Re: from the realms of private correspondence

>note that my geometric generalization of tenney's HD function so
>that it amounts to the LCM of the N integers when the N-ad is
>expressed in lowest harmonic terms (the LCM of great
>importance to measures like Euler's and, more recently,
>Marion's) allows us to forget that it's a metric altogether . . .
>and to forget about taxicab, since the construction of the
>rectangle for a dyad already contains the definition of taxicab
>inside it. of course, i have no idea what such a construction is
>called, but who cares?

IIRC LCM came up in '99 in the massive complexity thread. I
wonder what we said then...

...the only hits for "LCM" in my tuning-math inbox are from the
beginning of this year, when Gene was asking how similar Euler's
LCM/GCD was to harmonic entropy.

Here's selections from tuning...

[Graham]
Some people leave out multiples of 2 when calculating dissonances.
Then 15/8 and 6/5 come out the same. The LCM can be evaluated for
a whole chord, where C-E-G-B comes out unduly consonant. A way
round this problem is to calculate Euclidian distances on a
parallelogram lattice.

[Dave]
...applied to dyads where the ratio n:d is already in lowest terms,
the LCM is the same as the product n*d ... suffer from the possible
defect pointed out by Paul Erlich where for example, 15/4 has the
same complexity as 12/5, or 3/2 the same as 6/1. However the
disagreement with perceived dissonance may be taken care of by
TOLERANCE in the first case and SPAN in the second. So they may not
amount to a defect in the COMPLEXITY measure. Note also that the
LCM of a chord corresponds to the "guide tone"...

[Graham]
Harmonic entropy, which may or may not be related to dissonance, is
in fact related to the frequency an LCM above the VF. I believe
this is known as the "guide tone" in the literature. So, the lower
the guide tone, the lower the entropy. ... I don't know how to
generalise the harmonic entropy beyond dyads. However, my intuition
strongly suggests that it will still lead to the guide tone.

[Paul]
The guide tone is lower and simpler for utonal chords than otonal
chords. However, harmonic entropy will definitely be lower for
otonal chords. ... I don't see how LCM figures into it. If we can
somehow believe that for triads, the "area" is inversely
proportional to the geometric mean of the three integers most
simply expressing the chord (e.g., 4:5:6 or 10:12:15), then we
have a way to compute harmonic entropy for triads. But if you look
at a 2-d plot (such as a Dalitz plot) of all the triads, you'll see
that this "area" may have a very strange shape and might not be
possible to define with some generalization of mediants. Then
again, it might . . .

-Carl

>>note that my geometric generalization of tenney's HD function so
>>that it amounts to the LCM of the N integers when the N-ad is
>>expressed in lowest harmonic terms (the LCM of great
>>importance to measures like Euler's and, more recently,
>>Marion's) allows us to forget that it's a metric altogether . . .
>>and to forget about taxicab, since the construction of the
>>rectangle for a dyad already contains the definition of taxicab
>>inside it. of course, i have no idea what such a construction is
>>called, but who cares?

So the problem with lcm is the equal ranking of otonal and
utonal chords. The geometric mean doesn't have that problem.

What ever happened to the paper on Farey sets by Remy and Thiel?

-Carl

--- In harmonic_entropy@y..., "Carl Lumma" <clumma@y...>
wrote:

> What ever happened to the paper on Farey sets by Remy and
>Thiel?

i don't know . . . was that mentioned before?

>>What ever happened to the paper on Farey sets by Remy and
>>Thiel?
>
>i don't know . . . was that mentioned before?

You're the one who found it, hailing it as the answer to the
problem of the 3-term Farey series, but seemed to think that
if the validation excercise worked we wouldn't need the fancy
math.

See threads starting on message #s 387 and 456.

-Carl

--- In harmonic_entropy@y..., "Carl Lumma" <clumma@y...> wrote:
> >>What ever happened to the paper on Farey sets by Remy and
> >>Thiel?
> >
> >i don't know . . . was that mentioned before?
>
> You're the one who found it, hailing it as the answer to the
> problem of the 3-term Farey series, but seemed to think that
> if the validation excercise worked we wouldn't need the fancy
> math.
>
> See threads starting on message #s 387 and 456.
>
> -Carl

i want to see gene chime in on all this . . .

>>>>What ever happened to the paper on Farey sets by Remy and
>>>>Thiel?
//
>>>You're the one who found it, hailing it as the answer to the
>>>problem of the 3-term Farey series, but seemed to think that
>>>if the validation excercise worked we wouldn't need the fancy
>>>math.
//
>>See threads starting on message #s 387 and 456.
//
>i want to see gene chime in on all this . . .

Me too!

-Carl