Harmonic Entropy Questions

Is there a simple & straight-forward formula to calculate the harmonic entropy of a given cents-value interval? I'm thinking that I'd like to calculate the harmonic entropy for each interval in each EDO from 5 to 37, and maybe make some graphs to show how the intervals in each EDO are distributed over the harmonic entropy curve. I suppose doing a 2-octave graph would be pertinent as well, since many musicians like extended-voice chords.

Also, what exactly does it mean that a substantially-mistuned 3/2 has the same harmonic entropy as a purely-tuned 5/4?

-Igs

cityoftheasleep wrote:

> Is there a simple & straight-forward formula to calculate the
> harmonic entropy of a given cents-value interval?

It's a pain to calculate, but you can try my spreadsheet, which
looks up values from those Paul calculated:
http://lumma.org/music/theory/DyadicHarmonicEntropyCalc.xls

Paul usually only calculated in 1-cent increments, because
nowhere does the entropy function change much over the span
of a cent.

> Also, what exactly does it mean that a substantially-mistuned 3/2
> has the same harmonic entropy as a purely-tuned 5/4?

If you draw a straight line across any of the graphs we've
been looking at... if it crosses the h.e. curve six times,
that's six intervals with the same harmonic entropy. Does
that answer your question?

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> It's a pain to calculate, but you can try my spreadsheet, which
> looks up values from those Paul calculated:
> http://lumma.org/music/theory/DyadicHarmonicEntropyCalc.xls

Hey, that's even better than a formula! Thanks!

> > Also, what exactly does it mean that a substantially-mistuned 3/2
> > has the same harmonic entropy as a purely-tuned 5/4?
>
> If you draw a straight line across any of the graphs we've
> been looking at... if it crosses the h.e. curve six times,
> that's six intervals with the same harmonic entropy. Does
> that answer your question?
>

Not exactly. I was more wondering about the significance of intervals having the same h.e. I think I still haven't quite exactly grasped the concept, because the idea that ~679 cents has the same h.e. value as 386 cents is kind of weird to me. 386 cents is a local minimum, but ~679 cents is quite a ways from the minimum at 702 cents, does this mean that a fifth detuned by almost 20 cents is as harmonically stable as a pure major third? Or am I misinterpreting the concept?

-Igs

One of the features of recurrent sequences often is that you end up with scales that all have the same basic con/dis. This is my appeal to them
I too would be curious to see what results here if you have a scale where they are all the same HE and how many notes one could you find that worked. I wonder if there is a cross over here.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > It's a pain to calculate, but you can try my spreadsheet, which
> > looks up values from those Paul calculated:
> > http://lumma.org/music/theory/DyadicHarmonicEntropyCalc.xls
>
> Hey, that's even better than a formula! Thanks!
>
> > > Also, what exactly does it mean that a substantially-mistuned 3/2
> > > has the same harmonic entropy as a purely-tuned 5/4?
> >
> > If you draw a straight line across any of the graphs we've
> > been looking at... if it crosses the h.e. curve six times,
> > that's six intervals with the same harmonic entropy. Does
> > that answer your question?
> >
>
> Not exactly. I was more wondering about the significance of intervals having the same h.e. I think I still haven't quite exactly grasped the concept, because the idea that ~679 cents has the same h.e. value as 386 cents is kind of weird to me. 386 cents is a local minimum, but ~679 cents is quite a ways from the minimum at 702 cents, does this mean that a fifth detuned by almost 20 cents is as harmonically stable as a pure major third? Or am I misinterpreting the concept?
>
> -Igs
>

Igs wrote:

> Not exactly. I was more wondering about the significance of
> intervals having the same h.e. I think I still haven't quite
> exactly grasped the concept, because the idea that ~679 cents
> has the same h.e. value as 386 cents is kind of weird to me.
> 386 cents is a local minimum, but ~679 cents is quite a ways
> from the minimum at 702 cents, does this mean that a fifth
> detuned by almost 20 cents is as harmonically stable as a pure
> major third? Or am I misinterpreting the concept?

You can only misinterpret the concept. That's because there's
nothing to interpret. The harmonic entropy of an interval is
the number of bits needed to describe its relationship to the
harmonic series after it's been munged by noisy equipment.
If you try to understand it with phrases like "harmonically
stable" -- which are more ambiguous than the thing you're
trying to understand -- you'll never get anywhere. Doesn't
Lao Tzu have something about stripping away interpretations and
seeing the world as it really is? That's like, science.

-Carl

This is an interesting thought.

"If you draw a straight line across any of the graphs we've
been looking at... if it crosses the h.e. curve six times,
that's six intervals with the same harmonic entropy.
-Carl"

Has anyone made a tuning consisting of just the Farey Series N=79
minima for instance?

that would seem to be pretty obvious - and more or less a JI-ish tuning?

Chris

This is pretty much what i was suggesting except then you would have to look at all the cross relations otherwise you are just looking at it from one point. But is HE =the farey series?
Various people of the Wilsonian school have been playing with the co-prime grid which is the same thing . This is actually much closer to the Ancient Greek idea of mediants and i still think is a better approach than limits. Of course this would lead to a completely different way to evaluate temperaments and EDO if one was inclined that way.

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> This is an interesting thought.
>
> "If you draw a straight line across any of the graphs we've
> been looking at... if it crosses the h.e. curve six times,
> that's six intervals with the same harmonic entropy.
> -Carl"
>
> Has anyone made a tuning consisting of just the Farey Series N=79
> minima for instance?
>
> that would seem to be pretty obvious - and more or less a JI-ish tuning?
>
> Chris
>

On Sun, Aug 1, 2010 at 8:21 PM, Kraig Grady <kraiggrady@...> wrote:
>
> This is pretty much what i was suggesting except then you would have to look at all the cross relations otherwise you are just looking at it from one point. But is HE =the farey series?
> Various people of the Wilsonian school have been playing with the co-prime grid which is the same thing . This is actually much closer to the Ancient Greek idea of mediants and i still think is a better approach than limits. Of course this would lead to a completely different way to evaluate temperaments and EDO if one was inclined that way.

What do you mean by this? Can you elaborate?

-Mike

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Aug 1, 2010 at 8:21 PM, Kraig Grady <kraiggrady@...> wrote:
> >
> > This is pretty much what i was suggesting except then you would have to look at all the cross relations otherwise you are just looking at it from one point. But is HE =the farey series?
> > Various people of the Wilsonian school have been playing with the co-prime grid which is the same thing . This is actually much closer to the Ancient Greek idea of mediants and i still think is a better approach than limits. Of course this would lead to a completely different way to evaluate temperaments and EDO if one was inclined that way.
>
> What do you mean by this? Can you elaborate?
>
> -Mike
>

Seconded! What is the co-prime grid and how is it better than limits?

-Igs

Hi Kraig,

I didn't mean to imply HE = Farey series - I was just trying to
identify a particular graph, in my example where N=79, such as this
one
http://sonic-arts.org/td/erlich/hentropy6.gif

Now...

"then you would have to look at all the cross relations "

So then we are talking (in visual terms) of a 3-D surface grid, more or less.

There would seem to be two obvious ways to build that.

1. each minimum is used as a 1/1 of its own series going at right
angles to the HE graph I pointed out.

2. each minimum is used as its own identity in the same HE series
going at right angles to the HE graph I pointed out.

I can see where fractions become easier to deal with at this point -
to make a grid of relationships in 2 dimensions. Hmm or maybe not.

This should be doable in some fashion but every way I look at this my
solution is messy excepting a 3-D surface projection.

I think...

Its late for me

Chris

On Sun, Aug 1, 2010 at 8:21 PM, Kraig Grady <kraiggrady@...> wrote:
>
>
>
> This is pretty much what i was suggesting except then you would have to look at all the cross relations otherwise you are just looking at it from one point. But is HE =the farey series?
> Various people of the Wilsonian school have been playing with the co-prime grid which is the same thing . This is actually much closer to the Ancient Greek idea of mediants and i still think is a better approach than limits. Of course this would lead to a completely different way to evaluate temperaments and EDO if one was inclined that way.

Chris Vaisvil wrote:

> Has anyone made a tuning consisting of just the Farey Series N=79
> minima for instance?

Not sure what you mean. You can make a scale of ratios in
a Farey series of order 79. Such a scale would also be the
pitches of a 79-limit tonality diamond.

Alternatively, we can make a scale by listing harmonic entropy
minima. And harmonic entropy can be computed from on a Farey
series, and the order of the Farey series used can affect which
minima that come out. However, this converges as the order goes
to infinity, so usually harmonic entropy is computed using a
ridiculously high Farey order (like 65,000), but you could use
79 for kicks.

Did you mean either of these? The problem with the latter
approach is that we want the minima to occur in between all
the intervals in the scale, not just as measured from a
particular 1/1. Like if you just list

1/1 5/4 4/3 etc

the interval 4/3 - 5/4 = 16/15 may not be a minimum. That's
where regular temperaments come in: when you optimize one mode,
you optimize them all.

-Carl

Kraig Grady wrote:

> This is pretty much what i was suggesting except then you would
> have to look at all the cross relations otherwise you are just
> looking at it from one point.

I think you're referring to the case I mentioned, so the
interval between 4/3 and 5/4 (16/15) doesn't get overlooked?

> But is HE =the farey series?

No, but HE is based on the Farey series.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:

> Not sure what you mean. You can make a scale of ratios in
> a Farey series of order 79. Such a scale would also be the
> pitches of a 79-limit tonality diamond.

You need to take the Farey sequence together with its inversion and reduce to an octave.

{for some reason i am not getting my digest}
I was just asking about the connection between HE and the farey series i thought i might have missed something there.
then
If one looks at the ancient Greek scales one can see they are not concerned with limits at all but what type of ratios that fit in between another. If one focused on these mediants as to how early or late they occur as opposed to what limit they are it gives a different picture . these mediants are not just in half but with Ptolemy for instance often time in thirds.
It relates to the diophantine equation but i am in the dark a bit on that one. Anyway every adjacent term in the Farey series is separated by a superparticular ratio as can be seen in Erv's Epimore tree.
if we have a gap in a scale we either fill it with a run of intervals (MOS) we already have or like the greeks we can take a medient.
Both are musically useful methods

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Sun, Aug 1, 2010 at 8:21 PM, Kraig Grady <kraiggrady@...> wrote:
> >
> > This is pretty much what i was suggesting except then you would have to look at all the cross relations otherwise you are just looking at it from one point. But is HE =the farey series?
> > Various people of the Wilsonian school have been playing with the co-prime grid which is the same thing . This is actually much closer to the Ancient Greek idea of mediants and i still think is a better approach than limits. Of course this would lead to a completely different way to evaluate temperaments and EDO if one was inclined that way.
>
> What do you mean by this? Can you elaborate?
>
> -Mike
>

there are quite a few papers on the co-prime grid on the Wilson archive
http://anaphoria.com/wilson.html
but it is also the lambdoma and pascal (Mt. Meru) triangle. It is all interrelated.

It isn't that it is better , i just prefer it so it is better it for me.

--- In tuning@yahoogroups.com, "cityoftheasleep" <igliashon@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > On Sun, Aug 1, 2010 at 8:21 PM, Kraig Grady <kraiggrady@> wrote:
> > >
> > > This is pretty much what i was suggesting except then you would have to look at all the cross relations otherwise you are just looking at it from one point. But is HE =the farey series?
> > > Various people of the Wilsonian school have been playing with the co-prime grid which is the same thing . This is actually much closer to the Ancient Greek idea of mediants and i still think is a better approach than limits. Of course this would lead to a completely different way to evaluate temperaments and EDO if one was inclined that way.
> >
> > What do you mean by this? Can you elaborate?
> >
> > -Mike
> >
>
> Seconded! What is the co-prime grid and how is it better than limits?
>
> -Igs
>

What i meant by cross relations is that if we have say a third and a fifth that has the same HE we still have to look at what relation that third and fifth have to each other directly in term of HE. This is just conceptional as i don't work with it.

--- In tuning@yahoogroups.com, Chris Vaisvil <chrisvaisvil@...> wrote:
>
> Hi Kraig,
>
> I didn't mean to imply HE = Farey series - I was just trying to
> identify a particular graph, in my example where N=79, such as this
> one
> http://sonic-arts.org/td/erlich/hentropy6.gif
>
> Now...
>
> "then you would have to look at all the cross relations "
>
> So then we are talking (in visual terms) of a 3-D surface grid, more or less.
>
> There would seem to be two obvious ways to build that.
>
> 1. each minimum is used as a 1/1 of its own series going at right
> angles to the HE graph I pointed out.
>
>
> 2. each minimum is used as its own identity in the same HE series
> going at right angles to the HE graph I pointed out.
>
> I can see where fractions become easier to deal with at this point -
> to make a grid of relationships in 2 dimensions. Hmm or maybe not.
>
> This should be doable in some fashion but every way I look at this my
> solution is messy excepting a 3-D surface projection.
>
> I think...
>
> Its late for me
>
> Chris
>
>
> On Sun, Aug 1, 2010 at 8:21 PM, Kraig Grady <kraiggrady@...> wrote:
> >
> >
> >
> > This is pretty much what i was suggesting except then you would have to look at all the cross relations otherwise you are just looking at it from one point. But is HE =the farey series?
> > Various people of the Wilsonian school have been playing with the co-prime grid which is the same thing . This is actually much closer to the Ancient Greek idea of mediants and i still think is a better approach than limits. Of course this would lead to a completely different way to evaluate temperaments and EDO if one was inclined that way.
>

yes exactly.
any triad say ABC we would have to look at the HE of all three

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Kraig Grady wrote:
>
> > This is pretty much what i was suggesting except then you would
> > have to look at all the cross relations otherwise you are just
> > looking at it from one point.
>
> I think you're referring to the case I mentioned, so the
> interval between 4/3 and 5/4 (16/15) doesn't get overlooked?
>
> > But is HE =the farey series?
>
> No, but HE is based on the Farey series.
>
> -Carl
>

After composing with my scale and several others that have 2+
non-meantone-like 5th, I have noticed a pattern that when the notes are over 3/1
but less than 4/1 apart things begin to sound scrambled.

An easy example of such a fault...one of my own scales has a 5/4 in it but
also an 11/6 (4/3). Hence 5/4 * 3/1 = 15/4 and the nearest 11/6 (11/6 * the 2/1
octave) = 22/6....a difference of about 38.8596 cents and enough to give a large
sense of "commatic error". I gave my 9-tone scale to a professional guitarist
and that's the main thing he jumped on it about...that and the areas around the
2 new notes with consecutive semi-tones that tend to sound very tense when
played together.

Same goes with 18/11 which, when multiplied by 3 gives 54/11 which, when
divided by two octaves, gives about 11/9..which then clashes with 5/4, giving
about the same error as the 11/6 vs. 4/3 example. It seems that, quite simply
(error of alternative 5th / pure 5th) = (same error overtone to root mismatch
between those two tones in the scale 2/1 to 3/1 apart).

So would it be fair to say anything with an interval 7+ cents smaller or
larger than a perfect fifth is bound to have a clashing overtone/root
combination as a result?

It does seem to be a bit of an obstacle...especially considering how many
musicians use more than 2 octaves worth of frequency space in their songs (often
3 making overtones ideally matching up to about 8/1 or so). Any ideas how to
hack past this...or is this an inevitable side effect of NOT using near-perfect
fifths?