Re: [tuning] katapyknosis

> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning@yahoogroups.com>
> Sent: Monday, July 16, 2001 12:10 AM
> Subject: [tuning] Re: tenney distance function/katapyknosis
>
>
> > So does that mean that katapyknosis is simply
> > splitting intervals "in half"?
>
> No, it seems to apply to any scheme that divides an interval into
> parts that are in approximate small whole number ratios OF CENTS
> (logarithmic), but actually does it by using simple arithmetic (no
> logs or roots etc.) on the frequency ratios (linear), which is why it
> results in only an approximately equally (or rationally) divided
> interval in cents.

Dave, your explanation of the arithmetic mean method of
_katapyknosis_ is accurate, but the harmonic or geometric
means could also be used. Aristoxenos's approach was based
primarily on the *concept* (but not the actual practice)
of the geometric mean. These terms are explained well
in the Scala description which Dave quoted:

> /GEOMETRIC
>
> Uses the geometric mean, i.e. sqrt(a * b). This divides an interval
> equally.
>
> /HARMONIC
>
> Uses the harmonic mean, i.e. 2 * a * b / (a + b). Densifying a scale
> by inserting harmonic or arithmetic means is also known as
> katapyknosis.
>
> /ARITHMETIC
>
> Uses the arithmetic mean, i.e. (a + b) / 2. Densifying a scale by
> inserting harmonic or arithmetic means is also known as katapyknosis.

> > (64/63)+(63/62)+(93/92)+(92/91)+(91/90)=16/15
> >
> > Is that also katapyknosis? What is that called?
>
> Not sure, but it probably is katapyknosis.
>
> Of course all of the above should be multiplications, not additions,
> unless you put log() around every ratio.

Guys, you are missing an important point about _katapyknosis_.
It referred originally to the division of the small interval
at the bottom of a chromatic or enharmonic trichord, to form
a "proper" tetrachordal structure. This small interval,
which is always < (4/3)^(1/2) , is called the _pyknon_; its
division into two equal or quasi-equal parts is _katapyknosis_.

See my "Tutorial on Ancient Greek Tetrachord-theory"
<http://www.ixpres.com/interval/monzo/aristoxenus/tutorial.htm>.

Later, the term obviously referred to a more complicated
tuning structure, because Aristoxenos rails out against
the harmonists, theorists who used _katapyknosis_ to describe
the entire system. I take this to mean that the harmonists
were comparing different modes by using different "8ve" spaces
for all of them and retaining the *functional* values of
the notes for each mode in their diagrams, thus necessitating
_katapyknosis_ of various small intervals from one mode to
another in order to obtain the proper small ratios in each
shade of chromatic and enharmonic genos.

So AFAIK that example above doesn't qualify as _katapyknosis_.
For one thing, I think it has way too many notes.

> > Are there always
> > superparticular ratios when you do katapyknosis?
>
> I don't think so.

Superparticular ratios were important in the work of many
Greek theorists, but not all. Aristoxenos did not describe
his system in terms of ratios (of string-lengths) or any
other kind of physically measurable quantity at all, but
only in terms of string tension, and thus determined solely
by ear. For my initial assessment of Aristoxenos's work:
<http://www.ixpres.com/interval/monzo/aristoxenus/318tet.htm>.

> > If you had to explain katapyknosis clearly to a
> > bunch of high school kids, how might you do it?

I think what I wrote above helps a little.

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

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> I think what I wrote above helps a little.
> -monz

Absolutely! Thanks you and Dave Keenan both! Very helpful indeed.

A semantic note: the '+' should be '+'. Just because adding ratios
requires a multiply behind the scenes, doesn't mean that you are not
still adding the intervals intervals? No?

To me a 3/2 and a 3/2 is a 9/8. To our 3/2 we are adding another
3/2, no? So that is why i say '+', even though mathematically we
multiply to get to that 'sum'. Giving musical conceptions precidence
over mathmatics. Interestingly, that is what the Python script does as
well, it overloads the add attribute (__add__) to do Ratio addition. So
i am not alone in my idiosyncracies.

cheers,
-kevin parks
seoul, korea
kp87@lycos.com

--- In tuning@y..., kp712@y... wrote:
> > I think what I wrote above helps a little.
> > -monz
>
> Absolutely! Thanks you and Dave Keenan both! Very helpful indeed.
>
> A semantic note: the '+' should be '+'. Just because adding ratios
> requires a multiply behind the scenes, doesn't mean that you are not
> still adding the intervals intervals? No?

OK, so I'm a pedant. Yes, you can say you are adding intervals.
Intervals are musical entities. Rational numbers are mathematical
entities used for many other things besides describing musical
intervals in terms of frequency (or wavelength) ratios.

Even for musical intervals, we sometimes use these numbers in the
log-frequency domain to represent fractions of an octave (e.g. an
equal tempered semitone as 1/12 oct). In that case you will actually
_add_ the ratios.

> To me a 3/2 and a 3/2 is a 9/8. To our 3/2 we are adding another
> 3/2, no? So that is why i say '+', even though mathematically we
> multiply to get to that 'sum'.

But don't you think that could cause a lot of errors by your
aforementioned high-school students pushing the wrong buttons on their
calculators?

> Giving musical conceptions precidence over mathmatics.

No. As I showed above, you are merely giving one musical conception
precedence over another, and possibly introducing an opportunity for
confusion.

> Interestingly, that is what the Python script does as
> well, it overloads the add attribute (__add__) to do Ratio addition.

You mean it overloads the __add__ attribute to do _interval_ addition,
which in this case corresponds to ratio multiplication because
intervals are represented by their frequency ratio as opposed to
representing them as logarithmic fractions of an octave, or cents).

> So i am not alone in my idiosyncracies.

No. <sigh>

If you were to use some notation like

[3/2] + [3/2] = [9/4]
where [x] means "the musical interval whose frequency ratio is x"

then I could have no objection.

But in fact many of us prefer to use the ratio notation 2:3 (or 3:2)
for the interval of a just perfect fifth, and use 3/2 as the name of
the pitch which is a just perfect fifth above the tonic (called 1/1).
So I'd prefer
[2:3] + [2:3] = [4:9]

Then we have the problem with assuming octave equivalence and failing
to mention it. Just this morning I had email from someone reasonably
new to the field who was totally confused by an apparent contradiction
between two authors writing about the construction of the same simple
scale. One did the octave reductions explicitly, the other implicitly.
Neither of them mentioned what they were doing or why.

So one could write
[2:3] + [2:3] = [8:9]
where [x] means "a musical interval whose octave-reduced frequency
ratio is x".

See also http://dkeenan.com/Music/ANoteOnNotation.htm

-- Dave Keenan

> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning@yahoogroups.com>
> Sent: Monday, July 16, 2001 6:57 PM
> Subject: [tuning] Re: katapyknosis
>
> ...
>
> If you were to use some notation like
>
> [3/2] + [3/2] = [9/4]
> where [x] means "the musical interval whose frequency ratio is x"
>
> then I could have no objection.
>
> But in fact many of us prefer to use the ratio notation 2:3 (or 3:2)
> for the interval of a just perfect fifth, and use 3/2 as the name of
> the pitch which is a just perfect fifth above the tonic (called 1/1).
> So I'd prefer
> [2:3] + [2:3] = [4:9]

Hi Dave,

An excellent post explaining some of the unintended errors that
can creep into tuning calculations!

My preference: simply define the colon [:] as representing
rational intervals that can be added. That would eliminate
the need to use the brackets:
2:3 + 2:3 = 4:9

In the case of pitches, with the division line between the two
terms, one would still have to use the multiplication symbol
between two ratios, to show, for example, transposition from
one pitch-class to another.

These are just some "knee-jerk" responses. I actually agree
with your original dissenting opinion: it's much better to
simply make the mathematical descriptions as accurate and
explicit as possible. Thus my insistent us of, for example,
2^(32/55) instead of simply 32/55 to describe an EDO interval.

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

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
> My preference: simply define the colon [:] as representing
> rational intervals that can be added. That would eliminate
> the need to use the brackets:
> 2:3 + 2:3 = 4:9

Yes!

I notice that I myself often describe intervals using what looks
like a mixture of linear and logarithmic notation, with no ambiguity
whatsoever. e.g.

2:3 + 4.8 c

I certainly don't want to use a multiplication sign here.
The only way to make it consistent is to define

x:y = abs( log2(y/x) ) * 1200 c

or equivalently

x:y = log2( max(x,y)/min(x,y)) ) * 1200 c

So you might say that we have all been unconsciously doing just what
you propose, all along.

> In the case of pitches, with the division line between the two
> terms, one would still have to use the multiplication symbol
> between two ratios, to show, for example, transposition from
> one pitch-class to another.

That's what I'd prefer. But one _could_ redefine the / in the same way
as you defined the : above, but you'd have to explicitly say so, and
you'd then have the possible confusion with fractions of an octave
where it has the usual meaning.

> These are just some "knee-jerk" responses. I actually agree
> with your original dissenting opinion: it's much better to
> simply make the mathematical descriptions as accurate and
> explicit as possible. Thus my insistent us of, for example,
> 2^(32/55) instead of simply 32/55 to describe an EDO interval.

Yes. I write "32/55 oct" to mean the same interval, but in logarithmic
terms, where 1 oct = 1200 c.

So I can consistently write things like
32/55 oct + 4.8 c
whereas
2^(32/55) + 4.8c
is not consistent.
You'd have to write
2^(32/55) * 2^(4.8/1200)
which is a bit verbose (numerose?).

Thanks for this Monz.

-- Dave Keenan