talked about formally, to one degree or another, as sets --
I'm NOT saying that's how they are actually composed or heard in the
rough-and-tumble of creation or listening!] But in terms of the
moment-to-moment aural perceptions created, C1 and C2 have radically
different sounds. Speaking loosely, the "math" by which a lot of the
formal elements are sustained is the same -- the index set relationships
are identical, but the resulting "music," insofar as it can be separated
from the "math," is certainly different.

Another interesting question: does the combination S1 + S2 that permeates
both C1 and C2 result in an equal/unequal 24-tone chromatic basis, or does
it just sound like an "out of focus" 12-tone chromatic? It could go
either way I think, depending on your skill and intentions in working with
the material. The result could be intriguing and new, but if you don't
understand what a mod-24 system offers (there's a subtle distinction we
should be making between mod-n systems and n-tet tunings), even at some
intuitive level, the result might just be horribly embarrassing.

[This is all roughly the same as the age-old question about
"versions" -- just more radical. Bach writes a concerto for oboe and
violin in c minor. Later he "arranges" it for two violins and puts it in
d minor. Everything is nearly identical with the exception of giving the
oboe part to a fiddle and transposing the score up a major 2nd. Same
piece or different? Again, for many of the same reasons, both (I like the
oboe in there myself).]

Also note you could further play with the tuning for both C1
and C2: S1 and S2 don't HAVE to both be 12tets, and there's no reason one
couldn't treat a "megachromatic" basis as the sum of more than two
subchromatics S1+S2+S3+.... There's something to this, especially in
even-numbered universes. It's well-known that any sonority in 12tet, for
example, can be described and heard as a collection of notes selected from
two 6tet (whole tone) scales a half-step apart. Also, check out the
"Sieves" chapters in Xenakis' _Formalized Music_ where he constructs
scales using only elementary Boolean operations by only positing an
abstract "semitone" (no ratios here at all).

Bottom line: tuning is important, but it's only one of many aspects
holding a good piece of music together (I doubt anyone would argue with
this anyway -- and I hope Paul, whose work I greatly respect and admire,
now realizes I sign on to this truism as well). Finally, in only a tiny,
well-intentioned dig: once you tune, what you have is a set. Forte and
Babbitt don't really have much to do with it except for their initial
realizations and groundwork showing how far this idea can be pushed and
how widely it can be applied.

Best,

Steve

Stephen Soderberg
Music Division
Library of Congress

Thanks Stephen.

I may have a lot more to say later, but for now let me just thank you for
clarifying your point of view -- it's been too long since you've posted!

Just for a taste of what my response might be like, I'll respond to this:

>Also, check out the
>"Sieves" chapters in Xenakis' _Formalized Music_ where he >constructs
>scales using only elementary Boolean operations by only >positing an
>abstract "semitone" (no ratios here at all).

My problem with this is that these constructions seem very contrived, not at
all simple, and about as far from a "derivation" or "explanation" as you can
get. Much more simple and powerful descriptions of scales, I feel, come from
considering a few very basic considerations _including_ tuning, and if
understood correctly, the inherent flexibility of the tuning will come out
naturally.

For example, set theory alone will never be able to explain why 7 notes and
not some other number (you're not allowed to appeal to a 12-tone master set
since that was constructed later in history).

--- In tuning@y..., Stephen Soderberg <SSOD@L...> wrote:

/tuning/topicId_20848.html#20848

> Thanks for the mention, Paul. Though maybe a little more
> explanation is needed here, lest others leave with the
> impression that I could care less about how things are tuned.
> My recollection is that, yes, I have said (on more than one
> occasion) that "how you tune it" is "irrelevant." Paul took
> strong exception to that stance when I said it in passing on
> the tuning list, and I should have made more effort at that
> time to explain the context. So let me try to make up for
> that now -- Paul and I are not as far apart here as he might
> believe (the confusion is my fault).
>
> First, I have to admit to going for a little shock value in
> order to get others to think *beyond* tunings into the region
> of what compositional strategies might be implied (or maybe
> even compelled) by a tuning choice.
> ...
>
> Bottom line: tuning is important, but it's only one of many
> aspects holding a good piece of music together (I doubt anyone
> would argue with this anyway -- and I hope Paul, whose work
> I greatly respect and admire, now realizes I sign on to this
> truism as well). Finally, in only a tiny, well-intentioned dig:
> once you tune, what you have is a set. Forte and Babbitt don't
> really have much to do with it except for their initial
> realizations and groundwork showing how far this idea can be
> pushed and how widely it can be applied.

Hi Steve. Thanks very much for this great post. I've always
felt this way intuitively, and it's nice to see these thoughts
formalized as succinctly as you've done it here. Since I'm
fond of retuning old compositions, it definitely resonates
with my own practice.

PS - Mucho thanks for finding and providing me with Möllendorff's
little book. I translated the whole thing and made it available
with parallel German and English text (and MIDI-files of the
music examples) as part of my presentation. I'll send you a
copy snail-mail if you'd like. Anyone else, email me - I have
several left.

-monz
www.monz.org
"All roads lead to n^0"

Questions regarding the recent set-theory posts...

I'm curious as to how set-theory is defined on this list... for me any
examining of a scales structure to a certain extent implies the use of
set-theory. The concept of lattices (which I'm not all too familiar with)
is an examination of the tuning ( of the scale) looking at the structure of
the set is it not. My initial immersion into non-et's has given me great
troubles, due to the fact that there are different types of intervals, and
many more types of intervals. Also, the idea that a transposed scale or
mode within a given tuning may not equal the first. Slowly though I am
finding ways to use my set-theory based knowledge and modify it to help with
my non-et journeys. I would also like to point out that that the deriving
of scales from tunings can often be described by set-theoretic means. The
concepts of deep structures and maximally-even sets are drawn from
set-theoretic knowledge. I believe that these same principles apply to
other tunings, and scales.

I would like to propose that we select a few tunings, and examine their
structure over the list. I would love to see how the experienced memebers
of the list would look at a sets structure. I think that it would be rather
intersting also, to see the different approaches to examining a tuning.

Thank you,

Steven Kallstrom

Stephen!
Lou Harrison has written most of his compositions in Slendro and Pelog to be played in ANY
Slendro and Pelog. The actual scale can vary greatly. He chooses to call this the same
composition. There is no reason for composers not to think of different potential versions from
the get go. In Anaphoria, the composition "The Creation of the Worlds" has hundreds of different
versions depending on how one set up the generating lattice. See the two eight tone cycles
discussed in the later part of http://www.anaphoria.com/cps.PDF . Although the structure is the
same the people there choose their versions depending on the occasion the piece will function so
in a way they are BOTH the same and different compositions!

Stephen Soderberg wrote:

>
> The question then arises, are C1 and C2 the SAME composition or are they
> different compositions? I would answer (my point to all this) BOTH. In
> abstract set-theoretic terms, where it doesn't matter what the elements of
> a set actually are, both C1 and C2 "hang together" compositionally the
> exact same way.
>
> Everything is nearly identical with the exception of giving the
> oboe part to a fiddle and transposing the score up a major 2nd. Same
> piece or different? Again, for many of the same reasons, both (I like the
> oboe in there myself).]
> Best,
>
> Steve
>
> Stephen Soderberg
> Music Division
> Library of Congress

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

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

On 11 Apr 2001 tuning@yahoogroups.com wrote:

Steven Kallstrom:

> I'm curious as to how set-theory is defined on this list... for me any
> examining of a scales structure to a certain extent implies the use of
> set-theory. The concept of lattices (which I'm not all too familiar with)
> is an examination of the tuning ( of the scale) looking at the structure of
> the set is it not. My initial immersion into non-et's has given me great
> troubles, due to the fact that there are different types of intervals, and
> many more types of intervals. Also, the idea that a transposed scale or
> mode within a given tuning may not equal the first. Slowly though I am
> finding ways to use my set-theory based knowledge and modify it to help with
> my non-et journeys. I would also like to point out that that the deriving
> of scales from tunings can often be described by set-theoretic means. The
> concepts of deep structures and maximally-even sets are drawn from
> set-theoretic knowledge. I believe that these same principles apply to
> other tunings, and scales.

Steven -- go for it! I generally dislike the kind of answer that simply
gives a cite for a journal article or book and then tells the questioner
to go read it. But I don't think I have a choice here -- any attempt I
would make to give my personal views re your question would be lengthy and
likely degenerate into "Is too! -- Is not!" sidebars. There is a *huge*
amount of literature out there, hardly any of it getting much play on the
tuning list, that, given minimal math chops and a burning interest, you
can check out. My suggestion would be to get a copy of Music Theory
Spectrum (21:1, spring 1999) and work through the article "Scales, Sets,
and Interval Cycles: A Taxonomy" by John Clough, Nora Engebretson, and
Jonathan Kochavi. Then check out their bibliography for further readings.

Paul Erlich:
> Another reason I don't like maximal evenness -- why should you have to
> postulate a chromatic set in a musical style which doesn't make use of one?

I'd be interested to know where you heard that. (You don't "like" ME? --
what's that about??)

Me & Paul Erlich:

> >One way is by
> >stacking fifths until you get so close to an octave identity (at 7) that
> >you can call it quits.
>
> You get closer at 5 than at 7, but OK, I'll buy that. But where does the
> fifth itself come from? Tuning theory or set theory?

I don't know what glue I was sniffing when I wrote that, but maybe you
were sniffing the same stuff when you answered;-)... Obviously, when
stacking fifths, near closure comes at 12 for the chromatic. Stopping the
stack at 7 gives you the tritone, which many (e.g., see Richmond Browne,
"Tonal Implications of the Diatonic Set," _In Theory Only_ 5/6-7,
pp.3-21) posit as the most important means of orientation within the usual
diatonic since it is the rarest interval there.

Me & Paul Erlich:

> >Then there's maximal evenness, in whatever form
> >you wish to express it.
>
> That I won't buy -- it fails to explain why 7 out of 12 and not, say, 9 out
> of 16. And 7 out of 31 is not maximally even -- oops! (uh-oh, I'm getting
> McLarenish today)

What part of maximal evenness annoys you the most? -- It's certainly not a
rival theory whose acceptance implies that all your own excellent work
goes down the tubes! All it is is an important structural property
possessed by certain scales. I really don't get it.

At any rate, it has been proven by Clough & Douthett (work on ME goes
back at least to Clough & Myerson in 1985) that there is a
unique maximally even set for *any* given choice of cardinalities
d<c. For the three pairs you give above:

7 out of 12: 2212221
9 out of 16: 222122221

which are the usual diatonic and a "hyperdiatonic," respectively (both
maximally even), and

7 out of 31: 5454544

which may not be hyperdiatonic by C&D's definition (for any one of at
least 10 (mathematically equivalent) reasons), but it IS maximally even.

As for Xenakis' sieve theory, this is one of those things that threatens
to turn into "Is too!--Is not!" Obviously Paul and I don't share the
same criteria for relative importance within a given context. So why
don't we call it quits with Paul's agreement:

> OK, I applaud all creativity triggers.

Paul Erlich:

> Well then, you must think positing ratios or overtones as a basis is awfully
> flimsy. That serves marvelously to illustrate the point I was making in my
> original posts about how the relationship between mathematics and music is
> very controversial!

Please don't delete significant parts of quotes. If you go back and read
the whole statement you'll see that I was implying, if not outright
stating, something closer to this: if you get/create/originate/generate
some scale S by some "method," and it turns out that S has no significant
structural qualities/properties beyond its derivation, then it has nothing
more to recommend it *structurally* than any *random* scale. As it often
(not always) happens, the ability to be generated in a multiplicity of
ways gives certain scales (not only the usual diatonic) more "structural
potential" (i.e., compositional possibilities) than those which can only
be generated in one-and-only-one way. This is not to be confused with an
ostensibly randomly selected scale that might be "pleasing" for some
reason, and therefore we *find* or *make* ways to use it structurally.
E.g., the usual harmonic minor cannot, as far as I know, be "generated" in
the sense I mean. But it certainly is "pleasant" and ubiquitous. How do
you squeeze the structural juice out of a structural lemon like this? At
least one way is by (intuitively) mapping it as homologous to the usual
diatonic and squinting.

Best,

Steve

Stephen Soderberg
Music Division
Library of Congress

----- Original Message -----
From: Stephen Soderberg <SSOD@LOC.GOV>
To: <tuning@yahoogroups.com>
Sent: Wednesday, April 11, 2001 10:23 AM
Subject: [tuning] Re: Set theory

> What part of maximal evenness annoys you the most? -- It's certainly not a
> rival theory whose acceptance implies that all your own excellent work
> goes down the tubes! All it is is an important structural property
> possessed by certain scales. I really don't get it.

Here are a few of my worries concerning maximal evenness. The diatonic
set is maximally even, but it the major scale maximally even. Isn't it the
Dorian mode that is maximally even. From C to C, the C major scale isn't
maximally even any more is it? This has always been a concern of mine. I
realize that the collection is maximally even, but the arrangement isn't.
ME is still an interesting property, but there has to be more to it. The
way the elements are ordered in space and time is the most important I
think. Also, I think the tritone is the most important interval in 12tet,
the symmetrical division of the octave is probably the most important
interval in any even cardinal ET. I am curious as to how the lack of this
interval an an odd cardinal ET may affect its structure.

> Please don't delete significant parts of quotes. If you go back and read
> the whole statement you'll see that I was implying, if not outright
> stating, something closer to this: if you get/create/originate/generate
> some scale S by some "method," and it turns out that S has no significant
> structural qualities/properties beyond its derivation, then it has nothing
> more to recommend it *structurally* than any *random* scale. As it often
> (not always) happens, the ability to be generated in a multiplicity of
> ways gives certain scales (not only the usual diatonic) more "structural
> potential" (i.e., compositional possibilities) than those which can only
> be generated in one-and-only-one way. This is not to be confused with an
> ostensibly randomly selected scale that might be "pleasing" for some
> reason, and therefore we *find* or *make* ways to use it structurally.
> E.g., the usual harmonic minor cannot, as far as I know, be "generated" in
> the sense I mean. But it certainly is "pleasant" and ubiquitous. How do
> you squeeze the structural juice out of a structural lemon like this? At
> least one way is by (intuitively) mapping it as homologous to the usual
> diatonic and squinting.
>

The derivation of the harmonic minor and other 'diatonic' scales has
always intrigued me. I think the tritone plays a role here also, halfsteps
may also play a role, the ever great leading-tone. The aeolian mode has an
'out of place tritone' that leads nowhere. In A natural minor, the B leads
to C, with no emphasis, possibly because the entire tritone has yet to be
revealed in the scale. When the F is reached it proceeds to the G, which
has no pull, there is no leading-tone, no halfstep. By raising the G to a
G#, we have a new tritone, in a stronger position in the scale. The
tritone, in my opinion has to be the greatest interval in musical history,
by far the most important. The pull of a V-I cadence has never been the
same since the inception of the V7-I.

I like the ME property and am fascinated ny it, but I also think there
may be something to a disruption of the balance, throwing things off kilter
a little bit. You can measure a sets relationship to the ME property, I
assume. For instance, the melodic minor, and harmonic minor, are slightly
les ME, and the 7 tone chromatic is completely off balanced in a 12-tone
world. I bet, that all the sets in between slowly creep towards and away
from the ME property. There is a great article by Jay Rahn, that is related
to this idea entitled, "Coordination of Interval Sizes in Seven-Tone
Collections" Journal of Music Theory 35/1-2 (1991). I can summarize it in a
second post if anyone would like. It follows some of Clough, Douthett, and
Myerson's work, but is much simpler, and Rahn takes a non-mathematical
approach.

Thanks,

Steven Kallstrom

I wrote,

>> Another reason I don't like maximal evenness -- why should you have to
>> postulate a chromatic set in a musical style which doesn't make use of
one?

Stephen Soderberg wrote,

>I'd be interested to know where you heard that.

Heard what?

> (You don't "like" ME? --
> what's that about??)

I don't "like" maximal evenness -- I talked about this here on the list a
few days ago in a conversation with Dan Stearns. See below for more.

Stephen wrote,

>> >One way is by
>> >stacking fifths until you get so close to an octave identity (at 7) that
>> >you can call it quits.

I wrote,

>> You get closer at 5 than at 7, but OK, I'll buy that. But where does the
>> fifth itself come from? Tuning theory or set theory?

Stephen wrote,

>I don't know what glue I was sniffing when I wrote that, but maybe you
>were sniffing the same stuff when you answered;-)... Obviously, when
>stacking fifths, near closure comes at 12 for the chromatic. Stopping the
>stack at 7 gives you the tritone, which many (e.g., see Richmond Browne,
>"Tonal Implications of the Diatonic Set," _In Theory Only_ 5/6-7,
>pp.3-21) posit as the most important means of orientation within the usual
>diatonic since it is the rarest interval there.

I agree with Browne on that and other things in his paper, which is why I
cited it in my paper. But, glue or no glue, you didn't answer my question,
where does the fifth come from?

And the octave -- virtually all the set theory stuff assumes octave
equivalence -- but where do you suppose the octave comes from? Is it just a
total coincidence that every culture which recognizes a repetition of pitch
classes does so at an interval awfully close to a 2:1 frequency ratio --
that of the first overtone?

>What part of maximal evenness annoys you the most? -- It's certainly not a
>rival theory whose acceptance implies that all your own excellent work
>goes down the tubes! All it is is an important structural property
>possessed by certain scales. I really don't get it.

I just don't think it's all that important, or even psychoacoustically
meaningful in a lot of cases. In trying to understand what is important,
I've come to the conclusion that patterns which at first glance may appear
to be due to maximal evenness can be better explained in other ways. It is
the same with prime-limit and chords, if you've followed any of that
discussion.

Getting back to maximal evenness, let's say you hear an authentic
performance of simple piece of music from the 16th or 17th centuries. It
uses the diatonic scale, very nearly in 31-tone equal temperament. Does the
fact that the diatonic scale is not the ME 7 out of 31 collection have any
importance on how the music sounds?

And what if a scale is used where there is no clear choice as to an
embedding chromatic collection. What then?

>At any rate, it has been proven by Clough & Douthett (work on ME >goes
>back at least to Clough & Myerson in 1985) that there is a
>unique maximally even set for *any* given choice of cardinalities
>d<c.

Seems kind of obvious, doesn't it? Not sure what the point is.

>7 out of 31: 5454544

>which may not be hyperdiatonic by C&D's definition (for any one of at
>least 10 (mathematically equivalent) reasons), but it IS maximally >even.

Not sure what you're getting at. The maximally even 7 out of 31 is 5454544 .
. . which is maximally even . . . and?

You lost me in the last paragraph of you message.

Cheers,
Paul

Paul wrote,

> Getting back to maximal evenness, let's say you hear an authentic
> performance of simple piece of music from the 16th or 17th centuries. It
> uses the diatonic scale, very nearly in 31-tone equal temperament. Does
the
> fact that the diatonic scale is not the ME 7 out of 31 collection have any
> importance on how the music sounds?

A Maxiamlly Even Set has nothing to do with sounding good, just like
traditional rules of voice leading have nothing to do with what sounds good.
The concept of ME boils down to the fact that that set is highly symetrical
and highly structured. I believe that any diatonic structure, resulting in
'beautiful' music is not that far off from a maxiamlly even set. A non ET
set may not be ME, but it is probably very close. ME doesn't have anything
to do with accoustics, but the structure of a set, and it's subsets. Every
major scale in Werckmeister III is different, not ME, but closely related,
these invariants may provide some charm. This is like the comment I made
concerning the minor scale in a previous response to Stephen. I sometimes
envision the ME set as perfect, but I also consider perfection boring. No
theory can teach one to compose, and no theory can make something sound
good. All that theory can really do is hint at an ideal, and that is what
Maximal Evenness does, it hints at an ideally structured set. No, a
maximally even set is not in 'tune' (according to few), but does changing
some cents here and there ruin the concept of a maximally even set.
Mathematically yes, but practically no. I think that humans have innately
understood this principle, and it is interesting that mathematics has shown
that these naturally developed scales are superior in structure.

I do not think, though, that ME sets are all that superior (since
superior would require a definition). Also, compositionally symmetry is
frankly boring. ME means nothing to composition. We do not use ME sets as
maximally even, we do not distribute weight among the pitches even, we given
certain pitches prominence. This does not change the fact that the set has
this property though. It is only a property not a miracle.

I don't know why ME has been the onl'y property that has been talked
about here. Set-theory provides so much more wealthy information about sets
than simply whether or not it is maxiamally even (simple set-theory at
that). I think the concept of ME is complex and not as strong as say,
comparing the interval vectors of many sets, or say, looking at the subset
structures of different sets. There is much information that is much more
compelling.

Steven Kallstrom

Steven Kallstrom wrote,

>The diatonic
>set is maximally even, but it the major scale maximally even. Isn't it the
>Dorian mode that is maximally even.

You must be misunderstanding the definition of maximal evenness.

>From C to C, the C major scale isn't
>maximally even any more is it?

It is. In all the papers on maximal evenness, pitches are viewed in a
circle, not a line segment. So the white-note scale is maximally even no
matter how you rotate it.

>Also, I think the tritone is the most important interval in 12tet,
>the symmetrical division of the octave is probably the most important
>interval in any even cardinal ET. I am curious as to how the lack of this
>interval an an odd cardinal ET may affect its structure.

The diatonic scale has exactly one tritone even in an odd ET like 19 or 31.
The difference is that in 12 the tritone in question can belong to a distant
diatonic scale -- e.g. the B-F of the C major scale is the same as the B-E#
of the F# major scale. In odd ETs liek 19 and 31 a given tritone is unique
to a single diatonic scale.

>The derivation of the harmonic minor and other 'diatonic' scales has
>always intrigued me. I think the tritone plays a role here also, halfsteps
>may also play a role, the ever great leading-tone. The aeolian mode has an
>'out of place tritone' that leads nowhere. In A natural minor, the B leads
>to C, with no emphasis, possibly because the entire tritone has yet to be
>revealed in the scale.

?

>When the F is reached it proceeds to the G, which
>has no pull, there is no leading-tone, no halfstep.

I don't understand why you wouldn't just say the F proceeds to the E. That's
how I see it. And that's why the Aeolian mode proved useful for tonality:
the notes of the tritone resolve in contrary motion by halfsteps to notes of
the tonic triads.

But we haven't highlighted the tonic note itself the way a resolution in
major does, hence the following:

>By raising the G to a
>G#, we have a new tritone, in a stronger position in the scale. The
>tritone, in my opinion has to be the greatest interval in musical history,
>by far the most important. The pull of a V-I cadence has never been the
>same since the inception of the V7-I.

Yes, in A harmonic minor there are _two_ tritones that resolve. B-F resolves
to C-E, and D-G# resolves to C-A.

Steven Kallstrom wrote,

>A Maxiamlly Even Set has nothing to do with sounding good, just like
>traditional rules of voice leading have nothing to do with what sounds
good.

They don't?

>Every
>major scale in Werckmeister III is different, not ME

I think John Clough would disagree with you there!

>I think the concept of ME is complex and not as strong as say,
>comparing the interval vectors of many sets

I don't "like" interval vectors either :) That is, when a music teacher
pointed out the unique entries of the interval vector of the 12-tET diatonic
scale, and everyone was like, "gasp, wow, that's why the diatonic scale is
special", I felt like a major swindle was taking place.

-Paul

Paul wrote,

> You must be misunderstanding the definition of maximal evenness.

The definition of an ME set according to John Clough and Jack Douthett
is

'a set whose elements are distibuted as evenly as possible around the
chromatic circle.' "Maximally Even Sets" JMT 35, 1991

Following this definition my view is incorrect. The C Major Scale would
be ME in a 'chromatic circle'. I guess I am questioning this definition.
There is a difference when a set is ME in the overall pitch space, but, the
C Major scale is ordered is it not. From C to the C an octave above, the C
Major scale is not evenly distributed. So, I guess the difference comes
from the definition of what musical space we are talking about. Clough and
Douthett are obviously talking about a cyclical structure with octave
equivalence. I guess I am talking about a linear space. I believe, that
there must be some 'extra-musical' reason why we base our scale in the Major
mode as opposed to another mode. When I think about scales, I do not think
of an unordered set, but as an ordered or cyclical set, that ends or has a
repeating point. This is something that I think diatonic set theory has
ignored.

Paul wrote,

> The diatonic scale has exactly one tritone even in an odd ET like 19 or
31.
> The difference is that in 12 the tritone in question can belong to a
distant
> diatonic scale -- e.g. the B-F of the C major scale is the same as the
B-E#
> of the F# major scale. In odd ETs liek 19 and 31 a given tritone is unique
> to a single diatonic scale.
>

I guess I need to clarify. I think that an interval that symmetrically
divides the octave is a special interval, in 12tet this is the tritone. The
B-F of C and B-E# of F# are the same, but the interesting thing about them
is that they are different intervals diatonically, B-F a diminished fifth,
the other an augmented fourth. Odd numbered ET's do not have an interval
the symmetrically divides the octave. This is what I am curious about, how
not having an interval that symmetrically divides the octave.

I do want to thank you for the education, and I hope that I am not being
a pest. I have learned so much within just a day, and I am excited to
discuss these ideas. My university does little to help me in this area...
can't wait for graduate school. Thanks,

Steven Kallstrom

Hi Steven.

>I believe, that
>there must be some 'extra-musical' reason why we base our >scale in the
Major
>mode as opposed to another mode.

I don't think "we base our scale in the major mode" in any important sense.
If you're talking about the primacy of the major and minor modes, I don't
think it's extra-musical at all.

Before the advent of tonality in the 17th century, all modes (except
Locrian) of the diatonic scale were on more or less equal footing.

Tonality reduced the number of modes to two: major and minor.

You pretty much already know the reason for that -- the unique tritone
resolves in contrary motion by halfstep to notes of the tonic triad _only_
in those two modes.

>This is something that I think diatonic set theory has
>ignored.

Well, sure. I made sure in my paper to determine _which_ modes of the
decatonic scales would support a form of tonality.

>This is what I am curious about, how
>not having an interval that symmetrically divides the octave.

. . . affects music? Well, it depends. The stuff I use 22-tET for really
needs it. But diatonic stuff works perfectly well in 19-tET or 31-tET, so I
don't think that for diatonic music the half-octave is needed. Indeed, in
26-tET, the diatonic tritone and the half-octave are _different_ intervals!

I wrote,
>
> >A Maxiamlly Even Set has nothing to do with sounding good, just like
> >traditional rules of voice leading have nothing to do with what sounds
> good.
>

Paul wrote,
> They don't?

Debussy broke many rules of voice leading (parallelism) and he still sounds
good. They are only guides, nothing is written in stone, and nothing can
guarentee genuis.

I wrote,
> >Every
> >major scale in Werckmeister III is different, not ME
>

Paul wrote,
> I think John Clough would disagree with you there!

I would hope that he would disagree with me. They are ME for that
tuning, but I believe this is a flexible definition. I believe that
maximally even is ideally defined in ET, but it can easily be flexed into
non-ET. Since the steps are unequal, an equal distribution of pitches,
between the 12 pitches of the octave can only ideallt be achieved in ET. I
think that the slight changes of cents here and there don't change what we
are dealing with too much, to a limited degree. I've often thought about
apply the set relation ideas to intervals in non-ET. A fifith is a fitfh if
it is a few cent different. I think that there needs to be some
felexibility.

I wrote,
> >I think the concept of ME is complex and not as strong as say,
> >comparing the interval vectors of many sets

Paul wrote,
> I don't "like" interval vectors either :) That is, when a music teacher
> pointed out the unique entries of the interval vector of the 12-tET
diatonic
> scale, and everyone was like, "gasp, wow, that's why the diatonic scale is
> special", I felt like a major swindle was taking place.
>

I love interval vectors, and they make me happy :)... I can understand
you reaction to the diatonic interval vector. When you compoare it to
others though, you realize how special the set is! The interval vector
tells us the interval content of a set, but also the common notes under
transposition of that set. The diatonic Interval Vector shows us that there
is a hierarchy among its transpositions. You can transpose the scale by a
perfect fifth everytime and retain all tones but one. The only other 7-note
set that can do this is the 7-note chromatic. This IS important, no other
sets retain common notes like this... other sets have common note
structures, but none so perfect. Also, no other set contains such a high
degree of relations among its subsets either. My own music examines the
common note structures of other sets without such strong common note
structures. I personally think that it is very intriguing. Do all ME sets
enjoy such an interval structures, I would assume so, if they are in
even-ET's.

Thanks again,

Steven Kallstrom

Hi Steven.

>Debussy broke many rules of voice leading (parallelism) and he still sounds
>good. They are only guides, nothing is written in stone, and nothing can
>guarentee genuis.

Of course. But just because Debussy broke the traditional rules of voice
leading doesn't mean "they have nothing to do with what sounds good", as you
say. Just because they don't have everything to do with what sounds good
doesn't mean they have nothing to do with what sounds good.

>The only other 7-note
>set that can do this is the 7-note chromatic.

You're assuming 12-tET . . . a bad idea around here!

>Do all ME sets
>enjoy such an interval structures, I would assume so, if they >are in
>even-ET's.

Not all ME sets have such interval vectors. John Chalmers knows more. It
doesn't really interest me -- I can see all the properties you mentioned
from a different perspective.

-Paul

Paul!
It seems to me that the principles of voice leading had more to do with keeping as much
independence in the voices as possible more than any good or bad.

"Paul H. Erlich" wrote:

> Of course. But just because Debussy broke the traditional rules of voice
> leading doesn't mean "they have nothing to do with what sounds good", as you
> say. Just because they don't have everything to do with what sounds good
> doesn't mean they have nothing to do with what sounds good.
>
>
> -Paul

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

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

I wrote,

>>Of course. But just because Debussy broke the traditional rules of voice
>>leading doesn't mean "they have nothing to do with what sounds good", as
you
>>say. Just because they don't have everything to do with what sounds good
>>doesn't mean they have nothing to do with what sounds good.

Kraig wrote,

>It seems to me that the principles of voice leading had more to do with
keeping as much independence in the >voices as possible more than any good
or bad.

Doesn't keeping independence in the voices have something to do with
"sounding good?" Wouldn't a lack of independence, at least in certain
styles, "sound bad"?

Paul, here's a list of many of the important structural properties any
given scale may or may not possess as found in the Clough, Engebretson,
Kochavi article I ref'd earlier. (It's true that there are 12 properties,
but I don't really think this is an intentional 12-tet academicist plot.)

Features defined for rational OR irrational generators:
1- Generated by a single interval (G)
2- Well-formed (WF) [Carey and Clampitt, 1989]
3- Myhill Property (MP) [Clough and Myerson, 1986]
4- Distributionally Even (DE) [isolated for the first time in this paper]
Features defined for rational generators only:
5- Maximally Even (ME) [Clough and Douthett, 1991]
6- Deep (DP) [Gamer, 1967]
7- Diatonic (DT) [in the sense used here: Agmon, 1989]
8- Balzano (BZ) [Balzano, 1980]
Important features noted but not given full treatment here:
9- Inversional symmetry
10- Transpositional combination [Richard Cohn, 1991]
11- Prime cardinality
12- Cohn's property (maximally smooth cycles) [Cohn, 1996; Lewin, 1996]

The names are included because I consider them to be some of the
master-players in this game. The dates are included because, as someone
else recently only hinted at on the tuning list, much of their original
work shows up on the tuning list in slightly altered form under a
different rubric, sometimes attributed and sometimes not.

Now, so far I've got that you don't "like" maximal evenness, and, since
you don't "like" interval-class vectors, surely you don't "like" the deep
scale property which Steven Kallstrom brought up (since this property is
surely a slippery-slope into ic vectors; plus you've already verified that
unique interval multiplicity is a "so what?" for you). The reason I don't
consider all of this as empty sarcasm on my part is this:

If, following Clough et al, I say, "DP implies G, and I can prove it," and
you reply, "So what if you can? I don't *like* DP" or "I don't consider DP
all that important" -- one thing we're NOT having is a scholarly dispute.
Why? (Evidently this has to be spelled out!) Because you're response is
a communication STOPPER. You have neither convinced nor defeated my
premise (we never even got to "DP implies G") -- you've simply made me
speechless (and more than a little pissed). You've given us nothing more
to talk about than your own personal preferences. And any reasons you
give for these preferences have nothing to do with music theory, even
though it may appear that way to a novice; it's just theology.

So that I can avoid being blindsided by this frustrating situation in the
future, maybe you could sort through and tell me which of the above
properties you "like" and which you don't "like," and we'll have done with
the whole lot at one sitting.

As to the question you seem to put so much store in,

> where does the fifth come from?

... or the octave, or third, or comma, or a beat you can dance to.

Theorists, set or otherwise, (and most practicing composers and
performers) DON'T REALLY CARE where it *REALLY* comes from. "God made it,
I believe in it, and there's an end to it" is just as good an existential
justification as "People like it" or "Cultures instantiated it" or "It's a
simple ratio."

Theorists, set or otherwise, (and most practicing composers and
performers) only care that "it" (interval, scale, set class, procedure,
form,...) is THERE and that they can think up a multitude of creative ways
of GETTING MORE by studying it's structural characteristics in as
objective a way as possible.

One way to insure that we DON'T "get more" is to settle on only one True
Way and shut the door to other possibilities. Art itself is about
keeping the doors to possibility open, even (or especially) when everyone
else in the world is complaining about the draft.

I enjoy your theory room, Paul, as I enjoy those of others on the list.
But some (only a few) of these rooms come with a high admission price.
What I sometimes hear (unnecessarily) along with the theory is: "This is
MY room. If you like it, fine; if you don't, close the door on your way
out." I'll just keep wandering in and out the open door if you guys don't
mind (or even if you do).

If you want existential justification, see a priest, not a geneticist.

Steve

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
[(Charles Wuorinen:) Any advice for the young?
(Milton Babbitt:) In all honesty, no, no advice for the young or old
composer, if for no other reason than that I have given the latter such
poor advice on virtually every other consequential occasion. With less
honesty, and more high morality, I could urge the young composer to stick
to his guns, and probably shoot himself in the foot. Similarly, I could
proclaim that composing well is the best revenge; but if that be the case,
why am I (again in the name of modus tollens) so vengeful?]
vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv

Hi Stephen.

> Because you're response is
> a communication STOPPER. You have neither convinced nor defeated my
> premise (we never even got to "DP implies G") -- you've simply made me
> speechless (and more than a little pissed).

Why??

>You've given us nothing more
> to talk about than your own personal preferences.

Huh? You -- we -- can talk about anything we like.

> And any reasons you
> give for these preferences have nothing to do with music theory, even
> though it may appear that way to a novice; it's just theology.

What???

> As to the question you seem to put so much store in,

>> where does the fifth come from?

>... or the octave, or third, or comma, or a beat you can dance to.

>Theorists, set or otherwise, (and most practicing composers and
>performers) DON'T REALLY CARE where it *REALLY* comes from. "God >made it,
>I believe in it, and there's an end to it" is just as good an >existential
>justification as "People like it" or "Cultures instantiated it" or >"It's a
>simple ratio."

Can't I now call that a "communication stopper"? I mean, I don't care if DP
implies G (actually I never said that but for the sake of argument . . .),
you don't care (actually: "DON'T REALLY CARE", all caps) where the fifth
comes from . . . what's the difference? The difference is that I won't get
"more than a little pissed" . . . this should all be in the spirit of a fun
chess game.

>Theorists, set or otherwise, (and most practicing composers and
>performers) only care that "it" (interval, scale, set class, >procedure,
>form,...) is THERE and that they can think up a multitude of >creative ways
>of GETTING MORE by studying it's structural characteristics in as
>objective a way as possible.

I think most people on this list _do care_ where the fifth comes from and I
think the type of inquiry which I care about (which you call "theology" to
my complete surprise and slight dismay) is more likely to lead to genuine
musical innovation in the future than the type of theory under which you
would seek to class all "theorists" (but shouldn't).

>One way to insure that we DON'T "get more" is to settle on only one True
>Way and shut the door to other possibilities. Art itself is about
>keeping the doors to possibility open, even (or especially) when everyone
>else in the world is complaining about the draft.

I agree 100%. This sure doesn't sound like your attitude earlier in this
message, however.

In fact, a lot of the "set theory" people, you of course excluded,
contribute, perhaps unknowingly, to the attitude that 12-tET is the one True
Way. Unique interval vectors and so on, used to justify the "specialness" of
the diatonic scale in 12-tone tuning, provide yet another excuse for
academia to scoff at microtonality, and to completely ignore and even
ridicule the subject of tuning (witness several list members' recent
comments).

Our diatonic music has a rich heritage including hundreds of years during
which 12-tone equal temperament was considered _unacceptable_ on most
instruments. You can read this view in most treatises as late at the early
18th century, and the view persisted (though no longer as a consensus view)
well into the 19th century. Why is this simple fact never inquired into,
never pursued, USUALLY NEVER EVEN MENTIONED in an entire undergraduate or
even some graduate courses of study in music? As Joseph Pehrson observed,
this is unbelievable! The wool is being pulled over the eyes of our music
students. And all you have to do to appreciate the point is to listen to the
different tunings being used. Once you've heard that, the bubble is broken.

>I enjoy your theory room, Paul, as I enjoy those of others on the list.
>But some (only a few) of these rooms come with a high admission price.
>What I sometimes hear (unnecessarily) along with the theory is: "This is
>MY room. If you like it, fine; if you don't, close the door on your way
>out."

What do you mean, "close the door on your way out"?

>I'll just keep wandering in and out the open door if you guys don't
>mind (or even if you do).

That's all I'm doing too!

>If you want existential justification, see a priest, not a geneticist.

What is that supposed to mean?

One more thing on this issue, Stephen:

>You've given us nothing more
>to talk about than your own personal preferences.

Again, this remark mystifies me, but maybe it comes from the whole 'I "don't
like" X' bit. That bit of language was introduced by Dan Stearns to refer to
a certain set of observations I made regarding maximal evenness. I kept the
language -- but notice I always kept '"don't like"' in quotes -- as a
shorthand for the types of observations in question. It doesn't refer to my
personal preferences the way "I don't like caramel" would, nor does it refer
to my feelings toward any individual who may promote or simply work with the
concepts in question. So if you've really read my posts this months (which
happen to include a good bit of set-theory-related stuff, I think), I really
don't see how you could make the statement you're making above.

In friendship,
Paul

Paul wrote,
>
> In fact, a lot of the "set theory" people, you of course excluded,
> contribute, perhaps unknowingly, to the attitude that 12-tET is the one
True
> Way. Unique interval vectors and so on, used to justify the "specialness"
of
> the diatonic scale in 12-tone tuning, provide yet another excuse for
> academia to scoff at microtonality, and to completely ignore and even
> ridicule the subject of tuning (witness several list members' recent
> comments).
>
> Our diatonic music has a rich heritage including hundreds of years during
> which 12-tone equal temperament was considered _unacceptable_ on most
> instruments. You can read this view in most treatises as late at the early
> 18th century, and the view persisted (though no longer as a consensus
view)
> well into the 19th century. Why is this simple fact never inquired into,
> never pursued, USUALLY NEVER EVEN MENTIONED in an entire undergraduate or
> even some graduate courses of study in music? As Joseph Pehrson observed,
> this is unbelievable! The wool is being pulled over the eyes of our music
> students. And all you have to do to appreciate the point is to listen to
the
> different tunings being used. Once you've heard that, the bubble is
broken.

I have to agree with you Paul that many theorist use set-theory to
justify the superiority of the 12-tone set. Many have also applied these
theories to expanded ETs. I would like to formulate some of the strong
concepts in set-theory to help in the understanding of non-ET tunings. I
really don't have the time, right now, to investigate these ideas, but I am
sure there is similar work out there somewhere. How do you personally look
at the structures of tunings? I am also curious in examining the
interaction of different tunings, a 'poly-tunality' so to speak.
Unfortunately I have to do regular (not quite as fun) work so that I can
graduate. I am sure that there are pieces out there that use multiple
tunings, but I don't know them.

Tuning in the schools is not good at all. The Grout-Palisca History of
Western Music has a little section on Pythagorean tuning, but after that
there is very little. My personal interest in tuning started with my love
for Baroque music, and playing the harpsichord. I have given two
presentations on tuning for classes, one on early tunings, and another on
today's microtonality. The students couldn't really hear much of a
difference between ET and Werckmeister, or other late Baroque tunings. They
could only hear the difference in meantone tunings when I played in distant
keys. They showed only slight interest in some of the modern works that I
showed them, Partch, De Vries (from the great Annelie de Man CD) and
Corigliano's Chiaroscuro (a personal favorite). The only piece that the
commented on was Ben Johnston's (the name is escaping me right now) solo
clarinet work. The only other positive effect was playing Bach with a
violinist. He loved Werckmeister III because he didn't have to 'adjust' as
much as he does for the piano.

Anyway, just some personal comments, Thanks

Steven Kallstrom

Hi Steven.

>I have to agree with you Paul that many theorist use set-theory to
>justify the superiority of the 12-tone set. Many have also applied these
>theories to expanded ETs. I would like to formulate some of the strong
>concepts in set-theory to help in the understanding of non-ET tunings.

Good luck. But I fear you'll find much of the existing "set-theory" to come
up short in this regard.

>How do you personally look
>at the structures of tunings?

Do you mean tunings or scales? I generally look at tunings for the kinds of
structures they contain, both chords and scales. As for how I look at the
structures of scales, the criteria in my paper should give you a pretty good
idea. For example, if a scale is omnitetrachordal, then I'm disposed to
considering it a good scale melodically, whether or not it has any favorable
harmonic characteristics.

>I am also curious in examining the
>interaction of different tunings, a 'poly-tunality' so to speak.
>Unfortunately I have to do regular (not quite as fun) work so that I can
>graduate. I am sure that there are pieces out there that use multiple
>tunings, but I don't know them.

Yes -- for example, Johnny Reinhard calls himself a "polymicrotonal"
composer -- using, for example, several simple ETs at once. Even in 22-tET,
you can get "polymicrotonal" effects, as there are radically different
scales to be found within the 22-tET fabric.

>Tuning in the schools is not good at all. The Grout-Palisca History of
>Western Music has a little section on Pythagorean tuning, but after that
>there is very little.

Any mention that thirds were a lot better at one point?

>violinist. He loved Werckmeister III because he didn't have to 'adjust' as
>much as he does for the piano.

Cool!

The two might relate and then again they may not. completely subjective

"Paul H. Erlich" wrote:

> I wrote,
>
> >>Of course. But just because Debussy broke the traditional rules of voice
> >>leading doesn't mean "they have nothing to do with what sounds good", as
> you
> >>say. Just because they don't have everything to do with what sounds good
> >>doesn't mean they have nothing to do with what sounds good.
>
> Kraig wrote,
>
> >It seems to me that the principles of voice leading had more to do with
> keeping as much independence in the >voices as possible more than any good
> or bad.
>
> Doesn't keeping independence in the voices have something to do with
> "sounding good?" Wouldn't a lack of independence, at least in certain
> styles, "sound bad"?

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

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

I wrote,

> >Tuning in the schools is not good at all. The Grout-Palisca History of
> >Western Music has a little section on Pythagorean tuning, but after that
> >there is very little.
>

Paul wrote,
> Any mention that thirds were a lot better at one point?

The Grout-Palisca say the following:

Pythagorean made thirds and sixths sound rough (with discussion of the
monochord).

Mentions the birth of Just Intonation (from Bartolome Ramos de Pareja),
and the revealing of Ptolemy by Gaffurio and talks about the 'syntonic
diatonic'. This is discussed nd immediately tied to Musica Ficta. There is
a mention of Nicola Vicentino though (yeah).

There is then a slight mention of meantone tuning and how it gave way to
equal-temperament, and almost equal-temperaments.

The final mentions of tunings concern quarter-tones, and the sections on
Partch and Johnston, which talk of Partch's rejection of ET, and Johnston's
love og Just Intonation.

I guess this is better than nothing, but it is presented in a manner
that students easily forget about it. The recording anthology does little
to expose students to historically accurate performance. Students are
simply not made aware of tuning regarding performing practice or theory. I
think it could be done better. Although, I am not sure how. When I tried
to make classmates aware of tuning differences in some of Bach's music, they
heard nothing, even though Leonhardt's F Minor Prelude from book 2 is
amazingly brilliant in it's 'near' Werckmeister III. They can't even hear
the shifting colors in the modulatory sections. I know that it took me
time, but my friends are convinced that I am being 'elitist', and that I am
trying to make them feel 'stupid'. Really, they think there is no
difference. God help them.

Steven Kallstrom

Kraig wrote,

>The two might relate and then again they may not.

Do you think that composers sought to maintain the independence of voices
did so because it made the music sound good, or for some other reason?

composers where!

"Paul H. Erlich" wrote:

> Kraig wrote,
>
> >The two might relate and then again they may not.
>
> Do you think that composers sought to maintain the independence of voices
> did so because it made the music sound good, or for some other reason?
>
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

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

Kraig wrote,

>composers where!

Composers who sought to maintain the independence of voices, or used the
"rules of voice-leading" for whatever reason they did so . . . mostly
European composers before Debussy.

So did they do so because it sounded good, or for some other reason?

Steven Kallstrom wrote,

<<I am also curious in examining the interaction of different tunings,
a 'poly-tunality' so to speak. Unfortunately I have to do regular (not
quite as fun) work so that I can graduate. I am sure that there are
pieces out there that use multiple tunings, but I don't know them.>>

Hi Steven,

Almost all of my music works from this "'poly-tunality'" premise. "Day
Walks In", which I mention because it is available on the Xenharmonic
Alliance compilation CD, is constructed of various equal tempered,
rationally intoned, and neither nor, tuning schemes.

Also, while not the least bit academic (or set theory inclined), all
my tuning theory type endeavors here on this list are meant to
underscore these all-tunings, pitch continuum interests. And these
interests don't really have any hierarchical ties to JI, 12-tET, or
pure mathematics. So I doubt the "numbers" do either. But I do like
generalizations and schemes that enable various tuning methods to
coexist.

For me tuning theory is a creative pursuit that's half (amateur)
poetry and half (imaginary) problem solving.

(Music on the other hand...? Well falling asleep in the woods has
always been a risky roll of the dice when it comes to "enchantment",
but I think the job of music is much like that of invoking a dream --
explaining, describing or planing it just doesn't do the trick!)

--Dan Stearns

why do you want to know what i think when you have already made up your mind that what they think
is the only type of musical right thinking!

BTW thank you for answering my post from Robert Walker as to what i meant by using letters to
designate the squares!

"Paul H. Erlich" wrote:

> Kraig wrote,
>
> >composers where!
>
> Composers who sought to maintain the independence of voices, or used the
> "rules of voice-leading" for whatever reason they did so . . . mostly
> European composers before Debussy.
>
> So did they do so because it sounded good, or for some other reason?
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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>
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-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

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

Kraig wrote,

>why do you want to know what i think when you have already made up your
mind >that what they think is the only type of musical right thinking!

How absurd. It's that imaginary Paul Erlich again. Where do you get all
these misconceptions about me?

>BTW thank you for answering my post from Robert Walker as to what i meant
by >using letters to designate the squares!

If that's sincere, you're welcome. If not, I'm sorry, even though Robert
Walker seems to have found it very helpful. Do you have a problem with me
helping people to understand difficult concepts?

Paul,

Since we have discussing scales, and sets, and since I have mentioned
some of my problems with maximally even sets. I have a 'theoretical'
proposal get your opinion on. You can generalize the creation of harmonic
structure from a scale by following the guides of the diatonic scales. You
can arrange a 'scale' and say that its 'tonic' is the first, third and fifth
scale degree. I think you see where this is going. Well, last semester I
wrote a few short 'virginal style' harpsichord pieces. The harmonic
structures were based on a tone row, taking the first seven pitches as 'C
Major' and the next seven as 'G Major' (since based on a row the scales are
not identically in intervallic structure). The scales that were formed are
conjunct not linear. e.g. C F# A D Bb Eb F C. The tonic would then be, C A
Bb, the dominant Bb, F, F#. The results of this were actually very cool.
Stylistically modeled after Fitzwilliam pieces the music had a quasi-tonal
feel, the melodic aspect was difficult but made to work. I used common
scalar figures, 5 to 1 descents, turning the linear into conjunct. I guess
what I learned from this was that tonality is strongly related to
'information theory', pattern recognition, and familiarity. I do not know
if this technique will be seen in my works again, but it sure was fun.

I would like to here some comments on this procedure if possible, I think I
may try this is a few simple microtonal works.

Thanks,

Steven Kallstrom

--- In tuning@y..., "Steven Kallstrom" <skallstr@s...> wrote:

/tuning/topicId_20848.html#20861

> I would like to propose that we select a few tunings, and
examine their structure over the list. I would love to see how the
experienced memebers of the list would look at a sets structure. I
think that it would be rather intersting also, to see the different
approaches to examining a tuning.
>
> Thank you,
>
> Steven Kallstrom

Hello Steven...

I'm a little confused, but isn't that what we're doing all the time
with this LsssLss, etc. set analysis??

Incidentally, Orlando di Lasso.... L+ss

________ _____ ____ _
Joseph Pehrson

You have a problem with Moments of Symmetry (MOS) [Wilson,1975]

Stephen Soderberg wrote:

> Paul, here's a list of many of the important structural properties any
> given scale may or may not possess as found in the Clough, Engebretson,
> Kochavi article I ref'd earlier. (It's true that there are 12 properties,
> but I don't really think this is an intentional 12-tet academicist plot.)
>
> Features defined for rational OR irrational generators:
> 1- Generated by a single interval (G)
> 2- Well-formed (WF) [Carey and Clampitt, 1989]
> 3- Myhill Property (MP) [Clough and Myerson, 1986]
> 4- Distributionally Even (DE) [isolated for the first time in this paper]
> Features defined for rational generators only:
> 5- Maximally Even (ME) [Clough and Douthett, 1991]
> 6- Deep (DP) [Gamer, 1967]
> 7- Diatonic (DT) [in the sense used here: Agmon, 1989]
> 8- Balzano (BZ) [Balzano, 1980]
> Important features noted but not given full treatment here:
> 9- Inversional symmetry
> 10- Transpositional combination [Richard Cohn, 1991]
> 11- Prime cardinality
> 12- Cohn's property (maximally smooth cycles) [Cohn, 1996; Lewin, 1996]
>
> The names are included because I consider them to be some of the
> master-players in this game. The dates are included because, as someone
> else recently only hinted at on the tuning list, much of their original
> work shows up on the tuning list in slightly altered form under a
> different rubric, sometimes attributed and sometimes not.
>
> Now, so far I've got that you don't "like" maximal evenness, and, since
> you don't "like" interval-class vectors, surely you don't "like" the deep
> scale property which Steven Kallstrom brought up (since this property is
> surely a slippery-slope into ic vectors; plus you've already verified that
> unique interval multiplicity is a "so what?" for you). The reason I don't
> consider all of this as empty sarcasm on my part is this:
>
> If, following Clough et al, I say, "DP implies G, and I can prove it," and
> you reply, "So what if you can? I don't *like* DP" or "I don't consider DP
> all that important" -- one thing we're NOT having is a scholarly dispute.
> Why? (Evidently this has to be spelled out!) Because you're response is
> a communication STOPPER. You have neither convinced nor defeated my
> premise (we never even got to "DP implies G") -- you've simply made me
> speechless (and more than a little pissed). You've given us nothing more
> to talk about than your own personal preferences. And any reasons you
> give for these preferences have nothing to do with music theory, even
> though it may appear that way to a novice; it's just theology.
>
> So that I can avoid being blindsided by this frustrating situation in the
> future, maybe you could sort through and tell me which of the above
> properties you "like" and which you don't "like," and we'll have done with
> the whole lot at one sitting.
>
> As to the question you seem to put so much store in,
>
> > where does the fifth come from?
>
> ... or the octave, or third, or comma, or a beat you can dance to.
>
> Theorists, set or otherwise, (and most practicing composers and
> performers) DON'T REALLY CARE where it *REALLY* comes from. "God made it,
> I believe in it, and there's an end to it" is just as good an existential
> justification as "People like it" or "Cultures instantiated it" or "It's a
> simple ratio."
>
> Theorists, set or otherwise, (and most practicing composers and
> performers) only care that "it" (interval, scale, set class, procedure,
> form,...) is THERE and that they can think up a multitude of creative ways
> of GETTING MORE by studying it's structural characteristics in as
> objective a way as possible.
>
> One way to insure that we DON'T "get more" is to settle on only one True
> Way and shut the door to other possibilities. Art itself is about
> keeping the doors to possibility open, even (or especially) when everyone
> else in the world is complaining about the draft.
>
> I enjoy your theory room, Paul, as I enjoy those of others on the list.
> But some (only a few) of these rooms come with a high admission price.
> What I sometimes hear (unnecessarily) along with the theory is: "This is
> MY room. If you like it, fine; if you don't, close the door on your way
> out." I'll just keep wandering in and out the open door if you guys don't
> mind (or even if you do).
>
> If you want existential justification, see a priest, not a geneticist.
>
> Steve
>
> ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
> [(Charles Wuorinen:) Any advice for the young?
> (Milton Babbitt:) In all honesty, no, no advice for the young or old
> composer, if for no other reason than that I have given the latter such
> poor advice on virtually every other consequential occasion. With less
> honesty, and more high morality, I could urge the young composer to stick
> to his guns, and probably shoot himself in the foot. Similarly, I could
> proclaim that composing well is the best revenge; but if that be the case,
> why am I (again in the name of modus tollens) so vengeful?]
> vvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvvv
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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>
>
> Your use of Yahoo! Groups is subject to http://docs.yahoo.com/info/terms/

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

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

Kraig Grady wrote:

> You have a problem with Moments of Symmetry (MOS) [Wilson,1975]

I think Stephen covered that with

> > 2- Well-formed (WF) [Carey and Clampitt, 1989]

and

> > The dates are included because, as
> > someone
> > else recently only hinted at on the tuning list, much of their
> > original
> > work shows up on the tuning list in slightly altered form under a
> > different rubric, sometimes attributed and sometimes not.

Wilson's MOS is clearly Carey & Clampitt's WF in a slightly alter form
under a different rubric, completely unattributed, -14 years later. Also
notable by its absence is Rothenberg Propriety (1969?).

I'd also count being or including a Tonality Diamond (Partch?) or
Combination Product Set (Wilson, 1981) as "an important structural
property any given scale may or may not possess". But I don't expect
they're covered in the Clough, Engebretson, Kochavi article.

I know it's been established that nobody cares what anybody else thinks
around here. I will still say that it seems to me that, whatever the
importance of being Proper or an MOS, the later properties have little
more to add. And the concepts of Tonality Diamond and CPS are important
in innovative scale construction, but have the obvious defect that they
can't be applied to an equally tempered 7 from 12 scale. (A diatonic
scale could be 2)1.1.3.5.9, right?)

Graham

Hi Stephen.

You wrote,

>Theorists, set or otherwise, (and most practicing composers and
>performers) DON'T REALLY CARE where it *REALLY* comes from. "God >made it,
>I believe in it, and there's an end to it" is just as good an >existential
>justification as "People like it" or "Cultures instantiated it" or >"It's a
>simple ratio."

>Theorists, set or otherwise, (and most practicing composers and
>performers) only care that "it" (interval, scale, set class, >procedure,
>form,...) is THERE and that they can think up a multitude of >creative ways
>of GETTING MORE by studying it's structural characteristics in as
>objective a way as possible.

Here are some theorists who might be "more than a little pissed" by this
exclusive viewpoint:

Benedetti
Rameau
Helmholtz
Hindemith
Fokker
W. Carlos
Terhardt
Partch
Parncutt
J. Catler
Vogel
Blackwood
Sethares
E. Wilson
J. R. Pierce
Canright
D. Doty
S. Ramanathan
Lindley & Turner-Smith
Tenney
even Krumhansl

etc. etc.

"The names are included because I consider them to be some of the
master-players in this game."

Stephen, I have no idea why this degenerated into attacks involving doors
opening and slamming shut (???) Someone asked about the mathematics of music
and I wanted to be fair by letting them know that there were philosophies
very different from my own. That is where I mentioned you, and so you came
in to elaborate. I remarked that it was nice to hear from you, that it had
been too long. I also stated that everyone should do their own thinking and
experimenting and decide on their own concerning these different
philosophies. We've amply demonstrated the differences in at least two
philosophies. Or we could continue to do so -- for example:

Partchian: Triads are the first three identities of an otonal or utonal
series.

Agmonian(?): No, triads are the second-order ME 3-out-of-7-out-of-12.

Partchian: That's not the essence of triads.

Agmonian(?): Yes it is.

Partchian: No it's not.

Agmonian(?): You're wrong.

Partchian: You're wrong!

[Fistfight ensues] :) :) :)

What I don't understand is the vitriol. Your message yesterday left me
completely baffled and quite disturbed for the last 24 hours. I honestly
don't understand where any of that came from.

Hope today finds you in better spirits,
Paul

I would assume that they all would be pissed by you putting words in their mouths.
or are you the extreme authority of all the big players. gee you must be the condensation of them
all into one

"Paul H. Erlich" wrote:

>
> Here are some theorists who might be "more than a little pissed" by this
> exclusive viewpoint:
>
> Benedetti
> Rameau
> Helmholtz
> Hindemith
> Fokker
> W. Carlos
> Terhardt
> Partch
> Parncutt
> J. Catler
> Vogel
> Blackwood
> Sethares
> E. Wilson
> J. R. Pierce
> Canright
> D. Doty
> S. Ramanathan
> Lindley & Turner-Smith
> Tenney
> even Krumhansl
>
> etc. etc.
>
> "The names are included because I consider them to be some of the
> master-players in this game."
>
>
> Stephen, I have no idea why this degenerated into attacks involving doors
> opening and slamming shut (???) Someone asked about the mathematics of music
> and I wanted to be fair by letting them know that there were philosophies
> very different from my own. That is where I mentioned you, and so you came
> in to elaborate. I remarked that it was nice to hear from you, that it had
> been too long. I also stated that everyone should do their own thinking and
> experimenting and decide on their own concerning these different
> philosophies. We've amply demonstrated the differences in at least two
> philosophies. Or we could continue to do so -- for example:
>
> Partchian: Triads are the first three identities of an otonal or utonal
> series.
>
> Agmonian(?): No, triads are the second-order ME 3-out-of-7-out-of-12.
>
> Partchian: That's not the essence of triads.
>
> Agmonian(?): Yes it is.
>
> Partchian: No it's not.
>
> Agmonian(?): You're wrong.
>
> Partchian: You're wrong!
>
> [Fistfight ensues] :) :) :)
>
> What I don't understand is the vitriol. Your message yesterday left me
> completely baffled and quite disturbed for the last 24 hours. I honestly
> don't understand where any of that came from.
>
> Hope today finds you in better spirits,
> Paul

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

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

All right Kraig this is completely unfair. Stephen gave a list of theorists
who he thought would be "more than a little pissed" by my viewpoint of
seeking a basis for the octave and fifth. I responded with a list of
theorists who thought explanations involving ratios were important, as a
response to Stephen's assertion that theorists don't care about such
explanations. Do you actually read these posts or do you simply take every
possible opportunity to attack me, villify me, attribute beliefs to me which
I would never hold, and take my statements out of context?

i am sorry that i did not also direct it to steven also.

"Paul H. Erlich" wrote:

> All right Kraig this is completely unfair. Stephen gave a list of theorists
> who he thought would be "more than a little pissed" by my viewpoint of
> seeking a basis for the octave and fifth. I responded with a list of
> theorists who thought explanations involving ratios were important, as a
> response to Stephen's assertion that theorists don't care about such
> explanations. Do you actually read these posts or do you simply take every
> possible opportunity to attack me, villify me, attribute beliefs to me which
> I would never hold, and take my statements out of context?

-- 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..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_20848.html#20911

> ... where do you suppose the octave comes from? Is it just a
> total coincidence that every culture which recognizes a repetition
> of pitch classes does so at an interval awfully close to a 2:1
> frequency ratio -- that of the first overtone?

IMO (and yours too, I'm sure, Paul), it's not a coincidence at all.

In some cultures (or at least past cultures), the 3:2 or an
approximation of it serves the same function. For example, in
ancient Greek music theory the important repetitions of "pitch
classes" ocurred at the 3:2, with additional secondary repetitions
at the 2:1.

We've also noted here before that untrained singers who are
supposed to be singing in unision frequently match the pitches
of their melodies at a 3:2 or 4:3 instead of a 2:1.

It's obvious to me that pitch-class similarity has something
to do with (approximate) low-integer harmonic relationship.

-monz
http://www.monz.org
"All roads lead to n^0"

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

/tuning/topicId_20848.html#20911

Erlich:

> And the octave -- virtually all the set theory stuff assumes octave
> equivalence -- but where do you suppose the octave comes from? Is
it just a total coincidence that every culture which recognizes a
repetition of pitch classes does so at an interval awfully close to a
2:1 frequency ratio -- that of the first overtone?
>

You know... this is one reason that I'm beginning to suspect that ANY
set theory that doesn't take into account an underlying harmonic or
tuning basis is pretty specious...

________ ______ _____ ___
Joseph Pehrson

--- In tuning@y..., "Steven Kallstrom" <skallstr@s...> wrote:

/tuning/topicId_20848.html#20923

> I guess I need to clarify. I think that an interval that
symmetrically divides the octave is a special interval, in 12tet this
is the tritone. The B-F of C and B-E# of F# are the same, but the
interesting thing about them is that they are different intervals
diatonically, B-F a diminished fifth, the other an augmented fourth.

But the function of this tritone is basically the same in both cases,
right, Paul?? Isn't that what Richmond Browne is trying to get at??

________ _____ _____ _
Joseph Pehrson

--- In tuning@y..., "Steven Kallstrom" <skallstr@s...> wrote:

/tuning/topicId_20848.html#20926

>
> Paul wrote,
> > I don't "like" interval vectors either :) That is, when a music
teacher pointed out the unique entries of the interval vector of the
12-tET diatonic scale, and everyone was like, "gasp, wow, that's why
the diatonic scale is special", I felt like a major swindle was
taking
place.
> >
>

Steven Kallstrom:

> I love interval vectors, and they make me happy :)... I can
understand you reaction to the diatonic interval vector. When you
compare it to others though, you realize how special the set is!

Hello Paul Erlich and Steven Kallstrom!

Well, it's been, literally, YEARS since I have thought about
"interval vectors..." I don't know if that's good or bad! (probably
bad)

In any case, I also remember being taught that the diatonic
collection has a unique entry of interval vectors....

I can't find my notes on that topic, but when I plot out the diatonic
collection, I get:

[2,5,4,3,6,1]

Is that right?? It seems to have a familiar "ring" to it...

In any case, Paul, it sounds as though you believe there are other
tuning systems that have intervallic properties AS INTERESTING as the
diatonic collection. I would be interested in learning more about
that, if you have time...

Well, I must confess that though I went through George Perle's
_Serial Composition and Atonality_ with apparent ease, I had more
trouble with Allen Forte's _The Structure of Atonal Music_...

Well, not all of it is troublesome, but certainly a good portion of
it... I fear my overall math skills are not quite up to the task.

I was curious as to how many people on this list have actually been
successfully through that book.

I would not be at all surprised if there would be several...

Thanks!

_______ ______ ______ ____
Joseph Pehrson

Steven Kallstrom wrote,

>> I guess I need to clarify. I think that an interval that
>>symmetrically divides the octave is a special interval, in 12tet this
>>is the tritone. The B-F of C and B-E# of F# are the same, but the
>>interesting thing about them is that they are different intervals
>>diatonically, B-F a diminished fifth, the other an augmented fourth.

Joseph Pehrson wrote,

> But the function of this tritone is basically the same in both cases,
> right, Paul?? Isn't that what Richmond Browne is trying to get at??

Hmm . . . you'll have to ask Steven what he meant by the above. As for what
I think . . . well, in 12-tET you can use the tritone to modulate to a
distant key, while in 19-tET or 31-tET you can't, or at least not in the
same way. But in a single key or a small set of neighboring keys (such as
tonic and dominant keys), the tritone functions pretty much the same way in
all three tunings. At least if we're talking about common-practice diatonic
music.

--- In tuning@y..., Stephen Soderberg <SSOD@L...> wrote:

/tuning/topicId_20848.html#20970
>
> Features defined for rational OR irrational generators:
> 1- Generated by a single interval (G)
> 2- Well-formed (WF) [Carey and Clampitt, 1989]
> 3- Myhill Property (MP) [Clough and Myerson, 1986]
> 4- Distributionally Even (DE) [isolated for the first time in this
paper]
> Features defined for rational generators only:
> 5- Maximally Even (ME) [Clough and Douthett, 1991]
> 6- Deep (DP) [Gamer, 1967]
> 7- Diatonic (DT) [in the sense used here: Agmon, 1989]
> 8- Balzano (BZ) [Balzano, 1980]
> Important features noted but not given full treatment here:
> 9- Inversional symmetry
> 10- Transpositional combination [Richard Cohn, 1991]
> 11- Prime cardinality
> 12- Cohn's property (maximally smooth cycles) [Cohn, 1996; Lewin,
1996]

Thank you, Steven, for posting this... Of course, now we have to go
and hunt up all the articles... As I was mentioning to somebody at
the MicroFest, wouldn't it be great if there was an International
Microtonal Library, which would have ALL the McLaren-Op de Coul
indexed articles which they would fax for a fee?? Dream on.

About Erlich:

> You've given us nothing more to talk about than your own personal
preferences. And any reasons you give for these preferences have
nothing to do with music theory, even though it may appear that way
to a novice; it's just theology.
>

> If you want existential justification, see a priest, not a
geneticist.
>

I'm sorry to say that I found this entertaining... and, it's Easter
Sunday...

_______ _____ ____ _
Joseph Pehrson

Paul Erlich wrote:

> > And the octave -- virtually all the set theory stuff assumes octave
> > equivalence -- but where do you suppose the octave comes from?

Must set theory be independent of tuning theory to be valid? If a given tuning system assumes octave
equivalence, would not set theory also have to assume it in order to apply to that tuning?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

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

/tuning/topicId_20848.html#20978

>
> >Theorists, set or otherwise, (and most practicing composers and
> >performers) only care that "it" (interval, scale, set class,
>procedure, form,...) is THERE and that they can think up a multitude
of creative ways of GETTING MORE by studying it's structural
characteristics in as objective a way as possible.
>
> I think most people on this list _do care_ where the fifth comes
from and I think the type of inquiry which I care about (which you
call "theology" to my complete surprise and slight dismay) is more
likely to lead to genuine musical innovation in the future than the
type of theory under which you would seek to class all "theorists"
(but shouldn't).
>

Well, I would have to agree with Paul here... If we're going to
study "set theory," lets study "REAL" set theory, of MANY DIFFERENT
tuning systems!

_________ ______ _____
Joseph Pehrson

--- In tuning@y..., "Steven Kallstrom" <skallstr@s...> wrote:

/tuning/topicId_20848.html#20982

>I am also curious in examining the interaction of different tunings,
a 'poly-tunality' so to speak. Unfortunately I have to do regular
(not quite as fun) work so that I can graduate. I am sure that there
are pieces out there that use multiple tunings, but I don't know them.
>

The maestro of multiple tunings is our own Johnny Reinhard. One of
his main principles is that of "polymicrotonality" which he strongly
advocates. I would contact him, since I am certain he would know of
MANY works, including his own, which use such procedures, and he
could also describe the effects of such usage...

Regarding the "tuning in education" discussion, I would say that the
situation is even WORSE than the way Paul Erlich describes it. I
never encountered ONE MENTION of tuning even in any GRADUATE courses
in some pretty good institutions. NOT ONCE. In fact not even in
MEDIEVAL and Renaissance courses! I mean it! It's amazing.

The only person who taught ANYTHING about tuning was our wonderful
piano tuner at the University of Michigan, Kurt Pickett. I am
certain he is no longer living, but he would certainly have enjoyed
this list.

His course was the most practical thing I learned in school, too! :)

________ _____ ____ _____
Joseph Pehrson

Joseph wrote,

>In any case, Paul, it sounds as though you believe there are other
>tuning systems that have intervallic properties AS INTERESTING as the
>diatonic collection. I would be interested in learning more about
>that, if you have time...

It's not so much that . . . it's more that I think that the diatonic scale
is AS INTERESTING and AT LEAST AS MUSICAL when tuned in tunings other than
12-tET (especially other meantone tunings) -- you only get the [2,5,4,3,6,1]
when you consider the diatonic scale as coming from a 12-pitch universe . .
. and in general I find no compelling reason to do so.

--- End of reply to Joseph Pehrson ---

Note to others: please don't take my statements out of context. They are
intended to reply to particular queries of particular people and are stated
so as to make the most sense to said people with a minimum of verbiage.

I wrote,

> > And the octave -- virtually all the set theory stuff assumes octave
> > equivalence -- but where do you suppose the octave comes from?

David Finnamore wrote,

> Must set theory be independent of tuning theory to be valid?

Of course not! The philosophy I was replying to was the philosophy that "set
theory is valid, but tuning theory, or at least any tuning theory that
involves ratios, is either invalid, or its validity is irrelevant".

--- In tuning@y..., "monz" <MONZ@J...> wrote:

/tuning/topicId_20848.html#21052

>
> --- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:
>
> /tuning/topicId_20848.html#20911
>
> > ... where do you suppose the octave comes from? Is it just a
> > total coincidence that every culture which recognizes a repetition
> > of pitch classes does so at an interval awfully close to a 2:1
> > frequency ratio -- that of the first overtone?
>
>
> IMO (and yours too, I'm sure, Paul), it's not a coincidence at all.
>
> In some cultures (or at least past cultures), the 3:2 or an
> approximation of it serves the same function. For example, in
> ancient Greek music theory the important repetitions of "pitch
> classes" ocurred at the 3:2, with additional secondary repetitions
> at the 2:1.
>
> We've also noted here before that untrained singers who are
> supposed to be singing in unision frequently match the pitches
> of their melodies at a 3:2 or 4:3 instead of a 2:1.
>
> It's obvious to me that pitch-class similarity has something
> to do with (approximate) low-integer harmonic relationship.
>
>

Monz...

Paul obviously knows this! This part that you extracted illustrates
his emphasis of the point in the context of how it relates to SET
THEORY... (or doesn't relate to it...)

_________ _______ ______ ___
Joseph Pehrson

I wrote,
> Must set theory be independent of tuning theory to be valid?

Paul H. Erlich wrote:

> Of course not! The philosophy I was replying to was the philosophy that "set
> theory is valid, but tuning theory, or at least any tuning theory that
> involves ratios, is either invalid, or its validity is irrelevant".

Oh, I see. Wrong question, then. My assumptions
interfered with my perception of Stephen's assumptions.
:-) He's been mighty quiet since that post. Hope
you're still out there, Steve!

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

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

/tuning/topicId_20848.html#21124

> Joseph wrote,
>
> >In any case, Paul, it sounds as though you believe there are other
> >tuning systems that have intervallic properties AS INTERESTING as
the
> >diatonic collection. I would be interested in learning more about
> >that, if you have time...
>
> It's not so much that . . . it's more that I think that the diatonic
scale is AS INTERESTING and AT LEAST AS MUSICAL when tuned in tunings
other than 12-tET (especially other meantone tunings) -- you only get
the [2,5,4,3,6,1] when you consider the diatonic scale as coming from
a 12-pitch universe . . and in general I find no compelling reason to
do so.
>

Thanks, Paul, for this response. Actually, I have never spent much
time thinking about our "traditional" diatonic collection in anything
other than 12-tET... so thanks for the "revelations..."

(Well, maybe a little thinking about it in meantone...)
________ _____ _____ _
Joseph Pehrson