harmonic entropy

[Monzo:]
> intervals according to this formula
> are all harmonics of the subharmonic series
> of 1/1

[Lumma:]
> You shouldn't find it surprising, since
> Erlich's scheme cuts up ratio space using
> the Farey series, which is embedded in the
> diamond.

Absolutely true - my non-mathematical mind
hadn't realized that.

> I don't believe it measures consonance,
> it measures the 'field of attraction' of an
> interval

I know better than to speak for Erlich, but
having re-read his posting on HE, I'm pretty
sure you're right about this too :)

BTW, I'll have a nicely updated version of
that webpage in the near future, with a
commentary between Paul's posting that's
taken from a discussion I had with him.

- Monzo
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>> I don't believe it measures consonance,
>> it measures the 'field of attraction' of an
>> interval

>I know better than to speak for Erlich, but
>having re-read his posting on HE, I'm pretty
>sure you're right about this too :)

Well, the same model that is behind the harmonic entropy does, in a
sense, measure the 'field of attraction' because it tells you the most
likely rational interpretation of any interval. However, the harmonic
entropy itself is low only if there is one rational interpretation much
more likely than any other. As such, it is indeed supposed to model a
component of consonance.

> [Glen Peterson, TD 392.23]
>
> I know Partch speaks of relating the gravity of a
> ratio to it's limit. Hence the 300 cent note in our
> earlier example would be pulled 5/18 toward 13/11 and
> 13/18 toward 6/5. (The number 18 represents the two
> limits added together or 13 + 5.)
> I've played around with some numbers and come up with
> this table showing the relative gravitational pull of
> ratios of different limits:
>
> <etc.>

I haven't experimented with this to gain any empirical
knowledge myself, but years ago I wrote a computer program
to calculate Partch's 'Field of Attraction' for every
possible chord progression in the 19-Limit Tonality Diamond.
The printout gives the ratios and cents values in roughly
graphical page-layout for both the starting and ending
chords, with the interval size in cents for each 'permissible'
resolution. I followed Partch's guidelines exactly.

It's a fantastic reference for JI composers and theorists.

Paul Erlich expressed an interest in having a copy of
this a while back, so I suppose since I mentioned here
now I'll make it available to anyone who wants one.
I'll figure out how much it would cost to make a copy
- email me privately if interested.

-monz

Joseph L. Monzo Philadelphia monz@juno.com
http://www.ixpres.com/interval/monzo/homepage.html
|"...I had broken thru the lattice barrier..."|
| - Erv Wilson |
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In the octave-equivalent harmonic entropy curve, we saw that the harmonic
entropy was very nearly a monotonous function of the odd-limit up to a
certain limit. In the octave-specific case, could the Tenney complexity play
the same role as the odd-limit? Let's consider first the Farey-based results
such as that displayed at
http://www.egroups.com/files/tuning/perlich/ent_006.jpg. If we judge the
entropy relative to a baseline that reflects the downward slope of the
Farey-like results, does it agree with the Farey complexity? Perhaps I'll
have the insight at another time to attempt to judge this rigorously, but
for now I'll be content with a qualitative assessment. Let's rank the
simplest ratios in the first two octaves according to the product of
numerator and denominator (up to 105), which is of course the same as
ranking them according to Tenney complexity:

Ratio Product
1/1 1
2/1 2
3/1 3
4/1 4
3/2 6
5/2 10
4/3 12
7/2 14
5/3 15
5/4 20
7/3 21
8/3 24
7/4 28
6/5 30
10/3 30
11/3 33
7/5 35
9/4 36
8/5 40
7/6 42
11/4 44
9/5 45
13/4 52
11/5 55
8/7 56
15/4 60
12/5 60
9/7 63
13/5 65
11/6 66
14/5 70
10/7 70
9/8 72
11/7 77
13/6 78
16/5 80
12/7 84
17/5 85
11/8 88
18/5 90
10/9 90
13/7 91
19/5 95
11/9 99
17/6 102
13/8 104
15/7 105

Lo and behold, this is exactly the list of local minima in the graph, with
one exception -- the minimum marked ***, which is between 17/9 and 15/8,
appears in place of 13/7. But this is exactly the type of behavior we've
seen, involving little minima sliding up toward the unison or octave, when N
is too low and the harmonic entropy curve is still converging; and a "bump"
for 13/7 is visible on the graph, but fails to be a minimum due to the
graph's steep slope.

Furthermore, the products seem to be a monotonic decreasing function of the
valley depths relative to the overall slope, except that the valleys are an
extra amount deeper the farther to the right you go. This phenomenon, as
well the overall slope of the curve, is due to the diminishing density of
the Farey series as one moves to the right. We've seen that for lower N, we
get lower harmonic entropy values, and deeper valleys. As one moves to the
right of these graphs, the "effective N" gets lower, due to the lower
density of the ratios, and so the overall graph lowers, and the depth of the
valley for a given product of numerator and denominator decreases.

Open questions:

1. Can we calculate a smooth function that represents (models & explains)
the density of the Farey series as a function of pitch? Can we use some sort
of canonical probability distribution, parameterized by this density, to
represent (model & explain) the overall slope of the harmonic entropy
function of the Farey series?

2. Can we construct a Farey-like series that also obeys the partial ordering
that Pierre Lamonthe associates with the Stern-Brocot tree, but for which
this overall slope remains flat throughout? If we calculate harmonic entropy
using this series, I expect the local minima to still correspond to some
"limit" on Tenney complexity, but this time the height of the minimum should
correspond to the Tenney complexity for each ratio.

I wrote,

>2. Can we construct a Farey-like series that also obeys the partial
ordering
>that Pierre Lamonthe associates with the Stern-Brocot tree, but for which
>this overall slope remains flat throughout? If we calculate harmonic
entropy
>using this series, I expect the local minima to still correspond to some
>"limit" on Tenney complexity, but this time the height of the minimum
should
>correspond to the Tenney complexity for each ratio.

Perhaps the correct answer is the obvious guess: use a series defined by an
upper bound on Tenney complexity. Although my algorithm for calculating this
is very slow, I managed to go with 10 as the maximum complexity (meaning
numerator times denominator is less than or equal to 2^10 = 1024) and
calculate an s=1% curve with it. You can see it at
http://www.egroups.com/files/tuning/perlich/tenney1/t01_10.jpg. There
doesn't appear to be an overall slope, which is nice. I plotted the ratios
of the local minima of this curve (except for the "11/10") in
http://www.egroups.com/files/tuning/perlich/tenney1/tcmp1.jpg. Here you can
see that for the simplest ratios, the relationship between Tenney complexity
and harmonic entropy is amazingly linear. Then, as we hit the finite s and
the insufficient maximum complexity of the input series, the pattern begins
to break down.

So the naive guess is substantiated so far. A more convincing demonstration
would use a higher maximum complexity, a smaller s value (say 0.6%) as well
as an intermediate one, and an overall range much wider than one octave.
Anyone care to tackle the calculations (they seem to be beyond my computer's
resources for now . . .)?

I wrote,

>I plotted the ratios of the local minima of this curve (except for the
"11/10")

I forget to mention that these local minima, except for that one artifact,
are the first 13 ratios to appear in a Tenney complexity ranking of the
intervals in the first octave -- corroborating this from the previous post:

>I expect the local minima to still correspond to some
>"limit" on Tenney complexity

Good night!

>Perhaps the correct answer is the obvious guess: use a series defined by
>an upper bound on Tenney complexity.

That's what I was thinking.

>Although my algorithm for calculating this is very slow, I managed to go
>with 10 as the maximum complexity (meaning numerator times denominator is
>less than or equal to 2^10 = 1024) and calculate an s=1% curve with it.

Great!

>You can see it at: >http://www.egroups.com/files/tuning/perlich/tenney1/t01_10.jpg.

Wow. It's a Stern-Brocot tree starting with 1/1 and 2/1.

>There doesn't appear to be an overall slope, which is nice.

Yup.

>I plotted the ratios of the local minima of this curve (except for the
>"11/10") in http://www.egroups.com/files/tuning/perlich/tenney1/tcmp1.jpg.
>Here you can see that for the simplest ratios, the relationship between
>Tenney complexity and harmonic entropy is amazingly linear.

Zing. So for JI, perhaps the entropy component doesn't really give us
anything over van Eck's model -- maybe the rationals are just that well
distributed on the log-frequency line. It would be nice to try a few
different starting series and see what it takes for the entropy component
to give us a different ordering than the condition used to limit the
series (I never did get an answer on the ordering of Farey-series entropy).
Of course, it's still very natural to think of entropy in terms of the
periodicity model, and for irrational intervals it's wonderful.

*** And now, mediants ***

Paul, I asked you on my first visit what is meant by 'simplicity' when
we say that the mediant of two ratios is the simplest ratio between them.
You said perhaps n*d. Does anybody know? I seem to remember reading
a proof of this property of mediants -- on Cut the Knot? -- link would
be appreciated.

Anywho, take 4/1 and 1/1. Mediant 5/2 seems more complex than 3/1. Ouch!

-Carl

--- In tuning@egroups.com, Carl Lumma <
CLUMMA@N...> wrote:

> Zing. So for JI, perhaps the entropy component doesn't really give
us
> anything over van Eck's model

Huh?

-- maybe the rationals are just that well
> distributed on the log-frequency line.

wha?

It would be nice to try a few
> different starting series and see what it takes for the entropy
component
> to give us a different ordering than the condition used to limit the
> series

They all do, except Tenney and odd-limit.

>(I never did get an answer on the ordering
>of Farey-series entropy).

The answer would start with something like
100/1, 99/1, etc., with ratios like 100/99
beating out ratios like 7/5. Not very useful
information. However, the relevant point is
that whether you start with a Farey series,
or, I suspect, any other series that obeys the
condition

xy -wz = 1

for any two neighboring fractions w/x and y/
z, the local minima will be the set of ratios
below a certain Tenney complexity, with the
exact ordering departing from the Tenney
ordering only insofar as the density of the
series varies over the real number line. In
other words,

Farey in, Tenney out
Mann (sum limit) in, Tenney out
Pepper(n^2+d^2 limit) in, Tenney out
Stern-Brocot layer limit in, Tenney out

That's my conjecture.

> Of course, it's still very natural to think of entropy in terms of
the
> periodicity model, and for irrational intervals it's wonderful.
>
> *** And now, mediants ***
>
> Paul, I asked you on my first visit what is meant by 'simplicity'
when
> we say that the mediant of two ratios is the simplest ratio between
them.
> You said perhaps n*d. Does anybody know?

Carl, for any of the above definitions of
complexity, the mediant

(w+y)/(x+z)

gives you the least complex ratio between
w/x and y/z, IF

xy -wz = 1

>I seem to remember reading
> a proof of this property of mediants -- on Cut the Knot? -- link
would
> be appreciated.

?
>
> Anywho, take 4/1 and 1/1. Mediant 5/2 seems more complex than 3/1.
Ouch!

4/1 and 1/1 are not neighboring fractions in
any series produced by an upper limit on
complexity according to any of the above
definitions of complexity. Proof: 4*1 - 1*1 = 3,
not 1!

>>Zing. So for JI, perhaps the entropy component doesn't really give
>>us anything over van Eck's model
>
>Huh?

Looks like you can just use the widths between mediants to aproximate
harmonic entropy.

>>It would be nice to try a few different starting series and see what it
>>takes for the entropy component to give us a different ordering than the
>>condition used to limit the series
>
>They all do, except Tenney and odd-limit.

Odd-limit? That has density problems, doesn't it?
Ratios near the base-2's are closer together (Partch
added points to even out the 2nds in the diamond).

>>(I never did get an answer on the ordering
>>of Farey-series entropy).
>
>The answer would start with something like
>100/1, 99/1, etc., with ratios like 100/99
>beating out ratios like 7/5. Not very useful
>information.

Okay, but how have you been dealing with this
all along? Just dismissing it as an artifact
of the Farey series? Or is it because you've
only been reporting local minima, and they
never occur exactly on a ratio, and you just
figure they belong to simple ratios nearby?

>However, the relevant point is that whether you
>start with a Farey series, or, I suspect, any
>other series that obeys the condition
>
>xy -wz = 1
>
>for any two neighboring fractions w/x and y/z,
>the local minima will be the set of ratios
>below a certain Tenney complexity, with the
>exact ordering departing from the Tenney
>ordering only insofar as the density of the
>series varies over the real number line. In
>other words,
>
>Farey in, Tenney out
>Mann (sum limit) in, Tenney out
>Pepper(n^2+d^2 limit) in, Tenney out
>Stern-Brocot layer limit in, Tenney out
>
>That's my conjecture.

Damn, that's impressive. I wonder why it is?
And is there any way we can pick which one to
use for the final ordering?

>Carl, for any of the above definitions of
>complexity, the mediant
>
>(w+y)/(x+z)
>
>gives you the least complex ratio between
>w/x and y/z, IF
>
>xy -wz = 1

Thanks! I was missing that they had to be neighboring
fractions. That's nice, but not nearly as strong as I
thought. I wonder if there's a function that finds the
simplest (by some specific definintion) ratio between
any two others?

-Carl

Carl wrote,

>Looks like you can just use the widths between mediants to aproximate
>harmonic entropy.

Hardly.

>Odd-limit? That has density problems, doesn't it?
>Ratios near the base-2's are closer together (Partch
>added points to even out the 2nds in the diamond).

The other series have exactly the same "density problems". Actually those
aren't density problems at all, as I see it. They simply make the "minor
seconds" come out very dissonant, which is good. Anyway I'm extremely happy
with the way the octave-equivalent harmonic entropy curve came out, using
odd-limit (e.g., 223) and a convenient surrogate for mediants, the
"limit-weighted midpoint".

>Okay, but how have you been dealing with this
>all along? Just dismissing it as an artifact
>of the Farey series?

No.

>Or is it because you've
>only been reporting local minima,

Yes.

>and they
>never occur exactly on a ratio, and you just
>figure they belong to simple ratios nearby?

Well, there's no local minimum corresponding to 100/99, etc., which is the
main point. As for the simple ratios, they are typically within 1 cent of
the local minima, and I only use 1 cent resolution in my curves.

>Damn, that's impressive. I wonder why it is?

There's something "true" about n*d as a complexity measure.

>And is there any way we can pick which one to
>use for the final ordering?

The final ordering?

>>Looks like you can just use the widths between mediants to aproximate
>>harmonic entropy.
>
>Hardly.

For the rationals, I should have said. Why hardly? You just said
that it differs only when there's density problems with the initial
series.

>>Odd-limit? That has density problems, doesn't it?
>>Ratios near the base-2's are closer together (Partch
>>added points to even out the 2nds in the diamond).
>
>The other series have exactly the same "density problems".

You said odd-limit and Tenny-limit didn't have density problems,
but that everything else did. No?

>Anyway I'm extremely happy with the way the octave-equivalent harmonic
>entropy curve came out, using odd-limit (e.g., 223) and a convenient
>surrogate for mediants, the "limit-weighted midpoint".

I was impressed with the limit-weighted midpoint, but I don't find
octave-equivalent h.e. very interesting, or grounded in reality. I'd
much rather know the real h.e., and then impose octave-equivalence if
I needed it for convenience, later. Still, it is cool you could get
a one-footed bride out of h.e.

>>and they never occur exactly on a ratio, and you just
>>figure they belong to simple ratios nearby?
>
>Well, there's no local minimum corresponding to 100/99, etc., which is the
>main point. As for the simple ratios, they are typically within 1 cent of
>the local minima, and I only use 1 cent resolution in my curves.

Thanks for the confirmation -- and sorry, I wasn't accusing you of fudging
figures, or anything.

>>And is there any way we can pick which one to
>>use for the final ordering?
>
>The final ordering?

Well, you said all starting series would give local minima by Tenny limit,
just that the relative strengths of the minima would be different. So,
I was asking if there's any way of picking which starting series to "use"
to "get the final ordering". I agree that there's something "true" about
Tenney complexity, but I'd like a psychoacoustical reason, and Farey
complexity has potential in that area...

-Carl

>Uh huh. Why is it that I suspect you WANT the problem to be hard?

You're really doing Paul an injustice here, John.

>Don't get me wrong: I love a good challenge I can sink my teeth into.
>But, what are you going to do for 5-note chords? We're talking 4-D
>space now: you gonna Voronoi THAT? I don't think so!

Why not? Besides, I think there are much more efficient methods that
will give the same results as voronoi cells, just floating around out
there, waiting to be found.

>It strikes me that the method I propose, with suitable ratioing down
>of the effective volume of the difference notes, would yield interesting
>and possibly useful results. If I get time, I'll program it up and post
>some values.

Sure, it would be interesting. But we've already agreed that combination
tones cannot account for the effects we're attributing to the periodicity
mechanism. Paul's trying to model a very important, and long over-looked
part of the hearing, and I'm afraid combination tones just aren't on the
same boat. To be considered properly, they should be added as notes of
the chord under consideration, wether we're using the sensory dissonance
model or harmonic entropy, and weighted by their (usually very low)
amplitudes.

-Carl

Carl wrote,

>>>Looks like you can just use the widths between mediants to aproximate
>>>harmonic entropy.

I wrote,

>>Hardly.

Carl wrote,

>For the rationals, I should have said. Why hardly? You just said
>that it differs only when there's density problems with the initial
>series.

Well even accounting for that there's only an approximately monotone
function between mediant-to-mediant width and h.e. for the ratios which show
up as local minima. For others, forget about it.

>>>Odd-limit? That has density problems, doesn't it?
>>>Ratios near the base-2's are closer together (Partch
>>>added points to even out the 2nds in the diamond).
>
>>The other series have exactly the same "density problems".

>You said odd-limit and Tenny-limit didn't have density problems,
>but that everything else did. No?

Right -- that's why I put "density problems" in quotes, to emphasize that
I'm using _your_ definition of "density problems", quoted above.

>>Anyway I'm extremely happy with the way the octave-equivalent harmonic
>>entropy curve came out, using odd-limit (e.g., 223) and a convenient
>>surrogate for mediants, the "limit-weighted midpoint".

Plain old midpoints give virtually the same results, which is promising for
the idea of using Voronoi cells in higher dimensions.

>I was impressed with the limit-weighted midpoint, but I don't find
>octave-equivalent h.e. very interesting, or grounded in reality. I'd
>much rather know the real h.e., and then impose octave-equivalence if
>I needed it for convenience, later.

But how would you do that, fairly?

>Still, it is cool you could get
>a one-footed bride out of h.e.

I promised I would in the original collection of posts that Monz catalogued,
and though it was just a hunch, I was very glad it come up so cleanly.

>Well, you said all starting series would give local minima by Tenny limit,
>just that the relative strengths of the minima would be different. So,
>I was asking if there's any way of picking which starting series to "use"
>to "get the final ordering". I agree that there's something "true" about
>Tenney complexity, but I'd like a psychoacoustical reason, and Farey
>complexity has potential in that area...

The Farey-based harmonic entropy curve certainly seems a bit too lenient on
minor ninths, for example. A psychoacoustical reason? Well, I'd prefer to
think of it this way: Dave Keenan came up with a threefold criterion for
dissonance: complexity, tolerance, and range. We want to come up with a
complexity measure that is independent of range. Odd-limit is one, but is
only realistic when octave-equivalence is imposed on the problem. Otherwise,
what do you use? Some increasing function of n and d . . . Any other ideas?

In preparation for redoing Joseph's Tuning Lab experiment, I used the ratios
with n*d<10000 to seed a first-two-octaves harmonic entropy calculation,
with s=1%. The resulting harmonic entropy curve can be seen at
http://www.egroups.com/files/tuning/perlich/tenney1/t01_13p2877.jpg, in a
form suitable for use on mp3.com. Let's look again at the Tenney ranking,
with columns added for the results of this calculation:

Ratio Product Graph Entropy
1/1 1 loc. min. 2.2152
2/1 2 loc. min. 3.0576
3/1 3 loc. min. 3.4455
4/1 4 loc. min. 3.6805
3/2 6 loc. min. 3.9415
5/2 10 loc. min. 4.1779
4/3 12 loc. min. 4.2347
7/2 14 loc. min. 4.2965
5/3 15 loc. min. 4.3016
5/4 20 loc. min. 4.3769
7/3 21 loc. min. 4.3879
8/3 24 loc. min. 4.4049
7/4 28 loc. min. 4.4346
10/3 30 loc. min. 4.4312
6/5 30 loc. min. 4.4464
11/3 33 loc. min. 4.4445
7/5 35 loc. min. 4.4664
9/4 36 loc. min. 4.4588
8/5 40 loc. min. 4.4863
7/6 42 loc. min. 4.4763
11/4 44 loc. min. 4.4783
9/5 45 loc. min. 4.4908
13/4 52 loc. min. 4.4952
11/5 55 nothing 4.5053
8/7 56 locmin shifto 9/8 4.5016
15/4 60 nothing 4.5104
12/5 60 loc. min. 4.5111
9/7 63 loc. min. 4.5271
13/5 65 nothing 4.5302
11/6 66 locmin shifto 13/7 4.5309
no more local minima, all products 70 or higher.

The ones that did not get a local minumum but have a product below 70 are
very close to the octave or double octave, so have a steep local slope to
contend with (using a product limit higher than 10000 might have helped
too). Their entropy values are pretty much in line with the Tenney Harmonic
Distance. But the curve has absolutely no downward or upward trend even over
two octaves, so there won't be any bias to the tetrad results -- which I'll
post shortly.

>>>>Odd-limit? That has density problems, doesn't it?
>>>>Ratios near the base-2's are closer together (Partch
>>>>added points to even out the 2nds in the diamond).
>>
>>>The other series have exactly the same "density problems".
>
>>You said odd-limit and Tenny-limit didn't have density problems,
>>but that everything else did. No?
>
>Right -- that's why I put "density problems" in quotes, to emphasize that
>I'm using _your_ definition of "density problems", quoted above.

So you're saying that I'm using a wrong definition of density problems?
Why didn't you just say so? Incidentally, I don't see how the Tenney
series has the problem I mentioned.

>>I was impressed with the limit-weighted midpoint, but I don't find
>>octave-equivalent h.e. very interesting, or grounded in reality. I'd
>>much rather know the real h.e., and then impose octave-equivalence if
>>I needed it for convenience, later.
>
>But how would you do that, fairly?

With odd-limit.

>We want to come up with a complexity measure that is independent of range.
>Odd-limit is one, but is only realistic when octave-equivalence is imposed
>on the problem. Otherwise, what do you use? Some increasing function of n
>and d . . . Any other ideas?

Increasing function?? What about a numerator limit for proper fractions?

And, if the Tenney series has problems with range (and I'm pretty sure it
does), then why did you say earlier that the Tenney series doesn't have
density problems? Or am I still not getting what you originally meant by
the term (not crowding in the middle of the octave _or_ denisty changes
over range, apparently).

-Carl

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13159

The new entropy graph will work quite nicely, I believe! Thanks,
Paul. The experiment will be particularly interesting when the
tetrads are arranged in some kind of "sonic" order... I'm looking
forward to the developing results!

Thanks again!
____________ ____ __ __
Joseph Pehrson

Carl wrote,

>Incidentally, I don't see how the Tenney
>series has the problem I mentioned.

Why not?

>>>I was impressed with the limit-weighted midpoint, but I don't find
>>>octave-equivalent h.e. very interesting, or grounded in reality. I'd
>>>much rather know the real h.e., and then impose octave-equivalence if
>>>I needed it for convenience, later.
>
>>But how would you do that, fairly?

>With odd-limit.

Could you, eh, elaborate?

>>We want to come up with a complexity measure that is independent of range.
>>Odd-limit is one, but is only realistic when octave-equivalence is imposed
>>on the problem. Otherwise, what do you use? Some increasing function of n
>>and d . . . Any other ideas?

>Increasing function?? What about a numerator limit for proper fractions?

Isn't that the same as the Farey series?

>And, if the Tenney series has problems with range (and I'm pretty sure it
>does), then why did you say earlier that the Tenney series doesn't have
>density problems? Or am I still not getting what you originally meant by
>the term (not crowding in the middle of the octave _or_ denisty changes
>over range, apparently).

I meant density changes over range, and why do you think the Tenney series
has those problems? The Tenney-based entropy curves, even over two octaves,
have no disceranble trend.

[I wrote:]
>>Uh huh. Why is it that I suspect you WANT the problem to be hard?

[Carl Lumma:]
>You're really doing Paul an injustice here, John.

Maybe I am; sorry, Paul.

[JdL:]
>>Don't get me wrong: I love a good challenge I can sink my teeth into.
>>But, what are you going to do for 5-note chords? We're talking 4-D
>>space now: you gonna Voronoi THAT? I don't think so!

[Carl:]
>Why not? Besides, I think there are much more efficient methods that
>will give the same results as voronoi cells, just floating around out
>there, waiting to be found.

Maybe. And it's true that it's often easier to model multi-dimensional
things than to VISUALIZE them.

[JdL:]
>>It strikes me that the method I propose, with suitable ratioing down
>>of the effective volume of the difference notes, would yield
>>interesting and possibly useful results. If I get time, I'll program
>>it up and post some values.

[Carl:]
>Sure, it would be interesting. But we've already agreed that
>combination tones cannot account for the effects we're attributing to
>the periodicity mechanism. Paul's trying to model a very important,
>and long over-looked part of the hearing, and I'm afraid combination
>tones just aren't on the same boat.

Well you know, I'm not getting a Gestalt on what you and Paul are up to,
which most likely reflects a lack in my understanding. It doesn't look
like I'll have the time to rectify this any time soon, so go for it,
guys! Maybe it'll all be clear to me down the road sometime.
Meanwhile, there are things that make sense to me that I'll probably
spend the bulk of my time pursuing.

[Carl:]
>To be considered properly, they should be added as notes of the chord
>under consideration, wether we're using the sensory dissonance model or
>harmonic entropy, and weighted by their (usually very low) amplitudes.

All of which I would do, as I said.

JdL

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13191

Hi Paul...

I know we've been "forbidden" to speak of H.E. on this list... but
could you please, in layman's terms explain the general difference
between the Farey series method of H.E. evaluation and the Tenney
method? I wouldn't have posted about H.E. again, except that I think
there are probably a couple of other people who might benefit by
understanding this part of H.E.

_____________ _____ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph
Pehrson" <pehrson@p...> wrote:
> --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/13191
>
> Hi Paul...
>
> I know we've been "forbidden" to speak of H.E. on this list... but
> could you please, in layman's terms explain the general difference
> between the Farey series method of H.E. evaluation and the Tenney
> method? I wouldn't have posted about H.E. again, except that I
think
> there are probably a couple of other people who might benefit by
> understanding this part of H.E.

The Farey series is the set of all ratios of two
numbers (in lowest terms) where the larger
number is less than some limit (I often use
100). The "Tenney series" is the set of all
ratios of two numbers (in lowest terms)
where the product of the two numbers is
less than some limit (I've been using 10000
here). After the series in put in place, the
harmonic entropy calculation proceeds the
same way -- a bell curve around the true
interval is sliced among the ratios in the
series, and the entropy function is
calculated over the areas of the slices.

The two methods yield very similar curves,
with virtually the same local minima for a
given choice of s (width of the bell curve),
except that the Farey series results in a
curve with a steady downward slope as one
moves toward wider intervals, while the
Tenney series seems to result in a curve
with no inherent bias toward or against
wider intervals. The reason for this is that a
Farey series become less and less dense
(density evaluated logarithmically, i.e., per
octave or per semitone) as one moves to
wider and wider intervals, while the "Tenney
series" shows no such trend, at least over
the first two octave.

Interestingly, the ratios that show local
minima (and bumps that suggest local
minima would "grow" with a slightly smaller
s value) are neatly characterized by a
Tenney limit (about 67 for s=1%, about 108 for
s=0.6%) _regardless_ of whether a Farey or
Tenney series is used to "seed" the
calculation. So there seems to be some
universality to the Tenney complexity, or
product of numerator and denominator, as a
measure of diadic discordance.

>>Incidentally, I don't see how the Tenney
>>series has the problem I mentioned.
>
>Why not?

Is it easier to describe something you see than to describe everything
you can't see?

>>>>I was impressed with the limit-weighted midpoint, but I don't find
>>>>octave-equivalent h.e. very interesting, or grounded in reality. I'd
>>>>much rather know the real h.e., and then impose octave-equivalence if
>>>>I needed it for convenience, later.
>>
>>>But how would you do that, fairly?
>
>>With odd-limit.
>
>Could you, eh, elaborate?

I think of octave-equivalent formulations as being very "expensive"
generalizations. I guess I just don't see how the ability to measure
irrational intervals in an octave-equivalent way could be useful to
me. Give me an example of how you think it'd be useful, and I'll
see if I can do without it.

>>Increasing function?? What about a numerator limit for proper fractions?
>
>Isn't that the same as the Farey series?

Whoops! Got the meaning of "proper fraction" wrong. What I said is
the same as the Farey series. What I meant was a numerator limit for
fractions where the numerator is greater than the denominator.

>>And, if the Tenney series has problems with range (and I'm pretty sure it
>>does), then why did you say earlier that the Tenney series doesn't have
>>density problems? Or am I still not getting what you originally meant by
>>the term (not crowding in the middle of the octave _or_ denisty changes
>>over range, apparently).
>
>I meant density changes over range, and why do you think the Tenney series
>has those problems? The Tenney-based entropy curves, even over two octaves,
>have no disceranble trend.

Okay. So maybe I'm wrong. It just seems like, with Tenney order 35, say,
you'd get 35/1 and 7/5 -- just seemed like there'd be more ways to factor
a number into two terms when the difference between the terms was smaller.
Why I felt so sure at 4am this morning, I can't say.

-Carl

>Well you know, I'm not getting a Gestalt on what you and Paul are up to,
>which most likely reflects a lack in my understanding. It doesn't look
>like I'll have the time to rectify this any time soon, so go for it,
>guys! Maybe it'll all be clear to me down the road sometime.
>Meanwhile, there are things that make sense to me that I'll probably
>spend the bulk of my time pursuing.

John,

Why don't you read Paul's original post on the subject, from 1997? Last
I checked Monz still had it archived on his site somewhere...

-Carl

>Well you know, I'm not getting a Gestalt on what you and Paul are up to,
>which most likely reflects a lack in my understanding.

John-

BTW, I'm not up to sh__. I'm just hanging on for dear life.

-Carl

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, "Joseph

http://www.egroups.com/message/tuning/13227

Hi Paul...

Thanks for this help...

>
> The two methods yield very similar curves,
> with virtually the same local minima for a
> given choice of s (width of the bell curve),
> except that the Farey series results in a
> curve with a steady downward slope as one
> moves toward wider intervals, while the
> Tenney series seems to result in a curve
> with no inherent bias toward or against
> wider intervals. The reason for this is that a
> Farey series become less and less dense
> (density evaluated logarithmically, i.e., per
> octave or per semitone) as one moves to
> wider and wider intervals, while the "Tenney
> series" shows no such trend, at least over
> the first two octave.
>

So then is it because of the MULTIPLICATION that there is no bias
toward a certain kind of interval? Is it at all like multiplying
ratios... in that there is a logarithmic progression that keeps
things
consistent rather than an "arithmetic" one??

> Interestingly, the ratios that show local
> minima (and bumps that suggest local
> minima would "grow" with a slightly smaller
> s value) are neatly characterized by a
> Tenney limit (about 67 for s=1%, about 108 for
> s=0.6%) _regardless_ of whether a Farey or
> Tenney series is used to "seed" the
> calculation. So there seems to be some
> universality to the Tenney complexity, or
> product of numerator and denominator, as a
> measure of diadic discordance.

So again, this MULTIPLICATION method creates more consistent results
for evaluating degree of concordance/discordance... (??)

_______ ___ __ ____
Joseph Pehrson

[I wrote:]
>>Well you know, I'm not getting a Gestalt on what you and Paul are up
>>to, which most likely reflects a lack in my understanding. It doesn't
>>look like I'll have the time to rectify this any time soon, so go for
>>it, guys! Maybe it'll all be clear to me down the road sometime.
>>Meanwhile, there are things that make sense to me that I'll probably
>>spend the bulk of my time pursuing.

[Carl Lumma:]
>Why don't you read Paul's original post on the subject, from 1997?
>Last I checked Monz still had it archived on his site somewhere...

You're talking about the ones on harmonic entropy, at

http://www.ixpres.com/interval/td/entropy.htm

http://www.ixpres.com/interval/td/erlich/entropy.htm

Yeah, I've read'm! And I'm very interested in getting ahold of a
continuous function, such as this, for concordance and discordance; it
is a more sophisticated model than I'm currently using in my adaptive
methods. I don't always agree with the shape of the curve, but most of
the time it's pretty close to what I'd qualitatively pick (I've quibbled
about 9/7 vs. 11/9, and a few other things, recently).

But, as for the rationale for going into N-dimensional space for chords,
well, that escapes me. I take it as near given that you and Paul are
chasing something real, not "pentasystem", but I still don't make the
connection.

JdL

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> >>Incidentally, I don't see how the Tenney
> >>series has the problem I mentioned.
> >
> >Why not?
>
> Is it easier to describe something you see than to describe
everything
> you can't see?

uhh...doesn't the Tenney series have the same "unevenness" that you
noted for the odd-
limit series?
>
> >>>>I was impressed with the limit-weighted midpoint, but I don't
find
> >>>>octave-equivalent h.e. very interesting, or grounded in
reality. I'd
> >>>>much rather know the real h.e., and then impose
octave-equivalence if
> >>>>I needed it for convenience, later.
> >>
> >>>But how would you do that, fairly?
> >
> >>With odd-limit.
> >
> >Could you, eh, elaborate?
>
> I think of octave-equivalent formulations as being very "expensive"
> generalizations. I guess I just don't see how the ability to
measure
> irrational intervals in an octave-equivalent way could be useful to
> me. Give me an example of how you think it'd be useful, and I'll
> see if I can do without it.

Evaluating ETs, or finding optimal pentatonic scales in terms of
octave-invariant pitch
classes.
>
> >>Increasing function?? What about a numerator limit for proper
fractions?
> >
> >Isn't that the same as the Farey series?
>
> Whoops! Got the meaning of "proper fraction" wrong. What I said is
> the same as the Farey series. What I meant was a numerator limit
for
> fractions where the numerator is greater than the denominator.

That's still the Farey series. Maybe you mean denominator greater
than numerator? That
would become more dense the higher you went up, since
pitch space is
logarithmic. But I'm pretty sure it would yield the same
(Tenney-bounded) local minima.
Want me to check?

--- In tuning@egroups.com, "Joseph Pehrson" <josephpehrson@c...>
wrote:

>
> So then is it because of the MULTIPLICATION that there is no bias
> toward a certain kind of interval?

No bias toward large or small intervals.

> Is it at all like multiplying
> ratios... in that there is a logarithmic progression that keeps
> things
> consistent rather than an "arithmetic" one??

There may be something to that . . . I'll have to think about it.
>
> So again, this MULTIPLICATION method creates more consistent
results
> for evaluating degree of concordance/discordance... (??)

So it seems . . . I know Pierre Lamonthe has more to say on this
subject.

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:
>(I've quibbled
> about 9/7 vs. 11/9, and a few other things, recently).

I thought I'd satisfied you on that tack.
>
> But, as for the rationale for going into N-dimensional space for
chords,
> well, that escapes me. I take it as near given that you and Paul
are
> chasing something real, not "pentasystem", but I still don't make
the
> connection.

You yourself observed that you like 4:5:6:7 but don't like
1/7:1/6:1/5:1/4. If you stop at
diadic harmonic entropy, you'll never explain this. Chordal harmonic
entropy clearly will!

[I wrote:]
>>(I've quibbled about 9/7 vs. 11/9, and a few other things, recently).

[Paul E:]
>I thought I'd satisfied you on that tack.

Yes, in the more recent octave-invariant version. Sorry I didn't make
that clear.

[JdL:]
>> But, as for the rationale for going into N-dimensional space for
>>chords, well, that escapes me. I take it as near given that you and
>>Paul are chasing something real, not "pentasystem", but I still don't
>>make the connection.

[Paul:]
>You yourself observed that you like 4:5:6:7 but don't like
>1/7:1/6:1/5:1/4. If you stop at
>diadic harmonic entropy, you'll never explain this. Chordal harmonic
>entropy clearly will!

So will the method I proposed, which you called "cheap". My "cheap"
method makes sense to me, so I'm going to pursue it. (not saying
what you're after isn't worthwhile).

JdL

>>Why don't you read Paul's original post on the subject, from 1997?
>>Last I checked Monz still had it archived on his site somewhere...
>
>You're talking about the ones on harmonic entropy, at
>
> http://www.ixpres.com/interval/td/entropy.htm

Yup!

>But, as for the rationale for going into N-dimensional space for chords,
>well, that escapes me. I take it as near given that you and Paul are
>chasing something real, not "pentasystem", but I still don't make the
>connection.

No, it's not a pentasystem. :) Simply, I know of no reason why higher
dimensions are needed; it's just the only way we can think of right now.
One of my first trys on the problem was to map triadic space to 1-D -- but
it didn't work. Which isn't to say that it can't be done, of course.

In fact, let's see... There ought to be enough space on the real number
line to hold triples of natural numbers, as there's enough room for doubles
(the rationals), and going from ^2 to ^3 is fairly weak. If we assume
the continuum hypothesis, then the cardinality of the triads is the same as
the dyads and the natural numbers.

The idea behind harmonic entropy is that we try to do fourier transforms
on the complex sounds we hear, but that for whatever reason, our data
isn't perfect, and thus there are multiple possible transforms, and the
ambiguity as to which one is the best fit can be measured -- is measured,
and experienced as a type of discordance.

The trick is to figure out how much of whatever space you're working in
belongs to a given chord. We use mediants to section off the number line
around dyads. The problem with 1-D for triads is that there doesn't seem
to be a single "simplest" triad "between" two given triads. For example,
what's the simplest triad "between" 3:4:5 and 4:5:6? There doesn't seem
to be _one_, there seems to be many, implying higher-d (as on Paul's
voronoi plots).

-Carl

>>Is it easier to describe something you see than to describe
>>everything you can't see?
>
>uhh...doesn't the Tenney series have the same "unevenness"
>that you noted for the odd-limit series?

Sorry for the cryptic bit, I was just asking for an example.
Of course, I _could_ do that myself. . .

>>I think of octave-equivalent formulations as being very "expensive"
>>generalizations. I guess I just don't see how the ability to
>>measure irrational intervals in an octave-equivalent way could be
>>useful to me. Give me an example of how you think it'd be useful,
>>and I'll see if I can do without it.
>
>Evaluating ETs, or finding optimal pentatonic scales in terms of
>octave-invariant pitch classes.

Hrm. I think I'd rather just do that on one octave, and then just
expect a slight difference in actual use. Sounds like a cop-out,
maybe, but I'm suspicious of using octave-equivalent h.e. in this
way -- I'm not sure we're not using more accuracy than we have, know
what I mean? My fears may well be dispelled as I learn more about
octave-equivalent h.e.

>>>>Increasing function?? What about a numerator limit for proper
>>>>fractions?
>>>
>>>Isn't that the same as the Farey series?
>>
>>Whoops! Got the meaning of "proper fraction" wrong. What I said is
>>the same as the Farey series. What I meant was a numerator limit for
>>fractions where the numerator is greater than the denominator.
>
>That's still the Farey series. Maybe you mean denominator greater
>than numerator?

Do'h! Yes.

>That would become more dense the higher you went up, since pitch space
>is logarithmic. But I'm pretty sure it would yield the same (Tenney-bounded)
>local minima. Want me to check?

Actually, I'm beginning to believe in Tenney limit. If it wouldn't take
you long, check. But don't spend an hour.

-Carl

The Tenney value (n*d) of a dyad is simply the distance between the
fundamental and the guide tone, no? So in other words, it's how many GCFs of
the two wavelengths fit into one of their LCMs.

Gotta go,
Keenan P., stating what should be obvious...

Keenan,

Since you're walking in the way of numerical evidence, I propose some
subjects which could possibly interest you.

You wrote :

<< The Tenney value (n*d) of a dyad is simply the distance
between the fundamental and the guide tone, no? So in other
words, it's how many GCFs of the two wavelengths fit into
one of their LCMs. >>

[Look first]

Blending of subharmonics with square, sawtooth and triangular waves.

http://www.aei.ca/~plamothe/pix/sharmoniques.gif

You surely spot LCM. How to characterize subharmonic blending in comparison
of harmonic superposition ? Have only eyes possibility to detect such order
? Of course ears don't perceive phase, and here, appropriate choice of
amplitudes helps eyes to detect order. How it would be with symmetric chord
like (3 5 9 15) (1 3 5 15) (1 3 7 21) ?

[On Tenny value]

Let Q be the set of all dyads (n,d) corresponding to reduced fractions n/d
represented in the real plane.

Let consider two transformations to applied :

L : logarithm in base 2
R : 45 degree rotation around (1,1)

How act the inversible transformation R ?

R(n,d) = (n/d, n*d)

We have equivalent dyad (r,c) of the ratio and its complexity (Tenny value).

How act the inversible logarithmic transformation L ?

LR(n,d) = (Lr, Lc) = (Ln-Ld, Ln+Ld)

We have equivalent dyad (w,s) of the width and the sonance of this interval.

Let consider families of parallel straight lines joining intervals.

How much families of such parallels have parallelism invariance under the
logarithmic transformation ? How could you name these families of invariant
alignement ? Do you perceive yet GCFs and LCMs ?

Hoping you'll find there matter to think.

Pierre Lamothe

--- In tuning@egroups.com, Pierre Lamothe <plamothe@a...> wrote:
>

http://www.egroups.com/message/tuning/13317

>
> Blending of subharmonics with square, sawtooth and triangular
waves.
>
> http://www.aei.ca/~plamothe/pix/sharmoniques.gif
>
> You surely spot LCM. How to characterize subharmonic blending in
comparison
> of harmonic superposition ? Have only eyes possibility to detect
such order

This was EXTREMELY interesting... both the diagram and the
explanation...
___________ ____ __ __
Joseph Pehrson

--- In tuning@egroups.com, "John A. deLaubenfels" <
jdl@a...> wrote:
> [I wrote:]
> >>(I've quibbled about 9/7 vs. 11/9, and a few other things,
recently).
>
> [Paul E:]
> >I thought I'd satisfied you on that tack.
>
> Yes, in the more recent octave-invariant version. Sorry I didn't
make
> that clear.

It was the s=1.5%, rather than the octave invariance,
that made 9/7 come out more discordant than 11/9. If I
used Tenney limit instead of odd limit it would still
come out that way. But the other feature that bothered
you, that while 4:3 is more sensitive than 5:4 to
mistuning, 5:2 is more sensitive than 8:3, could only be
eliminated through octave invariance. Have you
reconsidered that second objection?

> So will the method I proposed, which you called "cheap". My "cheap"
> method makes sense to me, so I'm going to pursue it. (not saying
> what you're after isn't worthwhile).

And I'm not saying it won't be worthwhile for you to
pursue that method. I just would be wary of attaching
too much importance to the first-order difference
tones of the fundamentals relative to the other
combination tones of the fundamentals, as well as
combination tones formed between partials.

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...>
wrote:
>
> In fact, let's see... There ought to be enough space on the real
number
> line to hold triples of natural numbers, as there's enough room for
doubles
> (the rationals), and going from ^2 to ^3 is fairly weak. If we
assume
> the continuum hypothesis, then the cardinality of the triads is the
same as
> the dyads and the natural numbers.

You don't need to assume the continuum hypothesis.
But that's irrelevant -- what's relevant is that there are
two degrees of freedom in tuning a triad, therefore two
dimensions are needed to represent triadic space.
>
> The idea behind harmonic entropy is that we try to do fourier
transforms
> on the complex sounds we hear,

Not fourier transforms -- the cochlea does that -- but
periodicity detection and/or root finding. Everything
else in your post is right, though.

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...>
wrote:

> but I'm suspicious of using octave-equivalent h.e. in this
> way -- I'm not sure we're not using more accuracy than we have, know
> what I mean?

Not really . . .

> My fears may well be dispelled as I learn more about
> octave-equivalent h.e.

Notice that the 99-odd limit curve looks exactly like the
100-Farey limit curve near the unison (not a
coincidence).
>
> Actually, I'm beginning to believe in Tenney limit. If it wouldn't
take
> you long, check. But don't spend an hour.

Well, I'll probably let my computer spend an hour, if
that's what it takes -- no sweat off my back.

--- In tuning@egroups.com, "Keenan Pepper" <
mtpepper@p...> wrote:
> The Tenney value (n*d) of a dyad is simply the distance between the
> fundamental and the guide tone, no? So in other words, it's how
many GCFs of
> the two wavelengths fit into one of their LCMs.
>
> Gotta go,
> Keenan P., stating what should be obvious...

Yes Keenan, insightful as always. But the burning
question is, what is the generalization to triads and
tetrads? For both 4:5:6 and 10:12:15, the ratio between
the guide tone and fundamental, or LCM/GCF, is 60. For
4:5:6:7 and 1/7:1/6:1/5:1/4, it's 420. But are these pairs of
chords equally concordant?

My first response to this was "I think this would be very interesting if I
could understand it at all," but after careful scrutiny, I think I might be
getting it.

"Let Q be the set of all dyads (n,d) corresponding to reduced fractions n/d
represented in the real plane."

Another name for these is "prime nodes", correct?

"Let consider two transformations to applied :
L : logarithm in base 2
R : 45 degree rotation around (1,1)
How act the inversible transformation R ?
R(n,d) = (n/d, n*d)
We have equivalent dyad (r,c) of the ratio and its complexity (Tenny
value)."

I don't understand. (3,2) rotated 45 degrees about the point (1,1) is
(sqrt(2)/2+1,3*sqrt(2)/2+1), not (3/2,6). Have I misunderstood or did you
mean a different rotation/translation/mapping?

"How act the inversible logarithmic transformation L ?
LR(n,d) = (Lr, Lc) = (Ln-Ld, Ln+Ld)
We have equivalent dyad (w,s) of the width and the sonance of this
interval."

If you found a mapping that would take (n,d) to (n/d, n*d), this would work.
By "sonance" you mean of course the distance between the fundamental and the
guide tone measured in octaves.

"Let consider families of parallel straight lines joining intervals.
How much families of such parallels have parallelism invariance under the
logarithmic transformation ? How could you name these families of invariant
alignement ? Do you perceive yet GCFs and LCMs ?"

I might be able to answer this once we get the R(n,d) transformation figured
out.

Keenan,

I apologize for errors and thank you for alert scrutiny. It's ten years old
stuff inattentively written too late in the night.

You wrote :

<< Another name for these is "prime nodes", correct? >>

I suppose, but I'm not familiar with English terminology. What I can say is
that concerns relatively primes.

You wrote :

<< I don't understand. (3,2) rotated 45 degrees about the point (1,1) is
(sqrt(2)/2+1,3*sqrt(2)/2+1), not (3/2,6). Have I misunderstood or did
you mean a different rotation/translation/mapping? >>

There were several errors. Mainly I had reversed transformations L and R.
Rather than to detail this I would like start afresh. I'm not mathematician
and I don't like formalism that is not very clean. So it appears now like
that.

We'll use following symbolism :

(n,d) == (numerator, denominator)
(w,s) == (width, sonance)
(r,c) == (ratio, complexity) // complexity == Tenny

Let Q be the set of all dyads (n,d) corresponding to reduced fractions n/d
represented in the real plane.

Let us consider three transformations to applied.

L : logarithm in base 2
E : reverse of L // E(x) = 2^x and EL(x) = x
R : matrix ( 1 -1 )
( 1 1 ) // R(x,y) = (x-y, x+y)

How R transformation act ? With application on axes it becomes obvious.

R(0,0) = (0,0)
R(x,0) = (x,x)
R(0,y) = (-y,y)

There are : 45 degrees rotation around (0,0) // more significant aspect
stretching by factor sqrt(2) // what I had forgotten

By definition, (w,s) = RL(n,d) // log then rotation
(r,c) = ERL(n,d) // ... then exp

Thus

(w,s) = RL(n,d) = R(Ln,Ld) = (Ln-Ld, Ln+Ld)
(r,c) = ERL(n,d) = E(Ln-Ld, Ln+Ld) = E(L(n/d),L(n*d)) = (n/d, n*d)

So I repeat about parallelism invariance :

Let us consider families of parallel straight lines joining intervals.
How much families of such parallels have parallelism invariance under all
transformations ? How could you name these families of invariant alignement ?

You wrote :

<< By "sonance" you mean of course the distance between the
fundamental and the guide tone measured in octaves. >>

As such it's exact. But I would like to add a remark. In concrete
acoustical context, interval is defined as "distance" between two tones.
Set of intervals can be defined more abstractly in relation with internal
composition law and axioms applied. An interval is not fondamentally the
sensible correlate of an acoustical parameter like pitch with frequency.
It's not a "sensible" but an "intelligible". Pitch (Hz) can't be a
composable element in algebraic structure. With pitch, A+B or A*B have no
sense, but interval composed with interval give a compound interval. A deep
understanding of musical fonction of interval and macrotonal coherence
require cutting with sound parametrization. Interval act as sign in musical
communication and invariant properties have to be distinguished of
sensative one to modelize communication.

Hoping it's now error free.

Pierre Lamothe

[Paul E wrote:]
>>>You yourself observed that you like 4:5:6:7 but don't like
>>>1/7:1/6:1/5:1/4. If you stop at
>>>diadic harmonic entropy, you'll never explain this. Chordal harmonic
>>>entropy clearly will!

[I wrote:]
>>So will the method I proposed, which you called "cheap". My "cheap"
>>method makes sense to me, so I'm going to pursue it. (not saying
>>what you're after isn't worthwhile).

[Paul:]
>And I'm not saying it won't be worthwhile for you to
>pursue that method. I just would be wary of attaching
>too much importance to the first-order difference
>tones of the fundamentals relative to the other
>combination tones of the fundamentals, as well as
>combination tones formed between partials.

"Wary". On the one hand, I'm always "wary" that some technique that
I add to my methods might make the tuning worse rather than better.
But, perhaps unlike you, Paul, I expect to work by successive
refinement, rather than arriving after long study at some "perfect"
procedure. The reality of dynamic tuning is already on my desk, in
my ears; it pleases me greatly even in its imperfect state. There are
at least half a dozen things I'm aware of that are "wrong" with my
program; shall I therefore stop running it and listen to 12-tET music
till there is nothing more to do? That's not going to happen! Thanks
for the warning, though...

JdL

>>In fact, let's see... There ought to be enough space on the real
>>number line to hold triples of natural numbers, as there's enough
>>room for doubles (the rationals), and going from ^2 to ^3 is fairly
>>weak. If we assume the continuum hypothesis, then the cardinality
>>of the triads is the same as the dyads and the natural numbers.
>
>You don't need to assume the continuum hypothesis.

But then I would need to show a method for pairing triples with the
natural numbers. Or would I need that anyway?

>But that's irrelevant -- what's relevant is that there are
>two degrees of freedom in tuning a triad, therefore two
>dimensions are needed to represent triadic space.

Good point. While we're on it, could you explain again why you
want the axes at 60 degrees instead of 90?

>>The idea behind harmonic entropy is that we try to do fourier
>>transforms on the complex sounds we hear,
>
>Not fourier transforms -- the cochlea does that

It decomposes complex waves, but not necessarily into harmonics,
so that's not really true, but...

> -- but periodicity detection and/or root finding.

...I admit to an abuse of terminology, since the periodicity
mechanism doesn't actually decompose anything.

Thanks, as always, for reading.

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...>
wrote:
> >>In fact, let's see... There ought to be enough space on the real
> >>number line to hold triples of natural numbers, as there's enough
> >>room for doubles (the rationals), and going from ^2 to ^3 is
fairly
> >>weak. If we assume the continuum hypothesis, then the cardinality
> >>of the triads is the same as the dyads and the natural numbers.
> >
> >You don't need to assume the continuum hypothesis.
>
> But then I would need to show a method for pairing triples with the
> natural numbers. Or would I need that anyway?

We're way off topic now, right? But anyway, the method
for pairing triples with the natural numbers is the same
as the method for pairing doubles with the natural
numbers -- just map the doubles to the natural
numbers, and construct new doubles which consist of
one natural number which is a remapped double, and
another natural number which is just "itself" -- now
map these new doubles, which are really triples, to the
natural numbers the same way you mapped doubles
before.
>
> >But that's irrelevant -- what's relevant is that there are
> >two degrees of freedom in tuning a triad, therefore two
> >dimensions are needed to represent triadic space.
>
> Good point. While we're on it, could you explain again why you
> want the axes at 60 degrees instead of 90?

Because all three intervals in the triad should be on the
same distance scale. The diagram should simply rotate,
and not distort at all, if you redefine the axes to be the
outer interval and the lower interval, or the outer
interval and the upper interval, rather than the lower
interval and the upper interval. John Chalmers knows
this trick better than me -- see his triangular
tetrachordal diagrams (think of these as showing
triads within a fourth instead of triads within an
octave).
>
> >>The idea behind harmonic entropy is that we try to do fourier
> >>transforms on the complex sounds we hear,
> >
> >Not fourier transforms -- the cochlea does that
>
> It decomposes complex waves, but not necessarily into harmonics,
> so that's not really true, but...

Not necessarily into harmonics?

>> Good point. While we're on it, could you explain again why you
>> want the axes at 60 degrees instead of 90?
>
>Because all three intervals in the triad should be on the same
>distance scale. The diagram should simply rotate, and not distort
>at all, if you redefine the axes to be the outer interval and the
>lower interval, or the outer interval and the upper interval, rather
>than the lower interval and the upper interval. John Chalmers knows
>this trick better than me -- see his triangular tetrachordal diagrams
>(think of these as showing triads within a fourth instead of triads
>within an octave).

Ah, yes. I remember now, thanks. That Chalmers... it was those
graphs which first exposed me to propriety.

>>>Not fourier transforms -- the cochlea does that
>>
>>It decomposes complex waves, but not necessarily into harmonics,
>
>Not necessarily into harmonics?

The components do not necessarily belong to a single harmonic series,
which is a requirement of a transform being "fourier", true or false?

-Carl

Carl wrote,

>The components do not necessarily belong to a single harmonic series,
>which is a requirement of a transform being "fourier", true or false?

False. The components do not have to belong to a single harmonic series for
the transform to be "fourier". Only a periodic waveform will have a
perfectly harmonic series of partials, and it's easiest to do a fourier
transform on a periodic waveform, but one can do it for a non-periodic
waveform as well.

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:

http://www.egroups.com/message/tuning/13480

> Carl wrote,
>
> >The components do not necessarily belong to a single harmonic
series,which is a requirement of a transform being "fourier", true or
false?
>
> False. The components do not have to belong to a single harmonic
series forthe transform to be "fourier". Only a periodic waveform
will have a perfectly harmonic series of partials, and it's easiest
to do a fourier transform on a periodic waveform, but one can do it
for a non-periodic waveform as well.

Oh sure... all the early electronic music "cats" Cologne, etc., I
believe were always doing these analyses on their music, which was
certainly more complex than a single series....
____________ ____ __ __ _
Joseph Pehrson

>>The components do not necessarily belong to a single harmonic series,
>>which is a requirement of a transform being "fourier", true or false?
>
>False. The components do not have to belong to a single harmonic series for
>the transform to be "fourier". Only a periodic waveform will have a
>perfectly harmonic series of partials, and it's easiest to do a fourier
>transform on a periodic waveform, but one can do it for a non-periodic
>waveform as well.

Gee, my bad (I always thought that since Fourier proved you could
represent any periodic wave as a superposition of harmonics, that
this process was called a Fourier transform; that the process of
decomposing aperiodic waves was called something else...).

-Carl

> -----Original Message-----
> From: Joseph Pehrson [mailto:pehrson@pubmedia.com]
> Sent: Monday, September 25, 2000 2:07 PM
> To: tuning@egroups.com
> Subject: [tuning] Re: harmonic entropy
>
> --- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> http://www.egroups.com/message/tuning/13480
>
> > Carl wrote,
> >
> > >The components do not necessarily belong to a single harmonic
> series,which is a requirement of a transform being "fourier", true or
> false?
> >
> > False. The components do not have to belong to a single harmonic
> series forthe transform to be "fourier". Only a periodic waveform
> will have a perfectly harmonic series of partials, and it's easiest
> to do a fourier transform on a periodic waveform, but one can do it
> for a non-periodic waveform as well.
>
> Oh sure... all the early electronic music "cats" Cologne, etc., I
> believe were always doing these analyses on their music, which was
> certainly more complex than a single series....

We need to distinguish between the Fourier *series*, which is generated from
periodic inputs, and the Fourier *transform*, which has an arbitrary
complex-valued input (with some conditions about certain integrals being
finite). The so-called Fast Fourier Transform not only assumes periodic
inputs, they are also finite ... discrete points in and out, and a finite
number thereof.

Any good text on digital signal processing will tell you more than you want
to know about this stuff. It will also tell you how to do nifty things like
finite and infinite response digital filters.

The guys in Cologne most likely did not do much meaningful Fourier analysis
digitally. They predate the FFT by a decade or more, if my memory is
correct. What they did was most likely pass signals through banks of
*analog* filters, tuned to different frequencies, to do analysis of sounds.
--
M. Edward Borasky
mailto:znmeb@teleport.com
http://www.borasky-research.com

--- In tuning@egroups.com, "M. Edward Borasky" <znmeb@t...> wrote:

http://www.egroups.com/message/tuning/13522

> We need to distinguish between the Fourier *series*, which is
generated from periodic inputs, and the Fourier *transform*, which
has an arbitrary complex-valued input

Well... this certainly shows how much I know about *this* topic!

Thanks, Ed, for the clarifications. I guess what I had heard about
was the Fourier "transform," where electronic music composers were
subjecting complex inputs to Fourier analysis and coming out with
maps of reductive simultaneous harmonic series... or something of the
like. And, yes, it surely seems like it must be later than Cologne.
How about IRCAM??

Anyway, I'd better shut up about this... but the whole point was the
fact that I had heard that people had subjected complex inputs to
Fourier analysis, not just periodic inputs....

____________ ____ ___ __ _
Joseph Pehrson