pedegogical question

hello Group. As I'm grappling with terms in the tonalsoft
dictionary, was wondering why there seem to be a whole other separate
set of terms in discussing the origin scales, such as 'tonus, eptadem
minus' (which I've seeen several places, such as at
freedictionary.com), and with some accompanying interesting (but to
you guys inconsequential?) number patterns, such as that 5/3 = the
series of recipricols of triangular numbers
(1/1+1/3+1/6+1/10+1/15=5/3), or that 16/9 = the same series, up
through 1/36.

Is this an older view, or a competing view, of tuning math?

And being completely out of the knowledgable circuit on this (but
very curious), are your views (not that they're monolithic...but you
know what I mean) more or less in accord with the 'academic views'
on the subject?

thanks, Kelly

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:

> hello Group. As I'm grappling with terms in the tonalsoft
> dictionary, was wondering why there seem to be a whole other separate
> set of terms in discussing the origin scales, such as 'tonus, eptadem
> minus' (which I've seeen several places, such as at
> freedictionary.com), and with some accompanying interesting (but to
> you guys inconsequential?) number patterns, such as that 5/3 = the
> series of recipricols of triangular numbers
> (1/1+1/3+1/6+1/10+1/15=5/3), or that 16/9 = the same series, up
> through 1/36.
>
> Is this an older view, or a competing view, of tuning math?

You'll have to ask someone else about the history, but this number
pattern is interesting; it is connected to the commas with square
numerators, since if you take the associated infinite product (take
the sequencce from this series, and the ratios of those) you get the
infinite product (4/3)(9/8)(16/15) ... = 2 from your infinite series
1 + 1/3 + 1/6 + 1/10 + ... = 2. The successive terms of the sequence,
whether derived from the product or the sum, go 2-2/3, 2-2/5, 2-2/7 ...

You can find other associated series, products and sequences with
similar properties. For instance, 3/6+3/10+3/15+3/21+3/28+... = 3
leads to an infinite product giving us commas of the form
tri(n)/(tri(n)-1) where tri(n) is the nth triangular number. The
successive terms of the sequence are of the form 3-6/3, 3-6/4, 3-6/5,
3-6/6, 3-6/7 ...

I didn't know this stuff had a history; the only one who has ever
concerned himself with this that I know of is me, and if it is
connected to ancient music theory I'd be fascinated to learn about it.
It would all probably make a grand encyclopedia entry if we had that
information.

hi Kelly and Gene,

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:

> --- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> > hello Group. As I'm grappling with terms in the
> > tonalsoft dictionary, was wondering why there seem
> > to be a whole other separate set of terms in discussing
> > the origin scales, such as 'tonus, eptadem minus'
> > (which I've seeen several places, such as at
> > freedictionary.com), and with some accompanying
> > interesting (but to you guys inconsequential?) number
> > patterns, such as that 5/3 = the series of recipricols
> > of triangular numbers (1/1+1/3+1/6+1/10+1/15=5/3), or
> > that 16/9 = the same series, up through 1/36.
> >
> > Is this an older view, or a competing view, of tuning math?
>
> You'll have to ask someone else about the history,

well, normally, that could be me ... but i have no idea
what you (Kelly) are talking about.

> I didn't know this stuff had a history; the only one who
> has ever concerned himself with this that I know of is me,
> and if it is connected to ancient music theory I'd be
> fascinated to learn about it.
> It would all probably make a grand encyclopedia entry if
> we had that information.

yes. please, Kelly, tell us more.

PS -- freedictionary.com was useless for me. the
half-decent entry i found for "tonus" (mentioning the
"eptadem minus") was a Wikipedia ... but the entry in
the Tonalsoft Encyclopaedia is much better. ;-)

-monz

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
[snip]

> Is this an older view, or a competing view, of tuning math?
>
> And being completely out of the knowledgable circuit on this (but
> very curious), are your views (not that they're monolithic...but
you
> know what I mean) more or less in accord with the 'academic views'
> on the subject?
>
> thanks, Kelly

I think this is a good question, so I'll try to answer it and then
maybe people will correct me, and we'll get a good answer. My
guesses:

1. Most of the music theory establishment is not focused on
tuning questions, they take 12-eq temperment exclusively.
You can look at journals like "Journal of Music Theory"
or "Music Theory Online" or others that I don't know about...

2. The work done in tuning-math is largely original research;
I'm not sure to what extent it exists outside this group.
Paul Erlich is currently (?) still working on an expository
paper for Xenharmonikon.

3. A book was recently published "Mathematical Theory of
Tone Systems" ML 3807 H35 2003 (sorry, I didn't write down
the author's name) which referenced the work of a number of
the tuning-math people, and of course there is Dave Benson's
online text "Mathematics and Music" which mentions Periodicity
blocks, etc... Does that qualify as the "acedemic view"?

4. I don't know to what extent the work of Harry Partch is
taught in music schools.

that is my best guess, please feel free to improve it!
--Jeff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:

> 1. Most of the music theory establishment is not focused on
> tuning questions, they take 12-eq temperment exclusively.
> You can look at journals like "Journal of Music Theory"
> or "Music Theory Online" or others that I don't know about...

other relevant journals are "Music Theory Spectrum" and
"Music Analysis", and "19th Century Music" usually has
good analytical papers as well.

i subscribe to "Journal of Music Theory", and i can
tell you that while its authors do pretty much take
12-edo for granted, lately there has been much more
interest in JI and other tunings than there was just
a few years ago ... and since the issue in 1998 devoted
entirely to neo-Riemannian theory, in which the Tonnetz
plays a large part, use of lattice diagrams in the JMT
has postively exploded!

> 2. The work done in tuning-math is largely original research;
> I'm not sure to what extent it exists outside this group.
> Paul Erlich is currently (?) still working on an expository
> paper for Xenharmonikon.

this list, and even moreso the tuning-math list, is
admittedly a pretty closed group. (i refer to it as
one group instead of two because, without actually
examining the membership, i think it's a safe bet that
most tuning-math members are also on this list).

but the posts are archived here for posterity (or as
long as Yahoo exists, whichever comes first). so
outsiders can and do find our work when they do searches,
and little by little it's finding its way into doctoral
dissertations and other academic papers.

so it's still only a small trickle, but i'm confident
that conscious microtonality is more and more becoming
a part of the mainstream of music.

i say "conscious" because, especially in pop music,
microtonality is already ubiquitous ... but the performers
generally aren't aware of any of the theory of it.

> 3. A book was recently published "Mathematical Theory of
> Tone Systems" ML 3807 H35 2003 (sorry, I didn't write down
> the author's name) which referenced the work of a number of
> the tuning-math people,

wow, awesome! that book is by my good friend (and
fellow tuning list member) Jan Haluska! i didn't know
he was writing one. it's no wonder that he cites the
work of people who contribute here.

here's the complete citation:

The Mathematical Theory of Tone Systems
Series Volume: 262

Jan Haluska
Textbook | Print Published: 12/19/2003
Print ISBN: 0-8247-4714-3

World Price: $165.00

http://www.dekker.com/servlet/product/productid/4714-3/sub?n=i

> 4. I don't know to what extent the work of Harry Partch is
> taught in music schools.

basically, it isn't.

some classes may expose students to Partch's music,
but i doubt if there are many which really explore
his theory.

i know that there are a few exceptions ... especially
in England and Australia. but in general, no.

-monz

Hi Kelly and Gene,

Thanks, its interesting that the sum of the reciprocals of the
triangle numbers adds up to 2, never realised that.

It's a nice little maths puzzle to prove it, but you need
to know a couple of things first, sort of common knowledge
to mathematicians and a first year undergraduate would
be expected to know them, or perhaps a keen high school
mathematician intersted in properties of numbers.

You need to know the formula for a triangular
number, and you need to know about a technique called
mathematical induction. Maybe I can present these
so that everyone can join in.

A triangular number is a number of the form
1 + 2 + ...
e.g.
1 + 2 + 3 + 4 = 10.

The formula for the nth triangular number
is n (n+1) / 2

So for instance, the fifth triangular number
is 5 (5+1) / 2 = 15.

So Gene's observation is that
1 + 1/(1+2) + 1/(1+2+3) + ...
adds up to 2.

Of course you can't add up infinitely many
terms so what you do is to add up a finite
number at a time. That's the sequence of
partial sums as it is called.

Then writing tri(n) for n(n+1)/2, you find out that
1 + 1/tri(1) + 1/tri(2) + .. + 1/tri(n)
= 2 - 2/n

Well that is true for the first term:

1 = 2 - 1/1

You could go on and prove it for the second
term, then the third and so on but we
have infinitely many terms to prove it for,
so some shortcut is needed to get a finite
proof.

The way you prove these types of results
is by what mathematicians call mathematical
induction. You work out a method for
getting from the result for n to the
result for n+1.

So, given that

1 + 1/tri(1) + 1/tri(2) + .. + 1/tri(n)
= 2 - 2/n

which we already know for n = 1
you need to show that

1 + 1/tri(1) + 1/tri(2) + .. + 1/tri(n+1)
= 2 - 2/(n+1)

To prove that you need to show that
2 - 2/n + 1/tri(n+1)
= 2 - 2/(n+1).

If you work that out with a bit of algebraic
manipulation, you will find that it isn indeed
the case, and this proves the result for all n.

Another way of writing 2 - 2/n is as
2(n-1) / n

So the partial sums are

4/3, 6/4, 8/5, 10/6, 12/7, 14/8, 16/9, ...

(4/3, 3/2, 8/5, 5/3, 12/7, ...)

which gets closer and closer to 2.
In fact as n gets larger, it will get as close
as you like to 2, and will never exceed it.
That's what a mathematician means if they
say that a series sums to 2.

Then you can also make an infinite product working backwards
- products are more relevant for ratios as you multiply
them rather than add, so

4/3 * (3/2 / (4/3) ) *(8/5 / (3/2) ) * (10/6 / (8/5) ) * (12/7 / (10/6) ) ...

gives us the same numbers as the ratios on top and
bottom cancel, so e.g.

4/3 * (3/2 / (4/3) ) *(8/5 / (3/2) ) * (10/6 / (8/5) ) * (12/7 / (10/6) )
= 12/7

Multiplying out the numbers within the brackets, that's

4/3 * 9/8 * 16/15 * 25/24 * 36/35 * ...

which multiplies out to the same numbers and again gets closer
and closer to 2.

Obviously the next term in the product will be 49/48
then 64/65 then 81/80 - with the square numbers on the top.

It works because

2(n)/(n+1) / (2(n-1) / n)

= n^2 / (n^2 - 1)

..................

Actually this may well make an interesting infinite scale for a fractal
tunes.

I have an option in Fractal Tune Smithy already to make geometric series scales
based on equal temperament to play in, e.g.
0 cents 1200 cents 1800 cents 2100 cents 2250 cents 2325 cents 2362.5 cents 2381.25 cents,...
with each interval half the previous one.

It is done by entering e.g.

#g 1/2
for the scale

= each interval half the previous.

(see Scales | Special notations in the help).

However, one could easily follow the same idea with just intonation intervals and have
1 + 1/2 + 1/4 + ...
=
3/2 7/4 15/8 31/16 ...

and this triangular numbers one and
while at it I may as well let the user
specify any formula they like in n.

so
#j 2/(n(n+1))
would make 1 + 1/tri(1) + 1/tri(2) + ..
maybe I can allow use of 1/tri(n) as an abbreviation there.

Then
#j 1/2^n
would make 1 + 1/2 + 1/4 + ...

I'll give it a go and see if it comes up with
something interesting to listen to.

Robert

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
> --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
wrote:
>
> other relevant journals are "Music Theory Spectrum" and
> "Music Analysis", and "19th Century Music" usually has
> good analytical papers as well.
>

thanks; I'm going to have to do some more browsing at the library.

[snip]
> i say "conscious" because, especially in pop music,
> microtonality is already ubiquitous ... but the performers
> generally aren't aware of any of the theory of it.

Any pop songs that I know? I wasn't aware of this...!?

[snip]

> wow, awesome! that book is by my good friend (and
> fellow tuning list member) Jan Haluska! i didn't know
> he was writing one. it's no wonder that he cites the
> work of people who contribute here.

Well, I was meaning to use this as an example of someone
who wasn't on the list but yet was working with the same ideas...
I guess I don't know any examples, then.

--Jeff

--- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...> wrote:

> --- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:
>
> > --- In tuning@yahoogroups.com, "jjensen142000" <jjensen14@h...>
> > wrote:
> >
> > <snip>
> >
> > i say "conscious" because, especially in pop music,
> > microtonality is already ubiquitous ... but the performers
> > generally aren't aware of any of the theory of it.
>
> Any pop songs that I know? I wasn't aware of this...!?

i don't keep up with much of what's out now ... but
a superb example of a master microtonal pop performer
is Jimi Hendrix.

Hendrix played a lot of fast riffs in his solos, but
in between those riffs he held a lot of sustained notes,
and he bends the pitch of *every single one*.

the album to get if you really want to hear Hendrix
do his microtonal thing, is the posthumous "Blues" CD:

http://tinyurl.com/5nc4c

-monz

> 4. I don't know to what extent the work of Harry Partch is
> taught in music schools.

In my experience (undergrad level), only in the "Techniques of early 20th century
composers." Not in the theory classes, not in music history...

In fact, not a single time in my entire time in theory classes and history classes, was tuning
ever even mentioned! For an issue that dominated the theory and practice of music for a
vast majority of history, I find it strange. It makes you want to subscribe to Partch's view
of "the tyranny of equal temperment" or whatever. Are we living in musical "1984"?

Sorry for the OT rant :) I'm preaching to the choir I guess.

-Chris

The math you guys offered is beautiful...it's like music!

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
>
> > hello Group. As I'm grappling with terms in the tonalsoft
> > dictionary, was wondering why there seem to be a whole other
separate
> > set of terms in discussing the origin scales, such as 'tonus,
eptadem
> > minus' (which I've seeen several places, such as at
> > freedictionary.com), and with some accompanying interesting (but
to
> > you guys inconsequential?) number patterns, such as that 5/3 =
the
> > series of recipricols of triangular numbers
> > (1/1+1/3+1/6+1/10+1/15=5/3), or that 16/9 = the same series, up
> > through 1/36.
> >
> > Is this an older view, or a competing view, of tuning math?
>
> You'll have to ask someone else about the history, but this number
> pattern is interesting; it is connected to the commas with square
> numerators, since if you take the associated infinite product (take
> the sequencce from this series, and the ratios of those) you get
the
> infinite product (4/3)(9/8)(16/15) ... = 2 from your infinite series
> 1 + 1/3 + 1/6 + 1/10 + ... = 2. The successive terms of the
sequence,
> whether derived from the product or the sum, go 2-2/3, 2-2/5, 2-
2/7 ...
>
> You can find other associated series, products and sequences with
> similar properties. For instance, 3/6+3/10+3/15+3/21+3/28+... = 3
> leads to an infinite product giving us commas of the form
> tri(n)/(tri(n)-1) where tri(n) is the nth triangular number. The
> successive terms of the sequence are of the form 3-6/3, 3-6/4, 3-
6/5,
> 3-6/6, 3-6/7 ...
>
> I didn't know this stuff had a history; the only one who has ever
> concerned himself with this that I know of is me, and if it is
> connected to ancient music theory I'd be fascinated to learn about
it.
> It would all probably make a grand encyclopedia entry if we had that
> information.

hi Gene. Sorry...how do you get the series (4/3)(9/8)(16/15) ... = 2
mentioned below? -Kelly

... but this number
> pattern is interesting; it is connected to the commas with square
> numerators, since if you take the associated infinite product (take
> the sequencce from this series, and the ratios of those) you get
the
> infinite product (4/3)(9/8)(16/15) ... = 2 from your infinite series
> 1 + 1/3 + 1/6 + 1/10 + ... = 2. The successive terms of the
sequence,
> whether derived from the product or the sum, go 2-2/3, 2-2/5, 2-
2/7 ...
>
> You can find other associated series, products and sequences with
> similar properties. For instance, 3/6+3/10+3/15+3/21+3/28+... = 3
> leads to an infinite product giving us commas of the form
> tri(n)/(tri(n)-1) where tri(n) is the nth triangular number. The
> successive terms of the sequence are of the form 3-6/3, 3-6/4, 3-
6/5,
> 3-6/6, 3-6/7 ...
>
> I didn't know this stuff had a history; the only one who has ever
> concerned himself with this that I know of is me, and if it is
> connected to ancient music theory I'd be fascinated to learn about
it.
> It would all probably make a grand encyclopedia entry if we had that
> information.

--- In tuning@yahoogroups.com, "traktus5" <kj4321@h...> wrote:
> hi Gene. Sorry...how do you get the series (4/3)(9/8)(16/15) ... = 2
> mentioned below? -Kelly

Given a convergent sequence of positive terms s1, s2, ... you can
obtain a series by taking differences, s1 + (s2-s1) + (s3-s2) + ...
You can similarly obtain a corresponding infinite product by taking
ratios, s1 * s2/s1 * s3/s2 * ... The two are connected via logarithms,
being really two forms of the same thing. In this case we have a
convergent sequence whose nth term is 2-2/n, and we can take
differences and get (2-2/(n+1))-(2-2/n) = 1/(n(n+1)/2), or we can take
ratios and get (2-2/(n+1))/(2-2/n) = n^2/(n^2-1). Either way the
convergent sequence is the same.

--- In tuning@yahoogroups.com, "monz" <monz@t...> wrote:

/tuning/topicId_55708.html#55717
>
> > 4. I don't know to what extent the work of Harry Partch is
> > taught in music schools.
>
>
> basically, it isn't.
>
> some classes may expose students to Partch's music,
> but i doubt if there are many which really explore
> his theory.
>
> i know that there are a few exceptions ... especially
> in England and Australia. but in general, no.
>
>
>
> -monz

***I'm pretty sure that Dean Drummond teaches these concepts
sometimes at Montclair State College in New Jersey. Drummond has the
Partch instruments:

http://www.newband.org/instruments.htm

Jon Szanto would know more...

J. Pehrson

--- In tuning@yahoogroups.com, "chrisbryan82" <cnmbryan@s...> wrote:

/tuning/topicId_55708.html#55734

> > 4. I don't know to what extent the work of Harry Partch is
> > taught in music schools.
>
> In my experience (undergrad level), only in the "Techniques of
early 20th century
> composers." Not in the theory classes, not in music history...
>
> In fact, not a single time in my entire time in theory classes and
history classes, was tuning
> ever even mentioned! For an issue that dominated the theory and
practice of music for a
> vast majority of history, I find it strange. It makes you want to
subscribe to Partch's view
> of "the tyranny of equal temperment" or whatever. Are we living in
musical "1984"?
>
> Sorry for the OT rant :) I'm preaching to the choir I guess.
>
> -Chris

***I mentioned this a while ago, but actually I was exposed to Harry
Partch's theories at the University of Michigan. It was in a
composition seminar hosted by composer Leslie Bassett. One of the
composers in the class was from the University of Illinois and I
believe he either studied with Partch or with some of his disciples.
In any case, I saw the book _Genesis..._ and we went through some of
the theories in the class. It seemed pretty strange to me at the
time (say 1978...) but I could see there was concrete thinking and
detail behind these ideas, and I took away a respect for them... (I
don't remember if we played much of the music though... but we must
have... :)

J. Pehrson