Interval Database Experiences

Hi folks, I guess I mentioned a while ago I'd start an Interval
Databse, so let me mention my experiences, let's go...

I made an algorythm that made the following sequence by calculations;

num den ratio
1 1 1
2 1 2
3 2 1,5
4 3 1,33333333333333
5 3 1,66666666666667
5 4 1,25
7 4 1,75
6 5 1,2
7 5 1,4
8 5 1,6
9 5 1,8
ETC.....

I used and created a databse, I said that the division of my two
terms shouldn't be more than 2, and if a ratio value wasn't in the
database it should be inserted there. The denominator would increase
in value up to where I wanted... etc... etc... I also created a
column for the value in cents, and names if there were any (I got the
name from a couple of lists on the net).

Anyway, by that algorythm I coud easily calculate all these intervals
and I chose to do that up to a denominator of 1024, wich gave me all
the integer ratio intervals in an octave up to 2047/1024!!! That gave
me 318.965 numbers of intervals in na octave.

I was first doing that to convert a value in cents to a ratio
proportion, I guess the only way should be a database convertion. But
I realized that in these samples of intervals that I've got so far,
there are many small differences ( obviously, there are about 320
thousand, damn it ). For exemple, from 699.5 and 700.5 cents, this
universe of 1 cent arround the 12EQ fifth, I have 288 Intervals from
1411/942 (699.50282758137712 cents ) to 1187/792 (700.49711947475646
cents).

So at this point I started wondering about the practical use of a
bigger database, and even wonder about the use of big interval
ratios. I know that there are famous tempraments that user higher
number ratios [ Pythagorean for Exemple ( C# = 2187/2048 = 113,685
cents ) ].

But in my database I have 2 intervals between 113.68 and 113.69;

1007/943 = 113.68080879802572 cents &
236/221 = 113.6885879644916 cents

Now 236/221 is much simpler than 2187/2048 and the difference is only
0.0035... cents

Wich makes me believe that in this dimension, ( maybe about a couple
of cents ) a lot of big integer ratio proportions are more
theoretical than practical. That's when I talked to my accoustic
teacher, wich told me about the DLO effect, and the William's
syndrom. He also agreed when I mentioned the C# pythagorean example,
but he reminded me that a temperament is characterized by other
parameters that make it special than the possible similarity of a
simpler integer ratio, and that most people won't hear a difference
of a few cents.

That's when I felt like bringing this discussion to the group, (What
you guys have to say about that?) I hope you can all help me figuring
out how to categorize these samll differences... anyway, I'll save
the rest for a reply...

Cheers
Alex

Ps. Would anyone like to get the database?

hi alex,

most of your questions have been addressed at length in the old
discussions on the tuning list. perhaps you could take some time to
look over the tuning list archives, particularly posts by dave keenan.

my main response right now is, why would you restrict yourself to
rational numbers? while it is true that simple integer ratios are
audibly different from their neighbors, and that one can therefore
tune successive 3:2s and 4:3s to obtain (theoretically) a just
2187:2048, there are irrational numbers which are also very useful
for various tuning systems where simple integer ratios are
approximated "close enough" with various musically useful
relationships that would be impossible in just intonation.

Hy Wally,

well, as I said, I was first concerned on creating a tool ( a table)
where you could convert cents to a integer ratio, as you can't
calculate that I thought the only way would be to have it on a
database, then you could enter your value in cents and the database
would give the answer back. But I knew it would be incomplete because
I was still missing the irrational proportions, but I haven't
forgotten that, and I think I also have a way to do that.

But nevermind why I may have overdone this or that, I'm just a
begginer trying to learn the hard way, (that is doing myself),
because I don't know many experts around here in my city.

Anyway, I was just analising my data and then I realized that many
relationships may be more theorical than practical, I'm not assuming
I made a huge discovery, that's way I'm bringing it up in here, so
someone could lead me somewhere, and perhaps enlight my thoughts to
work on something more useful.

I still don't know what's the normal tuning error range that we can
actually notice, maybe around 2 cents? Someone mentioned about a db
concept, anyway, thanks for the information, I'll look it up on the
previous messages...

Cheers, Alex

--- In tuning-math@yahoogroups.com, "wallyesterpaulrus"
<wallyesterpaulrus@y...> wrote:
> hi alex,
>
> most of your questions have been addressed at length in the old
> discussions on the tuning list. perhaps you could take some time to
> look over the tuning list archives, particularly posts by dave
keenan.
>
> my main response right now is, why would you restrict yourself to
> rational numbers? while it is true that simple integer ratios are
> audibly different from their neighbors, and that one can therefore
> tune successive 3:2s and 4:3s to obtain (theoretically) a just
> 2187:2048, there are irrational numbers which are also very useful
> for various tuning systems where simple integer ratios are
> approximated "close enough" with various musically useful
> relationships that would be impossible in just intonation.

--- In tuning-math@yahoogroups.com, "Porres" <decuritiba@y...> wrote:

> I still don't know what's the normal tuning error range that we can
> actually notice, maybe around 2 cents?

as others have mentioned to you on other lists, it totally depends on
the context. it's quite rare for anyone to notice a melodic interval
difference of 5 cents or less; meanwhile, it's rather easy to tell if
a harmonic interval is in just intonation or 1 cent off from it, at
least if you hold the interval long enough. certain effects, such as
the combinational tones that jacques dudon exploits, can make tiny
differences easier to hear.

Great, great, now I'm getting it, t's just that I'm not familiar with
all this music, not to say the theory, and I'm also kinda far ( I do
live in Brazil ), and my folks around here don't care for that.
Anyway, I guess I'll have to work a lot more on it before I can bring
a new discussion around.

Cheers
Alex

> a harmonic interval is in just intonation or 1 cent off from it, at
> least if you hold the interval long enough. certain effects, such
as
> the combinational tones that jacques dudon exploits, can make tiny
> differences easier to hear.

--- In tuning-math@yahoogroups.com, "Porres" <decuritiba@y...> wrote:
> Hy Wally,
>
> well, as I said, I was first concerned on creating a tool ( a table)
> where you could convert cents to a integer ratio

There's no difficulty in doing that; you don't need a table.

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

> There's no difficulty in doing that; you don't need a table.

Oh really, how can I do that? I remember I saw someone asking about
this somewhere, and there was a reply that you couldn't actually do
it, that's how I first had the idea of doing what I did.

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

> When I want to look at intervals near a given one, I use an
algorithm
> which finds the next or previous interval in the nth row of the
Farey
> sequence (with denominators <= n, therefore.) These can then be
> filtered so as to only give the p-limit if that is what you are
> looking for.

Filtered? Do you mean when you're developing a synthesis or in which
case do you need to look for these intervals. Gee, I can't help but
feel so amateur, you all keep answering me and I only have more
questions, oh dear...

Have you all tuning enthusiast begun studying it without a great deal
of mathematical background, I've been interested in this issue for
quite a while, and it's hard to find a complete guide to tudy, I dig
the begginer stuff very well, but I seem to be stuck as to moving to
the next level, any hints, clues... ?

Thanks a bunch
Cheers
Alex!

--- In tuning-math@yahoogroups.com, "Porres" <decuritiba@y...> wrote:
> --- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
> wrote:
>
> > There's no difficulty in doing that; you don't need a table.
>
> Oh really, how can I do that?

Let <a[0], a[1], a[2], ...> be such that the a[i] are integers, and
a[i]>0 for i>0. For any finite subsequence, we define a corresponding
fraction as follows: let p[-2]=0, p[-1]=1, q[-2]=1, q[-1]=0, and for
i>=0, define p[i] and q[i] recursively by

p[i] = a[i] p[i-1] + p[i-2]

q[i] = a[i] q[i-1] + q[i-2]

Then we define <a[0], a[1], ... a[n]> = p[n]/q[n] to be the *nth
convergent* of the continued fraction <a[0], a[1], a[2], ..., a[n]>

Now suppose we have a value c given in cents, we convert it to a
frequency ratio f by setting f = 2^(c/1200). If "floor(x)" is the
largest integer less than or equal to x, we set f[0]=f and then

a[i] = floor(f[i])

f[i+1] = 1/(f[i] - a[i])

then for increasing n <a[0], ..., a[n]> rapidly approaches f, and
gives excellent approximations to it.

thanks, I've seen many things now for the topic of cents to ratio
convertion, I'll note them all down and show them to someone near me
that can 'translate' it, lucky I got a composition teacher which had
graduated in engeneering...

cheers
alex

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "Porres" <decuritiba@y...>
wrote:
> > --- In tuning-math@yahoogroups.com, "Gene Ward Smith"
<gwsmith@s...>
> > wrote:
> >
> > > There's no difficulty in doing that; you don't need a table.
> >
> > Oh really, how can I do that?
>
> Let <a[0], a[1], a[2], ...> be such that the a[i] are integers, and
> a[i]>0 for i>0. For any finite subsequence, we define a
corresponding
> fraction as follows: let p[-2]=0, p[-1]=1, q[-2]=1, q[-1]=0, and for
> i>=0, define p[i] and q[i] recursively by
>
> p[i] = a[i] p[i-1] + p[i-2]
>
> q[i] = a[i] q[i-1] + q[i-2]
>
> Then we define <a[0], a[1], ... a[n]> = p[n]/q[n] to be the *nth
> convergent* of the continued fraction <a[0], a[1], a[2], ..., a[n]>
>
> Now suppose we have a value c given in cents, we convert it to a
> frequency ratio f by setting f = 2^(c/1200). If "floor(x)" is the
> largest integer less than or equal to x, we set f[0]=f and then
>
> a[i] = floor(f[i])
>
> f[i+1] = 1/(f[i] - a[i])
>
> then for increasing n <a[0], ..., a[n]> rapidly approaches f, and
> gives excellent approximations to it.

Hi Alex,

Welcome to tuning-math. Here's what Gene was talking about,
implemented in an Excel spreadsheet.

http://dkeenan.com/Music/CentsToRatios.xls

You enter an interval in cents and it gives a (two-dimensional) series
of ever-better ratios with their errors in cents.

I remember when I first learned how to calculate convergents and
semi-convergents (which I don't think Gene mentioned), to get ratios
from cents, it seemed like magic.

On the question of how much error is acceptable in just intonation:

In my humble opinion, many folks find +-3 cent errors entirely
acceptable for most music intended to be played in just intonation,
and indeed this is a typical intonation error for many fixed-pitch
acoustic intruments, but for others you'd have to be within +-0.5
cents (and not tell them about it) before they would be happy. But
there is little point in accepting more than about 3 cents error, or
in striving for less than about 0.5 cents. But of course there are
extreme and unusual contexts in which these do make sense. There are
no sharp cutoffs.

Others might be amused to learn that George Secor and I, in our
deliberations on a universal microtonal notation system, have coined
the terms "mortal" and "olympian" (as both adjective and noun) for
these two extremes of accuracy.

Obviously a notation for JI can have fewer symbols if greater errors
can be tolerated. The general approach is to define symbols only for
the most common ratios and to reuse these symbols for less common
ratios which are sufficiently close.

But we'll never make everyone happy with one such notation. The
olympians would complain about the errors or ambiguities in a notation
designed for mere mortals, while the mortals would complain about the
number and complexity of the symbols in a system designed for olympians.

A system about midway between mortal and olympian we call "herculean".
A herculean notation will of course have too many symbols for mortals
and too much ambiguity for olympians. So we've decided to specify
three systems. The trick is in making the transition from one to the
other as seamless as possible; to have as much in common between the
three as possible.

I'm afraid we're progressing rather slowly (but surely) on this at
present because we have both had (and are still having) long periods
where (heaven forbid) we had to concentrate on our jobs and families.
We are also working with Manuel Op de Coul on a Scala implementation
of the notation system.

Regards,
-- Dave Keenan

Thnaks Kennan!!!

> I remember when I first learned how to calculate convergents and
> semi-convergents (which I don't think Gene mentioned), to get ratios
> from cents, it seemed like magic.

Gee, I only know about high school math, actually not even that cause
it's been so long... it's funny that in my musical training now I'm
having to develop this area too.

I just checked an explanation about continued fractions and it's
amazing, I see you're using convergents and semi-convergents on your
Excell table, it seemed at first kinda complicated, but I guess I'll
dig it...

but hey, what does it have to do with that pdf file ( Self Similar
Pitch Structures - Clampitt ) ??? I downloaded that once and couldn't
figure it out to, let me dig the traditional harmony first, I'm still
studying Bach's counterpoint technique.

> We are also working with Manuel Op de Coul on a Scala implementation
> of the notation system.

ha ha ha, good job on your notation research, specially by keeping up
the good humor, since you're involved in Scala, would you know of a
complete table of name intervals? I guess that would be useful...

Cheers
Alex

> Regards,
> -- Dave Keenan

Alex, see this:
http://www.xs4all.nl/~huygensf/doc/intervals.html
There's a link on the bottom to another page of Dave
for more info.

Manuel

Hi Manuel

I had seen that, actually I used it and put it into an eletronic
database, I'll read carefully Dave's page, I see you use that
terminology to name the intervals in Scala, so I guess that all these
interval names are enough for you guys, and that other names
belonging to not that important terminologies are irrelevent to you
people, is that right? Cause I was also told about books with huge
lists of interval names, that can't be found on the net or in the
form of an eletronic database...

Now what's your oppinion, should the terminology be redefined and
form a new tradition, like Dave is working on, or should one bother
on the collection of historycal names that are available out there?

Thanks,
Cheers

Alex!

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
> Alex, see this:
> http://www.xs4all.nl/~huygensf/doc/intervals.html
> There's a link on the bottom to another page of Dave
> for more info.
>
> Manuel

Alex wrote:
>Now what's your oppinion, should the terminology be redefined and
>form a new tradition, like Dave is working on, or should one bother
>on the collection of historycal names that are available out there?

No opinion about what should be done, but I felt more for making a
list of historical names, and Dave for making a consistent terminology.

>and that other names
>belonging to not that important terminologies are irrelevent to you
>people, is that right?

Well there's an infinite amount of interval ratios, there's no point in
trying to find a name for all of them.

Manuel

Hi Manuel

> Well there's an infinite amount of interval ratios, there's no
point in
> trying to find a name for all of them.

Sure, not all of them, just the ones that were historically
important, but then I guess you already used them all in Scala, right?

Well, so I got that list from the fokker site you gave me, and I put
it in a database, I just thought it'd be nice to have a database with
the names of intervals in tunings that were historically important, I
think it'd be good for comparisons... but I still wonder if that
fokker list is complete for this matter, any clues?

thanks a bunch
Alex.

>but I still wonder if that
>fokker list is complete for this matter, any clues?

I'm not aware of missing important intervals, and open
to any corrections.

Manuel

--- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
<manuel.op.de.coul@e...> wrote:
> >but I still wonder if that
> >fokker list is complete for this matter, any clues?
>
> I'm not aware of missing important intervals, and open
> to any corrections.
>
> Manuel

unfortunately, some of the "big names" in tuning history, such as
rameau, helmholtz, etc., have given different, mutually inconsistent
names to different intervals. in the following table, i attempted to
tabulate the most common names for important small 5-limit intervals,
and give the details of the temperaments that result when these
intervals vanish. some of the names of the latter were jovial
creations of this (tuning-math) list. this is meant to replace the
similar table on monz's "equal temperament" dictionary page:

/tuning/database?
method=reportRows&tbl=10&sortBy=3

----- Original Message -----
From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
To: <tuning-math@yahoogroups.com>
Sent: Friday, June 13, 2003 12:24 PM
Subject: [tuning-math] Re: Interval Database Experiences

> --- In tuning-math@yahoogroups.com, "Manuel Op de Coul"
> <manuel.op.de.coul@e...> wrote:
> > >but I still wonder if that
> > >fokker list is complete for this matter, any clues?
> >
> > I'm not aware of missing important intervals, and open
> > to any corrections.
> >
> > Manuel
>
> unfortunately, some of the "big names" in tuning history, such as
> rameau, helmholtz, etc., have given different, mutually inconsistent
> names to different intervals. in the following table, i attempted to
> tabulate the most common names for important small 5-limit intervals,
> and give the details of the temperaments that result when these
> intervals vanish. some of the names of the latter were jovial
> creations of this (tuning-math) list. this is meant to replace the
> similar table on monz's "equal temperament" dictionary page:
>
> /tuning/database?
> method=reportRows&tbl=10&sortBy=3

hi paul,

i thought i replaced the table on my webpage with
your database data. didn't i? ...?

-monz

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

> hi paul,
>
>
> i thought i replaced the table on my webpage with
> your database data. didn't i? ...?
>
>
>
>
> -monz

you never did, and more recently i'm waiting for the harmonic entropy
definition page to get updated as well (as per our e-mail) . . .

> From: "wallyesterpaulrus" <wallyesterpaulrus@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, June 15, 2003 1:38 AM
> Subject: [tuning-math] Re: Interval Database Experiences
>
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > hi paul,
> >
> >
> > i thought i replaced the table on my webpage with
> > your database data. didn't i? ...?
> >
> >
> >
> >
> > -monz
>
> you never did, and more recently i'm waiting for the harmonic entropy
> definition page to get updated as well (as per our e-mail) . . .

sorry about falling behind like this. i've been
really busy with other stuff. i'll try to chat with
you on Yahoo messenger at some point and we'll get
these done.

-monz

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

> > you never did, and more recently i'm waiting for the harmonic entropy
> > definition page to get updated as well (as per our e-mail) . . .

> sorry about falling behind like this. i've been
> really busy with other stuff. i'll try to chat with
> you on Yahoo messenger at some point and we'll get
> these done.

When you do that, could you also look at my question about the correct
defintion of "microtemperament"?

hi Gene,

> From: "Gene Ward Smith" <gwsmith@svpal.org>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, June 15, 2003 3:22 PM
> Subject: [tuning-math] Re: Interval Database Experiences
>
>
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > > you never did, and more recently i'm waiting for the harmonic entropy
> > > definition page to get updated as well (as per our e-mail) . . .
>
> > sorry about falling behind like this. i've been
> > really busy with other stuff. i'll try to chat with
> > you on Yahoo messenger at some point and we'll get
> > these done.
>
> When you do that, could you also look at my question about the correct
> defintion of "microtemperament"?

you were never very specific about what's wrong
with it.

the most foolproof way to get me to actually
update the page would be to write a commentary
that i could simply copy-and-paste and add as an
addendum at the bottom of the page.

-monz

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

> you were never very specific about what's wrong
> with it.

What I mean by a "microtemperament" is something like this: one where
all of the relevant consonant intervals are within a cent of being
pure. What you've defined is a regular temperament of dimension
greater than or equal to one, so I think it should not be used as a
definition of microtemperament.

I'd first define "regular temperament" as the expression of musical
intervals by means of a set of generators g1, ..., gn such that each
interval is represented as a number of the form g1^e1 ... gn^en, then
the dimension of the regular temperament as the number of generators
minus one. An equal temperament, with one generator, would be regular
of dimension zero; p-limit just intonation would be regular with
dimension pi(p), which is the number of primes <= p. A linear
temperament, or temperament of dimension one, has two generators.

--- In tuning-math@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...>
wrote:
> --- In tuning-math@yahoogroups.com, "monz" <monz@a...> wrote:
>
> > you were never very specific about what's wrong
> > with it.
>
> What I mean by a "microtemperament" is something like this: one
where
> all of the relevant consonant intervals are within a cent of being
> pure. What you've defined is a regular temperament of dimension
> greater than or equal to one, so I think it should not be used as a
> definition of microtemperament.

agreed, though what monz (actually graham) defined in the first
attempt is that the dimension must be *higher* than that of a linear
temperament. but he appears to be willing to take that back further
down on the page.

this definition page is a huge mess! the final quote on the page
should be attributed to graham, not to me. he's clearly not sticking
to his original attempt at a definition, so why should it be on the
top of the page? if this dictionary is to be of use to anyone, we
have to eliminate the nonsense or at least relegate it to some back
pages.

--- In tuning-math@yahoogroups.com, "Porres" <decuritiba@y...> wrote:
> I just checked an explanation about continued fractions and it's
> amazing, I see you're using convergents and semi-convergents on your
> Excell table, it seemed at first kinda complicated, but I guess I'll
> dig it...
>
> but hey, what does it have to do with that pdf file ( Self Similar
> Pitch Structures - Clampitt ) ??? I downloaded that once and couldn't
> figure it out to, let me dig the traditional harmony first, I'm still
> studying Bach's counterpoint technique.

Sorry to take so long to reply.

It is the same mathematics put to a different purpose. In Clampitt's
paper it is applied in the logarithmic pitch domain, whereas you're
using it in the linear frequency domain. Clampitt is finding ratios
representing fractions of an octave (or other interval of periodicity)
(i.e. degrees of equal temperaments) where you are finding ratios
representing frequency ratios (i.e. justly intoned intervals when the
numbers are small).

It's nice that the same mathematical tool has these two different
applications to tuning. You might say Clampitt is applying it to
melodic properties while you are applying it to harmonic ones.

> ha ha ha, good job on your notation research, specially by keeping up
> the good humor, since you're involved in Scala, would you know of a
> complete table of name intervals? I guess that would be useful...

I'm not really "involved in Scala", but the file 'intnam.par' (and its
equivalents in other languages) that comes with Scala, is very useful.
With a little work it can be imported into a spreadsheet and sorted by
interval size or whatever.

-- Dave Keenan