Yahoo Tuning Groups Ultimate Backup tuning-math Re: [tuning-math] Digest Number 862

> There are obviously different flavors of Z-relations. I have
> discovered that (in 24-et at least, for every subset 4 thru 9 and
> for 12, that I have tested) the Z-relations are based on the divisors
> of 24 (except for 24) (1,2,3,4,6,8,12) So there is some kind
> of "clumping" going on. In 19-et though, there are pairs and triples
> of z-relations and 3 is obviously not a divisor of 19. (I should point
> out that I am counting Z-related sets AFTER reducing for transposition
> and inversion). The final set counts are then based on unique
> interval vectors. If I am wasting my time looking for patterns of Z-
> relations, and of set counts based on unique interval vectors, you
> guys would tell me, right?

Hi again Paul - I know of three people who have tried solving this
problem. Steven Soderberg published an article in Music Theory Online
where he found a construction that caught a bunch of Z-related sets, but
not all. He also presented on the subject at a 2003 regional meeting of
the American Mathematical Society in Baton Rouge, you might be able to
find an abstract online. There's a guy called Nick Collins (google "nick
collins z-relation") who write some papers on this--he's a computer
researcher I think, who was working on this as a hobby. And David Lewin
has a couple of articles that generalise the Z-relation to various other
GIS's. His last article was titled something like "A commutative GIS
cannot have a 3-element Z-related pair, but a non-commutative GIS can".
"GIS" stands for generalised interval system by the way, and you can read
all about them in his book "Generalised Musical Intervals and
Transformations".

I think that Z-related groups of size 2,3,6,12 are more common in *all*
ets, not just ones for which those are a factor of the cardinality of the
aggregate set. I've got some statistics somewhere I pulled out from my
files, on which ets have Z-groupings of which sizes... I'll have a look
for it.

> Thanks Jon. You said it better than I did. Can you point me to these
> theorems regarding Z-relations?

I think you said you had the Rahn book - isn't it in there somewhere, in
the chapter on common-tone theorems?

> Jon, regarding your files: unfortunately I don't have the ability to
> scp files. And my email here only accepts attachments up to 10 mg.
> Any ideas? Could you burn your zipfile to a CD? Or you could ftp it
> to my website (uptownjazz.com). Whenever...

Maybe ftp'ing the files would work. Let's figure it out in 10 days or so,
sorry but I'm trying to finish something up, until then...

Carl wrote:

> Funny, sounds just like serial atonal music (to my novice ear).

Yeah funnily enough even stochastic music with the right parameters will
sound like serial atonal music with just a casual listen. Doesn't mean
they're not structured differently deep down though, just because we don't
pick up consciously on serial ordering.

>
> Is the final fugue of the WTC1 a serial piece? Why
> or why not?

I feel like this is a trick question, but: No, because it's not written
with a tone-row. Can you derive the countersubject from the subject via
serial procedures?

Gene wrote:

> It sounds ugly either way, but shouldn't that be "class-set",
> not "set-class"? And what is a "set-class correspondence"?

A pitch is, for example, {16}. It belongs to the pitch-class {4}.

A pitch-set is, e.g. {3,6,7}. It belongs to the set-class [014]. [014]
is an equivalence class of pitch sets, so its a pitch-set equivalence
class, or set-class for short.

> Same question re "set-class relationhip". How are these defined? Is
> this vertical, horizontal, or both?

I'm putting it glibly here, but: you pick a bunch of notes you feel belong
together. Determine the set-class they form. Pick another bunch of notes
you like. Determine that sc. Is it the same sc? Is it perhaps a subset or
superset of the first sc? Is it a Z-related sc? All those things can be
asserted as meaningful set-class relationships between the two passages.
The problem, as you can quickly see, is that with no strict definitions of
which notes "belong together", you can gerrymander your set-classes to
make whatever point you are trying to make. Generally analysts stick to
simultaneities or melodic fragments in one instrument or a combination of
those.

Regards to all --Jon

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:

> > Funny, sounds just like serial atonal music (to my novice ear).
>
> Yeah funnily enough even stochastic music with the right parameters
will
> sound like serial atonal music with just a casual listen.

But completely atonal stochastic music with triadic harmony will not
sound at all like what people think atonal music ought to sound like.

> > It sounds ugly either way, but shouldn't that be "class-set",
> > not "set-class"? And what is a "set-class correspondence"?
>
> A pitch is, for example, {16}. It belongs to the pitch-class {4}.
>
> A pitch-set is, e.g. {3,6,7}. It belongs to the set-class [014].
[014]
> is an equivalence class of pitch sets, so its a pitch-set
equivalence
> class, or set-class for short.

This is getting totally out of hand. A pitch denoted by an integer. A
pitch-class is denoted by an element of Z/12Z, as an integer reduced
to the range 0-11 mod 12. A set-class, short for set-pitch-class, is
a element of an equivalence class of sets of pitch-classes under the
dihedreal group acting as a permutation group on Z/12Z. What next?

In general, one might for "pitch" want an element of a fintely
generated free abelian group with a specififed mapping from the
positive rationals, to the reals, or both, determining what pitch it
is. In this case however we want a rank one group with a mapping T to
the reals, which sends 0 to the base pitch B, such that n-->T(n)/B is
a homomorphism. Then the number N such that T(N)/B = 2 is the number
of octave divisions. For instance in the above case, we can map n to
261.2*2^(n/12) Hz, defining a specific pitch for it. Is "pitch"
really the right word for this concept, given that it is only
actually a pitch after we've mapped it?

Next we can reduce this group modulo octaves, which means reducing a
group isomorphic to Z modulo N to obtain Z/NZ. If N is 12, the
residues mod 12 are {0,1,...11}; it seems to me that instead of the
equivalence classes of numbers mod 12 you really are working with the
residues in terms of nomenclature, so you could call them "residues"
and not "pitch classes". That is, clearly you don't really mean "the
pitch-class {4}" but the mod 12 residue 4, so why not call it that?
The pitch-class containing 4 is not {4}, but {... -20, -8, 4, 16,
30, ...}, an infinite set, so you really do seem to be trying for
this anyway.

Now you can take a permutation group G on the residues, and a set s
of residues, and define Pfred(s, G) as I did before--we associate a
number Ba(s) to s by taking the sum 2^i for i in s. Then Pfred(G,s) is
the least Ba(t) among all the sets t in the G-orbit of s. Whatever
name one gave this, it certainly shouldn't be something as confusing
as pitch-class-set!

🔗Carl Lumma <ekin@lumma.org>

>> Is the final fugue of the WTC1 a serial piece? Why
>> or why not?
>
>I feel like this is a trick question, but: No, because it's not written
>with a tone-row. Can you derive the countersubject from the subject via
>serial procedures?

No trick questions from me, unless I'm tricking myself!

So does the subject fail to be a tone row because some notes are used
more than once before all of them are used?

As for transforming it into the countersubject, can you give me two
subjects of the same length that cannot be transformed into one
another with serial procedures? I'll believe you if you say yes.

What are the allowed serial procedures?

transpostion?
contrary motion?
mirror inversion?
retrograde?
augmentation?
diminution?
others?

-Carl

🔗gooseplex <cfaah@eiu.edu>

Regarding these comments:

> 261.2*2^(n/12) Hz, defining a specific pitch for it. Is "pitch"
> really the right word for this concept, given that it is only
> actually a pitch after we've mapped it?
>
...

> it seems to me that instead of the
> equivalence classes of numbers mod 12 you really are
working with the
> residues in terms of nomenclature, so you could call them
"residues"
> and not "pitch classes".

...

Whatever
> name one gave this, it certainly shouldn't be something as
confusing
> as pitch-class-set!

Unfortunately, these are the names that have stuck; I'm afraid
that they are the accepted system of nomenclature for musical
set theory, no matter how wrong they are. Remember this
bastardization of mathematical set theory for supposed musical
purposes originally had nothing to do with tunings outside of 12
equal. It's not a 'theory' like a theory of harmony is a theory. It's
basically just a way to generate lots of information about groups
of notes, some of which often has no musical relevance
whatsoever.

I agree that the nomenclature is mostly misleading and silly. But
then I also feel that terms like "unison vector" are silly. We could
go on and on... the term "octave equivalence" isn't quite right, etc.
I address things like this in my own writing...

By all means, I encourage you to use your own terms if they
make more sense to you. Just include clear definitions in your
work and be aware of existing conventions when you do this.

It may also be a good idea to keep in mind that it does become
tiresome to have to wade through a bunch of re-definitions of
conventional terminology just to get at any meaning out of
someone's writing.

All best,
Aaron

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> > > Funny, sounds just like serial atonal music (to my novice ear).
> >
> > Yeah funnily enough even stochastic music with the right
parameters
> will
> > sound like serial atonal music with just a casual listen.
>
> But completely atonal stochastic music with triadic harmony will
not
> sound at all like what people think atonal music ought to sound
like.
>
> > > It sounds ugly either way, but shouldn't that be "class-set",
> > > not "set-class"? And what is a "set-class correspondence"?
> >
> > A pitch is, for example, {16}. It belongs to the pitch-class {4}.
> >
> > A pitch-set is, e.g. {3,6,7}. It belongs to the set-class [014].
> [014]
> > is an equivalence class of pitch sets, so its a pitch-set
> equivalence
> > class, or set-class for short.
>
> This is getting totally out of hand. A pitch denoted by an integer.
A
> pitch-class is denoted by an element of Z/12Z, as an integer
reduced
> to the range 0-11 mod 12. A set-class, short for set-pitch-class, is
> a element of an equivalence class of sets of pitch-classes under
the
> dihedreal group acting as a permutation group on Z/12Z. What next?
>
> In general, one might for "pitch" want an element of a fintely
> generated free abelian group with a specififed mapping from the
> positive rationals, to the reals, or both, determining what pitch
it
> is. In this case however we want a rank one group with a mapping T
to
> the reals, which sends 0 to the base pitch B, such that n-->T(n)/B
is
> a homomorphism. Then the number N such that T(N)/B = 2 is the
number
> of octave divisions. For instance in the above case, we can map n
to
> 261.2*2^(n/12) Hz, defining a specific pitch for it. Is "pitch"
> really the right word for this concept, given that it is only
> actually a pitch after we've mapped it?
>
> Next we can reduce this group modulo octaves, which means reducing
a
> group isomorphic to Z modulo N to obtain Z/NZ. If N is 12, the
> residues mod 12 are {0,1,...11}; it seems to me that instead of the
> equivalence classes of numbers mod 12 you really are working with
the
> residues in terms of nomenclature, so you could call
them "residues"
> and not "pitch classes". That is, clearly you don't really
mean "the
> pitch-class {4}" but the mod 12 residue 4, so why not call it that?
> The pitch-class containing 4 is not {4}, but {... -20, -8, 4, 16,
> 30, ...}, an infinite set, so you really do seem to be trying for
> this anyway.
>
> Now you can take a permutation group G on the residues, and a set s
> of residues, and define Pfred(s, G) as I did before--we associate a
> number Ba(s) to s by taking the sum 2^i for i in s. Then Pfred(G,s)
is
> the least Ba(t) among all the sets t in the G-orbit of s. Whatever
> name one gave this, it certainly shouldn't be something as
confusing
> as pitch-class-set!

Somehow, pitch-class-set is immediately clear to me, while the above
is practically indecipharable. And I'm no fan of PC set theory!

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:

> Unfortunately, these are the names that have stuck; I'm afraid
> that they are the accepted system of nomenclature for musical
> set theory, no matter how wrong they are.

I'm not saying they are wrong, but they are confusing in a rocco
way. Moreover calling it "set theory" is a little bizarre.

Remember this
> bastardization of mathematical set theory for supposed musical
> purposes originally had nothing to do with tunings outside of 12
> equal.

It isn't a bastardization of set theory. Set theory as an area of
study does not concern itself with finite sets; if you look at the
cardinality of such sets you get numbers, and therefore number
theory; and you can construct all the other animals of finite math,
such as permutation groups, designs, finite geometries, matroids and
so forth using finte sets. This is not set theory to a mathematician.
Algebra, combinatorics and number theory are the relevant areas, not
set theory. If a set isn't so big we aren't even sure if it can exist
without self-contradiction set theorists think of it as tiny.

> I agree that the nomenclature is mostly misleading and silly. But
> then I also feel that terms like "unison vector" are silly.

Point taken, but don't blame me for that one. :)

> It may also be a good idea to keep in mind that it does become
> tiresome to have to wade through a bunch of re-definitions of
> conventional terminology just to get at any meaning out of
> someone's writing.

True, but failing to use standard math terminology is also a matter
of redefinition.

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, jon wild <wild@f...> wrote:
>
> I think that Z-related groups of size 2,3,6,12 are more common in
*all*
> ets, not just ones for which those are a factor of the cardinality
of the
> aggregate set. I've got some statistics somewhere I pulled out from
my
> files, on which ets have Z-groupings of which sizes... I'll have a
look
> for it.

Hi Jon,

Thanks for all the info! But I'm starting to think that quantifying Z-
relations is a bit like trying to define the background complementary
space) of a fractal..can't be done. But I'm still going to pursue it
for a while. I know for example, that the 3 z-related pairs
in "Pentachords of 12-et" (I'm not saying C{12,5}!) are all based on
simple transforms and "Hexachords of 12-et" are all complementary
pairs. With regards to 24-et, I still think it is significant that
dodecachords have Z-relations based on all the divisors of 24 except
24 (1,2,3,4,6,8,12) but maybe I'm just being stubborn. Lower sets' z-
relations are also based on these divisors.

Of course, one could just scrap the study of Z-relations and go
straight to counting unique interval vectors. Which are actually
just measurement-vectors of the dyad-subsets in a set. One could
also count all the triad-subsets in a set, and so forth. And of
course there is the Steiner System S(5,6,12) from which the simple
sporadic Matthieu group M12 is based. One can analyze the interval
vectors directly and find, of course, 1-5 symmetry, etc.

What I really wonder about is this - is there any math (number theory
algebra or the like) that underlies BOTH the study of tuning (commas,
generators, vals) and permutations (counting sets, etc)? I would
be excited to see if there are any connections between the two
sides of music theory - or are they mutually exclusive?

Paul
Paul

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

🔗gooseplex <cfaah@eiu.edu>

Hi Gene,

> ... calling it "set theory" is a little bizarre.
> ...
> It isn't a bastardization of set theory. Set theory as an area of
> study does not concern itself with finite sets; ... This is not set
theory to a mathematician.
> Algebra, combinatorics and number theory are the relevant
areas, not
> set theory.

Excellent. You are absolutely correct. I remember the first time I
picked up a book on actual set theory after being introduced to
the 'musical' variety. I was almost at a complete loss to see the
connection. Gaussian modulo arithmetic and combinatorics are
the two areas where I see the most connection, but the way
these are applied does seem like a bastardization to me. So I
should have said something like 'bastardization of various
mathematical ideas' rather than referring directly to set theory.

I also wrote:

> > I agree that the nomenclature is mostly misleading and silly.
But
> > then I also feel that terms like "unison vector" are silly.
>
> Point taken, but don't blame me for that one. :)
>
> > It may also be a good idea to keep in mind that it does
become
> > tiresome to have to wade through a bunch of re-definitions of
> > conventional terminology just to get at any meaning out of
> > someone's writing.

You responded:
>
> True, but failing to use standard math terminology is also a
matter
> of redefinition.

This is also an excellent point. One issue I have grappled with is
the mathematical versus the musical definition of harmonics.
The mathematical 'harmonic series' as I understand it always
represents harmonics as 1/n, whereas in music we often talk
about harmonics as whole number multiples, or what would be
called in math the 'arithmetic series'. What is your take on this?

Aaron

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:

> This is also an excellent point. One issue I have grappled with is
> the mathematical versus the musical definition of harmonics.
> The mathematical 'harmonic series' as I understand it always
> represents harmonics as 1/n, whereas in music we often talk
> about harmonics as whole number multiples, or what would be
> called in math the 'arithmetic series'. What is your take on this?
>
> Aaron

The reason for this is historical. We say whole number multiples
today because everyone since Fourier talks about frequency
measurements. In the old days, it was string length measurements (or
still today, period or wavelength) where the numbers are *inversely
proportional* to the frequency numbers. So the harmonic series in the
old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same
harmonic series that today goes 1, 2, 3, 4 . . .

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

--- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
<paul.hjelmstad@u...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> wrote:
> > --- In tuning-math@yahoogroups.com, "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> >
> > > What I really wonder about is this - is there any math (number
> > theory
> > > algebra or the like) that underlies BOTH the study of tuning
> > (commas,
> > > generators, vals) and permutations (counting sets, etc)? I would
> > > be excited to see if there are any connections between the two
> > > sides of music theory - or are they mutually exclusive?
> >
> > I'd read up on groups.
>
Hate to sound dense, but could someone give me an example of group-
theoretical considerations in tuning theory (generators, vals,
commas, etc). If this ties into counting sets, all the better... the
only thing I can see is that it involves 1. modular arithmetic and 2.
generators. Is that it?

Thanx

🔗gooseplex <cfaah@eiu.edu>

>One issue I have grappled with is
> > the mathematical versus the musical definition of harmonics.
> > The mathematical 'harmonic series' as I understand it
always
> > represents harmonics as 1/n, whereas in music we often
talk
> > about harmonics as whole number multiples, or what would
be
> > called in math the 'arithmetic series'. What is your take on
this?
> >
> > Aaron
>
> The reason for this is historical. We say whole number
multiples
> today because everyone since Fourier talks about frequency
> measurements. In the old days, it was string length
measurements (or
> still today, period or wavelength) where the numbers are
*inversely
> proportional* to the frequency numbers. So the harmonic
series in the
> old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same
> harmonic series that today goes 1, 2, 3, 4 . . .

Right; I am aware of this. There is yet a gap between
mathematicians and musicians here since the series 1 2 3 4 ...
is not called a 'harmonic series' in mathematics. It is instead
called an 'arithmetic series'. This was my point. If we are
working with numbers all the time, shouldn't we adopt
terminology common to both mathematics _and music? For
example, we could call 1/1, 1/2, 1/3 ... a harmonic series and 1 2
3 4 ... an arithmetic series and be done with it ... almost. In music
we don't sum these series necessarily as is done in math, as
Gene pointd out. But at least we would be closer to common
agreement between these terms in both math and music.

Aaron

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
>
> >One issue I have grappled with is
> > > the mathematical versus the musical definition of harmonics.
> > > The mathematical 'harmonic series' as I understand it
> always
> > > represents harmonics as 1/n, whereas in music we often
> talk
> > > about harmonics as whole number multiples, or what would
> be
> > > called in math the 'arithmetic series'. What is your take on
> this?
> > >
> > > Aaron
> >
> > The reason for this is historical. We say whole number
> multiples
> > today because everyone since Fourier talks about frequency
> > measurements. In the old days, it was string length
> measurements (or
> > still today, period or wavelength) where the numbers are
> *inversely
> > proportional* to the frequency numbers. So the harmonic
> series in the
> > old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same
> > harmonic series that today goes 1, 2, 3, 4 . . .
>
>
> Right; I am aware of this. There is yet a gap between
> mathematicians and musicians here since the series 1 2 3 4 ...
> is not called a 'harmonic series' in mathematics. It is instead
> called an 'arithmetic series'. This was my point.

That's exactly what I explained above! Where did we lose each other?

> If we are
> working with numbers all the time, shouldn't we adopt
> terminology common to both mathematics _and music?

To some extent . . . but some people go too far trying to impose math
terminology on musicians whose terminology is already fine (and most
importantly, forms a common currency for musical communication) . . .

> For
> example, we could call 1/1, 1/2, 1/3 ... a harmonic series and 1 2
> 3 4 ... an arithmetic series and be done with it ... almost. In
music
> we don't sum these series necessarily as is done in math, as
> Gene pointd out. But at least we would be closer to common
> agreement between these terms in both math and music.
>
> Aaron

In music the numbers have to *represent* something. 1/1, 1/2, 1/3,
1/4 . . . *is* the musical harmonic series if the numbers are
representing periods, wavelengths, string lengths, air column
lengths, etc. Numbers alone do not represent anything musical, though
they are certainly a valid field of mathematical study.

🔗gooseplex <cfaah@eiu.edu>

> > > the harmonic
> > series in the
> > > old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same
> > > harmonic series that today goes 1, 2, 3, 4 . . .
> >
> >
> > Right; I am aware of this. There is yet a gap between
> > mathematicians and musicians here since the series 1 2 3 4
...
> > is not called a 'harmonic series' in mathematics. It is instead
> > called an 'arithmetic series'. This was my point.
>
> That's exactly what I explained above! Where did we lose each
other?

No, I haven't lost you, but my point must not be clear. So, I will try
to be clearer this time.

If we call 1,2,3,4 ... a harmonic series, then a mathematician will
say we are wrong; we are using the wrong term, because in
mathematics this is not called a harmonic series, it is called an
arithmetic series. To say that these two series are considered
the same because music history (as well as physics) says so -
this doesn't do anything reconcile the discrepancy between
mathematical and musical terminology, which is the point I am
trying to get at. If we as musicians start calling 1,2,3,4 an
arithmetic series, the reciprocal of the harmonic series, then a
mathematician will say that we are correct. Otherwise the gap
remains.

So, I suggest that if we are using terminology which is common
to both math and music, such as 'the harmonic series', we
should try to have as much agreement as possible, and on this
very basic point I feel that we could benefit from adopting the
common terms of a harmonic or arithmetic series from the
terminology of mathematics.

Sorry if this is still unclear.

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
>
> > > > the harmonic
> > > series in the
> > > > old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the same
> > > > harmonic series that today goes 1, 2, 3, 4 . . .
> > >
> > >
> > > Right; I am aware of this. There is yet a gap between
> > > mathematicians and musicians here since the series 1 2 3 4
> ...
> > > is not called a 'harmonic series' in mathematics. It is instead
> > > called an 'arithmetic series'. This was my point.
> >
> > That's exactly what I explained above! Where did we lose each
> other?
>
>
> No, I haven't lost you, but my point must not be clear. So, I will
try
> to be clearer this time.
>
> If we call 1,2,3,4 ... a harmonic series, then a mathematician will
> say we are wrong; we are using the wrong term, because in
> mathematics this is not called a harmonic series, it is called an
> arithmetic series.

That would be an awfully pedantic and solipsistic mathematician,
unless "we" patently failed to make it clear we were talking about
*frequencies* above (assuming, of course, that we were!) . . .

> So, I suggest that if we are using terminology which is common
> to both math and music, such as 'the harmonic series', we
> should try to have as much agreement as possible, and on this
> very basic point I feel that we could benefit from adopting the
> common terms of a harmonic or arithmetic series from the
> terminology of mathematics.

They don't apply. Mathematics deals with pure number. In music
numbers can signify many things, but have no meaning on their own. If
we decide to refer to "1/1 1/2 1/3 1/4" as the "harmonic sequence"
regardless of whether the numbers represent frequencies or periods or
wavelengths, we are stripping ourselves of the ability to make a
crucial distinction, and refer to an important musical referent.

You can replace "music" with "physics" in everything I've written on
this topic and it remains true. No physicist would accept what you
are proposing.

Are you aware of the history of the term "arithmetic
series", "arithmetic division", etc., in music?

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning-math@yahoogroups.com, "Paul Erlich" <perlich@a...>
wrote:

> > If we call 1,2,3,4 ... a harmonic series, then a mathematician
will
> > say we are wrong; we are using the wrong term, because in
> > mathematics this is not called a harmonic series, it is called an
> > arithmetic series.
>
> That would be an awfully pedantic and solipsistic mathematician,
> unless "we" patently failed to make it clear we were talking about
> *frequencies* above (assuming, of course, that we were!) . . .

A pedantic, solipsistic mathematician would say it is wrong because
it isn't a series at all, though usage is sometimes sloppy on this
point. A number theorist usually calls a sequence of the type
a, a+b, a+2b, a+3b, ... where a and b are integers an arithmetic
progression.

http://mathworld.wolfram.com/ArithmeticSeries.html
http://mathworld.wolfram.com/ArithmeticSequence.html
http://mathworld.wolfram.com/HarmonicSeries.html

🔗gooseplex <cfaah@eiu.edu>

--- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "gooseplex"
<cfaah@e...> wrote:
> >
> > > > > the harmonic
> > > > series in the
> > > > > old days *was* 1/1, 1/2, 1/3, 1/4, etc, and yet it's the
same
> > > > > harmonic series that today goes 1, 2, 3, 4 . . .
> > > >
> > > >
> > > > Right; I am aware of this. There is yet a gap between
> > > > mathematicians and musicians here since the series 1 2
3 4
> > ...
> > > > is not called a 'harmonic series' in mathematics. It is
instead
> > > > called an 'arithmetic series'. This was my point.
> > >
> > > That's exactly what I explained above! Where did we lose
each
> > other?
> >
> >
> > No, I haven't lost you, but my point must not be clear. So, I will
> try
> > to be clearer this time.
> >
> > If we call 1,2,3,4 ... a harmonic series, then a mathematician
will
> > say we are wrong; we are using the wrong term, because in
> > mathematics this is not called a harmonic series, it is called
an
> > arithmetic series.
>
> That would be an awfully pedantic and solipsistic
mathematician,
> unless "we" patently failed to make it clear we were talking
about
> *frequencies* above (assuming, of course, that we were!) . . .

Use of correct terminology alone hardly indicates solipsism or
pedantry. But does "we" in quotes mean that you are insulted
when I say "we"? presumably because "you" would not make
the same silly assumptions as "me" when discussing
something with a mathematician? Or perhaps the opposite is
true?

To be safe I will say "you" instead.

"You" will have to explain "yourself" to a mathematician when
"you" say that a harmonic series goes 1,2,3,4 ... because
according to the mathematician, the harmonic series does not
go 1,2,3,4 ... is goes 1/1, 1/2, 1/3 ... How much clearer does this
need to be?

> > So, I suggest that if we are using terminology which is
common
> > to both math and music, such as 'the harmonic series', we
> > should try to have as much agreement as possible, and on
this
> > very basic point I feel that we could benefit from adopting the
> > common terms of a harmonic or arithmetic series from the
> > terminology of mathematics.
>
> They don't apply. Mathematics deals with pure number. In
music
> numbers can signify many things, but have no meaning on
their own. If
> we decide to refer to "1/1 1/2 1/3 1/4" as the "harmonic
sequence"
> regardless of whether the numbers represent frequencies or
periods or
> wavelengths, we are stripping ourselves of the ability to make
a
> crucial distinction, and refer to an important musical referent.
>
> You can replace "music" with "physics" in everything I've written
on
> this topic and it remains true. No physicist would accept what
you
> are proposing.

I thought we have been discussing mathematicians, not
physicists, but I would guess that physicists would prefer to
speak for themselves. I would wager that all of the imaginary
physicists are not on "your" side. It's quite likely that there are
those who would agree with me and there are those who would
agree with "you".

> Are you aware of the history of the term "arithmetic
> series", "arithmetic division", etc., in music?

Certainly, but by all means feel free to enlighten me if you feel
that there is something I have missed.

Aaron

🔗Paul Erlich <perlich@aya.yale.edu>

> But does "we" in quotes mean that you are insulted
> when I say "we"?

No.

> "You" will have to explain "yourself" to a mathematician when
> "you" say that a harmonic series goes 1,2,3,4 ... because
> according to the mathematician, the harmonic series does not
> go 1,2,3,4 ... is goes 1/1, 1/2, 1/3 ...

Indeed -- except that, as Gene just pointed out, the latter is not
called the "harmonic series" by mathematicians until you sum up the
terms.

> I thought we have been discussing mathematicians, not
> physicists, but I would guess that physicists would prefer to
> speak for themselves.

OK, I guess so far only one has.

> I would wager that all of the imaginary
> physicists are not on "your" side.

Imaginary? Why imaginary?

> It's quite likely that there are
> those who would agree with me and there are those who would
> agree with "you".

The "harmonic series" is a frequent referent in physics texts and
journals that discuss acoustics or even nonlinear dynamical systems.

Physicists refer to a set of *periods* or *wavelengths* as 'harmonic'
if their proportions belong to the set {1/1, 1/2, 1/3, 1/4 . . .}

Physicists refer to a set of *frequencies* as 'harmonic' if their
proportions refer to the set {1, 2, 3, 4 . . .}

I challenge you to find a counterexample in any physics text or
journal article.

Let me now begin to collect supporting examples, which I hope not to
have to spend too much time doing:

http://www.phys.unsw.edu.au/~jw/harmonics.html
http://www.phys.unt.edu/~matteson/1251-001/mwf08.ppt
http://www.theconcertband.com/Music&Physics_harmonic_series.htm
http://www.colorado.edu/physics/phys4830/phys4830_fa01/lab/n1002.htm
http://www.physics.odu.edu/~hyde/Teaching/5
http://www.physics.northwestern.edu/classes/2001Spring/135-
1/Projects/3/sound.html

🔗gooseplex <cfaah@eiu.edu>

> > I would wager that all of the imaginary
> > physicists are not on "your" side.
>
> Imaginary? Why imaginary?

imaginary because you invoked an army of like-minded
physicists in order to rebut my point of view, and this seemed
rather fantastical and unnecessary to me.

> > It's quite likely that there are
> > those who would agree with me and there are those who
would
> > agree with "you".
>
> The "harmonic series" is a frequent referent in physics texts
and
> journals that discuss acoustics or even nonlinear dynamical
systems.
>
> Physicists refer to a set of *periods* or *wavelengths* as
'harmonic'
> if their proportions belong to the set {1/1, 1/2, 1/3, 1/4 . . .}
>
> Physicists refer to a set of *frequencies* as 'harmonic' if their
> proportions refer to the set {1, 2, 3, 4 . . .}
>
> I challenge you to find a counterexample in any physics text or
> journal article.
>
> Let me now begin to collect supporting examples, which I hope
not to
> have to spend too much time doing:
>
> http://www.phys.unsw.edu.au/~jw/harmonics.html
> http://www.phys.unt.edu/~matteson/1251-001/mwf08.ppt
>
http://www.theconcertband.com/Music&Physics_harmonic_serie
s.htm
>
http://www.colorado.edu/physics/phys4830/phys4830_fa01/lab/n
1002.htm
> http://www.physics.odu.edu/~hyde/Teaching/5
> http://www.physics.northwestern.edu/classes/2001Spring/135-
> 1/Projects/3/sound.html

Thank you. I appreciate these references. I was not discussing
the views of physicists to begin with, although the _applied
mathematics of physics in the area of acoustics is of course
completely relevant.

I'm sure this seems awfully stubborn, but for me all of this still
does not reconcile the semantic discrepancy as I outlined it
originally.

Without invoking physics, is there a way to find a better
agreement of terms?

🔗gooseplex <cfaah@eiu.edu>

> I challenge you to find a counterexample in any physics text or
> journal article.
>
> Let me now begin to collect supporting examples, which I hope
not to
> have to spend too much time doing:
>
> http://www.phys.unsw.edu.au/~jw/harmonics.html
> http://www.phys.unt.edu/~matteson/1251-001/mwf08.ppt
>
http://www.theconcertband.com/Music&Physics_harmonic_serie
s.htm
>
http://www.colorado.edu/physics/phys4830/phys4830_fa01/lab/n
1002.htm
> http://www.physics.odu.edu/~hyde/Teaching/5
> http://www.physics.northwestern.edu/classes/2001Spring/135-
> 1/Projects/3/sound.html

By the way, Paul, it wasn't necessary to go to the trouble. My
degrees are not in physics, but I studied physics and acoustics
as an undergraduate, and these continue to be interests of
mine. Sorry I forgot that you have a degree (or degrees?) in
physics. I teach electronic music here at EIU, in addition to
composition, ear training, music theory and music analysis. The
electronic music course is introductory and includes elementary
acoustics and psychoacoustics. So, you see, I am used to
refering to whole numbers representing frequency multiples as
'harmonics'; this language has just always bothered me a little.

Aaron

🔗Paul Erlich <perlich@aya.yale.edu>

🔗gooseplex <cfaah@eiu.edu>

--- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "gooseplex"
<cfaah@e...> wrote:
>
> > imaginary because you invoked an army of like-minded
> > physicists in order to rebut my point of view, and this seemed
> > rather fantastical and unnecessary to me.
>
> Fantastical and imaginary as in untrue?
>
> > By the way, Paul, it wasn't necessary to go to the trouble.
>
> If it seemed untrue/imaginary/fantastical before, but now
seems true,
> then certainly *something* changed your mind . . .

Ah, I see the problem now.

Paul, you have not told me anything I do not already know. You
started bringing in these legions of physicists to support your
argument from a physicists point of view and I told you that you
were missing my point completely. When I said that some
physicists would agree with me and some with you, I was talking
about agreement on the *real* issue at hand which is the
_mathematical _verses _the _musical _definition _of _the
_harmonic _series, which - I'm sorry - does not involve
acoustics!

Sheesh, this has become a royal waste of time...

Aaron

🔗Paul Erlich <perlich@aya.yale.edu>

Aaron, hopefully this post will make it up soon, because my posts
have not appeared in order. One crucial post I made before this one
has not yet appeared, and it seems we are on bad footing as a result.
Please be patient until that post appears.

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:
> --- In tuning-math@yahoogroups.com, "Paul Erlich"
> <perlich@a...> wrote:
> > --- In tuning-math@yahoogroups.com, "gooseplex"
> <cfaah@e...> wrote:
> >
> > > imaginary because you invoked an army of like-minded
> > > physicists in order to rebut my point of view, and this seemed
> > > rather fantastical and unnecessary to me.
> >
> > Fantastical and imaginary as in untrue?
> >
> > > By the way, Paul, it wasn't necessary to go to the trouble.
> >
> > If it seemed untrue/imaginary/fantastical before, but now
> seems true,
> > then certainly *something* changed your mind . . .
>
>
> Ah, I see the problem now.
>
> Paul, you have not told me anything I do not already know. You
> started bringing in these legions of physicists to support your
> argument from a physicists point of view and I told you that you
> were missing my point completely. When I said that some
> physicists would agree with me and some with you, I was talking
> about agreement on the *real* issue at hand which is the
> _mathematical _verses _the _musical _definition _of _the
> _harmonic _series, which - I'm sorry - does not involve
> acoustics!
>
> Sheesh, this has become a royal waste of time...
>
> Aaron

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

--- In tuning-math@yahoogroups.com, "gooseplex" <cfaah@e...> wrote:

> I'm sure this seems awfully stubborn, but for me all of this still
> does not reconcile the semantic discrepancy as I outlined it
> originally.
>
> Without invoking physics, is there a way to find a better
> agreement of terms?
>
> A

In case the two uses of the term appear to come into conflict with
one another, it's best to go over the historical explanation of where
both usages come from -- and then you're done. There is no real
conflict anyway because one definition applies to pure number and the
other to physical phenomena, so at worst you'll have to explain which
you're talking about (though it's usually obvious from the context).

This strategy is going to lead to far better communication than
attempting to alter the accepted definitions in either set of fields -
- in the latter case you're going to be misunderstood by everyone in
one of the sets of fields who comes across your writing without being
aware that you've altered the definitions, because your context will
indicate one definition but in fact you'll be using another.

If you want to spend your time re-writing the entire dictionary so
that no word has multiple meanings, that's noble. But you shouldn't
then *expect* the entire world to suddenly change the language
they've been speaking for hundreds of years.

If I have something to say that I feel is important, I say it in
terms that I feel will be understood correctly by the majority of
likely listeners. If I have time, I define lots of terms to avoid any
possible ambiguity. But we have to pick our battles. Changing
language, especially when its origin can be traced historically and
its development makes perfect sense and did not involve any errors,
is simply too vast a battle for me to contemplate, even if I know
I'll have you on my side.

That's my opinion and to me it seems like the *least* arrogant
approach, even though I'm sensing you have the opposite opinion of me
right now.

🔗gooseplex <cfaah@eiu.edu>

--- In tuning-math@yahoogroups.com, "Paul Erlich"
<perlich@a...> wrote:
> --- In tuning-math@yahoogroups.com, "gooseplex"
<cfaah@e...> wrote:
>
> > I'm sure this seems awfully stubborn, but for me all of this
still
> > does not reconcile the semantic discrepancy as I outlined it
> > originally.
> >
> > Without invoking physics, is there a way to find a better
> > agreement of terms?
> >
> > A
>
> In case the two uses of the term appear to come into conflict
with
> one another, it's best to go over the historical explanation of
where
> both usages come from -- and then you're done. There is no
real
> conflict anyway because one definition applies to pure number
and the
> other to physical phenomena, so at worst you'll have to explain
which
> you're talking about (though it's usually obvious from the
context).
>
> This strategy is going to lead to far better communication than
> attempting to alter the accepted definitions in either set of
fields -
> - in the latter case you're going to be misunderstood by
everyone in
> one of the sets of fields who comes across your writing without
being
> aware that you've altered the definitions, because your context
will
> indicate one definition but in fact you'll be using another.
>
> If you want to spend your time re-writing the entire dictionary so
> that no word has multiple meanings, that's noble. But you
shouldn't
> then *expect* the entire world to suddenly change the
language
> they've been speaking for hundreds of years.
>
> If I have something to say that I feel is important, I say it in
> terms that I feel will be understood correctly by the majority of
> likely listeners. If I have time, I define lots of terms to avoid any
> possible ambiguity. But we have to pick our battles. Changing
> language, especially when its origin can be traced historically
and
> its development makes perfect sense and did not involve any
errors,
> is simply too vast a battle for me to contemplate, even if I know
> I'll have you on my side.
>
> That's my opinion and to me it seems like the *least* arrogant
> approach, even though I'm sensing you have the opposite
opinion of me
> right now.

Paul, I agree with what you say here. In both speaking and
writing, I have not changed my own language simply to satisfy
myself rather than to communicate effectively, and I agree this is
the least arrogant thing to do.

There are in fact a couple of paragraphs about this in my book
(which I am still working on) which explain why 1,2,3,4 can be
called a harmonic series in music. I have done this with
mathematicians in mind, because my own understanding of the
relationship between mathematics and music (as opposed to
physics / acoustics and music) is outlined at the outset of the
book, and I want to treat the issue of harmonics with particular
care.

I couldn't find a better way to handle it than to give an explanation
not unlike the explanations you offered me. I brought it up here to
see if anyone else was similarly bothered by the mathematical
versus musical definition, and to see if someone else might
have an idea about how to reconcile things that hadn't occured to
me. Maybe there is no better way.

Also, when I mentioned that I had forgotten about your
background in physics, I meant that I had forgotten that it would
be totally natural for you to approach the issue from the point of
view of a physicist, and i didn't intend to imply that you were
being arrogant or any such thing. I hope we understand each
other better now.

Aaron

Re: [tuning-math] Digest Number 862

🔗jon wild <wild@fas.harvard.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Carl Lumma <ekin@lumma.org>

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

🔗gooseplex <cfaah@eiu.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Paul G Hjelmstad <paul.hjelmstad@us.ing.com>

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗gooseplex <cfaah@eiu.edu>

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗gooseplex <cfaah@eiu.edu>

🔗Paul Erlich <perlich@aya.yale.edu>

🔗gooseplex <cfaah@eiu.edu>