[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