tenney distance function

Anyone know what the James Tenney harmonic distance function is? I used
to have it written down somewhere, but i can't find it and i no where
near a library. Larry Polansky had uses it in his cute little MacRatio
app, but he doesn't say what it is in the about (which would have been
nice).

I know that there are lots of sophisticated HD functions out there but
the Tenney is simple (as i remember) and comes very close to what the
other functions give.

cheers,
kevin parks
seoul, korea

kp87@lycos.com

Kevin wrote:

> Anyone know what the James Tenney harmonic distance function is? I used
> to have it written down somewhere, but i can't find it and i no where
> near a library. Larry Polansky had uses it in his cute little MacRatio
> app, but he doesn't say what it is in the about (which would have been
> nice).

It's the logarithm of the lcm. Or, for 2^m*3^n..., |m| + |n|log2(3) + ...

I don't remember if the 2 term is included or not.

> I know that there are lots of sophisticated HD functions out there but
> the Tenney is simple (as i remember) and comes very close to what the
> other functions give.

Yes, nice and simple.

Graham

Kevin wrote:
> > Anyone know what the James Tenney harmonic distance function is?

Graham replied:
> It's the logarithm of the lcm. Or, for 2^m*3^n..., |m| + |n|log2(3)
+ ...
>
> I don't remember if the 2 term is included or not.

In other words Tenney's Harmonic Distance is simply the log (base 2)
of the product of the two sides of the ratio in lowest terms. This is
a big favourite on this list. Although if you only want to _compare_
ratios there's no need to take the log. Just use the product, which we
call complexity rather than distance. Of course the logs have the
advantage of being able to be added, e.g. when looking at distances on
a lattice.

One thing that bothers some of us about these measures is that they
don't take into account the apparent decrease in harmonic distance
when the smallest number in the ratio is a power of two; so called
"rootedness". And of course there's tolerance and span.

Factors of two are not removed in Tenney's. But of course if you do
remove factors of two, it would make sense to call it
"Octave-equivalent Harmonic Distance".

For a comparison of many harmonic complexity measures see my Excel
spreadsheet
http://dkeenan.com/Music/HarmonicComplexity.zip

For definitions of "harmonic distance", "katapyknosis" and many other
tuning terms, bookmark the Tuning Dictionary of the amazing Monz. We
couldn't live without it. And it's free!
http://www.ixpres.com/interval/dict/index.htm

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
/tuning/topicId_26175.html#26179

> For definitions of "harmonic distance", "katapyknosis"
> and many other tuning terms, bookmark the Tuning Dictionary
> of the amazing Monz. We couldn't live without it.
> And it's free!
> http://www.ixpres.com/interval/dict/index.htm

Thanks so very much for the plug and the kind words, Dave!

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

http://dkeenan.com/Music/HarmonicComplexity.zip
>
> For definitions of "harmonic distance", "katapyknosis" and many other
> tuning terms, bookmark the Tuning Dictionary of the amazing Monz. We
> couldn't live without it. And it's free!
> http://www.ixpres.com/interval/dict/index.htm

Thanks for the reply. I did check the dictionary, which is indeed
wonderful. However my hope was that someone could improve upon the
definition of katapyknosis there, since it is rather imprecise and
doesn't cover different types and instances of katapyknosis. Perhaps
several examples would help clear it up in my mind. The dictionary
says:

"The division of a musical interval by multiplication of the defining
integers and the insertion of arithmetic or harmonic means."

Here i take this "means" is not "way" but rather the "arithmetic mean",
as in average. I understand that 16/15 = 32/30, and that 32/31 + 31/30
= 16/15 = 32/30. So does that mean that katapyknosis is simply
splitting intervals "in half"? Is this 1/1 katapyknosis? Are there
other types of katapyknosis (ways of slicing intervals)?

for example:

(96/95) + (95/94) + (94/93) +(93/92) + (92/91) + (91/90) = 16/15 = 32/
30

(128/127) + (127/126) + (126/125) + (125/124) + (124/123) + (123/122) +
(122/121) + (121/120) = 16/15 = 32/30.

Also:
(64/63)+(63/62)+(93/92)+(92/91)+(91/90)=16/15

Is that also katapyknosis? What is that called? Additionally, how is
katapyknosis performed? What are the steps? Are there always
superparticular ratios when you do katapyknosis? If you had to explain
katapyknosis clearly to a bunch of high school kids, how might you do
it?

cheers,

-kevin parks

Wow, kevin! what a interesting question and oddly related
to..uhm..tuning. SO much more interesting than that interminable
discussion about that vitriolic crazy music list. I wish i knew the
answer, i'd elucidate katapyknosis for you and give some smart and
luminous examples, heck maybe it would get added to the monz's great
dictionary.

--- In tuning@y..., kp712@y... wrote:
> http://dkeenan.com/Music/HarmonicComplexity.zip
> >
> > For definitions of "harmonic distance", "katapyknosis" and many other
> > tuning terms, bookmark the Tuning Dictionary of the amazing Monz. We
> > couldn't live without it. And it's free!
> > http://www.ixpres.com/interval/dict/index.htm
>
> Thanks for the reply. I did check the dictionary, which is indeed
> wonderful. However my hope was that someone could improve upon the
> definition of katapyknosis there, since it is rather imprecise and
> doesn't cover different types and instances of katapyknosis. Perhaps
> several examples would help clear it up in my mind. The dictionary
> says:
>
> "The division of a musical interval by multiplication of the defining
> integers and the insertion of arithmetic or harmonic means."
>
>
> Here i take this "means" is not "way" but rather the "arithmetic mean",
> as in average. I understand that 16/15 = 32/30, and that 32/31 + 31/30
> = 16/15 = 32/30. So does that mean that katapyknosis is simply
> splitting intervals "in half"? Is this 1/1 katapyknosis? Are there
> other types of katapyknosis (ways of slicing intervals)?
>
> for example:
>
> (96/95) + (95/94) + (94/93) +(93/92) + (92/91) + (91/90) = 16/15 = 32/
> 30
>
> (128/127) + (127/126) + (126/125) + (125/124) + (124/123) + (123/122) +
> (122/121) + (121/120) = 16/15 = 32/30.
>
> Also:
> (64/63)+(63/62)+(93/92)+(92/91)+(91/90)=16/15
>
> Is that also katapyknosis? What is that called? Additionally, how is
> katapyknosis performed? What are the steps? Are there always
> superparticular ratios when you do katapyknosis? If you had to explain
> katapyknosis clearly to a bunch of high school kids, how might you do
> it?
>
> cheers,
>
> -kevin parks

--- In tuning@y..., kp712@y... wrote:
> Wow, kevin! what a interesting question and oddly related
> to..uhm..tuning. SO much more interesting than that interminable
> discussion about that vitriolic crazy music list. I wish i knew the
> answer, i'd elucidate katapyknosis for you and give some smart and
> luminous examples, heck maybe it would get added to the monz's great
> dictionary.

Hey yeah. Shouldn't that "Crazy growth!" thread be on the metatuning
list.

Can't someone answer poor Kevin's question. I thought katapyknosis was
a disease you got from cleaning out your nostrils after contact with a
member of the genus felis, until I discovered
Smirnoff ^H^H^H^H^H^H^H^H^HMonz's dictionary.

I just did a general web search on the term, using Yahoo search. I
guess you've already tried that.

From my reading of the 5 hits, it seems it's pretty general and covers
most or all of your examples.

> Here i take this "means" is not "way" but rather the "arithmetic
mean", as in average.

Certainly.

> I understand that 16/15 = 32/30, and that 32/31 + 31/30
= 16/15 = 32/30.

You mean 32/31 * 31/30.

>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.

>Is this 1/1 katapyknosis?

That's my guess.

> Are there other types of katapyknosis (ways of slicing intervals)?
for example:

(96/95) + (95/94) + (94/93) +(93/92) + (92/91) + (91/90) = 16/15 = 32/
30

(128/127) + (127/126) + (126/125) + (125/124) + (124/123) + (123/122)
+
(122/121) + (121/120) = 16/15 = 32/30.

I think those are all covered.

> Also:
> (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.

> Additionally, how is
katapyknosis performed? What are the steps?

If you can read Python (it's fairly "C"-like), this should tell you.
http://www.execpc.com/~wsannis/ratio.html
Then you can let us all know. :-)

> Are there always
superparticular ratios when you do katapyknosis?

I don't think so.

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

Sorry I can't help you there.

Here's an extract from the Scala help file that gives some clues:
-------------------------------------------------------------------
DOUBLE [interval-class] [expansion]

Double the number of pitches in the current scale by inserting a mean
between pitches separated by the given interval-class. The default
interval-class is 1, which means adjacent pitches. It may be any
integer
number. The default expansion factor is 2, i.e. doubling. To triple by
inserting two means, do DOUBLE 1 3. Expansion may be any positive
number.
The qualifiers select the pitch-averaging operation. Geometric mean is
the default. Qualifiers cannot be combined. See also SAMPLE.

/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.
---------------------------------------------------------------------

Regards,
-- Dave Keenan

--- In tuning@y..., kp712@y... wrote:
> Anyone know what the James Tenney harmonic distance function is?

Sure. Two notes having a musical ratio a:b, have Tenney Harmonic
Distance log(a*b) = log(a) + log (b). Logs to the base 2 are used, so
if you need to calculate these, use

log(a*b)/log(2)

or

log(a)/log(2) + log(b)/log(2)

> I know that there are lots of sophisticated HD functions out there
but
> the Tenney is simple (as i remember) and comes very close to what
the
> other functions give.

It's also better in some ways I've discussed. I find it accurate as
long as a*b is not too large (over 100 is too large).

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:

> Factors of two are not removed in Tenney's. But of course if you do
> remove factors of two, it would make sense to call it
> "Octave-equivalent Harmonic Distance".

I'd have a problem with that. Actually Polansky was using a more
reasonable version of Octave-equivalent Harmonic Distance, defined by
simply transposing the interval to within the first octave before
computing HD. So comparing 15:8 with 6:5 would still come out right,
while treating them as 15:1 and 3:5 wouldn't (since those have the
same HD, while 15:8 is clearly more dissonant than 6:5).

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > Factors of two are not removed in Tenney's. But of course if you
do
> > remove factors of two, it would make sense to call it
> > "Octave-equivalent Harmonic Distance".
>
> I'd have a problem with that. Actually Polansky was using a more
> reasonable version of Octave-equivalent Harmonic Distance, defined
by
> simply transposing the interval to within the first octave before
> computing HD. So comparing 15:8 with 6:5 would still come out right,
> while treating them as 15:1 and 3:5 wouldn't (since those have the
> same HD, while 15:8 is clearly more dissonant than 6:5).

Yes! Good point. Removing all factors of two is not of much use at
all.